Namespace ModularCurve 7,711 theorems
Landmarks here: Finiteness of the Galois invariants of the Eisenstein quotient of J₀(p) · Specialisation of the Eisenstein quotient away from p
— 2796 · B3 10 · ChainDirichlet 1 · CharPModel 88 · CharPReduction 7 · CompEq 1 · ComplexPlaceDictionary 21 · ComplexPlaceDictionaryOf 24 · ComponentChart 2 · CupPairing 9 · CuspSpace 7 · DRLevel 28 · DRModel 26 · DRModelPackage 90 · DRModelPackageLevel 187 · DRResolvedModelPackage 22 · DRResolvedModelPackageLevel 13 · DRResolvedModelPackageLevelRam 1 · FifteenA1 4 · FrobeniusQuadratic 1 · FullLevel 1518 · Gamma0Pair 1 · HahnSpecialise 3 · HpoolLevelRing 16 · IgusaCover 1 · IgusaScheme 132 · InLine 1 · IsDiamondPullbackModL 4 · IsFrickeAutFull 1 · IsGamma0PowAt 7 · IsGamma1Link 2 · IsGamma1Point 4 · IsInfReductionMap 9 · IsLevelPStructure 13 · IsModPFormFn 2 · IsModuliPlaceOf 2 · IsPlaceReductionModL 2 · JH 14 · JHNeronObjectAtP 153 · JHPlaceSpecialization 115 · JOne 20 · JOneES 2 · JZero 111 · JZeroGoodReductionSpecialization 1 · JZeroNeronIdentityComponent 13 · JZeroNeronIdentityComponentGood 1 · JZeroNeronObjectAtP 121 · JZeroNeronPrimaryTorsionCore 4 · JZeroNeronPrimaryTorsionFFModels 1 · JZeroNeronPrimaryTorsionFlag 18 · JZeroNeronPrimaryTorsionSheaf 2 · KatzGamma0Form 2 · KatzLevelPForm 8 · LambdaModularPolynomialData 4 · LambdaNodeLocalized 29 · LevelComponent 4 · LevelModuliPackageAbs 76 · LevelN 18 · LevelOneFibre 3 · LevelP 18 · LevelRelabelling 20 · MTorsionNeBot 1 · MazurII142 1 · ModularPolynomialData 58 · ModuliPoint 2 · MultCovering 118 · NodeLocalized 69 · PDPairing 7 · Period 17 · PhiGen 32 · PlaceSpecialization 585 · R2geoDet 1 · RigidWeierstrassData 2 · SSCarrier 1 · SSHeckeV2 24 · SSLevelDatum 16 · SerreImage 1 · SiegelUnit 17 · StarBank 11 · TateModule 1 · TatePoint 7 · TwoChart 7 · UVCrossingModel 111 · UniformizedHeckeCurve 4 · UnramifiedOutside 1 · XH 1 · XHDRLevel 54 · XHDRModelAtP 313 · XOne 18 · XOneGammaZeroP 24 · XOneP 376 · XZeroP 10 · XZeroPM 7
directly in ModularCurve 2796
- Galois-invariance of the cusp ∞ on X₀(N)_{ℚ̄}
ModularCurve.arithmeticGalois_smul_cuspInftyBar2 below · cited by 9 · depth 7 - The cusp ̄ 0 is fixed by the arithmetic Galois action
ModularCurve.arithmeticGalois_smul_cuspZeroBar4 below · cited by 1 · depth 7 - Cuspidal class survives in the Eisenstein quotient
ModularCurve.cuspidalClassSurvives_heckeModuleBar498 below · cited by 1 · depth 7 - The cusp ∞ is a degree-one place of X(N)_{ℚ̄}
ModularCurve.deg_cuspInftyBar3 below · cited by 11 · depth 7 - The cusp ̄ 0 has degree one
ModularCurve.deg_cuspZeroBar4 below · cited by 4 · depth 7 - landmark Finiteness of the Galois invariants of the Eisenstein quotient of J₀(p)
ModularCurve.eisensteinQuotientInvariantsFiniteAt_heckeModuleBar5,128 below · cited by 1 · depth 7 - Hecke operators on J₀(N) commute (all levels)
ModularCurve.heckeOperatorsCommuteBar224 below · cited by 145 · depth 7 - landmark Specialisation of the Eisenstein quotient away from p
ModularCurve.mazurQuotientSpecialization_heckeModuleBar2,179 below · cited by 1 · depth 7 - Rational moduli place on X₀(p) at a Vélu quotient
ModularCurve.moduliPointExists_jQuotVelu_of_mult_two323 below · cited by 1 · depth 7 - Tate cusp criterion for the Vélu p-isogeny quotient
ModularCurve.tateCuspCriterion_jQuotVelu24 below · cited by 1 · depth 7 - Arithmetic semilinear action commutes with geometric automorphisms
ModularCurve.arithmeticGalois_smul_geomAut1 below · cited by 11 · depth 8 - Coefficientwise Galois action preserves q-integrality
ModularCurve.arithmeticGalois_smul_mem_qIntegersBar_iff1 below · cited by 2 · depth 8 - Reduction of cuspidal class survival to three inputs
ModularCurve.cuspidalClassSurvives_heckeModuleBar_of_inputs0 below · cited by 1 · depth 8 - Nonvanishing of the cuspidal class at prime level
ModularCurve.cuspidalClass_ne_zero450 below · cited by 1 · depth 8 - The q-adic place at infinity has degree one
ModularCurve.deg_qInftyPlaceBar2 below · cited by 5 · depth 8 - The Eisenstein ideal annihilates the cuspidal class
ModularCurve.eisensteinIdeal_smul_cuspidalClass_heckeModuleBar229 below · cited by 1 · depth 8 - Eisenstein-ideal torsion meets the Eisenstein kernel trivially
ModularCurve.eisensteinKernelSubmodule_disjoint_eisensteinTorsion_heckeModuleBar0 below · cited by 1 · depth 8 - Finitely generated plus torsion gives the Eisenstein finiteness at p
ModularCurve.eisensteinQuotientInvariantsFiniteAt_heckeModuleBar_of_fg_of_isTorsion0 below · cited by 1 · depth 8 - Finite generation of the rational part of the Eisenstein quotient
ModularCurve.eisensteinQuotientRational_closure_fg_heckeModuleBar791 below · cited by 2 · depth 8 - Rational part of the Eisenstein quotient is torsion
ModularCurve.eisensteinQuotientRational_isTorsion_heckeModuleBar5,126 below · cited by 1 · depth 8 - Good-reduction specialisation of J₀(p) at ℓ with four predicates
ModularCurve.exists_jZeroGoodReductionSpecialization_doorPredicates2,177 below · cited by 1 · depth 8 - Existence of a symmetric modular polynomial Φ_ℓ
ModularCurve.exists_modularPolynomialData_evalSymm39 below · cited by 69 · depth 8 - Degree-one place above a root pair of Φₚ
ModularCurve.exists_place_of_modularPolynomial_eval_eq_zero160 below · cited by 2 · depth 8 - Toric data and 𝔪-dichotomy for J₀(Nq) at q
ModularCurve.exists_toricDichotomyData_jZero3,554 below · cited by 2 · depth 8 - Two hauptmodul relations pin C/D to four values
ModularCurve.fifteenIsogenyJ_of_hauptmodul_memberships16 below · cited by 1 · depth 8 - Degree of the diagonal in the Hecke exchange square
ModularCurve.finrankAlong_towerSubstBar_comp_heckeAlphaBar147 below · cited by 2 · depth 8 - Eichler–Shimura relation on p-power torsion of J₀(N)
ModularCurve.frobeniusQuadratic_JZero994 below · cited by 11 · depth 8 - Generation of j(qᵈ) by j(q) and j(q^N)
ModularCurve.functionFieldGeneration69 below · cited by 52 · depth 8 - Principal divisors on the function field of X₀(N) over ℚ̄
ModularCurve.hasPrincipalDivisors_modularFunctionFieldBar36 below · cited by 7 · depth 8 - Integrality of the degeneracy embedding ᾱ for prime ℓ
ModularCurve.heckeAlphaBarIntegral_of_prime47 below · cited by 35 · depth 8 - Commutativity of Hecke operators from the exchange identity
ModularCurve.heckeOperatorsCommuteBar_of_heckeExchangeAt66 below · cited by 1 · depth 8 - Roof generation for the Hecke exchange square
ModularCurve.heckeRoof_adjoin_range_union_eq_top52 below · cited by 5 · depth 8 - Peeling a prime qnot≡ 1mod p off the level
ModularCurve.isResiduallyModularOfLevel_div_of_mazurFamilies639 below · cited by 2 · depth 8 - Residual modularity at level N from lower-level torsion
ModularCurve.isResiduallyModularOfLevel_of_hasLowerLevelTorsion_of_isGoodPrimeFor1,424 below · cited by 1 · depth 8 - Mazur's specialisation input from per-prime good reduction
ModularCurve.mazurQuotientSpecialization_heckeModuleBar_of_doorV20 below · cited by 1 · depth 8 - Realising the irreducible mod p representation inside J₀(Nq)[𝔪]
ModularCurve.mazurRealizationFamily_of_modRepIsIrreducible_of_isUnramifiedAt1,286 below · cited by 1 · depth 8 - Symmetric modular polynomials Φ_ℓ exist for every prime
ModularCurve.modularPolynomialFamily39 below · cited by 18 · depth 8 - Modular polynomial vanishes at j(W) and j(W/⟨ Q⟩)
ModularCurve.modularPolynomial_eval_jInt_jQuotVelu_eq_zero101 below · cited by 2 · depth 8 - Simple root of Φₚ at the Vélu quotient j-invariant
ModularCurve.modularPolynomial_rootMultiplicity_jQuotVelu_eq_one239 below · cited by 1 · depth 8 - Uniqueness of a place over a simple root pair of Φₚ
ModularCurve.place_eq_of_modularPolynomial_rootMultiplicity_eq_one152 below · cited by 2 · depth 8 - Galois and Hecke actions on J₀(N) commute
ModularCurve.smulCommClass_JZero_of_heckeOperatorsCommuteBar15 below · cited by 63 · depth 8 - Triviality of J₀(p) for primes p<5
ModularCurve.subsingleton_jZero_of_lt_five460 below · cited by 4 · depth 8 - Finiteness of the modular function field tower along N ∣ M
ModularCurve.towerInclBar_finiteAlong50 below · cited by 5 · depth 8 - Finiteness of the degeneracy map q↦ q^ℓ to level M
ModularCurve.towerSubstBar_finiteAlong54 below · cited by 7 · depth 8 - Integrality of the degeneracy substitution q↦ q^ℓ
ModularCurve.towerSubstBar_isIntegral54 below · cited by 20 · depth 8 - Coefficientwise and geometric Galois actions on L· F₀ commute
ModularCurve.arithmeticRingAut_geomAut0 below · cited by 1 · depth 9 - Coefficient embedding sends j(q) over ℚ to j(q) over L
ModularCurve.coeffEmb_jq0 below · cited by 77 · depth 9 - Coefficient embedding commutes with the q^N-expansion of j
ModularCurve.coeffEmb_jqN0 below · cited by 23 · depth 9 - Coefficientwise maps commute with q↦ qⁿ
ModularCurve.coeffMap_qExpand0 below · cited by 194 · depth 9 - The cusp count of level p equals 2
ModularCurve.cuspCount_prime0 below · cited by 9 · depth 9 - The cusps ̄ 0 and ∞̄ are distinct for N>1
ModularCurve.cuspZeroBar_ne_cuspInftyBar9 below · cited by 15 · depth 9 - Multiplicativity of the Dedekind ψ function
ModularCurve.dedekindPsi_mul_of_coprime0 below · cited by 46 · depth 9 - Value of `dedekindPsi` at a prime: ψ(p)=p+1
ModularCurve.dedekindPsi_prime0 below · cited by 72 · depth 9 - Dedekind ψ at a prime power
ModularCurve.dedekindPsi_prime_pow0 below · cited by 33 · depth 9 - Places of K(j,j_N) have nonzero degree
ModularCurve.deg_ne_zero_modularFunctionFieldC86 below · cited by 6 · depth 9 - Degeneracy pushforward inputs at a prime level q
ModularCurve.degeneracyPushforwardInputs90 below · cited by 6 · depth 9 - Eigenform ideals above p lie in the support of J₀(N)
ModularCurve.eigenformSupportAt_jZero861 below · cited by 7 · depth 9 - An Eisenstein ideal element acting by num((p-1)/12) on J₀(p)
ModularCurve.eisensteinIdeal_image_cokernel_dvd_num_heckeModuleBar1,072 below · cited by 1 · depth 9 - Stabilisation of the chain P^m M on Eisenstein-quotient rational points
ModularCurve.eisensteinQuotientRationalLocalized_eisensteinPrimary_stabilizes16 below · cited by 1 · depth 9 - Uniform q^m-index bound on the localised Eisenstein quotient
ModularCurve.eisensteinQuotientRationalLocalized_qPowQuotient_le_h1Jtors16 below · cited by 1 · depth 9 - Locally torsion at Eisenstein maximal ideals implies torsion
ModularCurve.eisensteinQuotientRational_closure_locallyTorsion_isTorsion_heckeModuleBar1 below · cited by 1 · depth 9 - Rational points of the Eisenstein quotient are torsion
ModularCurve.eisensteinQuotientRational_isTorsion_heckeModuleBar_of_perPrimeFinite16 below · cited by 1 · depth 9 - Reduction mod ℓ is injective on prime-to-ℓ torsion of J₀(N)
ModularCurve.eq_zero_of_reductionModL_eq_zero_of_nsmul_eq_zero975 below · cited by 17 · depth 9 - Essential finite type of ℚ̄-function field of X₀(N)
ModularCurve.essFiniteType_modularFunctionFieldBar93 below · cited by 65 · depth 9 - Toric dichotomy data from a semistable specialisation at q
ModularCurve.existsToricDichotomyData_of_jZeroSemistableSpecialization237 below · cited by 5 · depth 9 - Eichler–Shimura: Galois representations from Hecke characters on VₚJ₀(N)
ModularCurve.exists_galoisRepAdic_charpoly_frobenius_of_heckeChar1,238 below · cited by 2 · depth 9 - Rational X₀(5) Hauptmodul value from a kernel quadratic
ModularCurve.exists_hauptmodulFive_of_kernelQuadratic0 below · cited by 1 · depth 9 - Hecke-equivariant Abel–Jacobi isomorphism Pic⁰ ≅ S₂(Γ₀(N))^∨/Λ_N
ModularCurve.exists_heckeEquivariant_addEquiv_pic0_complex_quotient_periodLattice673 below · cited by 2 · depth 9 - Divisorial Hecke ring of J₀(N) embeds in End_ℂS₂(Γ₀(N))
ModularCurve.exists_injective_ringHom_adjoin_heckeOperatorBar_cuspForm842 below · cited by 4 · depth 9 - Eigenform ideal in J₀(M) from a maximal Hecke ideal
ModularCurve.exists_isEigenformIdeal_heckeTorsion_jZero_ne_bot_of_isMaximal_heckeAlgebra_two891 below · cited by 1 · depth 9 - Residual modularity of level N yields an eigenform ideal
ModularCurve.exists_isEigenformIdeal_of_isResiduallyModularOfLevel4 below · cited by 3 · depth 9 - Localised Kummer package at an Eisenstein prime of J₀(p)
ModularCurve.exists_jKummerRow_admissibleChain_bounded_heckeModuleBar_v55,117 below · cited by 1 · depth 9 - Irreducible modular polynomial datum from degree ψ(N)
ModularCurve.exists_phiIrreducible_of_finrank_eq42 below · cited by 8 · depth 9 - Transfer of Hecke characters from S₂(Γ₀(N)) to VₚJ₀(N)
ModularCurve.exists_ringHom_rationalHeckeAlgebra_extends_heckeChar1,004 below · cited by 3 · depth 9 - Embedding of E[p] into the 𝔪-torsion of J
ModularCurve.exists_torsionEmbedding_of_congruences121 below · cited by 3 · depth 9 - ℚ̄-modular function field is a one-variable function field
ModularCurve.exists_transcendental_finiteDimensional_modularFunctionFieldBar141 below · cited by 31 · depth 9 - Torsion of Pic⁰ of the modular function field in characteristic ℓ
ModularCurve.fTorsionFor_pic0_residueField_modularFunctionFieldC161 below · cited by 1 · depth 9 - Finiteness of the degeneracy inclusion at prime level ℓ
ModularCurve.finiteAlong_heckeAlphaBar_of_prime46 below · cited by 34 · depth 9 - Finiteness along β̄ for prime level increase ℓ
ModularCurve.finiteAlong_heckeBetaBar_of_prime45 below · cited by 22 · depth 9 - ̄ F_ℓ is finite over ℚ̄(j)
ModularCurve.finiteDimensional_adjoin_coeffEmb_jq109 below · cited by 3 · depth 9 - Finiteness of the base-changed level-N function field over L(j)
ModularCurve.finiteDimensional_adjoin_coeffEmb_jq_full142 below · cited by 24 · depth 9 - Finiteness of the level-M modular function field over ℚ̄(j)
ModularCurve.finiteDimensional_adjoin_coeffEmb_jq_of_neZero72 below · cited by 34 · depth 9 - Geometric degree of X₀(ℓ)→ X(1) is ℓ+1
ModularCurve.finrank_adjoin_jqNModC_eq_of_prime75 below · cited by 5 · depth 9 - Eichler–Shimura relation for Fr_* on J₀(N) in characteristic ℓ
ModularCurve.frobenius_frobenius_sub_heckeOperatorModL_frobenius_add_smul_eq_zero111 below · cited by 8 · depth 9 - Level Mpᵃ⁺¹ divisor-expansion field is generated by j(q^pᵃ⁺¹)
ModularCurve.full_eq_adjoin_full_div_prime46 below · cited by 8 · depth 9 - Generation property equals collapse of the full modular function field
ModularCurve.functionFieldGeneration_iff_full_eq0 below · cited by 19 · depth 9 - Genus of X₀(p) for an odd prime p
ModularCurve.genus_modularFunctionFieldBar_eq_genusFormula_of_prime436 below · cited by 5 · depth 9 - Canonical divisors exist for X₀(N) over ℚ̄
ModularCurve.hasCanonicalDivisor_modularFunctionFieldBar137 below · cited by 53 · depth 9 - Principal divisors on the level-N modular function field over L
ModularCurve.hasPrincipalDivisors_laurentBaseChange_modularFunctionFieldFull35 below · cited by 2 · depth 9 - Unconditional principal divisors on ℚ̄-modular function fields
ModularCurve.hasPrincipalDivisors_modularFunctionFieldBar_unconditional75 below · cited by 248 · depth 9 - Additive maps commuting with all T_ℓ are T-linear
ModularCurve.heckeAlg_smul_comm_of_forall_gen0 below · cited by 2 · depth 9 - Integrality of the α-leg from a modular polynomial
ModularCurve.heckeAlphaBarIntegral_of_modularPolynomialData8 below · cited by 1 · depth 9 - Integrality of the degeneracy embedding β for prime ℓ
ModularCurve.heckeBetaBarIntegral_of_prime46 below · cited by 36 · depth 9 - Hecke correspondence at ℓ multiplies the cuspidal divisor by 1+ℓ
ModularCurve.heckeDivBar_cuspidalDivisor_of_prime180 below · cited by 3 · depth 9 - Uₚ fixes the cuspidal divisor at prime level
ModularCurve.heckeDivBar_cuspidalDivisor_self_of_prime151 below · cited by 3 · depth 9 - Hecke correspondence inputs at every level and prime
ModularCurve.heckeInputsAll93 below · cited by 53 · depth 9 - Hecke correspondence inputs at every level and prime ℓ
ModularCurve.heckeInputsAlong_of_prime92 below · cited by 28 · depth 9 - Commutativity of T_ℓ and T_{ℓ'} on J₀(N)(ℚ̄)
ModularCurve.heckeOperatorBar_comm_of_heckeExchangeAt65 below · cited by 1 · depth 9 - Prime level: Uₚ + w = 0 on J₀(p)
ModularCurve.heckeOperatorBar_self_add_frickeInvolutionBar_smul248 below · cited by 2 · depth 9 - Hecke correspondence scales the cuspidal class by 1+ℓ
ModularCurve.heckePic0Bar_cuspidalClass0 below · cited by 3 · depth 9 - Uₚ fixes the cuspidal class of J₀(p)
ModularCurve.heckePic0Bar_cuspidalClass_self1 below · cited by 3 · depth 9 - Galois equivariance of the Hecke correspondence on Pic⁰
ModularCurve.heckePic0Bar_smul13 below · cited by 1 · depth 9 - Finite generation of the Hecke subalgebra of End J₀(N)
ModularCurve.heckeSubalgebraBar_fg715 below · cited by 3 · depth 9 - Finiteness of the 𝔪-torsion of J₀(M)
ModularCurve.heckeTorsion_jZero_finite_of_natCast_mem714 below · cited by 16 · depth 9 - The base-changed modular function field is a curve over L
ModularCurve.isCurveOver_laurentBaseChange_modularFunctionFieldFull120 below · cited by 32 · depth 9 - The modular function field over ℚ̄ is a curve field
ModularCurve.isCurveOver_modularFunctionFieldBar120 below · cited by 186 · depth 9 - Modular function field is a curve over a perfect field
ModularCurve.isCurveOver_modularFunctionFieldC_of_perfectField122 below · cited by 125 · depth 9 - At prime level `frickeInvolutionFull` is a Fricke automorphism
ModularCurve.isFrickeAutFull_frickeInvolutionFull_prime48 below · cited by 34 · depth 9 - Integrality of j(q^{dℓ}) over fields containing j(qᵈ)
ModularCurve.isIntegral_jqNModC_mul3 below · cited by 5 · depth 9 - Integrality of ̄ j(q^N) over K(̄ j(q))
ModularCurve.isIntegral_jqNModC_of_modularPolynomialData3 below · cited by 5 · depth 9 - Finiteness of the n-torsion of the modular Jacobian
ModularCurve.jZeroTorsionFinite713 below · cited by 35 · depth 9 - Descent by one prime: j(q^M)∈ℚ(j(q),j(q^{Mp}))
ModularCurve.jqN_div_mem_modularFunctionField46 below · cited by 5 · depth 9 - Non-membership of j(q^pᵃ⁺²) over prime-power levels
ModularCurve.jqN_pow_not_mem_adjoin_full45 below · cited by 5 · depth 9 - Non-membership of j(qᵖ) in the level-M divisor-expansion field
ModularCurve.jqN_prime_not_mem_full59 below · cited by 8 · depth 9 - Kronecker congruence for prime-level modular polynomial data
ModularCurve.kroneckerCongruence49 below · cited by 19 · depth 9 - Base change of a Laurent-series subfield commutes with adjunction
ModularCurve.laurentBaseChange_adjoin0 below · cited by 38 · depth 9 - Base change of the modular function field along ℚ↪ L
ModularCurve.laurentBaseChange_modularFunctionField2 below · cited by 24 · depth 9 - Monotonicity of Laurent-coefficient base change
ModularCurve.laurentBaseChange_mono0 below · cited by 23 · depth 9 - Galois action on 𝔪-torsion of J₀(M) factors through a finite level
ModularCurve.mTorsionGaloisRep_jZero_galoisFactorsThroughFiniteLevel79 below · cited by 9 · depth 9 - Mazur's principle from the toric dichotomy when qnot≡ 1
ModularCurve.mazurPrinciple_of_ne_one_of_toricDichotomy1 below · cited by 1 · depth 9 - Minimal polynomial of j(q^M) splits into primitive slots
ModularCurve.minpoly_jqN_map_eq_prod_slots59 below · cited by 20 · depth 9 - Geometric function field of X₀(ℓ) as ℚ̄(j)(j_ℓ)
ModularCurve.modularFunctionFieldBar_eq_restrictScalars109 below · cited by 5 · depth 9 - Two generators suffice for the level-N modular function field
ModularCurve.modularFunctionFieldC_eq_modularFunctionFieldFullC112 below · cited by 164 · depth 9 - Descent closure: F_N = F_N^{full} from one-prime steps
ModularCurve.modularFunctionField_eq_full_of46 below · cited by 5 · depth 9 - Finite generation of the p-adic Tate module of J₀(N)
ModularCurve.moduleFinite_padicInt_tateModule_jZero458 below · cited by 18 · depth 9 - Existence of a semistable specialisation datum for J₀(Nq)
ModularCurve.nonempty_jZeroSemistableSpecialization3,551 below · cited by 1 · depth 9 - Existence of a modular polynomial datum at every level N
ModularCurve.nonempty_modularPolynomialData71 below · cited by 269 · depth 9 - Irreducible mod p representations give non-Eisenstein Hecke ideals
ModularCurve.not_isEventuallyEisenstein_of_modRepIsIrreducible67 below · cited by 3 · depth 9 - Irreducibility excludes eventually Eisenstein maximal ideals
ModularCurve.not_isEventuallyEisenstein_of_repClauses22 below · cited by 1 · depth 9 - Non-integrality of j at a prime of multiplicative reduction
ModularCurve.not_isIntegral_jInt_of_dvd_discriminant_not_dvd_c40 below · cited by 1 · depth 9 - ν₃(p)=2 if p≡ 1(mod 3), else 0
ModularCurve.nuThree_prime0 below · cited by 8 · depth 9 - Number of square roots of -1 in 𝔽ₚ
ModularCurve.nuTwo_prime0 below · cited by 8 · depth 9 - Coefficientwise injections preserve the order of a Laurent series
ModularCurve.order_coeffMap0 below · cited by 2 · depth 9 - Hecke stability of the period lattice at every level
ModularCurve.periodLatticeHeckeStable0 below · cited by 23 · depth 9 - Birational map from Ψ₁₅=0 to the curve 15a1
ModularCurve.psiFifteen_birational_identity0 below · cited by 1 · depth 9 - Substitution q↦ qⁿ commutes with base change of Laurent subfields
ModularCurve.qExpand_mem_laurentBaseChange2 below · cited by 12 · depth 9 - Raynaud clause from finite flat models of Eisenstein quotient torsion
ModularCurve.raynaudFor_of_le_finiteFlat_model_eisensteinQuotient11 below · cited by 1 · depth 9 - Reduction inputs modulo ℓ exist when ℓ∤ N
ModularCurve.reductionInputsModL_of_not_dvd740 below · cited by 54 · depth 9 - Eichler–Shimura congruence for the reduction map on J₀(N)
ModularCurve.reductionModL_heckeOperatorBar767 below · cited by 15 · depth 9 - Reduction intertwines arithmetic Frobenius with geometric Frobenius on J₀(N)
ModularCurve.reductionModL_smul_of_isFrobeniusAt756 below · cited by 13 · depth 9 - Relative degree ψ(N) of the field of divisor q-expansions
ModularCurve.relfinrank_full_eq_dedekindPsi69 below · cited by 39 · depth 9 - Relative degree of one prime-power step in the divisor-expansion tower
ModularCurve.relfinrank_full_eq_mul48 below · cited by 5 · depth 9 - Relative degree is preserved by constant field extension
ModularCurve.relfinrank_laurentBaseChange111 below · cited by 31 · depth 9 - Relative degree of ℚ(j,j_N) over ℚ(j)
ModularCurve.relfinrank_modularFunctionField0 below · cited by 7 · depth 9 - Degree of the q↦ q^ℓ degeneracy map on ℚ(j(qᵈ):d∣ N)
ModularCurve.relfinrank_qExpand_full71 below · cited by 9 · depth 9 - Residual realization attached to an occurring Hecke eigensystem
ModularCurve.residualRealization_of_occurs999 below · cited by 1 · depth 9 - Eichler–Shimura specialisation for J₀(N) at good primes
ModularCurve.specializationExists_JZero998 below · cited by 6 · depth 9 - Triviality of J₀(3)
ModularCurve.subsingleton_jZero_three446 below · cited by 3 · depth 9 - Triviality of J₀(2)
ModularCurve.subsingleton_jZero_two457 below · cited by 5 · depth 9 - Support transfer at level N for J₀(N)
ModularCurve.supportTransfer_jZero1,392 below · cited by 2 · depth 9 - Residue field at the q-adic place is generated by constants
ModularCurve.surjective_algebraMap_residueField_bar1 below · cited by 4 · depth 9 - Integrality of the level-raising inclusion for all N ∣ M
ModularCurve.towerInclBar_isIntegral49 below · cited by 19 · depth 9 - Surjectivity of the tower inclusion between mutually dividing levels
ModularCurve.towerInclBar_surjective_of_dvd_dvd0 below · cited by 4 · depth 9 - Transcendence of j in the base-changed modular function field
ModularCurve.transcendental_coeffEmb_jq109 below · cited by 79 · depth 9 - Transcendence of the q-expansion of j over any coefficient ring
ModularCurve.transcendental_jqModC0 below · cited by 398 · depth 9 - Transcendence of j(q^N) over ℚ
ModularCurve.transcendental_jqN1 below · cited by 9 · depth 9 - SL₂(ℤ)-invariance of E₄³/Δ
ModularCurve.E4_cube_div_discriminant_smul0 below · cited by 8 · depth 10 - Path periods generate the dual of S₂(Γ₀(N))
ModularCurve.addSubgroupClosure_range_periodAlong_eq_top8 below · cited by 2 · depth 10 - Base change of modular function field generated by j and j_N
ModularCurve.adjoin_jBar_jNBar_eq_top92 below · cited by 32 · depth 10 - Hecke relations on J₀(N) hold on S₂(Γ₀(N))
ModularCurve.aeval_heckeAlgebra_eq_zero_of_forall_smul_jZero_eq_zero714 below · cited by 2 · depth 10 - A rational polynomial vanishing at j(q) is zero
ModularCurve.aeval_jq_eq_zero0 below · cited by 2 · depth 10 - Constant term minus f is a non-unit at ∞
ModularCurve.algebraMap_coeff_zero_sub_not_isUnit_bar0 below · cited by 2 · depth 10 - Primes of residue characteristic p lie in the support of J₀(N)[p]
ModularCurve.annihilator_torsionBy_jZero_le_of_isPrime711 below · cited by 1 · depth 10 - Frobenius squared fixes supersingular places of F_N
ModularCurve.arithFrobC_smul_arithFrobC_smul_of_mem_ssPlaces387 below · cited by 13 · depth 10 - Arithmetic Frobenius preserves supersingular places
ModularCurve.arithFrobC_smul_mem_ssPlaces1 below · cited by 13 · depth 10 - Arithmetic Galois action commutes with the degeneracy inclusion α
ModularCurve.arithmeticGalois_smul_heckeAlphaBar0 below · cited by 15 · depth 10 - Arithmetic Galois action commutes with the β degeneracy embedding
ModularCurve.arithmeticGalois_smul_heckeBetaBar1 below · cited by 14 · depth 10 - Unramified points above j=0 on X₀(p) count ν₃(p)
ModularCurve.card_filter_ord_jBar_eq_one_eq_nuThree344 below · cited by 1 · depth 10 - Unramified places over j=1728 at odd prime level count ν₂(p)
ModularCurve.card_filter_ord_jBar_sub_1728_eq_one_eq_nuTwo342 below · cited by 1 · depth 10 - Fricke involution sends Ogg's unit to ℓ¹²u_ℓ⁻¹
ModularCurve.coe_frickeInvolutionFull_modularUnitSeries62 below · cited by 18 · depth 10 - Injectivity of the coefficient embedding ℚ((q)) → L((q))
ModularCurve.coeffEmb_injective1 below · cited by 19 · depth 10 - Coefficient extension commutes with q↦ qⁿ
ModularCurve.coeffEmb_qExpand1 below · cited by 66 · depth 10 - Coefficientwise map on Laurent series preserves injectivity
ModularCurve.coeffMap_injective0 below · cited by 32 · depth 10 - Reduction of integral q-expansions when ℓ ∤ N
ModularCurve.coeffMap_residue_mem_modularFunctionFieldFullC_of_not_dvd176 below · cited by 4 · depth 10 - q-expansion of the Hecke correspondence on differentials, ℓ∤ N
ModularCurve.coeff_diffQExpBar_heckeDiffBar_of_not_dvd151 below · cited by 4 · depth 10 - The q⁻¹ coefficient of ̄ j is 1
ModularCurve.coeff_jqModC_neg_one0 below · cited by 120 · depth 10 - Constants of the mod-p modular function field are K
ModularCurve.constantsAreBase_modularFunctionFieldC_of_perfectField122 below · cited by 17 · depth 10 - SL₂(ℤ)-invariance of the Hecke coset polynomial at ℓ
ModularCurve.cosetPoly_smul0 below · cited by 1 · depth 10 - Positivity of ψ(N) for N ≠ 0
ModularCurve.dedekindPsi_pos1 below · cited by 76 · depth 10 - All places of ℚ̄· F_M have degree one
ModularCurve.deg_eq_one_modularFunctionFieldBar143 below · cited by 249 · depth 10 - Degree of a canonical divisor on X₀(N) over ℚ̄ is 2g-2
ModularCurve.degree_canonicalDivisorOf_modularFunctionFieldBar171 below · cited by 31 · depth 10 - Maximality of the Hecke eigenvalue ideal over a finite field
ModularCurve.eigenIdeal_isMaximal0 below · cited by 2 · depth 10 - Only two cusps on X₀(ℓ)_ℚ̄ at prime level
ModularCurve.eq_cuspInftyBar_or_eq_cuspZeroBar75 below · cited by 24 · depth 10 - No p-power torsion in the Eichler–Shimura specialisation kernel
ModularCurve.eq_zero_of_torsion_of_mem_specializationKernel_jZero995 below · cited by 1 · depth 10 - Injectivity of evaluation at the formal j-invariant
ModularCurve.evalAtJGen_injective4 below · cited by 2 · depth 10 - Linearity of the period map at every level
ModularCurve.existsPeriodMapLinear5 below · cited by 9 · depth 10 - Boston–Lenstra–Ribet embedding of ρ into J[𝔪]
ModularCurve.exists_blrBlock_of_frobeniusQuadratic_of_dense3 below · cited by 1 · depth 10 - Existence of a complex place dictionary for X₀(N)
ModularCurve.exists_complexPlaceDictionary17 below · cited by 9 · depth 10 - Good constant reduction of X₀(N) at ℓ ∤ N
ModularCurve.exists_constantReduction_isGood_isPlaceReductionModL738 below · cited by 9 · depth 10 - Existence of an equivariant primitive of a weight-2 cusp form
ModularCurve.exists_hasEquivariantPrimitive0 below · cited by 23 · depth 10 - Fricke automorphism of ℚ(j(qᵈ):d∣ℓ) at prime level
ModularCurve.exists_isFrickeAutFull47 below · cited by 7 · depth 10 - Localised Kummer package at 2 with bounded admissible chains
ModularCurve.exists_jKummerRow_admissibleChain_bounded_heckeModuleBar_two_v54,950 below · cited by 1 · depth 10 - Two-exponent finite flat model of Eisenstein quotient torsion
ModularCurve.exists_le_finiteFlat_model_eisensteinQuotient_torsion_reductionModL_of_ne_two2,005 below · cited by 1 · depth 10 - q-expansion dictionary: ℂ⊗Ω_{reg}≅ S₂(Γ₀(N))
ModularCurve.exists_linearEquiv_tensor_regularDifferentialsBar_cuspForm643 below · cited by 3 · depth 10 - Eisenstein ideal element acting as n(p) on J₀(p)
ModularCurve.exists_mem_eisensteinIdeal_smul_eq_eisensteinNumerator_zsmul1,071 below · cited by 4 · depth 10 - j(q^N) is integral over ℤ[j(q)]
ModularCurve.exists_monic_evalAtJ_jqN_eq_zero41 below · cited by 3 · depth 10 - Mazur admissibility of the Eisenstein-primary q^m-torsion of J₀(p)
ModularCurve.exists_openAction_admissibleChain_eisensteinPrimaryTorsionBar1,063 below · cited by 3 · depth 10 - Existence of an irreducible symmetric modular polynomial Φ_ℓ
ModularCurve.exists_phiIrreducible_evalSymm40 below · cited by 29 · depth 10 - A prime other than q annihilates 𝔪-torsion in J₀(M)
ModularCurve.exists_prime_torsion_of_isMaximal716 below · cited by 6 · depth 10 - Weight-two Hecke algebra acts on J₀(N)
ModularCurve.exists_ringHom_heckeAlgebra_heckeOperatorBar713 below · cited by 9 · depth 10 - Widths, component map and glued specialisation for J₀(Nq) at q
ModularCurve.exists_width_comp_sp3,537 below · cited by 3 · depth 10 - Exit ideal proper iff contained in a maximal ideal
ModularCurve.exitIdeal_ne_top_iff_exists_maximal0 below · cited by 1 · depth 10 - Finiteness along the degeneracy inclusion at prime level
ModularCurve.finiteAlong_heckeAlphaBar_of_modularPolynomialData7 below · cited by 4 · depth 10 - Finiteness along the β-degeneracy map q↦ q^ℓ
ModularCurve.finiteAlong_heckeBetaBar_of_modularPolynomialData6 below · cited by 3 · depth 10 - Finiteness of K(j)bigl(j(q^N)bigr) over K(j)
ModularCurve.finiteDimensional_adjoin_jqNModC2 below · cited by 13 · depth 10 - Frobenius q↦ q^ℓ on the mod-ℓ modular function field has degree ℓ
ModularCurve.finrankAlong_frobeniusModL72 below · cited by 4 · depth 10 - Degree of the q↦ q^ℓ degeneracy map over L
ModularCurve.finrankAlong_heckeBetaBar147 below · cited by 34 · depth 10 - Degree ψ(N) of the level-N modular function field over K(j)
ModularCurve.finrank_adjoin_jqModC_modularFunctionFieldFullC_eq_dedekindPsi108 below · cited by 81 · depth 10 - Degree bound ψ(N) for K(j)(j(q^N)) over K(j)
ModularCurve.finrank_adjoin_jqNModC_le2 below · cited by 5 · depth 10 - Degree of ℚ(j)(j(q^N)) over ℚ(j) is ψ(N)
ModularCurve.finrank_adjoin_jqN_eq_dedekindPsi69 below · cited by 21 · depth 10 - Degree ℓ+1 of ℚ(j)(j(q^ℓ)) over ℚ(j)
ModularCurve.finrank_adjoin_jqN_eq_of_prime41 below · cited by 6 · depth 10 - Degree p for the next prime-power level of j
ModularCurve.finrank_adjoin_jqN_pow_succ_of_not_mem46 below · cited by 2 · depth 10 - Degree p+1 of j(qᵖ) over a field containing j(q)
ModularCurve.finrank_adjoin_jqN_prime_of_not_mem45 below · cited by 8 · depth 10 - Hecke polynomials killing regular differentials kill J₀(N)
ModularCurve.freeAlgebra_lift_heckeOperatorBar_eq_zero_of_lift_heckeDiffBar_eq_zero798 below · cited by 1 · depth 10 - The full Fricke map is an involution
ModularCurve.frickeInvolutionFull_apply_apply1 below · cited by 7 · depth 10 - Frobenius inputs on the mod-ℓ modular function field
ModularCurve.frobeniusInputsModL111 below · cited by 8 · depth 10 - Eichler–Shimura relation on the p-adic Tate module of J₀(N)
ModularCurve.frobeniusQuadratic_tateModule_jZero993 below · cited by 9 · depth 10 - At prime level the two modular function fields agree
ModularCurve.full_eq_of_prime2 below · cited by 14 · depth 10 - Function field generation at squarefree level
ModularCurve.functionFieldGeneration_of_squarefree57 below · cited by 2 · depth 10 - The level-3 modular function field over ℚ̄ has genus 0
ModularCurve.genus_modularFunctionFieldBar_three438 below · cited by 1 · depth 10 - Genus zero for the level-2 modular function field over ℚ̄
ModularCurve.genus_modularFunctionFieldBar_two451 below · cited by 2 · depth 10 - Existence of the P-primary Néron torsion sheaf when q ∣ n
ModularCurve.hasJZeroNeronPrimaryTorsionSheaf_of_dvd4,070 below · cited by 1 · depth 10 - Principal divisors of degree zero on L· F_N^{full}
ModularCurve.hasPrincipalDivisors_laurentBaseChange_modularFunctionFieldFull_unconditional74 below · cited by 10 · depth 10 - Principal divisors on the level-N modular function field
ModularCurve.hasPrincipalDivisors_modularFunctionFieldC_of_perfectField112 below · cited by 32 · depth 10 - Integrality of the β degeneracy map from a symmetric Φ_ℓ
ModularCurve.heckeBetaBarIntegral_of_modularPolynomialData7 below · cited by 1 · depth 10 - Base change commutes with the degeneracy map q↦ q^ℓ
ModularCurve.heckeBetaBar_coeffEmb2 below · cited by 7 · depth 10 - Commutativity of T_ℓ and T_{ℓ'} on divisors, given exchange
ModularCurve.heckeDivBar_comm_of_heckeExchangeAt8 below · cited by 1 · depth 10 - Hecke correspondence acts on the cuspidal divisor by 1+ℓ
ModularCurve.heckeDivBar_cuspidalDivisor2 below · cited by 1 · depth 10 - Uₚ fixes the cuspidal divisor from summed fibre data
ModularCurve.heckeDivBar_cuspidalDivisor_self_of_sum2 below · cited by 1 · depth 10 - Fibre identification Uₚ D + wₚ· D = σ^*ι_* D on X₀(p)
ModularCurve.heckeDivBar_self_add_frickeInvolutionBar_smul208 below · cited by 1 · depth 10 - Hecke relation on the cuspidal divisor descends to the cuspidal class
ModularCurve.heckePic0Bar_cuspidalClass_of_heckeDivBar0 below · cited by 1 · depth 10 - Hecke relations on S₂(Γ₀(N)) hold on J₀(N)
ModularCurve.heckeRelations_jZero843 below · cited by 1 · depth 10 - Modular function field in characteristic ℓ ∤ N is a curve
ModularCurve.isCurveOver_modularFunctionFieldC_of_good132 below · cited by 13 · depth 10 - The place at infinity is a cusp for j
ModularCurve.isCusp_cuspInftyBar5 below · cited by 26 · depth 10 - The cusp ̄ 0 is a pole of j
ModularCurve.isCusp_cuspZeroBar10 below · cited by 20 · depth 10 - Cusps are the places where j has negative order
ModularCurve.isCusp_iff_ord_neg1 below · cited by 19 · depth 10 - `frickeInvolutionFull` is a Fricke automorphism at level p²
ModularCurve.isFrickeAutFull_frickeInvolutionFull_sq64 below · cited by 2 · depth 10 - Integrality of Δ(q)/Δ(q^ℓ) over ℚ[j]
ModularCurve.isIntegral_adjoin_jq_modularUnitSeries30 below · cited by 16 · depth 10 - Inverse of Ogg's modular unit is integral over ℚ[j]
ModularCurve.isIntegral_adjoin_jq_modularUnitSeries_inv30 below · cited by 16 · depth 10 - Integrality of ̄ j(q^N) over K(̄ j(q))
ModularCurve.isIntegral_jqNModC_all43 below · cited by 23 · depth 10 - Integrality of j(q^N) over K(j(q)) at all levels
ModularCurve.isIntegral_jqNModC_all_of_modularPolynomialFamily4 below · cited by 2 · depth 10 - Degree-zero divisors on the level-one j-line are principal
ModularCurve.isPrincipal_of_degree_eq_zero_charLOne20 below · cited by 2 · depth 10 - Values of the j-coordinate at places of X₀(N)
ModularCurve.jCoordinate_spec_modularFunctionFieldBar198 below · cited by 11 · depth 10 - Mazur's dévissage inequality h¹+α≤ h⁰+δ at every level
ModularCurve.jZeroNeronTorsionSheaf_device_v51,468 below · cited by 1 · depth 10 - Common linear growth of δ(m) and trivial-step counts
ModularCurve.jZeroNeronTorsionSheaf_growth_v53,287 below · cited by 1 · depth 10 - Boundedness of h⁰(m) for the primary Néron torsion sheaf
ModularCurve.jZeroNeronTorsionSheaf_h0_bounded_v52,186 below · cited by 2 · depth 10 - Kronecker congruence at level N, norm form
ModularCurve.kroneckerCongruence_norm_heckeBetaBar151 below · cited by 3 · depth 10 - Base change of the full divisor-expansion field equals L(j,j_M)
ModularCurve.laurentBaseChange_adjoin_pair52 below · cited by 22 · depth 10 - Base change of the full modular function field to L
ModularCurve.laurentBaseChange_modularFunctionFieldFull2 below · cited by 9 · depth 10 - ℚₚ-independence of Hecke operators on the rational Tate module
ModularCurve.linearIndependent_rationalHeckeRep_of_linearIndependent712 below · cited by 3 · depth 10 - Quadratic relation for every g on J[𝔪]
ModularCurve.mTorsionGaloisRep_quadratic_of_frobeniusQuadratic_of_frobeniusPowerDense0 below · cited by 2 · depth 10 - Toric exclusion with scalar Frobenius forces q=0 or q=1
ModularCurve.mazurPrinciple_toric_exclusion_of_scalar0 below · cited by 1 · depth 10 - q-expansion principle at level one: invariance implies f∈ℚ[j]
ModularCurve.mem_adjoin_jq_of_hasSum_of_slash_invariant10 below · cited by 5 · depth 10 - Igusa: K(j(q),j(q^N)) contains every j(qᵈ), ℓ∤ N
ModularCurve.modularFunctionFieldC_eq_modularFunctionFieldFullC_of_charP_pos109 below · cited by 22 · depth 10 - K(j(q),j(q^N)) equals K(j(qᵈ):d∣ N) in characteristic zero
ModularCurve.modularFunctionFieldC_eq_modularFunctionFieldFullC_of_charZero70 below · cited by 13 · depth 10 - Extensionality of ℚ-algebra maps on ℚ(j(qᵈ):d∣ N)
ModularCurve.modularFunctionFieldFull_algHom_ext0 below · cited by 17 · depth 10 - Level-N modular function field equals its full divisor field
ModularCurve.modularFunctionField_eq_full69 below · cited by 30 · depth 10 - Kronecker's congruence for Φ_ℓ modulo ℓ
ModularCurve.modularPolynomial_kronecker48 below · cited by 15 · depth 10 - Ogg's modular unit lies in the full level-N function field
ModularCurve.modularUnitSeries_mem_modularFunctionFieldFull31 below · cited by 80 · depth 10 - Toric torsion inequality at a q'-new eigenform, q' not≡ 1
ModularCurve.natCard_toricTorsion_le_of_not_exists_hasLowerLevelTorsion_of_attachedOddBlr_sqf_five_of_six_mul_dvd_of_neZero12,128 below · cited by 1 · depth 10 - Frobenius node permutation is an involution
ModularCurve.nodePerm_arithFrobC_nodePerm_of_forall_smul_smul_eq0 below · cited by 2 · depth 10 - Existence of modular polynomial data at squarefree levels
ModularCurve.nonempty_modularPolynomialData_of_squarefree40 below · cited by 19 · depth 10 - Order at the cusp ∞ is the q-order
ModularCurve.ord_cuspInftyBar1 below · cited by 20 · depth 10 - The q-expansion of j has order -1 at ∞
ModularCurve.ord_cuspInftyBar_coeffEmb_jq2 below · cited by 50 · depth 10 - Order of j(qᵈ) at the cusp at infinity
ModularCurve.ord_cuspInftyBar_coeffEmb_qExpand4 below · cited by 14 · depth 10 - Order of j at the cusp 0 equals -N
ModularCurve.ord_cuspZeroBar_coeffEmb_jq7 below · cited by 11 · depth 10 - Order of j(qᵇ) at the cusp ̄ 0
ModularCurve.ord_cuspZeroBar_coeffEmb_qExpand6 below · cited by 4 · depth 10 - Ramification over j=0 divides 3 for odd level
ModularCurve.ord_jBar_dvd_three_of_odd227 below · cited by 2 · depth 10 - Ramification over j=1728 divides 2 for odd level
ModularCurve.ord_jBar_sub_1728_dvd_two_of_odd227 below · cited by 2 · depth 10 - Unramifiedness of jmath̄ - c away from 0 and 1728, odd level
ModularCurve.ord_jBar_sub_eq_one_of_ne_zero_of_ne_of_odd225 below · cited by 2 · depth 10 - Coefficientwise base change preserves the order of a Laurent series
ModularCurve.order_coeffEmb0 below · cited by 10 · depth 10 - Substitution q ↦ q^N multiplies the order by N
ModularCurve.order_qExpand0 below · cited by 19 · depth 10 - Eichler–Shimura: period pair map onto parabolic homomorphisms
ModularCurve.periodHomPair_range_eq_parabolicHoms575 below · cited by 3 · depth 10 - Hecke equivariance of the period map in weight two
ModularCurve.periodMap_heckeTLin3 below · cited by 7 · depth 10 - Period map intertwines U_q with the cohomological operator
ModularCurve.periodMap_heckeULin3 below · cited by 5 · depth 10 - Injectivity of the period map on weight-2 cusp forms
ModularCurve.periodMap_injective6 below · cited by 2 · depth 10 - Period map of a weight-2 cusp form is parabolic
ModularCurve.periodMap_mem_parabolicHoms4 below · cited by 3 · depth 10 - Places of the level-N modular function field have degree one
ModularCurve.place_deg_eq_one_of_isAlgClosed88 below · cited by 91 · depth 10 - Rationalised Eichler–Shimura: VₚJ₀(M) free of rank two
ModularCurve.rationalRankTwoCyclotomic_family993 below · cited by 9 · depth 10 - Relative degree ψ(N) of k(̄ j,̄ j_N) over k(̄ j)
ModularCurve.relfinrank_adjoin_jqModC_modularFunctionFieldC_eq_dedekindPsi105 below · cited by 55 · depth 10 - Adjoining a new prime level multiplies the degree by ℓ+1
ModularCurve.relfinrank_full_mul_prime53 below · cited by 1 · depth 10 - Relative degree ψ(p²) of the level-p² divisor-expansion field
ModularCurve.relfinrank_full_sq59 below · cited by 3 · depth 10 - Relative degree over the j-line under base change of constants
ModularCurve.relfinrank_laurentBaseChange_modularFunctionFieldFull111 below · cited by 32 · depth 10 - Degree ℓ+1 of the full level-ℓ modular function field
ModularCurve.relfinrank_modularFunctionFieldFull_prime111 below · cited by 3 · depth 10 - Galois and Hecke actions commute on Tₚ J₀(N)
ModularCurve.rep_tateModule_jZero_comm16 below · cited by 7 · depth 10 - Inertia at ℓ ∤ Np acts trivially on Tₚ J₀(N)
ModularCurve.rep_tateModule_jZero_eq_self_of_mem_inertiaSubgroupIn979 below · cited by 3 · depth 10 - Finiteness of supersingular places of the modular function field
ModularCurve.ssPlaces_finite120 below · cited by 41 · depth 10 - The polar divisor of jmath̄ has degree ψ(N)
ModularCurve.sum_neg_ord_jBar_eq_dedekindPsi188 below · cited by 5 · depth 10 - Zeros of jmath̄-j₀ on X₀(N) total ψ(N)
ModularCurve.sum_ord_jBar_sub_eq_dedekindPsi144 below · cited by 12 · depth 10 - Coefficient law for θ = q d/dq on Laurent series
ModularCurve.theta_coeff73 below · cited by 35 · depth 10 - Leibniz rule for θ = q d/dq on Laurent series
ModularCurve.theta_mul73 below · cited by 16 · depth 10 - Toric dichotomy for J₀(Nq) at the monodromy toric part
ModularCurve.toricDichotomy_toricMonodromyPart_jZero3,554 below · cited by 2 · depth 10 - Frobenius acts as q T_q on the monodromy toric part of J₀(Nq)
ModularCurve.toricFrobeniusHecke_toricMonodromyPart_jZero5,206 below · cited by 3 · depth 10 - Frob_q² = q² on the monodromy toric part of J₀(Nq)
ModularCurve.toricFrobeniusSq_toricMonodromyPart_jZero3,553 below · cited by 2 · depth 10 - Special-fibre n-torsion is finite and bounded by generic torsion
ModularCurve.torsionCard_le_of_goodReduction_of_charP_of_not_dvd1,780 below · cited by 1 · depth 10 - Transcendence of the q-expansion j(q) over ℚ
ModularCurve.transcendental_jq1 below · cited by 56 · depth 10 - Order of the cuspidal class divides ℓ-1
ModularCurve.addOrderOf_cuspidalClass_dvd109 below · cited by 1 · depth 11 - Squared arithmetic Frobenius fixes supersingular places, q ∤ N
ModularCurve.arithFrobC_smul_arithFrobC_smul_of_mem_ssPlaces_of_not_dvd384 below · cited by 4 · depth 11 - Arithmetic Galois action fixes ι_L(F₀) pointwise
ModularCurve.arithmeticGalois_smul_coeffEmb0 below · cited by 13 · depth 11 - Places above j=j₀ count moduli points with that j-invariant
ModularCurve.card_eq_natCard_moduliPoint_j_eq_of_EMD181 below · cited by 4 · depth 11 - Cayley–Hamilton identity for all σ on J[𝔪]
ModularCurve.cayleyHamilton_forall_of_frobeniusQuadratic_of_dense1 below · cited by 2 · depth 11 - Fricke involution sends the ∞-expansion to the 0-expansion
ModularCurve.coe_frickeInvolutionFull_eq_of_hasSum_of_gamma0_invariant58 below · cited by 2 · depth 11 - ℓ-th powers of reduced Laurent series and t↦ t^ℓ
ModularCurve.coeffMap_pow_char_eq_qExpand_of_frobenius1 below · cited by 2 · depth 11 - Coefficient transposition under the bivariate swap
ModularCurve.coeff_coeff_swapBivar0 below · cited by 4 · depth 11 - q-expansion of the Hecke differential operator for ℓ ∣ N
ModularCurve.coeff_diffQExpBar_heckeDiffBar_of_dvd148 below · cited by 3 · depth 11 - SL₂(ℤ)-invariance plus holomorphic q-expansion forces constancy
ModularCurve.coeff_eq_zero_of_hasSum_of_slash_invariant0 below · cited by 1 · depth 11 - Uniqueness of the place with ̃ j polar and ̃ jₚ̃ j⁻ᵖ vanishing
ModularCurve.cuspChartInftyZero_place_unique106 below · cited by 1 · depth 11 - Uniqueness of the place where jmath̃ jmath̃ₚ⁻ᵖ vanishes
ModularCurve.cuspChartZeroInfty_place_unique98 below · cited by 1 · depth 11 - Dedekind ψ at squarefree level
ModularCurve.dedekindPsi_of_squarefree2 below · cited by 5 · depth 11 - Γ₀(N)-invariance of Δ(τ)/Δ(Nτ)
ModularCurve.discriminant_div_discriminant_heckeDiagMatrix_smul1 below · cited by 9 · depth 11 - Dimension of parabolic homomorphisms for Γ₀(N)
ModularCurve.eichlerShimura_dim_parabolic574 below · cited by 1 · depth 11 - Admissibility of the Eisenstein-primary torsion of J₀(p)
ModularCurve.eisensteinPrimaryTorsion_isMazurAdmissible_heckeModuleBar1,059 below · cited by 5 · depth 11 - Assembly of `EMD` from five local inputs
ModularCurve.emd_of_beta_docks0 below · cited by 3 · depth 11 - Killing the pole at q=0 by a polynomial in j(q)
ModularCurve.exists_aeval_jq_sub_holomorphicAtInfty0 below · cited by 1 · depth 11 - Atkin–Lehner automorphism wₚ over ℚ̄
ModularCurve.exists_algEquiv_modularFunctionFieldBar_atkinLehner74 below · cited by 4 · depth 11 - The period lattice of X₀(N) spans S₂(Γ₀(N))^∨ over ℝ
ModularCurve.exists_basis_periodLattice_linearIndependent_real_span_eq_top567 below · cited by 20 · depth 11 - Rational weight-2 cusp forms as Kähler differentials
ModularCurve.exists_coeffMap_diffQExpBar_eq_qExpansion108 below · cited by 4 · depth 11 - Cyclic p-subgroups as places above j₀, p odd
ModularCurve.exists_elliptic_cycSub_orbitMap_prime_of_ne_two296 below · cited by 2 · depth 11 - Frobenius shift permutes supersingular node pairs
ModularCurve.exists_equiv_forall_apply_fst_snd_eq_fst_fst_of_forall_mem_iff_mem_ssNodePairsQExp4 below · cited by 6 · depth 11 - Finite flat Hopf model of the ℓ^k-torsion of J₀(p)
ModularCurve.exists_finiteFlat_model_jZero_torsion_reductionModL_eq_zero1,771 below · cited by 3 · depth 11 - Rational Tate module of J₀(N) free of rank two
ModularCurve.exists_heckeEquivariant_linearEquiv_rationalTateModule_jZero_fun_two744 below · cited by 3 · depth 11 - Hecke-equivariant comparison TₚJ₀(N)≅mathbb Zₚ⊗ H₁
ModularCurve.exists_heckeEquivariant_linearEquiv_tateModule_jZero_padicInt_tensor_periodLattice711 below · cited by 9 · depth 11 - Hecke-equivariant Abel–Jacobi map for J₀(N)(ℚ̄)
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_jZero_quotient_periodLattice710 below · cited by 11 · depth 11 - Existence of the Atkin–Lehner automorphism wₚ at level Np
ModularCurve.exists_isAtkinLehnerAutFull_of_prime_of_not_dvd73 below · cited by 96 · depth 11 - Existence of a Fricke automorphism at prime level
ModularCurve.exists_isFrickeAut43 below · cited by 1 · depth 11 - Existence of a Fricke automorphism at level p²
ModularCurve.exists_isFrickeAutFull_sq63 below · cited by 1 · depth 11 - Comparison of the H=top and Igusa integral models
ModularCurve.exists_iso_xHDRLevel_top_drLevel_epsInf_pointEquivPlace191 below · cited by 1 · depth 11 - Néron object of J₀(N₀p) at p with its bridges
ModularCurve.exists_jZeroNeronObjectAtP_and_bridge4,509 below · cited by 4 · depth 11 - Prime-to-q monodromy lies in the toric locus
ModularCurve.exists_jZeroSemistableSpecialization_monodromy_mem_toricLocus3,551 below · cited by 1 · depth 11 - Toric 𝔪-torsion of J₀(Nq) lies in monodromy part
ModularCurve.exists_jZeroSemistableSpecialization_toricLocus_heckeTorsion_le_toricMonodromyPart3,551 below · cited by 1 · depth 11 - Existence of a Kronecker-congruent modular polynomial at a prime
ModularCurve.exists_kroneckerCongruence_of_prime52 below · cited by 27 · depth 11 - Lifting ℓ-power torsion in the Eisenstein kernel through reduction
ModularCurve.exists_le_mem_eisensteinKernelSubmodule_torsionBy_reductionModL_eq1,884 below · cited by 1 · depth 11 - Upper bound for the Eisenstein ideal index at level p
ModularCurve.exists_mem_eisensteinIdeal_heckeProj_eq_eisensteinNumerator593 below · cited by 2 · depth 11 - Lifting m-torsion along the Eisenstein quotient map
ModularCurve.exists_mem_torsionBy_eisensteinQuotientMk_eq414 below · cited by 1 · depth 11 - Elements of ℂF_N are ratios of forms on Γ₀(N)
ModularCurve.exists_modularForm_mul_qExpansion_eq_of_mem_laurentBaseChange14 below · cited by 19 · depth 11 - Values of integral functions at places where j is A-valued
ModularCurve.exists_ord_sub_pos_of_integral_affineBaseFin0 below · cited by 11 · depth 11 - Chart dichotomy for ̄ j in terms of ord_w
ModularCurve.exists_ord_sub_pos_or_exists_ord_inv_sub_pos_of_dataAll113 below · cited by 4 · depth 11 - Existence of a model prolongation tuple at a place above q
ModularCurve.exists_placeSpecialization_prolongationTuple_isModel_regularityLaw_nodeValueLaw_orderLawFixed1,659 below · cited by 4 · depth 11 - Normalised Hahn-series embeddings induce places of F_N
ModularCurve.exists_place_of_emb97 below · cited by 5 · depth 11 - Analytic functions on X₀(N) are realizations of ℂF_N
ModularCurve.exists_realize_eventuallyEq_of_meromorphic152 below · cited by 1 · depth 11 - Reduction of J₀(N) modulo ℓ∤ N
ModularCurve.exists_reductionModL_jZero_jZeroC993 below · cited by 1 · depth 11 - Gauss reduction of X₀(N) at a place above ℓ ∤ N
ModularCurve.exists_regularProlongation_modularFunctionFieldBar109 below · cited by 9 · depth 11 - Riemann's inequality for the function field of X₀(N)
ModularCurve.exists_riemannConstant_modularFunctionFieldBar210 below · cited by 2 · depth 11 - A-values at places of functions integral over A[̄ j⁻¹]
ModularCurve.exists_sub_mem_nonunits_of_integral_affineBaseInf0 below · cited by 7 · depth 11 - Functions with order zero at all cuspidal places: nonzero limits
ModularCurve.exists_tendsto_realize_smul_of_forall_ord_eq_zero16 below · cited by 1 · depth 11 - Deligne–Rapoport model at p ∥ M with wₚ pinned generically
ModularCurve.exists_xHDRModelAtP_atkinLehner_generic1,218 below · cited by 11 · depth 11 - Finiteness of the component group for positive weights
ModularCurve.finite_componentGroup_of_pos6 below · cited by 8 · depth 11 - Finitely many cyclic subgroups of order N on an elliptic curve over ℚ̄
ModularCurve.finite_cycSub2 below · cited by 4 · depth 11 - Degree [K(j)(j(qᵈ)):K(j)]=ψ(d) over any field
ModularCurve.finrank_adjoin_jqNModC_eq_dedekindPsi_of_socket88 below · cited by 9 · depth 11 - Relative degree [ℚ(j)(j_N):ℚ(j)]=ψ(N) at squarefree level
ModularCurve.finrank_adjoin_jqN_eq_of_squarefree54 below · cited by 1 · depth 11 - Degree p of j(q^{p^2}) over a field containing j(qᵖ)
ModularCurve.finrank_adjoin_jqN_sq_of_not_mem46 below · cited by 1 · depth 11 - Hecke relations on S₂(Γ₀(N)) descend to J₀(N)
ModularCurve.freeAlgebra_lift_heckeOperatorBar_eq_zero_of_lift_cuspForm_eq_zero712 below · cited by 2 · depth 11 - Fricke involution over ℚ̄ exchanges j(qᵃ) and j(qᵇ)
ModularCurve.frickeInvolutionBar_coeffEmb_qExpand0 below · cited by 43 · depth 11 - The geometric Fricke involution squares to the identity
ModularCurve.frickeInvolutionBar_frickeInvolutionBar3 below · cited by 39 · depth 11 - Determinant ℓ of Frobenius in any rank-two Hecke basis on VₚJ₀(N)
ModularCurve.frobenius_coordDet_eq_of_basis_rationalTateModule_jZero915 below · cited by 1 · depth 11 - At squarefree level, divisor expansions are generated by prime ones
ModularCurve.full_eq_adjoin_primes46 below · cited by 4 · depth 11 - Divisor expansions at level p² generate ℚ(j,jₚ,j_{p²})
ModularCurve.full_sq_eq_adjoin46 below · cited by 3 · depth 11 - Divisor expansions at level p²ℓ: generation by prime chains
ModularCurve.full_sq_mul_prime_eq_adjoin46 below · cited by 1 · depth 11 - Function field generation at prime level
ModularCurve.functionFieldGeneration_of_prime0 below · cited by 2 · depth 11 - Genus of the mod-ℓ modular function field equals that over ℚ̄
ModularCurve.genusFF_modularFunctionFieldFullC_eq_genusFF_modularFunctionFieldBar729 below · cited by 20 · depth 11 - Genus of the modular function field equals 1+ψ/12-ν₂/4-ν₃/3-c_∞/2
ModularCurve.genus_modularFunctionFieldBar_eq_genusFormula499 below · cited by 14 · depth 11 - Atkin–Lehner automorphism exchanges the two degeneracy legs
ModularCurve.geomAut_atkinLehner_comp_legs1 below · cited by 38 · depth 11 - Divisible subgroup killed by the good eigenideal lowers the level
ModularCurve.goodEigensystemOccursAt_of_divisible787 below · cited by 4 · depth 11 - Geometric fibre count for the 2-primary Eisenstein torsion sheaf
ModularCurve.hasJZeroNeronTorsionSheaf_two_fibreCount_of_dvd_eisensteinNumerator_v54,074 below · cited by 1 · depth 11 - Mazur's principle at p for J₀(N₀p)
ModularCurve.hasLowerLevelTorsion_jZero_of_isPeuRamifieeAt5,914 below · cited by 1 · depth 11 - Mazur's principle at p for J₀(N₀p), p ≥ 5
ModularCurve.hasLowerLevelTorsion_jZero_of_isPeuRamifieeAt_of_five_le5,915 below · cited by 1 · depth 11 - Principal divisors on the degeneracy roof k(̃ j,̃ j_N,̃ j_q,̃ j_{Nq})
ModularCurve.hasPrincipalDivisors_charLDegeneracyRoof79 below · cited by 29 · depth 11 - Formal q-series j(q) sums to E₄³/Δ
ModularCurve.hasSum_jq_qParam4 below · cited by 11 · depth 11 - q-expansion of Δ(Nτ)/Δ(τ) realised at period one
ModularCurve.hasSum_modularUnitSeries_inv_qParam6 below · cited by 4 · depth 11 - q-expansion of Δ(τ)/Δ(Nτ) at the cusp ∞
ModularCurve.hasSum_modularUnitSeries_qParam6 below · cited by 11 · depth 11 - q_ℓ-expansion of F(ℓτ) is A(q^{ℓ^2})
ModularCurve.hasSum_qParam_heckeDiagMatrix_smul0 below · cited by 4 · depth 11 - Twisted q-expansion at period ℓ under τ↦(τ+b)/ℓ
ModularCurve.hasSum_qParam_heckeMatrix_smul0 below · cited by 5 · depth 11 - Cauchy product of Laurent q-expansions on H
ModularCurve.hasSum_qParam_mul_laurent1 below · cited by 24 · depth 11 - q-expansion of Ogg's unit at the cusp 0
ModularCurve.hasSum_smul_modularUnitSeries_inv_qParam6 below · cited by 5 · depth 11 - q_N-expansion of Ogg's unit at the cusp 0
ModularCurve.hasSum_smul_modularUnitSeries_qParam6 below · cited by 3 · depth 11 - Fricke conjugation swaps the two degeneracy maps at level p²
ModularCurve.heckeAlphaBar_frickeInvolutionBar_sq148 below · cited by 1 · depth 11 - Hecke and diamond inputs for the q-expansion model of X_H(M)
ModularCurve.heckeDiamondInputsHAll66 below · cited by 48 · depth 11 - Composite of two Hecke divisor correspondences through the roof level
ModularCurve.heckeDivBar_heckeDivBar_of_heckeExchangeAt7 below · cited by 1 · depth 11 - Hecke inputs at the q-degeneracy roof from separability
ModularCurve.heckeInputsFibre_of_separable_phi_map62 below · cited by 1 · depth 11 - Scalar transport: elements projecting to integers act by multiplication
ModularCurve.heckeModuleBar_smul_eq_zsmul_of_heckeProj_eq744 below · cited by 1 · depth 11 - Vanishing of mathfrak P_q^M-torsion in J₀(p) when q ∤ n(p)
ModularCurve.heckeTorsion_eisensteinMaximalIdeal_pow_eq_bot_of_not_dvd_eisensteinNumerator1,237 below · cited by 1 · depth 11 - Inertia degrees are one over an algebraically closed constant field
ModularCurve.inertiaDegAlong_eq_one_laurentBaseChange93 below · cited by 13 · depth 11 - K(j(q),j(q^N)) is a curve over K when j(q^N) is separable
ModularCurve.isCurveOver_modularFunctionFieldC_of_isSeparable_jqNModC112 below · cited by 4 · depth 11 - The full modular function field is a curve over K
ModularCurve.isCurveOver_modularFunctionFieldFullC111 below · cited by 67 · depth 11 - Locally bounded level-N modular functions are integral over ℂ[j]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_of_forall_isBoundedUnder_realize18 below · cited by 1 · depth 11 - Integrality over ℚ[j] of a Γ₀(ℓ)-invariant q-expansion
ModularCurve.isIntegral_adjoin_jq_of_hasSum_of_gamma0_invariant22 below · cited by 3 · depth 11 - Separability of j(q^N) over K(j) in good characteristic
ModularCurve.isSeparable_jqNModC_of_good91 below · cited by 16 · depth 11 - Separability of j(q^M) over K(j(q)) for M invertible
ModularCurve.isSeparable_jqNModC_of_natCast_ne_zero42 below · cited by 60 · depth 11 - Pinned α bounded by trivial steps of an admissible chain
ModularCurve.jZeroNeronTorsionSheaf_alpha_le_filtAlpha_v51,990 below · cited by 1 · depth 11 - Linear growth of the toric defect at 2
ModularCurve.jZeroNeronTorsionSheaf_growth_two_v53,213 below · cited by 1 · depth 11 - Linear growth of δ_m and α_m for the J₀(p) torsion sheaf
ModularCurve.jZeroNeronTorsionSheaf_inv_linearGrowth_v53,057 below · cited by 1 · depth 11 - j-series over ℂ equals E₄³/Δ in ℂ((q))
ModularCurve.jqModC_eq_qExpansion_E4_cube_div_discriminant10 below · cited by 51 · depth 11 - j-series lies in the integral form ratios over any K, Γ
ModularCurve.jqModC_mem_intFormRatiosC2 below · cited by 242 · depth 11 - Frobenius collapse: ̄ j(q^{Nℓ^k})=̄ j(q^N)^{ℓ^k}
ModularCurve.jqNModC_mul_pow_eq_pow3 below · cited by 2 · depth 11 - j(qᵖ) is not a rational function of j(q)
ModularCurve.jqNModC_prime_not_mem_adjoin_of_charZero48 below · cited by 8 · depth 11 - j(q^r) is not rational in j and the j(qᵖ), p∈ S
ModularCurve.jqN_prime_not_mem_adjoin48 below · cited by 4 · depth 11 - j(q^{p^2}) lies outside ℚ(j,j(qᵖ),j(q^s)_{s∈ S})
ModularCurve.jqN_sq_not_mem_adjoin49 below · cited by 2 · depth 11 - Dedekind ψ dominates its argument
ModularCurve.le_dedekindPsi0 below · cited by 2 · depth 11 - Hecke divisor of a single place, transported along sp
ModularCurve.mapDomain_heckeDivBar_single1 below · cited by 8 · depth 11 - Finite and toric parts transport from Jₜop(N₀p) to J₀(N₀p)
ModularCurve.map_finPts_jHNeronObjectAtP_top_eq_and_map_toricPts_eq_of_pic0Congr_of_bridge136 below · cited by 1 · depth 11 - Integrality over A[̄ j] forces coefficients into A
ModularCurve.mem_integralCoeffs_of_integral_affineBaseFin114 below · cited by 21 · depth 11 - Integrality over A[1/j] forces Laurent coefficients into A
ModularCurve.mem_integralCoeffs_of_integral_affineBaseInf114 below · cited by 12 · depth 11 - Weight-2 cusp form q-expansion forces a regular differential
ModularCurve.mem_regularDifferentialsBar_of_coeffMap_diffQExpBar_eq_qExpansion266 below · cited by 4 · depth 11 - Ogg's unit Δ(q)/Δ(q^ℓ) lies in ℚ(j,j_ℓ)
ModularCurve.modularUnitSeries_mem_modularFunctionField30 below · cited by 4 · depth 11 - Normalised Hahn-series embeddings at a place above j₀
ModularCurve.natCard_normalized_algHom_jBar_eq_toNat_ord151 below · cited by 7 · depth 11 - Interchange inequality for toric parts of the 𝔪-torsion
ModularCurve.natCard_toricTorsion_le_of_not_exists_hasLowerLevelTorsion_of_isMaximal_of_attachedOddBlr_sqf_five_of_six_mul_dvd_of_neZero12,127 below · cited by 1 · depth 11 - n-torsion of J₀(N) over ℚ̄ has order n^{rank Λ_N}
ModularCurve.natCard_torsion_jZero_eq_pow_finrank_periodLattice712 below · cited by 12 · depth 11 - Existence of the Néron primary-torsion core for J₀(p)
ModularCurve.nonempty_jZeroNeronPrimaryTorsionCore_of_dvd3,635 below · cited by 1 · depth 11 - Finite flat Hopf models over mathbb Z_{(ℓ)} exist
ModularCurve.nonempty_jZeroNeronPrimaryTorsionFFModels11 below · cited by 2 · depth 11 - Existence of a flag for the Eisenstein-primary torsion sheaf
ModularCurve.nonempty_jZeroNeronPrimaryTorsionFlag1,089 below · cited by 4 · depth 11 - Existence of admissible invariants for a Néron core
ModularCurve.nonempty_jZeroNeronPrimaryTorsionInvPins1,488 below · cited by 2 · depth 11 - Semistable specialisation datum for J₀(Nq) with Néron clauses
ModularCurve.nonempty_jZeroSemistableSpecialization_neronClauses3,550 below · cited by 5 · depth 11 - Every nonnegative q-coefficient of j is at least 1
ModularCurve.one_le_coeff_jq0 below · cited by 4 · depth 11 - Order -1 of j(q^N) at the cusp zero
ModularCurve.ord_cuspZeroBar_coeffEmb_jqN7 below · cited by 2 · depth 11 - Order zero at cuspidal places for nonvanishing cusp limits
ModularCurve.ord_eq_zero_of_jq_not_mem_of_realize_tendsto159 below · cited by 1 · depth 11 - Order at the q-adic place equals q-expansion order
ModularCurve.ord_qInftyPlaceBar0 below · cited by 3 · depth 11 - The q-expansion of j has order -1
ModularCurve.order_jqModC0 below · cited by 48 · depth 11 - Injectivity of the Eichler–Shimura period pair map
ModularCurve.periodHomPair_injective6 below · cited by 4 · depth 11 - Additivity of the weight-two period map
ModularCurve.periodMap_add3 below · cited by 3 · depth 11 - Period map equals the period character of any primitive
ModularCurve.periodMap_eq_periodHom2 below · cited by 10 · depth 11 - Homogeneity of the weight-two period map
ModularCurve.periodMap_smul3 below · cited by 4 · depth 11 - Canonical transport J_H(M,top)→ J₀(M) is Galois-, Hecke- and diamond-compatible
ModularCurve.pic0Congr_jH_top_jZero_galois_hecke_diamond_compat187 below · cited by 1 · depth 11 - A Hahn-series embedding induces at most one place
ModularCurve.place_eq_of_induces0 below · cited by 5 · depth 11 - Frobenius-stability of the supersingular j-set
ModularCurve.pow_mem_ssJSet_iff0 below · cited by 20 · depth 11 - Frobenius identity ̄ j(q^ℓ) = ̄ j(q)^ℓ in characteristic ℓ
ModularCurve.qExpand_jqModC_eq_pow_unconditional1 below · cited by 62 · depth 11 - Period pairs give parabolic homomorphisms on Γ₀(N)
ModularCurve.range_periodHomPair_le_parabolicHoms8 below · cited by 2 · depth 11 - Realization of a Laurent series agrees with g/h
ModularCurve.realize_eq_div0 below · cited by 23 · depth 11 - Inertia acts trivially after reduction of Pic⁰
ModularCurve.reductionModL_smul_eq_self_of_mem_inertiaSubgroupIn755 below · cited by 9 · depth 11 - Degree of ℚ(j)(jₚ : p ∈ S) over ℚ(j)
ModularCurve.relfinrank_adjoin_primes53 below · cited by 1 · depth 11 - Relative degree ψ(N) at squarefree level
ModularCurve.relfinrank_full_of_squarefree53 below · cited by 2 · depth 11 - Finiteness of the supersingular j-invariants
ModularCurve.ssJSet_finite6 below · cited by 54 · depth 11 - Prime-to-ℓ torsion lifts along reduction mod ℓ
ModularCurve.surjOn_reductionModL_torsion_of_not_dvd1,760 below · cited by 3 · depth 11 - Evaluation symmetry over ℚ((q)) forces symmetry of Φ
ModularCurve.swapBivar_eq_of_evalSymm0 below · cited by 21 · depth 11 - Counting places where jmath̄ vanishes, for odd N
ModularCurve.three_mul_card_eq_dedekindPsi_add_of_forall_mem_iff_pos_ord_jBar299 below · cited by 1 · depth 11 - Moduli points with j = 0: 3#=ψ(N)+2ν₃(N)
ModularCurve.three_mul_natCard_moduliPoint_j_eq_zero_eq_dedekindPsi_add_two_mul_nuThree24 below · cited by 3 · depth 11 - Two-component exhaustion at an ℓ-adic place of X₀(Nℓ)
ModularCurve.twoComponentExhaustion_valuation_mul_lt_one_of_ord_inv_sub_pos218 below · cited by 2 · depth 11 - Two-component exhaustion at level Nℓ: product of values is a non-unit
ModularCurve.twoComponentExhaustion_valuation_mul_lt_one_of_ord_sub_pos218 below · cited by 2 · depth 11 - Counting places above j=1728 on X₀(N), N odd
ModularCurve.two_mul_card_eq_dedekindPsi_add_of_forall_mem_iff_pos_ord_jBar_sub_1728299 below · cited by 1 · depth 11 - Moduli points with j=1728: 2 #=ψ(N)+ν₂(N)
ModularCurve.two_mul_natCard_moduliPoint_j_eq_1728_eq_dedekindPsi_add_nuTwo22 below · cited by 3 · depth 11 - Coefficients in the maximal ideal force the value there
ModularCurve.valuation_lt_one_of_ord_sub_pos_of_coeff_lt_one117 below · cited by 4 · depth 11 - Small Fourier coefficients force pole-chart values into the maximal ideal
ModularCurve.valuation_lt_one_of_sub_mem_nonunits_of_coeff_lt_one_inf131 below · cited by 4 · depth 11 - Γ_H(M)≤Γ_{H'}(M/p) for H' the reduction of H
ModularCurve.GammaH_le_GammaH_div_infSubgroup0 below · cited by 22 · depth 12 - Abel's theorem for X₀(N): necessity, on H
ModularCurve.abelJacobi_mem_periodLattice_of_meromorphicOrderAt_eq_card_stabilizer18 below · cited by 1 · depth 12 - j(q^N) is a root of the modular polynomial over ℚ(j)
ModularCurve.aeval_jqN_toAdjoin0 below · cited by 2 · depth 12 - Arithmetic Frobenius acts on places as geometric Frobenius
ModularCurve.arithFrobC_smul_eq_frobOnPlacesGeomLevel0 below · cited by 106 · depth 12 - The partial Atkin–Lehner automorphism wₚ is an involution
ModularCurve.atkinLehnerInvolutionFull_apply_apply74 below · cited by 13 · depth 12 - Second Atkin–Lehner q-expansion pin from the first
ModularCurve.atkinLehner_qExpand_pin_of_pin231 below · cited by 4 · depth 12 - Number of cusps of X₀(N) over ℚ̄
ModularCurve.card_eq_cuspCount_of_forall_mem_iff_ord_jBar_neg156 below · cited by 2 · depth 12 - Places where jmath̄-j₀ vanishes count embedding classes
ModularCurve.card_eq_natCard_quot_samePlace_of_forall_mem_iff_pos_ord176 below · cited by 1 · depth 12 - Primitive coset representatives are counted by Dedekind's ψ
ModularCurve.card_primCosetReps_eq_dedekindPsi0 below · cited by 3 · depth 12 - No coefficients below q⁻ᵇ in j(q)ᵇ
ModularCurve.coeff_jqModC_pow_of_lt0 below · cited by 12 · depth 12 - The coefficient of q⁻ᵇ in j(q)ᵇ is 1
ModularCurve.coeff_jqModC_pow_self0 below · cited by 12 · depth 12 - Vanishing component group for at most one edge
ModularCurve.componentGroup_subsingleton0 below · cited by 1 · depth 12 - Mod m Eisenstein recurrences force σ' coefficients
ModularCurve.dvd_sub_sigmaPrimeTo_mul_of_eisenstein_eigen_mod0 below · cited by 2 · depth 12 - EMD holds at every level N and every j₀
ModularCurve.emd_holds335 below · cited by 2 · depth 12 - Uniqueness of the coset factorisation over K((t))
ModularCurve.eq_cosetTwoVarPoly_of_forall_isRoot0 below · cited by 3 · depth 12 - Vanishing of all real periods forces f=0
ModularCurve.eq_zero_of_forall_re_period_eq_zero2 below · cited by 6 · depth 12 - Abelian scheme model of J₀(p) over ℤ_{(ℓ)}
ModularCurve.exists_abelianSchemePropertyBundle_model_jZero1,730 below · cited by 4 · depth 12 - Lifting automorphisms of F₀ to the compositum L· F₀
ModularCurve.exists_algEquiv_laurentBaseChange_cover0 below · cited by 21 · depth 12 - Degeneracy push-forwards J_H(M) → J_{H'}(M/p) on divisor classes
ModularCurve.exists_degPts_mk_eq_mk_pushforwardAlong33 below · cited by 1 · depth 12 - P-primary q^m-torsion as I^m-torsion, I=(q)+Pᶜ
ModularCurve.exists_eisensteinPrimaryTorsionBar_eq_torsionBySet_span_sup_pow714 below · cited by 3 · depth 12 - Normalised embeddings at j₀ versus roots of Φ_N(j₀+t,Y)
ModularCurve.exists_emb_equiv_rootsAt167 below · cited by 2 · depth 12 - Finite flat Hopf model of J₀(p)[ℓ^k] for ℓ∤ p
ModularCurve.exists_finiteFlat_model_jZero_torsion1,773 below · cited by 5 · depth 12 - Finiteness of the zero locus of jmath̄ - j₀
ModularCurve.exists_finset_ord_jBar_sub_pos144 below · cited by 5 · depth 12 - Gauss normalisation of q-expansions on X₀(N)
ModularCurve.exists_forall_coeff_smul_mem_and_exists_inv_coeff_mem_of_forall_ord_neg113 below · cited by 4 · depth 12 - Finite-level approximation of decomposition by Frobenius times inertia
ModularCurve.exists_frobeniusAt_pow_mul_inertia_fixing_of_mem_decompositionSubgroup2 below · cited by 10 · depth 12 - Base change of J₀(N) from ℚ̄ to ℂ
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_jZero_pic0_complex697 below · cited by 1 · depth 12 - Hecke-equivariant Abel–Jacobi injection for Pic⁰ of X₀(N)
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_pic0_complex_quotient_periodLattice673 below · cited by 1 · depth 12 - Existence of a Fricke automorphism from symmetric irreducible Φ
ModularCurve.exists_isFrickeAut_of_modularPolynomialData1 below · cited by 1 · depth 12 - Slash by Γ₀(M) has integral q-expansion after scaling
ModularCurve.exists_isIntegralQExp_smul_slash_of_mem_Gamma028 below · cited by 14 · depth 12 - The period map is a parabolic homomorphism on Γ₀(N)
ModularCurve.exists_isParabolicHom_apply_eq_period2 below · cited by 7 · depth 12 - Igusa's model of X₀(M) as the H=top two-chart model
ModularCurve.exists_iso_igusaScheme_xHDRLevel_X_gammaH_top188 below · cited by 1 · depth 12 - Localised Kummer rows from integral Kummer data
ModularCurve.exists_jKummerRow_addEquiv_fppfCohomology_of_localizedKummerData_of_dvd0 below · cited by 2 · depth 12 - The j-invariant as a transcendental-residue witness
ModularCurve.exists_mem_integers_transcendental_residue_finrank_eq_of_regularProlongation_modularFunctionFieldBar116 below · cited by 3 · depth 12 - Exponentiating an invariant differential of the third kind on H
ModularCurve.exists_meromorphic_smul_eq_mul_of_slashInvariant_residue1 below · cited by 4 · depth 12 - Meromorphic Γ₀(N)-invariant functions are quotients of modular forms
ModularCurve.exists_modularForm_eventuallyEq_div_of_meromorphic0 below · cited by 1 · depth 12 - j(qᵈ) as a ratio of weight-12 forms on Γ₀(N)
ModularCurve.exists_modularForm_mul_qExpansion_eq_coeffEmb_qExpand_jq13 below · cited by 5 · depth 12 - Modular functions are locally quotients g/h of modular forms
ModularCurve.exists_modularForm_realize_eventuallyEq_div16 below · cited by 5 · depth 12 - Multiplicative-type subgroup inside the I^m-torsion of J₀(p)
ModularCurve.exists_multiplicativeTypeNat_torsionBySet_pow_inertiaSubgroupIn1,977 below · cited by 2 · depth 12 - Order of the Eisenstein-primary q^m-torsion is a q-power
ModularCurve.exists_natCard_eisensteinPrimaryTorsionBar_eq_pow714 below · cited by 2 · depth 12 - Mazur-type bound on T^L/((q^m)+(P^L)^M) for large M
ModularCurve.exists_natCard_heckeLatticeAlgebra_quotient_span_pow_sup_pow_le_natCard_eisensteinPrimaryTorsionBar_quotient_mul_pow2,327 below · cited by 1 · depth 12 - Class counts agree under the embedding–moduli dictionary at j₀
ModularCurve.exists_natCard_quot_samePlace_eq_natCard_quot_sameOrbit_of_EMD0 below · cited by 1 · depth 12 - Upper transfer bound for I^m-torsion of J₀(N)
ModularCurve.exists_natCard_torsionBySet_jZero_le_sq_natCard_torsionBySet_heckeLatticeAlgebra_quotient_mul_pow776 below · cited by 1 · depth 12 - Divisibility of Pic⁰ of the reduced level-N modular function field
ModularCurve.exists_nsmul_eq_pic0_modularFunctionFieldC_residueField916 below · cited by 1 · depth 12 - Right multiplication permutes the Γ₀(ℓ) coset slots
ModularCurve.exists_perm_gamma0_cosetReps0 below · cited by 4 · depth 12 - Existence of an irreducible modular polynomial at every level
ModularCurve.exists_phiIrreducible69 below · cited by 20 · depth 12 - Relative Jacobian of X₀(p) over ℤ_{(ℓ)}
ModularCurve.exists_relJacobian_jZero1,823 below · cited by 4 · depth 12 - Conjugating Γ₀(N) past diag(N,1) with equal denominators
ModularCurve.exists_sl2_heckeDiagMatrix_smul_eq0 below · cited by 5 · depth 12 - Every cusp place of X₀(N) arises from a slot
ModularCurve.exists_slot_of_isCusp151 below · cited by 8 · depth 12 - T = ℤ + I_{Eis} at every level
ModularCurve.exists_sub_C_mem_eisensteinIdeal0 below · cited by 2 · depth 12 - Interchange inequality for toric monodromy at q and q'
ModularCurve.exists_submodule_finrank_span_toricMonodromyPart_le_of_not_exists_hasLowerLevelTorsion_of_attachedOddBlr_sqf_five_of_six_mul_dvd_of_neZero12,125 below · cited by 1 · depth 12 - Finiteness of the q-expansion function field over K(j)
ModularCurve.exists_transcendental_finiteDimensional_qExpFunctionFieldC_of_isAlgClosed8 below · cited by 77 · depth 12 - Deligne–Rapoport model at p ∥ M with pinned Atkin–Lehner map
ModularCurve.exists_xHDRModelAtP_atkinLehner_generic_chart1,217 below · cited by 6 · depth 12 - Finiteness of the four-generator roof over k(j,j_N)
ModularCurve.finiteAlong_heckeAlphaC59 below · cited by 34 · depth 12 - Finiteness of the q-twisting degeneracy embedding
ModularCurve.finiteAlong_heckeBetaC59 below · cited by 17 · depth 12 - Finiteness of L-algebra maps between base-changed q-expansion fields
ModularCurve.finiteAlong_laurentBaseChange_qExpFunctionFieldC2 below · cited by 16 · depth 12 - Finiteness and separability of the modular function field over K(j)
ModularCurve.finiteDimensional_and_isSeparable_adjoin_jGeomGen_of_isSeparable_jqNModC1 below · cited by 30 · depth 12 - Finiteness of Eisenstein P^m-torsion on J₀(p)
ModularCurve.finite_torsionBySet_eisensteinMaximalIdeal_pow714 below · cited by 2 · depth 12 - Both degeneracy embeddings have degree p+1 when p ‖ M
ModularCurve.finrankAlong_eq_add_one_and_finrankAlong_eq_add_one_of_coe_eq_qExpand247 below · cited by 7 · depth 12 - dim_ℝH¹ₚₐᵣ(Γ₀(N),ℝ)≤ 2dim_ℂS₂(Γ₀(N))
ModularCurve.finrank_parabolicHoms_le_two_mul_finrank_cuspForm562 below · cited by 6 · depth 12 - dim_K H¹ₚₐᵣ(Γ₀(N),K) ≤ 2g(N)
ModularCurve.finrank_parabolicHoms_le_two_mul_genusFormula12 below · cited by 3 · depth 12 - Pole-or-finite chart dichotomy for places of the modular function field
ModularCurve.forall_ord_jBar_sub_le_zero_or_exists_ord_pos144 below · cited by 3 · depth 12 - The full Fricke involution equals its own inverse
ModularCurve.frickeInvolutionFull_symm2 below · cited by 1 · depth 12 - Squared geometric Frobenius fixes supersingular places over k̄
ModularCurve.frobOnPlacesGeomLevel_frobOnPlacesGeomLevel_eq_self_of_mem_ssPlaces_of_isAlgClosed378 below · cited by 47 · depth 12 - Determinant ℓ of Frobenius on Vₚ J₀(N) for ℓ ≠ p
ModularCurve.frobenius_coordDet_eq_of_basis_rationalTateModule_jZero_of_ne914 below · cited by 2 · depth 12 - Frobenius identity j(q^ℓ)=j(q)^ℓ in characteristic ℓ
ModularCurve.frobenius_identity_geom_unconditional1 below · cited by 28 · depth 12 - Generic-point comparison of the two pinned curve models at level N₀p
ModularCurve.fromSpecStalk_genericPoint_comp_eq_of_xHDRModelAtP_top_drModelPackageLevel0 below · cited by 1 · depth 12 - Genus of the level-N modular function field in characteristic ℓ≥ 5
ModularCurve.genusFF_modularFunctionFieldFullC_eq_genusFF_modularFunctionFieldBar_of_five_le710 below · cited by 2 · depth 12 - Genus lower bound for X₀(N) in characteristic 2 or 3
ModularCurve.genusFormula_le_genusFF_modularFunctionFieldFullC_of_lt_five441 below · cited by 1 · depth 12 - Canonical-divisor genus equals adelic genus for X₀(N) over ℚ̄
ModularCurve.genus_eq_genusFF_modularFunctionFieldBar181 below · cited by 50 · depth 12 - Geometric integrality of the generic fibre of the two-chart model
ModularCurve.geometricallyIntegral_pullback_snd_toBase_twoChartIntegralModel_qExpFunctionFieldC_rat3 below · cited by 3 · depth 12 - Two-adic Eisenstein torsion sheaf pinned to reduction mod 2
ModularCurve.hasJZeroNeronTorsionSheaf_two_residue_iff_reductionModL_of_dvd_eisensteinNumerator_v54,071 below · cited by 1 · depth 12 - Finite flat rank-two module at q forces lower-level 𝔪-torsion
ModularCurve.hasLowerLevelTorsion_of_finiteFlat_rankTwo_heckeTorsion5,566 below · cited by 2 · depth 12 - Finite-flat level lowering at q ≥ 5
ModularCurve.hasLowerLevelTorsion_of_finiteFlat_rankTwo_heckeTorsion_of_five_le5,567 below · cited by 1 · depth 12 - qE₄³/Δ is the sum of its q-series
ModularCurve.hasSum_jNum_qParam3 below · cited by 1 · depth 12 - Products of convergent q-expansions on H
ModularCurve.hasSum_qParam_mul0 below · cited by 9 · depth 12 - Pinned automorphism intertwines degeneracy maps and divisor-class pullback
ModularCurve.heckeBetaHBar_pins_and_smul_pullbackAlongHom_of_qExpand_pins5 below · cited by 1 · depth 12 - Hecke action on the component group is multiplication by n
ModularCurve.heckeComponentAction_eq_nsmul_of_offDiagDivides0 below · cited by 4 · depth 12 - Atkin–Lehner relation U_q D + w_q· D = β^*α_*D on divisors
ModularCurve.heckeDivBar_self_add_atkinLehner_smul186 below · cited by 3 · depth 12 - Descent of the fibre Hecke correspondence to Pic⁰
ModularCurve.heckeDivFibreDescends_of_separable_phi_map59 below · cited by 3 · depth 12 - Hecke action on J₀(N)_{ℚ̄} is Galois-equivariant
ModularCurve.heckeGen_smul_galois_smul237 below · cited by 12 · depth 12 - The seven Hecke inputs for X_H(M) over any base field
ModularCurve.heckeInputsHAlong36 below · cited by 12 · depth 12 - Uₚ + wₚ = β^*α_* on J₀(N₀p)
ModularCurve.heckeOperatorBar_self_add_atkinLehner_smul225 below · cited by 2 · depth 12 - Surjectivity of the projection from the abstract Hecke algebra
ModularCurve.heckeProj_surjective0 below · cited by 8 · depth 12 - Frobenius permutes the supersingular q-expansion places
ModularCurve.image_qExpFrobeniusPlaceModL_ssPlacesQExp_eq2 below · cited by 26 · depth 12 - Index ℓ+1 of the upper Hecke subgroup of Γ₀(N)
ModularCurve.index_heckeUpper0 below · cited by 7 · depth 12 - Finite flatness of [ℓ^k] on a model of J₀(p)
ModularCurve.isFinite_and_flat_schemeNsmul_pow_of_jZeroC_points263 below · cited by 11 · depth 12 - Integrality of Y⁶̂ j⁴(̂ j-1728)³ over ℂ[̂ j]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_of_mul_thetaL_eq_qExpansion82 below · cited by 1 · depth 12 - Integrality over k[j] descends along a field embedding
ModularCurve.isIntegral_adjoin_of_isIntegral_adjoin_coeffMap0 below · cited by 9 · depth 12 - Points of J₀(p) over a residue field above ℓ are torsion
ModularCurve.isOfFinAddOrder_jZeroC_residueField195 below · cited by 1 · depth 12 - Any algebra image of j(q^N) is a root of Φ
ModularCurve.isRoot_map_Phi_apply_jBar92 below · cited by 2 · depth 12 - Separability of the full modular function field over L(j)
ModularCurve.isSeparable_adjoin_coeffEmb_jq_full142 below · cited by 5 · depth 12 - Separability of j(q^N) over K(j(q))
ModularCurve.isSeparable_jqNModC_of_modularPolynomialData3 below · cited by 7 · depth 12 - Separability of j(q^N) over K(j(q)) from Φ_N over K(X)
ModularCurve.isSeparable_jqNModC_of_separable_phi_map2 below · cited by 5 · depth 12 - Pull-back along φ⁻¹ intertwines the two Abel–Jacobi dictionaries
ModularCurve.jZeroNeronObjectAtP_pts_pic0Congr_eq_pts_comp_pullbackHom_of_modelIso_levelData66 below · cited by 1 · depth 12 - Divisibility of J₀(N): multiplication by m≠0 is surjective
ModularCurve.jZero_zsmul_surjective413 below · cited by 5 · depth 12 - Non-polynomiality of j(qᵖ) in j(q) forces non-rationality
ModularCurve.jqNModC_prime_not_mem_adjoin_of_forall_aeval_ne48 below · cited by 3 · depth 12 - Equal kernels: Hecke algebras on S₂(Γ₀(p)) and on J₀(p)
ModularCurve.ker_heckeEvalForms_latticeRestrict_eq_ker_heckeEvalBar845 below · cited by 7 · depth 12 - Divisor-generator criterion for the base-changed modular function field
ModularCurve.laurentBaseChange_le_of_divisor_generators_mem0 below · cited by 6 · depth 12 - Uniqueness of Laurent q-expansions at period h
ModularCurve.laurent_qParam_coeff_unique1 below · cited by 20 · depth 12 - Hecke specialisation extends from places to all divisors
ModularCurve.mapDomain_heckeDivBar_eq_of_forall_single0 below · cited by 5 · depth 12 - Specialisation of T_ℓ on divisors: Eichler–Shimura relation
ModularCurve.mapDomain_heckeDivBar_single_eq_heckeFibreGeomLevel170 below · cited by 1 · depth 12 - Degeneracy restriction commutes with the Hecke correspondence at ℓ≠ q
ModularCurve.mapDomain_restrictAlong_degeneracy_heckeDivBar_comm_of_ne204 below · cited by 2 · depth 12 - Frobenius on reductions of integral Laurent series: ̄ s(q)^ℓ=̄ s(q^ℓ)
ModularCurve.map_intCast_pow_char_eq_qExpand0 below · cited by 20 · depth 12 - Rationality of q-expansions of ratios of forms on Γ₀(N)
ModularCurve.mem_laurentBaseChange_of_coeffMap_eq_qExpansion_div101 below · cited by 1 · depth 12 - q-expansion principle for X₀(ℓ), ℓ prime
ModularCurve.mem_modularFunctionField_of_hasSum_of_gamma0_invariant22 below · cited by 2 · depth 12 - Laurent series fixed by q↦ζ q lie in K((qⁿ))
ModularCurve.mem_range_qExpand_of_qTwist_eq0 below · cited by 4 · depth 12 - Φ_N(j,Y) as minimal polynomial of j(q^N) over ℚ(j)
ModularCurve.minpoly_jqN_eq_toAdjoin0 below · cited by 14 · depth 12 - Kirchhoff closed form for the component group order
ModularCurve.natCard_componentGroup_eq_kirchhoffCount4 below · cited by 2 · depth 12 - Eisenstein-primary q^m-torsion quotient has order q^α
ModularCurve.natCard_eisensteinPrimaryTorsionBar_quotient_eq_pow_alpha_of_multiplicativeTypeNat29 below · cited by 2 · depth 12 - T/I ≅ ℤ/n at prime level (Mazur II.9.7)
ModularCurve.natCard_heckeLatticeAlgebra_quotient_eisensteinIdeal_eq_eisensteinNumerator1,158 below · cited by 1 · depth 12 - Two-sided linear growth of toric I^m-torsion on J₀(p)
ModularCurve.natCard_jZeroToricTorsion_inf_torsionBySet_pow_linearGrowth762 below · cited by 2 · depth 12 - Fibre of Y₀(N) over j(W) as order-N points modulo Aut(W)
ModularCurve.natCard_moduliPoint_j_eq_eq_natCard_quot_addOrderOf_eq1 below · cited by 8 · depth 12 - Normalised Hahn-series embeddings inducing a place above j₀
ModularCurve.natCard_normalized_algHom_hahnSeries_jBar_sub_eq_toNat_ord0 below · cited by 1 · depth 12 - Simple zeros of ̄ j on level-N curve count ν₃(N)
ModularCurve.natCard_ord_jBar_eq_one_eq_nuThree239 below · cited by 2 · depth 12 - Simple zeros of ̄ j-1728 number ν₂(N)
ModularCurve.natCard_ord_jBar_sub_1728_eq_one_eq_nuTwo239 below · cited by 2 · depth 12 - Orbits of cyclic N-subgroups count moduli points over j(E₀)
ModularCurve.natCard_quot_sameOrbit_cycSub_eq_natCard_moduliPoint_j_eq2 below · cited by 1 · depth 12 - I^m-torsion of J₀(p) at most the square of its toric part
ModularCurve.natCard_torsionBySet_pow_le_sq_natCard_jZeroToricTorsion_inf_mul1,816 below · cited by 2 · depth 12 - Toric and mod-2 reduction bound for I^m-torsion of J₀(p)
ModularCurve.natCard_torsionBySet_pow_two_le_natCard_jZeroToricTorsion_inf_mul_natCard_map_reductionModL_mul_pow3,208 below · cited by 1 · depth 12 - Inhabitedness of the Deligne–Rapoport package at level N₀q
ModularCurve.nonempty_dRModelPackageLevel1,209 below · cited by 2 · depth 12 - Existence of a Néron identity component for J₀(p)
ModularCurve.nonempty_jZeroNeronIdentityComponent3,575 below · cited by 1 · depth 12 - A supersingular place above every supersingular j-invariant
ModularCurve.nonempty_ssPlaces_fibre165 below · cited by 2 · depth 12 - Order bound at a cusp for the coefficient of ω_f = y dj
ModularCurve.one_sub_ord_le_ord_of_coeffMap_mul_thetaL_eq_qExpansion98 below · cited by 1 · depth 12 - Regularity of x dj at a place where ord j ≠ 0
ModularCurve.ordDiff_smul_D_coeffEmb_jq_nonneg_iff159 below · cited by 1 · depth 12 - Order of jmath̄ at a zero divides 3
ModularCurve.ord_jBar_dvd_three311 below · cited by 4 · depth 12 - Ramification over j = 1728 divides 2 on X₀(N)
ModularCurve.ord_jBar_sub_1728_dvd_two234 below · cited by 4 · depth 12 - Order one for jmath̄ - c when c ≠ 0, 1728
ModularCurve.ord_jBar_sub_eq_one_of_ne_zero_of_ne337 below · cited by 3 · depth 12 - Degree ψ(d) and generation at every d ∣ M
ModularCurve.package_of_socket57 below · cited by 2 · depth 12 - The segment period equals F(γ i)-F(i)
ModularCurve.period_apply_eq_sub_of_hasEquivariantPrimitive0 below · cited by 13 · depth 12 - Frobenius and q ↦ qᵖ on Laurent series in characteristic p
ModularCurve.pow_char_eq_map_frobenius_qExpand0 below · cited by 14 · depth 12 - The q-expansion Frobenius acts bijectively on places
ModularCurve.qExpFrobeniusPlaceModL_bijective2 below · cited by 49 · depth 12 - Frobenius pullback of places equals arithmetic Frobenius twist
ModularCurve.qExpFrobeniusPlaceModL_eq_qExpArithFrobC_smul2 below · cited by 19 · depth 12 - Bijectivity of the mod-ℓ Frobenius push-forward on Pic⁰
ModularCurve.qExpFrobeniusPushforwardModL_bijective_of_transcendental43 below · cited by 2 · depth 12 - q-expansion function field of Γ₀(M) equals ℚ(j(qᵈ):d∣ M)
ModularCurve.qExpFunctionFieldC_rat_gamma0_eq_modularFunctionFieldFull186 below · cited by 102 · depth 12 - q-expansion of Δ as integral series qprod(1-qⁿ)²⁴
ModularCurve.qExpansion_discriminant_eq_map_X_mul_dedekindEtaUnit1 below · cited by 66 · depth 12 - Ratios of q-expansions of Γ₀(N)-forms lie in ℂ· F_N
ModularCurve.qExpansion_div_mem_laurentBaseChange149 below · cited by 4 · depth 12 - Reduction mod ℓ commutes with T_q for q≠ℓ
ModularCurve.reductionModL_heckeOperatorBar_of_ne938 below · cited by 8 · depth 12 - Places of normalised Hahn-series embeddings differ by monodromy
ModularCurve.samePlace_iff_exists_monodromy177 below · cited by 2 · depth 12 - Quadratic Galois relation on Eisenstein torsion of J₀(p)
ModularCurve.smul_smul_sub_smul_add_eq_zero_of_mem_torsionBySet_eisensteinMaximalIdeal1,054 below · cited by 1 · depth 12 - Existence of a supersingular j-invariant in characteristic q
ModularCurve.ssJSet_nonempty40 below · cited by 51 · depth 12 - Ramification indices along β sum to ℓ+1
ModularCurve.sum_ramificationIndexAlong_heckeBetaBar_of_deg_eq_one170 below · cited by 7 · depth 12 - Surjectivity of reduction SL₂(ℤ)toSL₂(ℤ/N)
ModularCurve.surjective_specialLinearGroup_map_zmod0 below · cited by 33 · depth 12 - Galois-stable divisors have K-rational symmetric coefficients
ModularCurve.symVec_mem_of_stable199 below · cited by 6 · depth 12 - Eichler–Shimura inequality: 2dim S₂(Γ₀(N))≤dim parabolic homs
ModularCurve.two_mul_finrank_cuspForm_le_finrank_parabolicHoms11 below · cited by 2 · depth 12 - Index of Γ₀(N) equals ψ(N)
ModularCurve.Gamma0_index3 below · cited by 47 · depth 13 - Atkin–Lehner automorphism at p ∥ M commutes with diamonds
ModularCurve.algEquiv_diamondAutHBar_comm_of_qExpand_of_diamondAutHBar_div277 below · cited by 2 · depth 13 - Automorphisms fixing the level-M/p subfield are the identity
ModularCurve.algEquiv_eq_refl_of_forall_coe_eq_infSubgroup226 below · cited by 4 · depth 13 - Arithmetic Frobenius squared fixes the supersingular places
ModularCurve.arithFrobC_smul_arithFrobC_smul_eq_self_of_mem_ssPlaces384 below · cited by 37 · depth 13 - Arithmetic Frobenius preserves the supersingular places
ModularCurve.arithFrobC_smul_mem_ssPlaces_univ1 below · cited by 38 · depth 13 - Unipotence of inertia on prime-to-p torsion of J₀(p)
ModularCurve.arithmeticGalois_smul_sub_mem_inertiaInvariantPoints1,510 below · cited by 2 · depth 13 - Toric exclusion: scalar inertia forces n(σ)=1
ModularCurve.atP_toric_exclusion0 below · cited by 1 · depth 13 - Count of supersingular j-invariants in characteristic q
ModularCurve.card_eq_of_ssJSet36 below · cited by 10 · depth 13 - Fibre counts for j on the modular curve of level N
ModularCurve.card_fibres_jqModC_modularFunctionFieldFullC_eq362 below · cited by 3 · depth 13 - Poles of j count the cusps of level N
ModularCurve.card_poles_jqModC_modularFunctionFieldFullC_eq_cuspCount123 below · cited by 3 · depth 13 - Monic fibres have exactly degΦ roots over ̄ K
ModularCurve.card_roots_fibrePoly_of_monic0 below · cited by 2 · depth 13 - Degeneracy roof at (N,q) equals full level-Nq function field
ModularCurve.charLDegeneracyRoof_eq_modularFunctionFieldFullC_mul113 below · cited by 60 · depth 13 - Characters of the degree-zero lattice are evaluations at node data
ModularCurve.characterLattice_evalHom_surjective_and_trivial_iff_const0 below · cited by 2 · depth 13 - The degree-zero lattice ℤ[S]⁰ is free of rank |S|-1
ModularCurve.characterLattice_free_and_finrank_eq0 below · cited by 2 · depth 13 - Localised modular ring of level Nq gives a valuation dichotomy
ModularCurve.coe_mem_modularLocalized_or_coe_inv_mem_modularLocalized_mul_of_not_dvd153 below · cited by 2 · depth 13 - q-expansion of the normalised derivative equals θ of the q-expansion
ModularCurve.coe_qExpansion_normalizedDerivOfComplex74 below · cited by 1 · depth 13 - Compatibility of rational diamond actions at levels M and M/p
ModularCurve.coe_ringAut_gamma0_apply_eq_of_coe_eq_infSubgroup105 below · cited by 2 · depth 13 - Weight-two cusp forms on Γ₀(2) and Γ₀(3) vanish
ModularCurve.cuspForm_gamma0_two_eq_zero_of_le_three11 below · cited by 1 · depth 13 - The j-line place at j=1728 has degree 1
ModularCurve.deg_jLinePlace17280 below · cited by 2 · depth 13 - The place j=0 of the j-line has degree 1
ModularCurve.deg_jLinePlaceZero0 below · cited by 2 · depth 13 - Degeneracy pull-back inputs hold at every prime level q
ModularCurve.degeneracyPullbackInputs_of_prime90 below · cited by 3 · depth 13 - Degeneracy pushforward inputs at prime index q
ModularCurve.degeneracyPushforwardInputs_of_prime90 below · cited by 2 · depth 13 - Degeneracy relation β_*∘ Uₚ = p α_* on J₀
ModularCurve.degeneracyPushforwardPair_one_heckeOperatorBar_self203 below · cited by 6 · depth 13 - Degeneracy relation α_*Uₚ=Tₚα_*-β_* at a prime p∤ N₀
ModularCurve.degeneracyPushforwardPair_zero_heckeOperatorBar_self228 below · cited by 6 · depth 13 - Square law ⟨ d⟩ θ²=id for the Atkin–Lehner automorphism
ModularCurve.diamondAutHBar_algEquiv_algEquiv_eq_self_of_qExpand_of_diamondAutHBar_div_of_unitsMap_mul_eq_one277 below · cited by 1 · depth 13 - Diamond automorphism as base change of the rational action
ModularCurve.diamondAutHBar_apply_coeffEmb_eq_coeffEmb_ringAut_apply105 below · cited by 7 · depth 13 - Injectivity of the q-expansion map on differentials
ModularCurve.diffQExpBar_injective_of_neZero145 below · cited by 2 · depth 13 - Eisenstein numerator as a gcd, for odd primes
ModularCurve.eisensteinNumerator_eq_gcd0 below · cited by 1 · depth 13 - Inertia at q≠ p acts on Eisenstein displacements cyclotomically
ModularCurve.eisensteinTorsionBar_inertia_smul_sub_eq_nat_smul1,976 below · cited by 3 · depth 13 - A place with ordᵥ(jmath̃-c)>0 is the place jmath̃=c
ModularCurve.eq_charLGeomPlaceOfPoint_of_ord_pos4 below · cited by 33 · depth 13 - Full-level modular function field is essentially of finite type
ModularCurve.essFiniteType_modularFunctionFieldFullC73 below · cited by 34 · depth 13 - Local constancy modulo periods of the Abel fibre sum
ModularCurve.eventually_abelFibreSum_sub_mem_periodLattice17 below · cited by 1 · depth 13 - Atkin–Lehner automorphism at ℓ exchanging α and β
ModularCurve.exists_algEquiv_atkinLehner_heckeAlphaHBar_heckeBetaHBar30 below · cited by 5 · depth 13 - Diamond ⟨ e⟩⁻¹ on the j-finite chart algebra
ModularCurve.exists_algEquiv_chartAlgFin_gammaH_infSubgroup_forall_coeffEmb_eq_diamondAutHBar_symm142 below · cited by 1 · depth 13 - Constant term as ℤ₍ₚ₎-point of the pole chart
ModularCurve.exists_algHom_chartAlgInf_ratLocalizedAt_apply_eq_coeff_zero0 below · cited by 6 · depth 13 - Cusp-label embedding of the base-changed modular function field
ModularCurve.exists_algHom_laurentBaseChange_slot150 below · cited by 2 · depth 13 - Prescribing a root of Φ_N by an L-algebra map
ModularCurve.exists_algHom_of_isRoot145 below · cited by 1 · depth 13 - Chart functions of the two-chart model have ℤ₍ₚ₎-integral q-expansions
ModularCurve.exists_coeffMap_eq_coe_of_mem_chartAlg_twoChartIntegralModel_qExpFunctionFieldC2 below · cited by 32 · depth 13 - Geometric generic fibre of the two-chart integral model
ModularCurve.exists_curveModel_iso_genericFibre_galoisCompat_chartPin_twoChartIntegralModel4 below · cited by 7 · depth 13 - Special fibre of the two-chart integral model of X(Γ) at p ∤ M
ModularCurve.exists_curveModel_iso_pullback_toBase_twoChartIntegralModel_qExpFunctionFieldC_readChart_of_not_dvd896 below · cited by 2 · depth 13 - Regular differentials on X₀(N)_ℚ̄ are weight-two cusp forms
ModularCurve.exists_cuspForm_coeffMap_diffQExpBar_eq_qExpansion_of_mem_regularDifferentialsBar450 below · cited by 2 · depth 13 - Places above j₀ and orbits of cyclic N-subgroups
ModularCurve.exists_elliptic_cycSub_orbitMap332 below · cited by 2 · depth 13 - ℚ is algebraically closed in the q-expansion field
ModularCurve.exists_eq_algebraMap_of_isAlgebraic_qExpFunctionFieldC1 below · cited by 8 · depth 13 - Frobenius-equivariant bijection: supersingular places and supersingular moduli points
ModularCurve.exists_equiv_ssPlaces_ssLocus_frobenius_equivariant_univ373 below · cited by 1 · depth 13 - Local-local model of the Tₚ-nilpotent part of J₀(M)[p]
ModularCurve.exists_finiteFlat_local_local_model_jZero_torsion_heckeNilpotent1,918 below · cited by 2 · depth 13 - Outside a finite set, places of k(jmath̄,jmath̄_N) are determined by their centre
ModularCurve.exists_finset_place_eq_of_ord_jqModC_sub_pos142 below · cited by 4 · depth 13 - Integral q-expansion from A-integral j-values at poles
ModularCurve.exists_forall_coeff_smul_mem_of_forall_ord_neg112 below · cited by 9 · depth 13 - A Fricke automorphism of the level-N modular function field
ModularCurve.exists_frickeAlgEquiv_modularFunctionFieldBar79 below · cited by 5 · depth 13 - j(q^ℓ) as a ratio of weight-12 forms on Γ₀(ℓ)
ModularCurve.exists_gamma0_qExpansion_div_eq_jqNModC12 below · cited by 5 · depth 13 - Height-bounded linear system cutting out Riemann–Roch spaces on X₀(N)
ModularCurve.exists_height_system_modularFunctionFieldBar216 below · cited by 1 · depth 13 - Finite surjective morphism of two-chart integral models for Γ≤Γ'
ModularCurve.exists_hom_twoChartIntegralModel_qExpFunctionFieldC_pinned_of_le127 below · cited by 2 · depth 13 - Ramification index equals ord_w(p(j)) away from 0,1728,∞
ModularCurve.exists_irreducible_ramificationIndex_eq_ord_aeval_of_restrict_ne_jLinePlaces8 below · cited by 2 · depth 13 - Existence of a diamond pull-back action for arbitrary H
ModularCurve.exists_isDiamondPullbackModL_of_isAlgClosed289 below · cited by 11 · depth 13 - Integral q-expansions up to Mᵃ under Γ₀(M)-translation
ModularCurve.exists_isIntegralQExp_level_pow_smul_slash_of_mem_Gamma0102 below · cited by 19 · depth 13 - Bounded denominators for rational q-expansions on Γ₁(M)
ModularCurve.exists_isIntegralQExp_smul_of_ratCast_qExpansion24 below · cited by 43 · depth 13 - Diamond automorphisms of the two-chart integral model of X_H(M)
ModularCurve.exists_iso_twoChartIntegralModel_qExpFunctionFieldC_gammaH_diamond5 below · cited by 1 · depth 13 - Semistable specialisation of J₀(Mq') with supersingular nodes
ModularCurve.exists_jZeroSemistableSpecialization_ssPlaces_monodromy3,551 below · cited by 1 · depth 13 - Inertia at p acts on q-power torsion of J₀(p) through one element
ModularCurve.exists_mem_inertiaSubgroupIn_forall_jZeroTorsion_pow_smul_eq_imp1,530 below · cited by 2 · depth 13 - Riemann–Roch spaces strictly grow at each place in large degree
ModularCurve.exists_mem_riemannRochSpace_notMem_sub_single_of_le_degree164 below · cited by 4 · depth 13 - Exponentiating a form with simple poles and integer residues
ModularCurve.exists_meromorphic_logDeriv_eq_of_int_residue0 below · cited by 1 · depth 13 - Diamond action of Γ₀(M) on the q-expansion function field
ModularCurve.exists_monoidHom_gamma0_algEquiv_qExpFunctionFieldC_gammaH_of_charZero30 below · cited by 10 · depth 13 - Bound for I^m-torsion in the kernel of reduction above 2
ModularCurve.exists_natCard_torsionBySet_pow_inf_ker_reductionModL_le_natCard_heckeLatticeAlgebra_quotient_two_mul_pow2,553 below · cited by 1 · depth 13 - Rank-two Hecke freeness of the period lattice up to index
ModularCurve.exists_nsmul_eq_smul_add_smul_periodLattice602 below · cited by 2 · depth 13 - A semistable witness package for J₀(Nq) at q∤ N
ModularCurve.exists_placeSpecialization_prolongationTuple_width_comp_sp_gluedSpecialization_placeWidthChar_regularityLaw_nodeValueLaw3,321 below · cited by 2 · depth 13 - Uniform exponent bound on the p-power torsion killed by Tₚ²-(p+1)²
ModularCurve.exists_pow_smul_eq_zero_of_heckeOperatorBar_heckeOperatorBar_eq_smul1,258 below · cited by 3 · depth 13 - Relative Jacobian of X₀(p) over ℤ_{(ℓ)}, Abel–Jacobi normalised
ModularCurve.exists_pts_heckeRingAction_relJacobian_jZero_of_representsRelSubPic_of_ratCurveModel_of_abelJacobi1,167 below · cited by 3 · depth 13 - Rational Atkin–Lehner automorphism at p ∥ M
ModularCurve.exists_ratAlgEquiv_atkinLehner_gammaH_qExpand_diamondAutHBar69 below · cited by 22 · depth 13 - Rationality of q-expansions under slashing by Γ₀(M)
ModularCurve.exists_ratCast_qExpansion_slash_of_mem_Gamma018 below · cited by 7 · depth 13 - Rational presentation of ℚ̄· F_N over ℚ̄[j]
ModularCurve.exists_rational_presentation_modularFunctionFieldBar145 below · cited by 2 · depth 13 - Gauss prolongation of X₀(Nq) at a place above q∤ N
ModularCurve.exists_regularProlongation_modularFunctionFieldBar_mul_of_not_dvd117 below · cited by 7 · depth 13 - Relative Jacobian of X₀(p) over ℤ_{(ℓ)} from finite-map data
ModularCurve.exists_relJacobian_jZero_of_smoothProperModel_of_finiteMapData_of_ratCurveModel1,527 below · cited by 1 · depth 13 - Smooth proper ℤ_{(ℓ)}-model of X₀(p) with finite-map data
ModularCurve.exists_smoothProperModel_jZero_relCurve_finiteMapData_ratCurveModel1,158 below · cited by 4 · depth 13 - Rank-two bound: Hecke quotients against I^m-torsion of J₀(N)
ModularCurve.exists_sq_natCard_heckeLatticeAlgebra_quotient_le_natCard_torsionBySet_mul_pow779 below · cited by 1 · depth 13 - Multiplicative-type submodule and pairing of Eisenstein torsion (q ≠ 2)
ModularCurve.exists_submodule_multiplicativeTypeNat_heckeTorsion_span_sup_pairing_heckeLatticeAlgebra_quotient_of_ne_two2,326 below · cited by 1 · depth 13 - Deuring's inequality for reduced q-expansion function fields
ModularCurve.exists_transcendental_finiteDimensional_qExpFunctionFieldC_residueField6 below · cited by 36 · depth 13 - Fibre of Φ₂ over j(E) splits over the Vélu 2-quotients
ModularCurve.fibrePoly_phiTwo_j_eq_prod_veluQuotient2_j8 below · cited by 1 · depth 13 - Finiteness of the level-N modular function field over κ(j)
ModularCurve.finiteDimensional_adjoin_jqModC44 below · cited by 24 · depth 13 - K is finite over L(j) for the q-expansion field
ModularCurve.finiteDimensional_adjoin_of_coe_eq_coeffEmb_jq_of_eq_laurentBaseChange4 below · cited by 212 · depth 13 - Degree of the q-expansion function field over the j-line
ModularCurve.finiteDimensional_and_finrank_adjoin_jqModC_qExpFunctionFieldC_le_index119 below · cited by 74 · depth 13 - Finite type of the two chart algebras over ℤ₍ₚ₎
ModularCurve.finiteType_chartAlgFin_and_chartAlgInf_twoChartIntegralModel_qExpFunctionFieldC124 below · cited by 22 · depth 13 - Counting Frobⁿ-fixed places of the modular function field
ModularCurve.finite_fixedPoints_frobeniusPlaceModL_iterate_and_card_eq117 below · cited by 3 · depth 13 - Finiteness of H¹_{fppf}(Specℤ,mathcal J_m) for primary-torsion cores
ModularCurve.finite_fppfCohomology_one_jZeroNeronPrimaryTorsionCore1,481 below · cited by 1 · depth 13 - Finiteness of zeros of j - a on the level-N modular function field
ModularCurve.finite_setOf_ord_jGeomGen_sub_pos116 below · cited by 31 · depth 13 - Finiteness of the supersingular places of X(Γ)
ModularCurve.finite_ssPlacesQExp59 below · cited by 20 · depth 13 - Degeneracy embedding of level N into level Nℓ has degree ℓ+1
ModularCurve.finrankAlong_heckeAlphaC_residueField_eq_add_one762 below · cited by 4 · depth 13 - Degree ℓ+1 of the degeneracy map α on function fields
ModularCurve.finrankAlong_heckeAlphaHBar206 below · cited by 10 · depth 13 - Degree p of the forgetful degeneracy map when p ∣ M
ModularCurve.finrankAlong_heckeAlphaHBar_of_dvd215 below · cited by 5 · depth 13 - Degree of the second degeneracy embedding β on X_H(M)
ModularCurve.finrankAlong_heckeBetaHBar249 below · cited by 3 · depth 13 - Degree over L(j) bounded by the index of Γ'
ModularCurve.finrank_adjoin_jqModC_laurentBaseChange_qExpFunctionFieldC_le_index112 below · cited by 26 · depth 13 - Degree of K(j)(j(qᵈ)) over K(j) equals ψ(d)
ModularCurve.finrank_adjoin_jqNModC_eq_dedekindPsi_of_charZero88 below · cited by 3 · depth 13 - Degree ψ(N) of the j-cover on function fields
ModularCurve.finrank_jAdjoin_modularFunctionField_eq_dedekindPsi70 below · cited by 2 · depth 13 - Parabolic homomorphisms of Γ₀(N): bound by 2g
ModularCurve.finrank_parabolicHoms_gamma0_le_two_mul_genusFormula22 below · cited by 1 · depth 13 - 𝔪-torsion is entirely toric at level Nq'q
ModularCurve.finrank_span_toricMonodromyPart_eq_finrank_of_not_exists_hasLowerLevelTorsion3,556 below · cited by 1 · depth 13 - Toric part comparison at q versus q' when p ∣ q-1
ModularCurve.finrank_span_toricMonodromyPart_le_finrank_span_of_dvd_sub_one_of_not_exists_hasLowerLevelTorsion_sqf_five_of_six_mul_dvd_of_neZero10,542 below · cited by 1 · depth 13 - Toric part of the 𝔪-torsion has rank at most n
ModularCurve.finrank_span_toricMonodromyPart_le_of_not_dvd_sub_one_of_attachedBlr5,236 below · cited by 1 · depth 13 - Fricke-type automorphism of J₀(N) commutes with Galois
ModularCurve.galois_smul_ofAlgAut_smul_of_fricke93 below · cited by 3 · depth 13 - Genus of X₀(Nq) versus its two-component special fibre
ModularCurve.genusFF_modularFunctionFieldBar_mul_add_one_eq_of_ssPlaces783 below · cited by 20 · depth 13 - Injectivity of the width-weighted Gram map
ModularCurve.gramMap_injective0 below · cited by 0 · depth 13 - Gram matrix of difference characters at a marked point
ModularCurve.gramMatrixOf_diffChar_marked_apply0 below · cited by 1 · depth 13 - Existence of at-q Néron data for J₀(Nq)
ModularCurve.hasJZeroNeronAtPDataCore5,563 below · cited by 1 · depth 13 - Unconditional integrality of the degeneracy leg `heckeAlphaC`
ModularCurve.heckeAlphaCIntegral_unconditional73 below · cited by 18 · depth 13 - Unconditional integrality of the degeneracy map heckeBetaC
ModularCurve.heckeBetaCIntegral_unconditional82 below · cited by 17 · depth 13 - Divisor exchange identity for β_ℓ, β_{ℓ'} at distinct primes
ModularCurve.heckeBetaExchangeAt_of_primes_of_ne194 below · cited by 1 · depth 13 - Exchange identity for the Hecke square at two distinct primes
ModularCurve.heckeExchangeAt_of_primes_of_ne191 below · cited by 4 · depth 13 - Hecke operator acts on the period class by a_ℓ(f)
ModularCurve.heckeOperatorHom_periodMap_of_isNormalizedEigenform21 below · cited by 1 · depth 13 - U_q scales the period homomorphism of an eigenform by a_q
ModularCurve.heckeOperatorHom_periodMap_of_isNormalizedEigenform_of_dvd21 below · cited by 1 · depth 13 - Vanishing of 𝔪-torsion under Eisenstein annihilation
ModularCurve.heckeTorsion_eq_bot_of_eisensteinAnnihilates0 below · cited by 1 · depth 13 - Hecke stability of the toric m-torsion of J₀(p)
ModularCurve.hecke_smul_mem_jZeroToricTorsion238 below · cited by 1 · depth 13 - Inertia degree one along maps into L·ℚ(X(Γ))
ModularCurve.inertiaDegAlong_eq_one_laurentBaseChange_qExpFunctionFieldC136 below · cited by 8 · depth 13 - Affine geometric places are stable under the Frobenius map
ModularCurve.isAffineGeomPlace_frobOnPlacesGeomLevel0 below · cited by 31 · depth 13 - Integrality of Y^{2N}ĵ^{N+1}(ĵ-1728)^N over ℂ[ĵ⁻¹]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_inv_of_mul_thetaL_eq_qExpansion89 below · cited by 1 · depth 13 - Integrality of j(qᵈ) over a field containing j(q^{dℓ})
ModularCurve.isIntegral_jqNModC_of_mul3 below · cited by 3 · depth 13 - Igusa irreducibility: characteristic-p fibres of the two-chart model
ModularCurve.isIntegral_pullback_toBase_twoChartIntegralModel_qExpFunctionFieldC_of_charP322 below · cited by 10 · depth 13 - Characteristic-zero fibres of the two-chart integral model are integral
ModularCurve.isIntegral_pullback_toBase_twoChartIntegralModel_qExpFunctionFieldC_of_charZero2 below · cited by 4 · depth 13 - Coefficient automorphisms stabilise the supersingular node pairs
ModularCurve.isNodeStable_nodePairsOfPlaces_arithFrobC_coeffSemilinearAut19 below · cited by 3 · depth 13 - Torsion of J₀(N) over fields algebraic over 𝔽_ℓ
ModularCurve.isOfFinAddOrder_jZeroC_of_isAlgebraic194 below · cited by 1 · depth 13 - Igusa good reduction for the two-chart ℤ₍ₚ₎-model of X(Γ)
ModularCurve.isProper_and_smooth_and_geometricallyIntegral_twoChartIntegralModel_qExpFunctionFieldC_of_not_dvd931 below · cited by 17 · depth 13 - Places of ℚ̄(X₀(N)) over ℚ̄ are rational
ModularCurve.isRational_place_modularFunctionFieldBar121 below · cited by 26 · depth 13 - Descent: j(q^M)∈ K(j(q),j(q^{Mp}))
ModularCurve.jqNModC_mem_modularFunctionFieldC_mul_prime53 below · cited by 1 · depth 13 - ̄ j(q^{Nℓ}) = ̄ j(q^N)^ℓ in characteristic ℓ
ModularCurve.jqNModC_mul_eq_pow3 below · cited by 5 · depth 13 - ̄ j(q^{Nℓ}) lies in K(̄ j(q),̄ j(q^N))
ModularCurve.jqNModC_mul_mem3 below · cited by 1 · depth 13 - Non-membership of j(qᵖ) in the level-M field, p ∤ M
ModularCurve.jqNModC_prime_not_mem_fullC51 below · cited by 1 · depth 13 - Base change of the full modular function field to K
ModularCurve.laurentBaseChange_modularFunctionFieldFull_eq_modularFunctionFieldFullC0 below · cited by 21 · depth 13 - Wild different over j=0 in characteristics 2 and 3
ModularCurve.le_six_mul_sum_ordDiff_D_jqModC_of_lt_five389 below · cited by 1 · depth 13 - Mutual integrality of j(q) and j(qᵖ) on two-chart models
ModularCurve.mem_chartAlgFin_and_forall_mem_chartAlgInf_exists_mul_mem_of_coe_eq_coeffEmb_jq_qExpand85 below · cited by 9 · depth 13 - Ratios of integral q-expansions form a field
ModularCurve.mem_qExpFunctionFieldC_rat_iff_mem_intFormRatiosC0 below · cited by 9 · depth 13 - Order of j-j(τ) equals half the stabiliser order
ModularCurve.meromorphicOrderAt_E4_cube_div_discriminant_sub_eq_card_stabilizer_div_two2 below · cited by 6 · depth 13 - Base change of the level-N modular function field to ℚ̄
ModularCurve.modularFunctionFieldBar_eq_modularFunctionFieldC74 below · cited by 10 · depth 13 - Divisor expansions j(qᵈ) lie in the Γ₀(M) q-expansion field
ModularCurve.modularFunctionFieldFullC_le_qExpFunctionFieldC_gamma05 below · cited by 65 · depth 13 - Component-group order as |det| of a difference Gram matrix
ModularCurve.natCard_componentGroup_eq_natAbs_det_diffChar1 below · cited by 1 · depth 13 - Cosets of Γ₀(N) fixed by ST number ν₃(N)
ModularCurve.natCard_fixedPoints_ST_cosets_Gamma0_eq_nuThree0 below · cited by 4 · depth 13 - Cosets of Γ₀(N) fixed by S number ν₂(N)
ModularCurve.natCard_fixedPoints_S_cosets_Gamma0_eq_nuTwo0 below · cited by 3 · depth 13 - Places above a finite j-value as ℚ̄-points of the coordinate ring
ModularCurve.nonempty_equiv_place_pos_ord_algHom_integralClosure143 below · cited by 2 · depth 13 - Existence of a good Néron identity component for J₀(p)
ModularCurve.nonempty_jZeroNeronIdentityComponentGood3,574 below · cited by 2 · depth 13 - Existence of supersingular places on X(Γ) in characteristic p
ModularCurve.nonempty_ssPlacesQExp93 below · cited by 25 · depth 13 - No isolated points on characteristic-p fibres of the two-chart model
ModularCurve.not_isOpen_singleton_pullback_toBase_twoChartIntegralModel_qExpFunctionFieldC_of_charP140 below · cited by 1 · depth 13 - Involutivity of the Fricke automorphism on J₀(N)
ModularCurve.ofAlgAut_smul_ofAlgAut_smul_of_fricke93 below · cited by 1 · depth 13 - Order of jmath̄ at its zeros divides three, under a branch bound
ModularCurve.ord_jBar_dvd_three_of_pos_of_forall_isRoot_hasRamBound300 below · cited by 1 · depth 13 - Hecke operators and the Fricke-twisted Weil pairing on J₀(N)
ModularCurve.pair_heckeOperatorBar_eq_pair_fricke_heckeOperatorBar267 below · cited by 3 · depth 13 - Segment periods along a Γ₀(N)-orbit differ by a period
ModularCurve.periodAlong_smul_sub_periodAlong_eq_period2 below · cited by 3 · depth 13 - Eichler–Shimura: period pair map injective with parabolic image
ModularCurve.periodHomPair_eichlerShimura576 below · cited by 4 · depth 13 - Period map intertwines w_q with matrix conjugation
ModularCurve.periodMap_atkinLehnerLin_apply3 below · cited by 1 · depth 13 - Period map intertwines the level-lowering trace with the transfer
ModularCurve.periodMap_traceLin4 below · cited by 1 · depth 13 - Irreducibility of the modular polynomial at every level
ModularCurve.phiIrreducible_all72 below · cited by 11 · depth 13 - Ramification over the j-line divides the width at supersingular places
ModularCurve.placeRamificationJ_dvd_jWidthChar_three_of_mem_ssPlaces385 below · cited by 9 · depth 13 - Ramification over the j-line divides the char-2 width
ModularCurve.placeRamificationJ_dvd_jWidthChar_two_of_mem_ssPlaces385 below · cited by 9 · depth 13 - Ramification over the j-line divides the j-width
ModularCurve.placeRamificationJ_dvd_jWidth_of_mem_ssPlaces353 below · cited by 22 · depth 13 - Width divides off-diagonal Hecke correspondence coefficients in characteristic q'
ModularCurve.placeWidthChar_dvd_correspondence_heckeAlphaC_heckeBetaC_single_of_ne_of_prime702 below · cited by 3 · depth 13 - Weighted symmetry of the degree-s Hecke correspondence matrix
ModularCurve.placeWidthChar_mul_correspondence_heckeAlphaC_heckeBetaC_single_comm_of_prime604 below · cited by 3 · depth 13 - Places of the characteristic-ℓ degeneracy roof have degree one
ModularCurve.place_deg_eq_one_charLDegeneracyRoof108 below · cited by 3 · depth 13 - Places of L·ℚ(X(Γ)) have degree one
ModularCurve.place_deg_eq_one_laurentBaseChange_qExpFunctionFieldC135 below · cited by 30 · depth 13 - Frobenius invariance of the supersingular j-set
ModularCurve.pow_mem_ssJSet_iff_of_perfectField0 below · cited by 37 · depth 13 - Frobenius inputs and degree ℓ for ̄ F/Frob(̄ F)
ModularCurve.qExpFrobeniusInputsModL_and_finrankAlong_of_transcendental42 below · cited by 11 · depth 13 - Geometric Frobenius as p-th power of the inverse arithmetic Frobenius
ModularCurve.qExpFrobeniusModL_eq_inv_qExpArithFrobC_smul_pow0 below · cited by 5 · depth 13 - Frobenius push-forward on Pic⁰ computed by Φ_* on divisors
ModularCurve.qExpFrobeniusPushforwardModL_mk_eq_mk_of_eq_mapDomain43 below · cited by 5 · depth 13 - q↦ qᵖ sends the level M/p function field into level M
ModularCurve.qExpand_mem_xHFunctionField_of_mem_div3 below · cited by 6 · depth 13 - q-expansion of E₄ equals 1+240sumσ₃(n)qⁿ
ModularCurve.qExpansion_E4_eq_map_eisenstein40 below · cited by 20 · depth 13 - q-expansion of F((aτ+b)/d) as a coset substitution
ModularCurve.qExpansion_cosetTranslate_eq_cosetSubst3 below · cited by 4 · depth 13 - q-expansion of Δ as a formal product
ModularCurve.qExpansion_discriminant_eq_X_mul_tprod0 below · cited by 1 · depth 13 - Uniqueness of q-expansion coefficients at period h
ModularCurve.qParam_coeff_unique0 below · cited by 3 · depth 13 - Ramification index over j=0 equals ord_w(j)
ModularCurve.ramificationIndex_eq_ord_of_restrict_eq_jLinePlaceZero41 below · cited by 2 · depth 13 - Ramification index over j=1728 equals ord_w(j-1728)
ModularCurve.ramificationIndex_eq_ord_sub_of_restrict_eq_jLinePlace172841 below · cited by 2 · depth 13 - Rational points versus Eisenstein quotient: kernel and cokernel torsion
ModularCurve.rationalPoints_eisensteinQuotient_ker_and_coker_torsion_primeCompl_unconditional239 below · cited by 1 · depth 13 - Prime-power step for fields of divisor q-expansions of j
ModularCurve.relfinrank_fullC_mul_prime_pow51 below · cited by 1 · depth 13 - Both degeneracy inclusions multiply ψ by the same index
ModularCurve.relfinrank_modularFunctionFieldFullC_mul_dedekindPsi120 below · cited by 20 · depth 13 - Inertia-fixed q-power torsion is toric up to bounded index
ModularCurve.relindex_jZeroToricTorsion_pos_and_le_natCard_jZeroTorsion1,786 below · cited by 1 · depth 13 - A place of the modular function field restricts to j=1728 iff ord_w(j-1728)>0
ModularCurve.restrict_eq_jLinePlace1728_iff41 below · cited by 2 · depth 13 - Places of F_N above j=∞ are the poles of j
ModularCurve.restrict_eq_jLinePlaceInfty_iff41 below · cited by 2 · depth 13 - Place of X₀(N) lies over j=0 iff ord_w(j)>0
ModularCurve.restrict_eq_jLinePlaceZero_iff41 below · cited by 2 · depth 13 - Orbit relation on cyclic subgroups is equality when c₄,c₆≠ 0
ModularCurve.sameOrbit_iff_eq_of_c4_ne_zero_of_c6_ne_zero2 below · cited by 1 · depth 13 - Same induced place iff related by a monodromy twist
ModularCurve.samePlace_iff_exists_hahnTwist176 below · cited by 1 · depth 13 - Igusa separability of the level-q degeneracy maps mod ℓ
ModularCurve.separableAlong_heckeAlphaC_heckeBetaC134 below · cited by 28 · depth 13 - Separability of the coset polynomial for a simple-pole series
ModularCurve.separable_cosetTwoVarPoly0 below · cited by 2 · depth 13 - Cusp width and orders of j(q), j(q^N) at a place
ModularCurve.slot_ord_of_algHom_laurentBaseChange78 below · cited by 4 · depth 13 - Places of X₀(N) labelled by pairs (a,b)
ModularCurve.slot_place_eq_iff_modEq78 below · cited by 2 · depth 13 - Existence of a supersingular place on the level-N modular curve
ModularCurve.ssPlaces_nonempty206 below · cited by 48 · depth 13 - Orders of jmath̄ - a sum to ψ(N)
ModularCurve.sum_ord_jGeomGen_sub_eq_dedekindPsi161 below · cited by 11 · depth 13 - Transcendence of j(q^N) over the constants
ModularCurve.transcendental_jqNModC0 below · cited by 22 · depth 13 - Transcendence of j over the base domain A
ModularCurve.transcendental_of_coe_eq_coeffEmb_jq0 below · cited by 202 · depth 13 - j=0 lies in the supersingular set for q<5
ModularCurve.zero_mem_ssJSet_of_lt_five0 below · cited by 29 · depth 13 - The q-old character lattice dies modulo 𝔪
ModularCurve.SW_local_old_smul_top_eq_top_of_not_hasLowerLevelTorsion_aux20 below · cited by 2 · depth 14 - Order of the cuspidal class on X₀(ℓ)
ModularCurve.addOrderOf_cuspidalClass_eq_eisensteinNumerator148 below · cited by 1 · depth 14 - Counting Hasse-supersingular j-invariants in characteristic q
ModularCurve.card_eq_of_ssJSetHasse22 below · cited by 1 · depth 14 - Eichler–Deuring count of supersingular places at level N
ModularCurve.card_eq_ssCountFormula_of_ssPlaces408 below · cited by 3 · depth 14 - Places over j=j₀ count pairs (E,C_N)
ModularCurve.card_places_modularFunctionFieldFullC_over_eq_natCard_moduliPoint_j_eq334 below · cited by 4 · depth 14 - #P¹(ℤ/N)=ψ(N)
ModularCurve.card_projectiveLine_zmod2 below · cited by 2 · depth 14 - Index of Γ₀(N) is unchanged in GL₂(ℝ)
ModularCurve.card_quotient_gamma0_eq_index149 below · cited by 3 · depth 14 - Coset count for Γ₀(N) bounded by ψ(N)
ModularCurve.card_quotient_gamma0_le_dedekindPsi149 below · cited by 4 · depth 14 - Partial Atkin–Lehner involution takes u_Q to Q¹²u_Q⁻¹
ModularCurve.coe_atkinLehnerInvolutionFull_modularUnitSeries105 below · cited by 9 · depth 14 - Partial Atkin–Lehner involution inverts Ogg's unit
ModularCurve.coe_atkinLehnerInvolutionFull_modularUnitSeries_of_not_dvd100 below · cited by 8 · depth 14 - Diamond automorphisms at levels M and M/p agree
ModularCurve.coe_diamondAutHBar_eq_coe_diamondAutHBar_div_of_coe_eq110 below · cited by 4 · depth 14 - Diamonds commute with the degeneracy map q ↦ qᵖ
ModularCurve.coe_diamondAutHBar_eq_qExpand_coe_diamondAutHBar_div_of_coe_eq_qExpand67 below · cited by 3 · depth 14 - First seven coefficients of the q-expansion of j
ModularCurve.coeff_jNum_le_six0 below · cited by 1 · depth 14 - Surjectivity from hitting all crossing-coordinate classes
ModularCurve.comp_surjective_of_forall_exists_eq_crossingCoord0 below · cited by 2 · depth 14 - Constants of ℚ̄-modular function field are the base
ModularCurve.constantsAreBase_modularFunctionFieldBar119 below · cited by 11 · depth 14 - Degree at least ψ(M) over K(j) for X₀(M)
ModularCurve.dedekindPsi_le_finrank_adjoin_qExpFunctionFieldC_gamma0113 below · cited by 8 · depth 14 - Multiplying the level by a prime in Dedekind's ψ
ModularCurve.dedekindPsi_mul_prime3 below · cited by 20 · depth 14 - Riemann's inequality for the function field of X₀(N) over ℚ̄
ModularCurve.degree_add_one_sub_genusFF_le_finrank_riemannRochSpace185 below · cited by 10 · depth 14 - Determinant equals the exponent of σ on p-th roots of unity
ModularCurve.det_eq_natCast_of_forall_rootsOfUnity_of_det_frobenius_eq_natCast28 below · cited by 1 · depth 14 - Diamond automorphisms fix functions of level Γ₀(N)
ModularCurve.diamondAutHBar_apply_eq_self_of_coe_eq_coeffEmb_of_mem_gamma0108 below · cited by 14 · depth 14 - Multiplicativity of the diamond automorphisms ⟨ d⟩^*
ModularCurve.diamondAutHBar_mul_and_diamondAutHBar_one108 below · cited by 7 · depth 14 - Elementary divisibility: m ∣ (p²-1)/24
ModularCurve.dvd_sq_sub_one_div_of_isCoprime_of_not_dvd0 below · cited by 1 · depth 14 - The Eisenstein ideal annihilates the cuspidal class
ModularCurve.eisensteinIdeal_smul_cuspidalClass266 below · cited by 1 · depth 14 - Inertia at 2 acts by cyclotomic exponent on Eisenstein torsion displacements
ModularCurve.eisensteinTorsionBar_inertia_smul_sub_eq_nat_smul_two1,972 below · cited by 1 · depth 14 - A place with a pole at jmath̃ is the place at infinity
ModularCurve.eq_charLGeomPlaceEquiv_placeInfty_of_ord_neg6 below · cited by 10 · depth 14 - A place of the j-line is v₁₇₂₈ iff ordᵥ(j-1728)>0
ModularCurve.eq_jLinePlace1728_iff_ord_jGen_sub_pos40 below · cited by 1 · depth 14 - A place of the j-line is v_∞ iff j has a pole
ModularCurve.eq_jLinePlaceInfty_iff_ord_jGen_neg40 below · cited by 1 · depth 14 - A place of the j-line is v₀ iff ordᵥ(j)>0
ModularCurve.eq_jLinePlaceZero_iff_ord_jGen_pos40 below · cited by 1 · depth 14 - Uniqueness of the Laurent-series root of the modular equation Φₚ
ModularCurve.eq_qExpand_jqModC_of_isRoot_map_modularPolynomial25 below · cited by 1 · depth 14 - Supersingular j-invariants in 𝔽₉ vanish in characteristic 3
ModularCurve.eq_zero_of_mem_ssJSet_three0 below · cited by 28 · depth 14 - Characteristic 2: only j=0 in 𝔽₄ is supersingular
ModularCurve.eq_zero_of_mem_ssJSet_two0 below · cited by 28 · depth 14 - Injectivity of P ↦ P(j(q)) on low-order coefficients
ModularCurve.eq_zero_of_natDegree_le_of_coeff_evalAtJ_eq_zero0 below · cited by 2 · depth 14 - The q-expansion function field is essentially of finite type
ModularCurve.essFiniteType_qExpFunctionFieldC_of_isAlgClosed10 below · cited by 18 · depth 14 - Hecke-equivariant bounded-kernel map on Eisenstein torsion killed by reduction
ModularCurve.exists_addMonoidHom_inf_ker_reductionModL_eisensteinTorsionBar_heckeLatticeAlgebra_quotient_two_pow_natCard_ker_le2,547 below · cited by 1 · depth 14 - Atkin–Lehner toggle automorphism of the full function field over K
ModularCurve.exists_algEquiv_atkinLehner_fullC_of_prime_of_not_dvd113 below · cited by 11 · depth 14 - Diamond automorphism restricted to the pole chart algebra
ModularCurve.exists_algEquiv_chartAlgInf_forall_coeffEmb_eq_diamondAutHBar_symm142 below · cited by 3 · depth 14 - Igusa reduction of the two chart rings, packaged
ModularCurve.exists_algEquiv_residueField_tensor_chartAlg_twoChartIntegralModel_qExpFunctionFieldC_chartRing890 below · cited by 8 · depth 14 - Poles of j lie in the orbit of the q-adic place
ModularCurve.exists_algEquiv_smul_qInftyPlaceBar_eq_of_ord_jqModC_neg137 below · cited by 1 · depth 14 - Base change to ℚ̄ of both j-charts
ModularCurve.exists_algEquiv_tensor_chartAlg_chartRing_laurentBaseChange1 below · cited by 3 · depth 14 - Roof prolongation at level Nℓ over a level-N reduction datum
ModularCurve.exists_charLDegeneracyRoof_regularProlongation_heckeCompat_of_ne_of_residue_jq_jqN768 below · cited by 4 · depth 14 - Integral q-expansions: integrality over R((q)) forces R-coefficients
ModularCurve.exists_coeffMap_eq_of_isIntegralElem_of_exists_eq_div_of_injective0 below · cited by 3 · depth 14 - Degeneracy conjugation: (a,pb;c/p,d)∈Γ_{H'}(M/p)
ModularCurve.exists_conj_mem_GammaH_div0 below · cited by 1 · depth 14 - Cusp forms from ℚ̄-rational modular functions via ι₀(x)(vartheta j)^m
ModularCurve.exists_cuspForm_qExpansion_eq_coeffMap_mul_thetaL_pow_of_isIntegral96 below · cited by 1 · depth 14 - Integrality criterion making X θ j a weight-two cusp form
ModularCurve.exists_cuspForm_qExpansion_eq_mul_thetaL_of_isIntegral95 below · cited by 1 · depth 14 - Existence of a Deligne–Rapoport model package with q-expansion pin
ModularCurve.exists_dRModelPackage_ffPin1,050 below · cited by 1 · depth 14 - A weight-2m divisor on X₀(N)_ℚ̄ with integrality
ModularCurve.exists_divisor_degree_weight_and_isIntegral_of_mem_riemannRochSpace454 below · cited by 1 · depth 14 - Jacobi inversion: effective representatives for classes of J₀(M)
ModularCurve.exists_effective_pic0Mk_sub_eq_of_genusFF_le_degree189 below · cited by 8 · depth 14 - Orbit correspondence for cyclic N-subgroups over a given j₀
ModularCurve.exists_elliptic_cycSub_orbitMap_of_props199 below · cited by 1 · depth 14 - Everywhere-regular functions on X₀(N)_k are constant
ModularCurve.exists_eq_algebraMap_of_forall_ord_nonneg162 below · cited by 6 · depth 14 - Node-compatible sections in general position are constant
ModularCurve.exists_eq_algebraMap_of_hasValue_smul_of_generalPosition0 below · cited by 8 · depth 14 - ℚ is algebraically closed in the modular function field
ModularCurve.exists_eq_algebraMap_of_isAlgebraic_modularFunctionFieldFull1 below · cited by 3 · depth 14 - Unitary multiplier with non-zero cusp limits forces constancy
ModularCurve.exists_eq_const_of_norm_multiplier_eq_one0 below · cited by 1 · depth 14 - Frobenius-equivariant dictionary over an elliptic centre j=0,1728
ModularCurve.exists_equiv_ssPlaces_ssLocus_fibre_of_elliptic_centre_univ336 below · cited by 1 · depth 14 - Frobenius-equivariant bijection of supersingular places and moduli points
ModularCurve.exists_equiv_ssPlaces_ssLocus_fibre_of_generic_centre_univ342 below · cited by 1 · depth 14 - Eichler–Shimura congruence on a finite flat model of J₀(M)[p]
ModularCurve.exists_finiteFlat_model_jZero_torsion_heckeNilpotent_eichlerShimura1,914 below · cited by 1 · depth 14 - Almost all affine places of the level-N fibre are smooth
ModularCurve.exists_finset_forall_isCentreOf_unique_ord_eq_one146 below · cited by 1 · depth 14 - Eichler–Shimura over a DVR with Galois-equivariant quotient of Tₚ J₀(N)
ModularCurve.exists_galoisRepAdic_charpoly_frobenius_of_heckeChar_tateModule_quotient1,238 below · cited by 1 · depth 14 - Generic fibre of the two-chart integral model, with Galois compatibility
ModularCurve.exists_genericFibreIso_twoChartIntegralModel_chartPin_and_galoisCompat0 below · cited by 3 · depth 14 - Eisenstein-primary projectors on q^m-torsion, compatible in m
ModularCurve.exists_heckeAlg_tower_smul_smul_eq_and_smul_eq_iff_mem_eisensteinPrimaryTorsionBar2 below · cited by 1 · depth 14 - Hecke operator T_q as endomorphism of the relative Jacobian
ModularCurve.exists_heckeEndomorphism_relJacobian_of_representsRelSubPic_of_ratCurveModel591 below · cited by 2 · depth 14 - Exponential decay of cusp-regular differentials pulled back to H
ModularCurve.exists_isBigO_slash_realize_mul_deriv_realize_of_forall_ordDifferential_nonneg17 below · cited by 1 · depth 14 - Diamond action on the function field of X₁(M) and Γ_H-invariants
ModularCurve.exists_isDiamondPullbackModL_bot_forall_coe_mem_gammaH_iff288 below · cited by 4 · depth 14 - Existence of a Fricke automorphism at every level N
ModularCurve.exists_isFrickeAutFull_of_neZero77 below · cited by 59 · depth 14 - Integrality of q-expansions under the Atkin–Lehner matrix at ℓ
ModularCurve.exists_isIntegralQExp_smul_atkinLehnerSlash_of_even28 below · cited by 6 · depth 14 - Integrality at all cusps: Mᵃ clears denominators of f∣γ
ModularCurve.exists_isIntegral_level_pow_mul_qExpansion_slash_coeff97 below · cited by 7 · depth 14 - Structures on the λ-adic Tate module of J₀(M)
ModularCurve.exists_module_padicInt_tateModule_jZero_galoisRep_isAdicContinuous_heckeRep893 below · cited by 2 · depth 14 - Bounded denominators for rational q-expansions of level N
ModularCurve.exists_ne_zero_forall_intCast_mul_qExpansion_coeff_of_gamma_invariant22 below · cited by 1 · depth 14 - Level-one supersingular node pairs are indexed by j-invariants
ModularCurve.exists_nodePairsOfPlaces_arithFrobC_eq_nodePairsOf22 below · cited by 3 · depth 14 - Places over j=j₀ as cyclic N-subgroups of E₀
ModularCurve.exists_orbitMap_cyclicAddSubgroup_places_modularFunctionFieldFullC332 below · cited by 11 · depth 14 - Hecke-balanced pairing with multiplicative-type left kernel (q odd)
ModularCurve.exists_pairing_heckeTorsion_span_sup_heckeLatticeAlgebra_quotient_of_multiplicativeTypeNat_maximal_of_ne_two2,321 below · cited by 1 · depth 14 - A Laurent expansion with a uniformiser defines a place
ModularCurve.exists_place_of_ringHom_laurentSeries0 below · cited by 2 · depth 14 - A place of X₀(Nq) above a cusp where t_∞ reduces to 1
ModularCurve.exists_place_restrictAlong_heckeAlphaBar_eq_and_hasValue_tInfty204 below · cited by 8 · depth 14 - Bounded exponent on p-power torsion from Tate module vanishing
ModularCurve.exists_pow_smul_eq_zero_of_forall_tateModule_eq_zero919 below · cited by 1 · depth 14 - Points, reduction and Hecke action on a representing relative Jacobian
ModularCurve.exists_pts_relJacobian_jZero_of_representsRelSubPic_of_ratCurveModel1,162 below · cited by 1 · depth 14 - Rationality of q-expansions preserved under Γ₀(N)
ModularCurve.exists_ratCast_qExpansion_comp_smul_of_mem_Gamma016 below · cited by 1 · depth 14 - Descent to a finite field of traces, with semisimple image
ModularCurve.exists_semisimple_descent_of_trace_det_mem_range_finite4 below · cited by 3 · depth 14 - E₄³/Δ separates SL₂(ℤ)-orbits on H
ModularCurve.exists_smul_eq_of_E4_cube_div_discriminant_eq1 below · cited by 2 · depth 14 - Supersingular datum and Cartier anchors at levels Nq Rightarrow N
ModularCurve.exists_ssLevelDatum_heckeLaws_cartierAnchor_edgeHecke_and_vertexHecke3,385 below · cited by 1 · depth 14 - Maximal multiplicative-type submodule of Eisenstein torsion absorbs inertia displacements
ModularCurve.exists_submodule_multiplicativeTypeNat_maximal_heckeTorsion_span_sup_inertiaSubgroupIn1,977 below · cited by 1 · depth 14 - Gauss valuation ring and coefficientwise reduction of q-expansion fields
ModularCurve.exists_valuationSubring_ringHom_laurentSeries_qExpFunctionFieldC_of_liesOverPrime5 below · cited by 7 · depth 14 - Fibre polynomial factors as Frobenius times Verschiebung
ModularCurve.fibrePoly_eq_of_kroneckerCongruence0 below · cited by 6 · depth 14 - Finiteness of F_N^{full} over ℚ(j)
ModularCurve.finiteDimensional_adjoin_jFull_modularFunctionFieldFull117 below · cited by 29 · depth 14 - Finiteness and degree bound descend under coefficientwise field maps
ModularCurve.finiteDimensional_and_finrank_adjoin_le_of_eq_coeffMap0 below · cited by 4 · depth 14 - Finiteness and separability over K(j(q^N))
ModularCurve.finiteDimensional_and_isSeparable_adjoin_jqNModC_of_natCast_ne_zero98 below · cited by 3 · depth 14 - Finiteness of H¹_{fppf}(Specℤ,mathcal J_m) for odd q
ModularCurve.finite_fppfCohomology_one_jZeroNeronPrimaryTorsionCore_of_ne_two1,464 below · cited by 1 · depth 14 - Finitely many Γ₀(N)-orbits of zeros and poles of F-t
ModularCurve.finite_image_orbitRel_meromorphicOrderAt_sub_ne_zero0 below · cited by 1 · depth 14 - Roof substitution leg has the degree of β_ℓ
ModularCurve.finrankAlong_towerSubstBar_roof147 below · cited by 1 · depth 14 - Degree ψ(N) of the modular function field over K(j_N)
ModularCurve.finrank_adjoin_jqNModC_modularFunctionFieldFullC_eq_dedekindPsi119 below · cited by 3 · depth 14 - Ribet's exchange inequality: dim X^{old}+dim Y_{q'}≤dim Y_q
ModularCurve.finrank_quotient_old_add_ribbon_le_finrank_quotient_ribbon_of_twoPlaceTorsionDatum_of_le_invariants3,576 below · cited by 1 · depth 14 - Rank inequality between old-plus-ribbon and ribbon parts at two places
ModularCurve.finrank_quotient_old_add_ribbon_le_finrank_quotient_ribbon_of_twoPlaceTorsionDatum_of_le_invariants_of_two_mul_dvd_of_neZero3,576 below · cited by 1 · depth 14 - Toric 𝔪-torsion bounded by old plus ribbon dimensions
ModularCurve.finrank_span_toricMonodromyPart_le_finrank_quotient_old_add_ribbon_of_ssLevelDatum3,563 below · cited by 1 · depth 14 - Toric monodromy bound by old and ribbon parts
ModularCurve.finrank_span_toricMonodromyPart_le_finrank_quotient_old_add_ribbon_of_ssLevelDatum_of_two_mul_dvd_of_neZero3,563 below · cited by 1 · depth 14 - From generators T_ℓ to the whole Hecke algebra
ModularCurve.forall_heckeAlg_exists_hom_mul_and_pts_smul_eq_comp_of_forall_heckeGen0 below · cited by 1 · depth 14 - Characteristic 2: ord_P(dj)≥ 14 when ord_P(j)=12
ModularCurve.fourteen_le_ordDiff_D_jqModC_of_ord_eq_twelve352 below · cited by 1 · depth 14 - Fricke involution on Ogg's modular unit over ℚ̄
ModularCurve.frickeInvolutionBar_coeffEmb_modularUnitSeries109 below · cited by 12 · depth 14 - Geometric Frobenius sends the place jmath̃=a to jmath̃=a^q
ModularCurve.frobOnPlacesGeomLevel_charLGeomPlaceOfPoint42 below · cited by 28 · depth 14 - Frobenius on places fixes the q-adic infinite place
ModularCurve.frobOnPlacesGeomLevel_qInftyPlaceBar0 below · cited by 1 · depth 14 - Riemann–Roch for the modular function field over ℚ̄
ModularCurve.functionFieldRiemannRoch_modularFunctionFieldBar151 below · cited by 32 · depth 14 - Side pairing on the boundary of a tiled fundamental set
ModularCurve.gammaFundamentalSet_boundary_sidePairing_of_slash_eq_add0 below · cited by 6 · depth 14 - Geometric integrality of the rational fibre of the two-chart model
ModularCurve.geometricallyIntegral_baseChangeToBase_twoChartIntegralModel_rat853 below · cited by 3 · depth 14 - Same-ideal at-q Néron data yields the core data
ModularCurve.hasJZeroNeronAtPDataCore_of_hasJZeroNeronAtPDataSameIdeal0 below · cited by 1 · depth 14 - Existence of at-q Néron data for J₀(Nq), version 2.2
ModularCurve.hasJZeroNeronAtPDataOrdV225,561 below · cited by 1 · depth 14 - Degree-zero principal divisors on K(j,j_N)
ModularCurve.hasPrincipalDivisors_modularFunctionFieldC_of_isSeparable_jqNModC104 below · cited by 1 · depth 14 - q-expansion at the coset point (aτ+b)/d
ModularCurve.hasSum_cosetSubst_coeff_mul_qParam_pow0 below · cited by 1 · depth 14 - Two β-substitutions generate the roof field at Nℓℓ'
ModularCurve.heckeBetaRoof_adjoin_range_union_eq_top90 below · cited by 1 · depth 14 - Diagonal identity β̄^*ᾱ_* = Uₚ + σ_* for p ∤ N
ModularCurve.heckeDiagonalIdentity_of_prime_of_not_dvd187 below · cited by 1 · depth 14 - Transposed Hecke correspondence inputs at level N for every prime ℓ
ModularCurve.heckeInputsAllTranspose93 below · cited by 2 · depth 14 - Hecke inputs at the degeneracy roof for invertible q
ModularCurve.heckeInputsFibre_of_natCast_ne_zero68 below · cited by 11 · depth 14 - Fricke automorphism intertwines the Hecke correspondence with its transpose
ModularCurve.heckePic0BarTranspose_fricke_smul80 below · cited by 1 · depth 14 - Igusa's lower bound for the mod-ℓ q-expansion field
ModularCurve.index_gammaH_le_finrank_adjoin_jqModC_qExpFunctionFieldC_residueField216 below · cited by 18 · depth 14 - Inertia degree one along α̃_q over an algebraically closed field
ModularCurve.inertiaDegAlong_heckeAlphaC_eq_one109 below · cited by 6 · depth 14 - Inertia degree along the β-leg is one over ̄ k
ModularCurve.inertiaDegAlong_heckeBetaC_eq_one112 below · cited by 2 · depth 14 - Every place is affine or a pole of ̄ j
ModularCurve.isAffineGeomPlace_or_ord_jGeomGen_lt_zero72 below · cited by 22 · depth 14 - Finite flatness of [n] on base changes of J
ModularCurve.isFinite_and_flat_schemeNsmul_baseChange_of_jZeroC_points271 below · cited by 1 · depth 14 - Integrality over K[j(q)] inside the modular function field
ModularCurve.isIntegral_adjoin_mk_coeffMap79 below · cited by 3 · depth 14 - Integrality bounds from regularity of x djmath̄ on X₀(N)_ℚ̄
ModularCurve.isIntegral_and_isIntegral_of_smul_D_mem_regularDifferentialsBar427 below · cited by 2 · depth 14 - Separability over K(j) inside K((q)) for perfect K
ModularCurve.isSeparable_adjoin_jqModC_of_isAlgebraic1 below · cited by 15 · depth 14 - Coupled Kronecker dichotomy at the two degeneracy restrictions
ModularCurve.kroneckerCentreDichotomy63 below · cited by 1 · depth 14 - Coordinatewise Kronecker dichotomy at the two degeneracy places
ModularCurve.kroneckerCoordinatewiseDichotomy67 below · cited by 2 · depth 14 - Pair form of Kronecker's congruence at level (N,ℓ)
ModularCurve.kroneckerPairIntegral81 below · cited by 1 · depth 14 - Base change of the q-expansion function field
ModularCurve.laurentBaseChange_qExpFunctionFieldC_eq0 below · cited by 28 · depth 14 - Generating the level-Mp roof function field over L
ModularCurve.laurentBaseChange_xHFunctionField_sup_adjoin_qExpand_xHTopFunctionFieldC175 below · cited by 1 · depth 14 - Relative degree at least ℓ+1 for level Mℓ
ModularCurve.le_relrank_xHFunctionField_xHTopFunctionFieldC_of_not_dvd1 below · cited by 2 · depth 14 - Integral q-expansion quotients lie in the localised modular ring
ModularCurve.mem_modularLocalized_mul_of_not_dvd_of_exists_coeffMap_mul_eq144 below · cited by 1 · depth 14 - Supersingular j-invariants transfer along maps from algebraically closed fields
ModularCurve.mem_ssJSet_map_of_isAlgClosed18 below · cited by 27 · depth 14 - Modular equation as minimal polynomial of α(j) over β
ModularCurve.minpoly_heckeAlphaBar_along_heckeBetaBar148 below · cited by 5 · depth 14 - Modular polynomial as minimal polynomial of j(q^N) over L(j(q))
ModularCurve.minpoly_jqNModC_eq145 below · cited by 9 · depth 14 - Minimal polynomial of j(q^M) over K(j) as a slot product
ModularCurve.minpoly_jqNModC_map_eq_prod_slots51 below · cited by 6 · depth 14 - Characteristic-ℓ divisor expansions generate k(̃ j,̃ j_N)
ModularCurve.modularFunctionFieldFullC_eq_modularFunctionFieldC_residueField751 below · cited by 7 · depth 14 - Order of the component group as a Gram determinant
ModularCurve.natCard_componentGroup_eq_natAbs_det0 below · cited by 1 · depth 14 - Number of ⟨ T⟩-orbits on SL₂(ℤ)/Γ₀(N)
ModularCurve.natCard_orbitRelQuotient_zpowers_T_gamma0_eq_cuspCount5 below · cited by 3 · depth 14 - Good-reduction Néron identity component of J₀(p) from the Deligne–Rapoport model
ModularCurve.nonempty_jZeroNeronIdentityComponentGood_of_dRModelPackage_of_ffPin3,335 below · cited by 1 · depth 14 - Existence of good Néron identity component data when J₀(p) vanishes
ModularCurve.nonempty_jZeroNeronIdentityComponentGood_of_subsingleton1 below · cited by 1 · depth 14 - Semistable specialization datum for J₀(Nq) with Néron clauses
ModularCurve.nonempty_jZeroSemistableSpecialization_neronClauses_nodes3,550 below · cited by 1 · depth 14 - Leg-two input (v2) for the Deligne–Rapoport package
ModularCurve.nonempty_legTwoInputV21,395 below · cited by 1 · depth 14 - Existence of a supersingular two-level degeneracy datum
ModularCurve.nonempty_ssLevelDatum307 below · cited by 4 · depth 14 - Positivity of the characteristic-q width at supersingular places
ModularCurve.one_le_placeWidthChar_of_mem_ssPlaces395 below · cited by 15 · depth 14 - At the cusp of the level-one j-line, ord is the q-adic order
ModularCurve.ord_charLGeomPlaceEquiv_placeInfty_eq_order9 below · cited by 14 · depth 14 - Order zero of the units Δ(q)/Δ(q^δ) outside the cusps
ModularCurve.ord_coeffEmb_modularUnitSeries_eq_zero_of_not_isCusp63 below · cited by 11 · depth 14 - ord of j-1728 at the place j=1728 is 1
ModularCurve.ord_jLinePlace1728_jGen_sub40 below · cited by 1 · depth 14 - j is a uniformiser at the place j=0
ModularCurve.ord_jLinePlaceZero_jGen40 below · cited by 1 · depth 14 - Ramification of j over j=0 on X₀(N) in characteristic 3
ModularCurve.ord_jqModC_census_of_char_three344 below · cited by 3 · depth 14 - Zeros of j on X₀(N) in characteristic 2: ramification census
ModularCurve.ord_jqModC_census_of_char_two340 below · cited by 1 · depth 14 - Adjointness of degeneracy maps for the divisorial Weil pairing
ModularCurve.pair_degeneracyPullbackPair_eq_pair_degeneracyPushforwardPair141 below · cited by 1 · depth 14 - i times the Petersson functional lies in the period lattice iff all periods have integral real part
ModularCurve.petersson_mem_periodLattice_iff_re_period_int576 below · cited by 2 · depth 14 - Irreducibility of prime-level modular polynomial data
ModularCurve.phiIrreducible_of_prime42 below · cited by 2 · depth 14 - Level-one place specialisation over an arbitrary residue field
ModularCurve.placeSpecialization_exists_level_one_of_surjective227 below · cited by 6 · depth 14 - Cross identity w(β W)e_α(W)=w(α W)e_β(W) on the degeneracy roof
ModularCurve.placeWidthChar_restrictAlong_mul_ramificationIndexAlong_heckeAlphaC_heckeBetaC_cross_of_prime593 below · cited by 4 · depth 14 - Supersingular j-invariants satisfy j^{q^2}=j
ModularCurve.pow_q_sq_eq_self_of_mem_ssJSet6 below · cited by 103 · depth 14 - Mordell's relation τ(p²) = τ(p)² - p¹¹
ModularCurve.qCoeff_discriminant_prime_sq_eq_sq_sub_pow_eleven5 below · cited by 2 · depth 14 - Frobenius push-forward on Pic⁰ is arithmetic Frobenius
ModularCurve.qExpFrobeniusPushforwardModL_eq_qExpArithFrobC_smul47 below · cited by 2 · depth 14 - Substitution q↦ q^ℓ on ratios of integral forms
ModularCurve.qExpand_image_intFormRatiosC_subset1 below · cited by 22 · depth 14 - The cusp coordinate t=j(qᵖ)/jᵖ at the prime p
ModularCurve.qExpand_jq_div_pow_mem_chartAlgInf_and_coeff_zero_and_mem_nonunits_gauss79 below · cited by 6 · depth 14 - Ratios of q-expansions on Γ_H(N) lie in ℂ· F
ModularCurve.qExpansion_div_mem_laurentBaseChange_xHFunctionField52 below · cited by 16 · depth 14 - Off-diagonal roof places: e_α· r equals characteristic j-width
ModularCurve.ramificationIndexAlong_heckeAlphaC_mul_placeRamificationJ_eq_jWidthChar_of_restrictAlong_ne_of_prime701 below · cited by 1 · depth 14 - Degree of j(q^N) bounds the index of Γ∩Γ₀(N)
ModularCurve.relIndex_gamma0_le_relrank_adjoin_insert_jqNModC6 below · cited by 11 · depth 14 - Degree of the q-expansion field of X_H(M) over that of X₀(M)
ModularCurve.relfinrank_qExpFunctionFieldC_gamma0_gammaH_eq_index_of_charZero236 below · cited by 13 · depth 14 - Relative rank of q-expansion fields at most the index
ModularCurve.relrank_adjoin_qExpansion_div_le_relIndex0 below · cited by 12 · depth 14 - Separatedness pins the reduction of a B-point
ModularCurve.schemeHomOver_residue_eq_ptsSp_reductionModL_of_isSeparated0 below · cited by 1 · depth 14 - Different exponent ≥ 7 at a characteristic-3 place with ord_P(j)=6
ModularCurve.seven_le_ordDiff_D_jqModC_of_ord_eq_six353 below · cited by 1 · depth 14 - Γ₀(ℓ)-invariance of the sharp eta quotient
ModularCurve.sharpUnitInvariant6 below · cited by 2 · depth 14 - Necessity in the sharp-unit criterion at prime level ℓ
ModularCurve.sharpUnitNecessary35 below · cited by 1 · depth 14 - Smoothness of the characteristic-p fibre of the two-chart model
ModularCurve.smoothOfRelativeDimension_one_pullback_snd_toBase_twoChartIntegralModel_qExpFunctionFieldC_of_charP915 below · cited by 1 · depth 14 - Supersingular count formula equals genus defect g(Nq)-2g(N)+1
ModularCurve.ssCountFormula_eq_genus7 below · cited by 1 · depth 14 - Deuring's criterion: the two supersingular j-sets coincide
ModularCurve.ssJSet_eq_ssJSetHasse12 below · cited by 14 · depth 14 - Triviality of Pic⁰ in characteristic ℓ for genus-zero X₀(p)
ModularCurve.subsingleton_jZeroC_residueField_of_subsingleton_jZero949 below · cited by 1 · depth 14 - Zeros of ̄ j - j₀ count the degree over ℚ̄(̄ j)
ModularCurve.sum_ord_jBar_sub_eq_finrank186 below · cited by 1 · depth 14 - Monicity of the transpose under coefficient degree bounds
ModularCurve.swapBivar_monic_of_coeff_bounds5 below · cited by 2 · depth 14 - Tate modules have no Tₚ²=(p+1)² eigenvectors
ModularCurve.tateModule_eq_zero_of_forall_heckeOperatorBar_heckeOperatorBar_eq_smul1,212 below · cited by 1 · depth 14 - Order-three elliptic count for Γ₀(N): 3ε₃=ψ(N)+2ν₃(N)
ModularCurve.three_mul_card_orbitRelQuotient_zpowers_S_mul_T_gamma0_eq6 below · cited by 1 · depth 14 - Count of moduli points with j=0: ψ(N)+2ν₃(N)
ModularCurve.three_mul_natCard_moduliPoint_j_eq_zero_eq_dedekindPsi_add_two_mul_nuThree_of_ne_zero22 below · cited by 3 · depth 14 - Transcendence and finiteness under constant field extension of F₀
ModularCurve.transcendental_and_finiteDimensional_adjoin_laurentBaseChange_of_coe_eq_coeffEmb0 below · cited by 7 · depth 14 - Transcendence of j and finiteness over K(j)
ModularCurve.transcendental_and_finiteDimensional_adjoin_laurentBaseChange_qExpFunctionFieldC_of_coe_eq_jqModC120 below · cited by 61 · depth 14 - Denominator-cleared form of the genus formula for X₀(N)
ModularCurve.twelve_mul_genusFormula0 below · cited by 3 · depth 14 - Counting ⟨ S⟩-orbits on SL₂(ℤ)/Γ₀(N)
ModularCurve.two_mul_card_orbitRelQuotient_zpowers_S_gamma0_eq6 below · cited by 1 · depth 14 - Points of X₀(N) with j-invariant 1728
ModularCurve.two_mul_natCard_moduliPoint_j_eq_1728_eq_dedekindPsi_add_nuTwo_of_ne_zero20 below · cited by 3 · depth 14 - Compositum of level M/p q-expansions and their q↦ qᵖ translates
ModularCurve.xHFunctionFieldBar_div_sup_adjoin_qExpand_eq_xHFunctionFieldBar179 below · cited by 3 · depth 14 - q-expansion field of X_H(M) generated by j(qᵖ)
ModularCurve.xHFunctionFieldBar_div_sup_adjoin_qExpand_jqModC_eq_xHFunctionFieldBar178 below · cited by 4 · depth 14 - Integrality of X⁶j⁴(j-1728)³ bounds orders of hΔ by gE₄²E₆
ModularCurve.analyticOrderAt_le_of_isIntegral_adjoin_coeffEmb_jq14 below · cited by 1 · depth 15 - Arithmetic Frobenius sends the place j=a to j=a^q
ModularCurve.arithFrobC_smul_charLGeomPlaceOfPoint17 below · cited by 15 · depth 15 - Arithmetic Frobenius sends a place to its coefficientwise twist
ModularCurve.arithFrobC_smul_eq_of_apply_eq_coeffMap_frobenius_univ1 below · cited by 1 · depth 15 - Modular functions with K-rational q-expansion: fixed, with equivariant values
ModularCurve.arithmeticGalois_smul_eq_self_and_evalAt_smul_of_coe_mem_fieldOver7 below · cited by 12 · depth 15 - Eichler–Deuring supersingular count in characteristics 2 and 3
ModularCurve.card_eq_ssCountFormula_of_ssPlaces_of_lt_five357 below · cited by 7 · depth 15 - Chart rings over ℚ̄ lie in the mathbb Z₍ₚ₎-span
ModularCurve.chartRing_laurentBaseChange_le_span_coeffEmb_chartAlg0 below · cited by 5 · depth 15 - Fricke involution sends Δ(q)/Δ(q^N) to N¹² times its inverse
ModularCurve.coe_frickeInvolutionFull_modularUnitSeries_of_neZero82 below · cited by 5 · depth 15 - Valuation property of the localized modular ring at level q
ModularCurve.coe_mem_modularLocalized_or_coe_inv_mem_modularLocalized123 below · cited by 3 · depth 15 - Commutativity of the divisor correspondences α_*β^* at two primes
ModularCurve.correspondence_heckeBetaC_heckeAlphaC_correspondence_heckeBetaC_heckeAlphaC_comm258 below · cited by 1 · depth 15 - Uₛ plus Atkin–Lehner equals b^*a_* on divisors
ModularCurve.correspondence_heckeBetaC_heckeAlphaC_single_add_single_autOnPlaces_eq_pullbackAlong_pushforwardAlong208 below · cited by 3 · depth 15 - Degeneracy pair: finite separable of degree s+1, supersingularity preserved and reflected
ModularCurve.degeneracyPair_finiteSeparableDeg_ssPlaces_preserved_reflected402 below · cited by 14 · depth 15 - Degeneracy pair at level Ms: degree s+1 and place transport
ModularCurve.degeneracyPair_finrankAlong_and_place_transports257 below · cited by 3 · depth 15 - Degeneracy pushforwards commute with α_*β^* at ℓ ≠ s
ModularCurve.degeneracyPair_pushforwardAlong_correspondence_heckeBetaC_heckeAlphaC_comm_of_ne_of_not_dvd237 below · cited by 1 · depth 15 - Kronecker congruence: Δ̄^{q-1} times the supersingular product is 1
ModularCurve.delta_pow_mul_prod_jqModC_sub_pow_eq_one98 below · cited by 9 · depth 15 - The p-adic Eisenstein torsion of J₀(p) vanishes
ModularCurve.eisensteinTorsionBar_self_eq_bot1,072 below · cited by 1 · depth 15 - Joint injectivity of polar coefficients on section pairs in general position
ModularCurve.eq_zero_of_forall_sum_mul_taylorCoeff_mul_pow_eq_zero_of_generalPosition12 below · cited by 1 · depth 15 - Integrality over ℂ[j⁻¹] forces cuspidal decay of gE₄²E₆
ModularCurve.eventually_norm_slash_le_of_isIntegral_adjoin_coeffEmb_jq_inv15 below · cited by 1 · depth 15 - Uniqueness of the Kronecker remainder modulo p
ModularCurve.existsUnique_kroneckerRemainder0 below · cited by 11 · depth 15 - Relative Pic⁰ of the Deligne–Rapoport model: group law and points
ModularCurve.exists_abelJacobi_pts_relativeGroupLaw_of_dRModelPackage_of_representsRelSubPic1,214 below · cited by 1 · depth 15 - Hecke coordinate on multiplicative-type subgroups of J₀(p)[P^m] at 2
ModularCurve.exists_addMonoidHom_heckeLatticeAlgebra_quotient_two_pow_natCard_ker_le_of_multiplicativeTypeNat_le_eisensteinTorsionBar2,544 below · cited by 1 · depth 15 - Base change of an Atkin–Lehner automorphism exchanges the degeneracy maps
ModularCurve.exists_algEquiv_comp_heckeAlphaBar_eq_heckeBetaBar2 below · cited by 5 · depth 15 - Fricke and Atkin–Lehner involutions of the degeneracy roof
ModularCurve.exists_algEquiv_modularFunctionFieldC_swap_and_charLDegeneracyRoof_swap147 below · cited by 4 · depth 15 - Igusa reduction: finite chart of the Kroneckerian model
ModularCurve.exists_algEquiv_residueField_tensor_chartAlgFin_twoChartIntegralModel_qExpFunctionFieldC_chartRing888 below · cited by 1 · depth 15 - Igusa's theorem, pole chart: reduction of 𝒪_∞
ModularCurve.exists_algEquiv_residueField_tensor_chartAlgInf_twoChartIntegralModel_qExpFunctionFieldC_chartRing888 below · cited by 1 · depth 15 - Base change of the modular function field inside κ((q))
ModularCurve.exists_algHom_tensorProduct_modularFunctionFieldC_injective1 below · cited by 1 · depth 15 - J[𝔪] is a direct sum of copies of ρ
ModularCurve.exists_blrDecomposition_heckeTorsion_of_span_eq_top_of_frobeniusQuadratic_of_dense2 below · cited by 4 · depth 15 - Cartier anchor of the toric part of J₀(Lp)
ModularCurve.exists_cartierAnchor_toricMonodromyPart_ssHeckeFamilyC3,376 below · cited by 3 · depth 15 - Character detecting inertia eigenvectors in Eisenstein torsion, q odd
ModularCurve.exists_character_generator_heckeTorsion_span_sup_inertiaSubgroupIn_of_ne_two2,072 below · cited by 2 · depth 15 - A component map on inertia invariants of J₀(p)
ModularCurve.exists_componentHom_extension_of_dRModelPackage_of_abelJacobi_of_ffPin2,509 below · cited by 1 · depth 15 - Principal multiple of the cuspidal divisor gives an invariant (Δ/Δ(ℓ ·))^m branch
ModularCurve.exists_continuous_pow_eq_of_isPrincipal_smul_cuspidalDivisor134 below · cited by 1 · depth 15 - Crossing presentation at the j=0 node of X₀(3)
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring_of_q_eq_three181 below · cited by 3 · depth 15 - Crossing presentation at the characteristic-two supersingular node
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring_of_q_eq_two182 below · cited by 3 · depth 15 - Riesz representation for the weight-2 Petersson pairing
ModularCurve.exists_cuspForm_petersson_eq7 below · cited by 2 · depth 15 - Weight-2m cusp forms from integrality over ℂ[j] and ℂ[1/j]
ModularCurve.exists_cuspForm_qExpansion_eq_mul_thetaL_pow_of_isIntegral95 below · cited by 1 · depth 15 - Cyclic N-subgroups as K-embeddings of the level-N function field
ModularCurve.exists_equiv_algHom_modularFunctionFieldFullC_of_transcendental_j305 below · cited by 4 · depth 15 - Equivariant reduction of ofJ(t) at a place over j₀
ModularCurve.exists_equivariant_torsion_reduction_ofJ45 below · cited by 8 · depth 15 - Integral lift of level-N modular functions over a number field
ModularCurve.exists_fieldOver_lift_isIntegral_of_isIntegral743 below · cited by 6 · depth 15 - Descent of a level-M modular function and a residue value
ModularCurve.exists_finiteDimensional_mem_fieldOver_and_redRestrict_eq_level114 below · cited by 5 · depth 15 - Eichler–Shimura relation on a finite flat model of J₀(M)[p]
ModularCurve.exists_finiteFlat_model_jZero_torsion_heckeNilpotent_frobenius_verschiebung_reductionModL1,913 below · cited by 1 · depth 15 - Integral Eisenstein family on Γ₁(M) permuted by Γ₀(M)
ModularCurve.exists_gamma1_eisenstein_isIntegralQExp_and_slash_eq2 below · cited by 6 · depth 15 - Gauss valuation subring of L·ℚ(X(Γ)) inside L((q))
ModularCurve.exists_gaussValuationSubring_laurentBaseChange_qExpFunctionFieldC0 below · cited by 69 · depth 15 - Regular branch reductions at an mathbb F_{q²}-point with no pole
ModularCurve.exists_hasValue_and_hasValue_frobNodePair_of_forall_pole_not_centred270 below · cited by 3 · depth 15 - Agreeing branch values at a supersingular node
ModularCurve.exists_hasValue_frobNodePair_of_forall_pole_not_centred535 below · cited by 13 · depth 15 - Hecke endomorphisms of the integral model of J₀(p)
ModularCurve.exists_heckeEndomorphism_of_dRModelPackage_of_representsRelSubPic291 below · cited by 1 · depth 15 - Inertia-invariant classes of J₀(M) lift to invariant divisors
ModularCurve.exists_inertiaStable_degZero_pic0Mk_eq178 below · cited by 1 · depth 15 - Diamond action of Γ₀(M) on the q-expansion function field
ModularCurve.exists_isDiamondPullbackModL_bot_of_natCast_ne_zero109 below · cited by 1 · depth 15 - Galois model over K(j) with ramification dividing 6, characteristic 3
ModularCurve.exists_isGalois_ord_jqModC_dvd_six_of_char_three333 below · cited by 1 · depth 15 - Galois model over K(j) with ramification at j=0 dividing 12
ModularCurve.exists_isGalois_ord_jqModC_dvd_twelve_of_char_two332 below · cited by 1 · depth 15 - Integrality of 1/wₚ(j) over R[1/j] up to a unit
ModularCurve.exists_isIntegral_adjoin_inv_jq_mul_inv_atkinLehnerInvolutionFull80 below · cited by 1 · depth 15 - Integrality at the Fricke image, up to a power of N
ModularCurve.exists_isIntegral_level_pow_mul_qExpansion_slash_fricke_coeff96 below · cited by 2 · depth 15 - Periods of Γ₀(N) as edge integrals of integral parabolic characters
ModularCurve.exists_isParabolicHom_sum_intCast_mul_edgeIntegral_eq_period3 below · cited by 1 · depth 15 - At-p Néron ordinary datum for J₀(N₀p) with p-new toric part
ModularCurve.exists_jZeroNeronAtPDataOrdV22_pNew5,560 below · cited by 1 · depth 15 - Ogg's modular unit mod p for p<5
ModularCurve.exists_laurentSeries_int_modularUnitSeries_coeffMap_eq_jqModC_pow_of_lt_five1 below · cited by 7 · depth 15 - Reduction mod p of Δ(q)/Δ(qᵖ) as a supersingular product
ModularCurve.exists_laurentSeries_int_modularUnitSeries_coeffMap_eq_prod_ssJSet259 below · cited by 9 · depth 15 - Places over a centre from the roots of Φ_N
ModularCurve.exists_map_roots_places_of_card_roots_eq_dedekindPsi_univ278 below · cited by 1 · depth 15 - Edge integrals of a parabolic character are periods
ModularCurve.exists_mem_periodLattice_eq_sum_intCast_mul_edgeIntegral_of_isParabolicHom2 below · cited by 2 · depth 15 - Interpolation on the full level-N modular curve with simple poles
ModularCurve.exists_mem_riemannRochSpace_ord_sub_eq_one_hasValue_modularFunctionFieldFullC164 below · cited by 4 · depth 15 - Δ∣₁₂diag(p,1) is a modular form on Γ₀(p)
ModularCurve.exists_modularForm_coe_eq_discriminant_slash_heckeDiagMatrix0 below · cited by 5 · depth 15 - Weight-two Eisenstein series on Γ₀(p) with prescribed q-expansion
ModularCurve.exists_modularForm_qCoeff_eq_eisensteinTwoCoeff3 below · cited by 5 · depth 15 - Diamond action on the mod-ℓ q-expansion function field of X_H(M)
ModularCurve.exists_monoidHom_gamma0_algEquiv_qExpFunctionFieldC_zmod106 below · cited by 6 · depth 15 - Descent by Fourier coefficients to K(j,fᵥ)
ModularCurve.exists_mvPolynomial_mul_aeval_fricke_eq_of_qExpansion_coeff_mem15 below · cited by 1 · depth 15 - Fppf H⁰,H¹ bounds for a generic-degree-2 Hopf points sheaf
ModularCurve.exists_natCard_fppfCohomology_of_sectionsEquiv_algHom_two50 below · cited by 3 · depth 15 - Bounded denominators for Fricke functions of level N
ModularCurve.exists_ne_zero_forall_mul_qExpansion_coeff_fricke_mem_adjoin1 below · cited by 1 · depth 15 - Fractions over a subfield of constants at a place
ModularCurve.exists_ne_zero_mul_eq_isIntegral_of_mem_closure_of_mem_valuationSubring72 below · cited by 2 · depth 15 - Number-field presentation of functions on X₀(Nq)
ModularCurve.exists_numberField_presentation_level114 below · cited by 3 · depth 15 - Base change of the two-chart model to ℚ̄
ModularCurve.exists_ofGenerator_baseChangeIso_chartPin_and_placeCompat2 below · cited by 3 · depth 15 - Cyclic N-subgroups parametrise places and moduli points over j(E₀)
ModularCurve.exists_orbitMap_places_moduliPoint_arithFrobC_compat_univ332 below · cited by 1 · depth 15 - Points dictionaries modulo ℓ for the relative Pic⁰ of X₀(p)
ModularCurve.exists_pointsDict_pullback_snd_ratLocalizedAt_of_dRModelPackage_of_representsRelSubPic1,835 below · cited by 2 · depth 15 - Rationality of q-expansions at the Atkin–Lehner translates
ModularCurve.exists_ratCast_qExpansion_comp_smul_of_mem_Gamma0_of_dvd16 below · cited by 1 · depth 15 - Gauss regular prolongation of a valuation ring to L· F(Γ)
ModularCurve.exists_regularProlongation_laurentBaseChange_qExpFunctionFieldC_residue_mul_eq1 below · cited by 15 · depth 15 - Regular prolongation and place map for X₀(M) at ℓ ∤ M
ModularCurve.exists_regularProlongation_placeMap_modularFunctionFieldFullC_of_not_dvd737 below · cited by 5 · depth 15 - Divisors of degree at least the genus admit nonzero sections
ModularCurve.exists_section_of_genusFF_le_degree186 below · cited by 3 · depth 15 - Coefficient automorphisms extend to the roof, intertwining both Hecke legs
ModularCurve.exists_semilinearAut_intertwinesAlong_heckeAlphaC_heckeBetaC_coeffSemilinearAut0 below · cited by 2 · depth 15 - Two valuation rings of ℚ(X₀(Np)) lying over p
ModularCurve.exists_valuationSubring_pair_modularFunctionFieldFull_mul_of_not_dvd123 below · cited by 23 · depth 15 - ℤ-sections of G have finite index in J₀(p)(ℚ)
ModularCurve.finiteIndex_closure_range_sections_addSubgroupOf_fixedPoints_of_compMap4 below · cited by 1 · depth 15 - Finiteness of fixed points of the squared Frobenius push-forward
ModularCurve.finite_fixedPoints_frobeniusPushforwardModL_comp_self210 below · cited by 2 · depth 15 - Finiteness of places fixed by the squared Frobenius
ModularCurve.finite_setOf_frobOnPlacesGeomLevel_frobOnPlacesGeomLevel_eq_self123 below · cited by 11 · depth 15 - Degree one along the equal-level tower inclusion
ModularCurve.finrankAlong_towerInclBar_of_eq147 below · cited by 1 · depth 15 - Relative degree of full modular function fields at a prime
ModularCurve.finrank_modularFunctionFieldFull_mul_prime_eq_of_coe_eq71 below · cited by 5 · depth 15 - Degree of level Mℓ over level M along q↦ q^ℓ
ModularCurve.finrank_modularFunctionFieldFull_mul_prime_eq_of_coe_eq_qExpand72 below · cited by 4 · depth 15 - Rational p-adic Tate module of J₀(N) has dimension 2g
ModularCurve.finrank_rationalTateModule_jZero_eq_two_mul_finrank_regularDiffs469 below · cited by 1 · depth 15 - Toric 𝔪-torsion bounded via a character lattice sandwich
ModularCurve.finrank_span_toricMonodromyPart_le_finrank_quotient_add_of_anchor_of_ker_le_range2 below · cited by 2 · depth 15 - Hecke action on the generic fibre of relative Pic⁰
ModularCurve.forall_exists_schemeHomOver_baseChange_rat_isHom_pts_smul_of_dRModelPackage1,207 below · cited by 1 · depth 15 - Mutual pole-chart visibility for j and jₚ
ModularCurve.forall_mem_chartAlgInf_jFull_exists_mul_mem_and_symm_of_coe_eq_qExpand85 below · cited by 5 · depth 15 - Fricke involution carries the level-one inclusion to q-rescaling
ModularCurve.frickeInvolutionBar_comp_heckeAlphaBar_one79 below · cited by 8 · depth 15 - At level one the full and two-generator modular function fields agree
ModularCurve.full_one_eq1 below · cited by 1 · depth 15 - Coprime expansion of the genus defect g(MN)-2g(N)+1
ModularCurve.genusFormula_mul_expand4 below · cited by 2 · depth 15 - Good-prime data for relative Pic⁰ of the DR model at ℓ ∤ p
ModularCurve.goodPrime_relativePic0_of_dRModelPackage_of_representsRelSubPic1,837 below · cited by 1 · depth 15 - T_ℓ multiplies the cuspidal class by 1+ℓ
ModularCurve.heckeOperatorBar_cuspidalClass264 below · cited by 1 · depth 15 - Uₚ fixes the cuspidal class of J₀(p)
ModularCurve.heckeOperatorBar_cuspidalClass_self264 below · cited by 1 · depth 15 - Norm transform endomorphism realises T_q on ℚ̄-points
ModularCurve.heckeOperatorBar_points_eq_comp_of_transform359 below · cited by 3 · depth 15 - Index of H{±1} bounds the degree of F_H/F₀
ModularCurve.index_le_relfinrank_qExpFunctionFieldC_gamma0_gammaH_of_charZero41 below · cited by 3 · depth 15 - Inertia at q acts by n on the reduction kernel
ModularCurve.inertia_smul_eq_nsmul_of_mem_heckeTorsion_span_sup_of_reductionModL_eq_zero1,977 below · cited by 5 · depth 15 - Inertia acts by the cyclotomic character on t· v in the Eisenstein torsion tower
ModularCurve.inertia_smul_smul_eq_nsmul_of_latticeRestrict_heckeEvalForms_mem_span_sup775 below · cited by 2 · depth 15 - Petersson product as a side-pairing sum of periods
ModularCurve.integral_petersson_gammaFundamentalSet_eq_sum_conj_period_mul_edgeIntegral5 below · cited by 1 · depth 15 - Base change of the X_H(M) q-expansion field is a curve over L
ModularCurve.isCurveOver_and_essFiniteType_laurentBaseChange_xHFunctionField45 below · cited by 188 · depth 15 - Finite flat multiplication by n transported along a group isomorphism
ModularCurve.isFinite_and_flat_schemeNsmul_of_schemeHomOver_iso_of_jZeroC_points269 below · cited by 1 · depth 15 - Cuspidal places of X₀(q): the ∞/0 dichotomy
ModularCurve.isInftySide_or_isZeroSide_of_isCuspidal164 below · cited by 9 · depth 15 - Atkin–Lehner involution preserves integrality over R[j]
ModularCurve.isIntegral_adjoin_jq_atkinLehnerInvolutionFull77 below · cited by 4 · depth 15 - Principality of n·((̄ 0)-(∞̄)) on X₀(ℓ)
ModularCurve.isPrincipal_eisensteinNumerator_smul_cuspidalDivisor124 below · cited by 1 · depth 15 - Igusa good reduction: regular one-dimensional charts at p ∤ M
ModularCurve.isRegularLocalRing_localization_tensor_chartAlg_twoChartIntegralModel_qExpFunctionFieldC_of_charP913 below · cited by 1 · depth 15 - The q-expansion of j is nonzero over any nontrivial ring
ModularCurve.jqModC_ne_zero0 below · cited by 12 · depth 15 - j(q^N) lies in the localised modular ring of level Nq
ModularCurve.jqNModC_mem_modularLocalized_mul_of_not_dvd130 below · cited by 1 · depth 15 - Kronecker's congruence for modular polynomials of prime level
ModularCurve.kroneckerCongruence_of_prime58 below · cited by 5 · depth 15 - Inertia eigenvectors force membership in the stable lattice ideal
ModularCurve.latticeRestrict_heckeEvalForms_mem_span_sup_of_inertia_smul_smul_eq_nsmul_of_ne_two2,320 below · cited by 1 · depth 15 - Degeneracy compositum for Γ_{H'}(N)∩Γ₀(Nq) over a base field L
ModularCurve.laurentBaseChange_xHFunctionField_sup_adjoin_qExpand_eq_laurentBaseChange_xHTopFunctionFieldC177 below · cited by 1 · depth 15 - Reduction commutes with the divisorial Hecke correspondence
ModularCurve.mapDomain_heckeDivBar_single_eq_heckeDivFibre_of_regularProlongation238 below · cited by 3 · depth 15 - Coefficientwise maps commute with the q-expansion of j(q^N)
ModularCurve.map_jqNModC0 below · cited by 5 · depth 15 - Supersingular j-values persist over algebraically closed extensions
ModularCurve.mem_ssJSet_algebraMap_of_pow_eq_of_ne_two13 below · cited by 4 · depth 15 - Roots of the level-ℓ modular equation preserve supersingularity
ModularCurve.mem_ssJSet_of_mem_roots_fibrePoly254 below · cited by 10 · depth 15 - Supersingular places of the level-one j-line are the supersingular points
ModularCurve.mem_ssPlaces_one_iff_exists_charLGeomPlaceOfPoint_eq6 below · cited by 36 · depth 15 - Base change preserves the degeneracy inclusion ̄ F_N≤̄ F_M
ModularCurve.modularFunctionFieldBar_le1 below · cited by 3 · depth 15 - Uniqueness of the modular polynomial at prime level
ModularCurve.modularPolynomialData_phi_unique_of_prime46 below · cited by 2 · depth 15 - Integrality of Ogg's unit and p¹²u⁻¹ over ℤ[j]
ModularCurve.modularUnitSeries_mem_chartAlgFin_int94 below · cited by 12 · depth 15 - Counting m-torsion of J₀(N₀p) via supersingular places
ModularCurve.natCard_jZeroTorsion_mul_eq_pow_of_ssPlaces1,629 below · cited by 3 · depth 15 - Generic j-fibres of Γ₀(N)-moduli have ψ(N) points
ModularCurve.natCard_moduliPoint_j_eq_eq_dedekindPsi_of_ne_zero13 below · cited by 2 · depth 15 - Degree of the fibre polynomial of a monic Φ
ModularCurve.natDegree_fibrePoly0 below · cited by 3 · depth 15 - Node pairs from places agree with node pairs from j-values
ModularCurve.nodePairsOfPlaces_map_charLGeomPlaceOfPoint_eq_nodePairsOf0 below · cited by 1 · depth 15 - j=1728 is Hasse-supersingular iff q≡ 3 (mod 4)
ModularCurve.ofNat1728_mem_ssJSetHasse_iff1 below · cited by 6 · depth 15 - j=1728 is supersingular iff q ≡ 3 (mod 4)
ModularCurve.ofNat1728_mem_ssJSet_iff15 below · cited by 21 · depth 15 - Geometric j is a uniformiser where j ≠ 0, 1728
ModularCurve.ord_jGeomGen_sub_algebraMap_eq_one_of_evalAt_eq358 below · cited by 6 · depth 15 - Divisor correspondence agrees with classifying endomorphism on ℚ̄-points
ModularCurve.pic0Correspondence_pts_eq_comp_of_poincare_pullbackAlong_iso359 below · cited by 1 · depth 15 - Ramification over the j-line divides the j-width
ModularCurve.placeRamificationJ_dvd_jWidth_of_ord_pos353 below · cited by 34 · depth 15 - Width one at off-diagonal affine places of level Ms
ModularCurve.placeWidthChar_eq_one_of_restrictAlong_ne535 below · cited by 1 · depth 15 - Width-weighted symmetry of the two Hecke correspondence orientations
ModularCurve.placeWidthChar_mul_correspondence_heckeBetaC_heckeAlphaC_single_apply_eq_of_prime596 below · cited by 3 · depth 15 - Base change to ℚ̄ respects restriction of places
ModularCurve.pointEquivPlace_comp_eq_restrictAlong_of_baseChange41 below · cited by 4 · depth 15 - Frobenius-fixed coefficients: s(q)ᵖ = s(qᵖ)
ModularCurve.pow_char_eq_qExpand_of_coeff_fixed0 below · cited by 5 · depth 15 - Old and ribbon terms bounded by the first-place torsion subgroup
ModularCurve.pow_finrank_quotient_old_add_ribbon_le_natCard_twoPlaceTorsionDatum_fst_W3,546 below · cited by 2 · depth 15 - Mod p q-expansion field of X_H(M) lies in that of X_{H'}(M/p)
ModularCurve.qExpFunctionFieldC_gammaH_le_qExpFunctionFieldC_gammaH_infSubgroup130 below · cited by 16 · depth 15 - j(qᵈ) lies in ℚ(j(q^{dℓ}),j(q^{dℓ'}),j(q^{dℓℓ'}))
ModularCurve.qExpand_jq_mem_adjoin_of_primes_of_ne72 below · cited by 1 · depth 15 - Width transport along the degeneracy pair at supersingular places
ModularCurve.ramificationIndexAlong_mul_placeWidthChar_eq_placeWidthChar_restrictAlong_degeneracyPair502 below · cited by 3 · depth 15 - Width transport along both degeneracy maps at every place
ModularCurve.ramificationIndexAlong_mul_placeWidth_eq_placeWidth_restrictAlong440 below · cited by 5 · depth 15 - Reduction mod ℓ on modular divisor classes is surjective
ModularCurve.reductionModL_surjective917 below · cited by 11 · depth 15 - Hecke correspondence at ℓ preserves supersingular places
ModularCurve.restrictAlong_heckeAlphaC_mem_ssPlaces_of_restrictAlong_heckeBetaC_mem_ssPlaces266 below · cited by 2 · depth 15 - Restriction along β of the Fricke translate equals restriction along α
ModularCurve.restrictAlong_heckeBetaBar_frickeInvolutionBar_smul81 below · cited by 13 · depth 15 - Roots of the fibre polynomial: {a^ℓ}+ℓ·{a^{1/ℓ}}
ModularCurve.roots_fibrePoly0 below · cited by 1 · depth 15 - Prime-to-p torsion extending over A equals toric torsion
ModularCurve.setOf_mem_jZeroTorsion_and_exists_schemeHomOver_eq_coe_jZeroToricTorsion9 below · cited by 1 · depth 15 - Sharp-unit necessity for primes ℓ ≡ 1 (mod 12)
ModularCurve.sharpUnitNecessary_of_mod_twelve_eq_one26 below · cited by 1 · depth 15 - `SharpUnitNecessary` at primes ℓnot≡ 1 (mod 12)
ModularCurve.sharpUnitNecessary_of_prime16 below · cited by 1 · depth 15 - Non-vanishing of z dj in the modular function field
ModularCurve.smul_D_jqModC_ne_zero7 below · cited by 3 · depth 15 - Eichler–Deuring mass formula in Hasse-invariant form
ModularCurve.sum_inv_jWidth_of_ssJSetHasse19 below · cited by 3 · depth 15 - Bilinear relation between periods and edge integrals on X₀(N)
ModularCurve.sum_period_mul_edgeIntegral_eq_zero4 below · cited by 1 · depth 15 - Frobenius cannot act by ± p on the q-adic Tate module of J₀(N₀)
ModularCurve.tateModule_eq_zero_of_forall_frobenius_smul_eq_mul_smul1,161 below · cited by 1 · depth 15 - Vanishing of Tate-module elements fixed by σ²
ModularCurve.tateModule_eq_zero_of_forall_frobenius_smul_smul_eq1,023 below · cited by 2 · depth 15 - Naturality of θ = q d/dq under coefficient base change
ModularCurve.thetaL_coeffMap_eq_coeffMap_single_mul_derivative0 below · cited by 20 · depth 15 - W(𝔪) is toric at the second place
ModularCurve.twoPlaceTorsionDatum_snd_W_le_toric_of_not_hasLowerLevelTorsion_of_W_le_invariants3,571 below · cited by 2 · depth 15 - Vanishing Hasse invariant at j=0 iff q≡ 2(mod 3)
ModularCurve.zero_mem_ssJSetHasse_iff1 below · cited by 6 · depth 15 - j=0 is supersingular in characteristic q≥ 5 iff q≡ 2(mod 3)
ModularCurve.zero_mem_ssJSet_iff15 below · cited by 23 · depth 15 - The place j=a has residue field K
ModularCurve.algebraMap_residueField_charLGeomPlaceOfPoint_surjective0 below · cited by 5 · depth 16 - Interior order comparison from integrality of X⁶ j^{4m}(j-1728)^{3m}
ModularCurve.analyticOrderAt_le_of_isIntegral_adjoin_coeffEmb_jq_pow14 below · cited by 2 · depth 16 - The ℓ-roof equals the level-Nℓ modular function field
ModularCurve.charLDegeneracyRoof_eq_modularFunctionFieldC_mul113 below · cited by 4 · depth 16 - Constants of the base-changed modular function field are L
ModularCurve.constantsAreBase_laurentBaseChange_modularFunctionFieldFull119 below · cited by 17 · depth 16 - Multiplicativity of the cusp count ν_∞
ModularCurve.cuspCount_mul_of_coprime0 below · cited by 3 · depth 16 - Degeneracy pair at level Ms: finiteness, separability, supersingular places
ModularCurve.degeneracyPair_finite_separable_identity_ssPlaces178 below · cited by 1 · depth 16 - Discriminant of the Tate curve equals qprod(1-qⁿ)²⁴
ModularCurve.delta_tateLaurent6 below · cited by 3 · depth 16 - Unique centre with uniformiser gives partial_YΦ̄_N(c)≠ 0
ModularCurve.derivative_evalEval_ne_zero_of_isCentreOf_unique_of_ord_jGeomGen_sub_eq_one144 below · cited by 4 · depth 16 - Nonvanishing derivatives of Φ_ℓ at (b,b^ℓ) in characteristic ℓ
ModularCurve.derivative_evalEval_ne_zero_of_kroneckerCongruence_of_pow_sq_ne0 below · cited by 4 · depth 16 - Nonvanishing of partialⱼΦ̄_N at a unique centre with uniformiser
ModularCurve.derivative_swapBivar_evalEval_ne_zero_of_isCentreOf_unique_of_ord_jNGeomGen_sub_eq_one164 below · cited by 4 · depth 16 - Eisenstein kernel annihilates the cuspidal class on J₀(p)
ModularCurve.eisensteinKernelKillsCuspidalClass_heckeModuleBar0 below · cited by 2 · depth 16 - The Eisenstein numerator is coprime to p
ModularCurve.eisensteinNumerator_coprime0 below · cited by 2 · depth 16 - Γ₀(p)-invariance of p E₂(pz)-E₂(z)
ModularCurve.eisensteinTwoSlash_slash_eq_self0 below · cited by 1 · depth 16 - Sharp Riemann–Roch on the full modular function field
ModularCurve.ell_eq_degree_add_one_sub_genusFF_modularFunctionFieldFullC163 below · cited by 19 · depth 16 - Uniqueness of the common root of Φ_ℓ and Φ_N
ModularCurve.eq_jqNModC_mul_sq_of_eval2_modularPolynomial_eq_zero_of_coprime27 below · cited by 1 · depth 16 - Uniqueness of the common root jmath̃(q^{ℓ^2}) of two modular equations
ModularCurve.eq_jqNModC_sq_of_eval2_modularPolynomial_eq_zero_of_eval2_swap_eq_zero27 below · cited by 1 · depth 16 - Real periods detect vanishing of weight-2 cusp forms
ModularCurve.eq_zero_of_forall_re_periodOf_eq_zero3 below · cited by 11 · depth 16 - Evaluation at places commutes with coefficientwise constant extension
ModularCurve.evalAt_eq_apply_evalAt_of_coe_eq_coeffMap6 below · cited by 5 · depth 16 - Level-N and level-q modular equations among j,j_N,j_q,j_{Nq}
ModularCurve.evalModularPair_jFun_jNFun_jQFun_jNQFun_eq_zero0 below · cited by 4 · depth 16 - Decay of g(E₄²E₆)^m against hΔ^m at the cusps
ModularCurve.eventually_norm_slash_le_of_isIntegral_adjoin_coeffEmb_jq_inv_pow15 below · cited by 1 · depth 16 - Unique extension of automorphisms along a constant field change
ModularCurve.existsUnique_algEquiv_qExpFunctionFieldC_coe_apply_eq_coeffMap0 below · cited by 3 · depth 16 - Universal property of the modular function field K(̃ j,̃ j_N)
ModularCurve.existsUnique_algHom_modularFunctionFieldC_apply_jqModC_eq_of_eval2_eq_zero5 below · cited by 2 · depth 16 - Explicit isomorphism of the Mazur–Rapoport and cycle component groups
ModularCurve.exists_addEquiv_appendixComponentGroup_x0MqResolvedTable_apply_eq_componentGroupProj0 below · cited by 2 · depth 16 - Hecke coordinate on inertia displacements in J₀(p)[P^m]
ModularCurve.exists_addSubgroup_le_eisensteinTorsionBar_inertia_smul_sub_mem_addMonoidHom_heckeLatticeAlgebra_quotient_natCard_ker_le2,543 below · cited by 1 · depth 16 - Base change to a place above p preserves normality of both charts
ModularCurve.exists_algHom_tensor_chartAlg_twoChartIntegralModel_qExpFunctionFieldC_injective_isIntegrallyClosed324 below · cited by 2 · depth 16 - Modular function field is finite separable over K(j)
ModularCurve.exists_algebra_ratFunc_modularFunctionFieldC_finite_isSeparable78 below · cited by 9 · depth 16 - A character on the stable Eisenstein torsion of J₀(p)
ModularCurve.exists_character_free_heckeTorsion_span_sup_inertiaSubgroupIn_of_ne_two2,319 below · cited by 1 · depth 16 - Eₚ₊₁ mod p is non-vanishing at supersingular places
ModularCurve.exists_coe_eq_qP_mul_thetaL_jqModC_zpow_and_stackOrd_eq_zero499 below · cited by 4 · depth 16 - Hasse invariant as (thetajmath̄)^{-(p-1)/2} on level N
ModularCurve.exists_coe_eq_thetaL_jqModC_zpow_and_stackOrd_eq471 below · cited by 9 · depth 16 - Finite support of the weight floor on K(jmath̄,jmath̄_N)
ModularCurve.exists_divisor_forall_eq_weightFloor_fieldC123 below · cited by 17 · depth 16 - Polar-coefficient form of general position for section pairs
ModularCurve.exists_eq_algebraMap_of_forall_taylorCoeff_mul_pow_eq_zero_of_generalPosition3 below · cited by 1 · depth 16 - Embeddings of the full level-N modular function field over transcendental j₀
ModularCurve.exists_equiv_algHom_modularFunctionFieldFullC_isRoot_of_transcendental113 below · cited by 7 · depth 16 - Eichler–Shimura relation on a finite flat model of J₀(M)[p]
ModularCurve.exists_finiteFlat_model_jZero_torsion_heckeNilpotent_eichlerShimuraDual_reductionModL1,909 below · cited by 1 · depth 16 - Frobenius-semilinear level-N model of the generic elliptic curve
ModularCurve.exists_frobeniusSemilinear_torsionModel_ofJ_univ312 below · cited by 1 · depth 16 - Existence of a peaked auxiliary form on Γ₁(N)
ModularCurve.exists_gamma1_peaked_auxiliary_form26 below · cited by 1 · depth 16 - Lifting integrality over κ_A[jmath̄] to Gauss-integral q-expansions
ModularCurve.exists_gaussIntegral_lift_isIntegral_of_isIntegral_qExpFunctionFieldC_residueField_of_not_dvd849 below · cited by 2 · depth 16 - Equal branch values above a supersingular node of X₀(q)
ModularCurve.exists_hasValue_frobNodePair_of_isIntegral_modularLocalizedAtPoint_of_pow_eq443 below · cited by 2 · depth 16 - Both branch reductions share a value at the node (a,a^q)
ModularCurve.exists_hasValue_frobNodePair_of_mem_modularLocalizedAtPoint53 below · cited by 6 · depth 16 - Diamond operators act faithfully on reduced q-expansions
ModularCurve.exists_intSeriesC_mul_ne_of_gamma0Units_not_mem3 below · cited by 7 · depth 16 - Integral points above ζ with extendable Hecke translate
ModularCurve.exists_integralPoints_through_of_torsion_over_p244 below · cited by 1 · depth 16 - Existence of diamond automorphisms of F(Γ₁(M))
ModularCurve.exists_isDiamondAut29 below · cited by 15 · depth 16 - Occurrence of a weight-two mod p eigensystem in J₀(N)
ModularCurve.exists_isMaximal_heckeTorsion_jZero_ne_bot_of_ringHom_heckeAlgebra_two910 below · cited by 1 · depth 16 - Good-reduction specialisation of J₀(M) above ℓ∤ M
ModularCurve.exists_jZeroGoodReductionSpecialization_of_not_dvd1,042 below · cited by 1 · depth 16 - Galois-equivariant functions with independent residue pairs at level Nq
ModularCurve.exists_linearIndependent_residuePair_forall_arithmeticGalois_smul_eq_of_finiteDimensional_mul2 below · cited by 1 · depth 16 - Galois descent: S-fixed basis of a Riemann–Roch space
ModularCurve.exists_linearIndependent_riemannRochSpace_forall_arithmeticGalois_smul_eq2 below · cited by 4 · depth 16 - Monic integral relation for Δ(q)/Δ(qᵖ) over ℤ[j]
ModularCurve.exists_monic_int_relation_modularUnit26 below · cited by 1 · depth 16 - Gauss integrality forces 𝔭-integrality at height-one primes above q
ModularCurve.exists_mul_eq_of_height_one_of_natCast_mem_level203 below · cited by 3 · depth 16 - Residue fields of the two Gauss valuations at level p
ModularCurve.exists_mul_eval_sub_eval_mem_nonunits_of_mem_gaussValuationSubring_one_mul126 below · cited by 1 · depth 16 - Modular unit mod q is a degree q-1 polynomial in j
ModularCurve.exists_natDegree_eq_sub_one_and_modularUnit_intCast_eq_aeval_jqModC_of_charP16 below · cited by 3 · depth 16 - Bounded denominators for the Fricke transform of a Γ₁(N)-form
ModularCurve.exists_ne_zero_isIntegral_mul_qExpansion_slash_fricke_coeff60 below · cited by 1 · depth 16 - A generator for Eisenstein torsion inertia-eigenvectors at odd q
ModularCurve.exists_nsmul_generator_heckeTorsion_span_sup_of_inertia_smul_eisensteinMaximalIdeal_smul_eq_nsmul_of_ne_two2,070 below · cited by 1 · depth 16 - Places of X₀(M), X₀(Ms) and the two degeneracy laws
ModularCurve.exists_orbitMap_cyclicAddSubgroup_places_restrictAlong_heckeAlphaC_heckeBetaC_eq437 below · cited by 2 · depth 16 - Multiplicative–étale filtration of 𝔪-torsion on J₀(N) at p
ModularCurve.exists_ordinaryFiltration_heckeTorsion_jZero_of_heckeGen_notMem_of_ne_two1,930 below · cited by 1 · depth 16 - The β-fibre over a cusp: ramification 1 and ℓ
ModularCurve.exists_pair_fiberAlong_heckeBetaBar_of_ord_neg164 below · cited by 3 · depth 16 - A Fricke involution on supersingular places swapping the Hecke legs
ModularCurve.exists_perm_ssPlaces_correspondence_heckeBetaC_heckeAlphaC_perm_eq_correspondence_heckeAlphaC_heckeBetaC335 below · cited by 1 · depth 16 - A place above w along β where t₀ reduces to 1
ModularCurve.exists_place_restrictAlong_heckeBetaBar_eq_and_hasValue_tZero202 below · cited by 4 · depth 16 - The two prolongations of X₀(Np) above p∤ N
ModularCurve.exists_regularProlongation_pair_valuationSubring_eq_or_eq_of_not_dvd122 below · cited by 8 · depth 16 - Mod p reduction on a Gauss valuation subring of ℚ((q))
ModularCurve.exists_ringHom_laurentSeries_zmod_of_gaussValuationSubring0 below · cited by 2 · depth 16 - Change of coefficient field for q-expansion function fields
ModularCurve.exists_ringHom_qExpFunctionFieldC_coe_eq_coeffMap0 below · cited by 7 · depth 16 - Elliptic elements of Γ₀(Ms) factor through diag(1,s)
ModularCurve.exists_smul_one_add_smul_eq_diagonal_mul_of_mem_Gamma00 below · cited by 1 · depth 16 - Uniformisers with correction divisors at supersingular places and their Frobenius translates
ModularCurve.exists_unifFst_unifSnd_correctionDivisor_laws_of_ssPlaces218 below · cited by 1 · depth 16 - Finite-dimensionality of Ω(D) on the modular function field
ModularCurve.finiteDimensional_omegaSpace183 below · cited by 2 · depth 16 - Finiteness of the weight divisor Riemann–Roch space
ModularCurve.finiteDimensional_riemannRochSpace_weightDivisor134 below · cited by 1 · depth 16 - Degree inequality for reduced q-expansion function fields
ModularCurve.finrank_adjoin_qExpFunctionFieldC_le_of_valuationSubring1 below · cited by 3 · depth 16 - Multiplicity one bound: dim J₀(M)[𝔪]≤ 2
ModularCurve.finrank_heckeTorsion_jZero_le_two_of_isAbsolutelyIrreducible2,935 below · cited by 1 · depth 16 - Degree of level N₀ℓ q over level N₀ q
ModularCurve.finrank_modularFunctionFieldFull_mul_eq_of_coe_eq72 below · cited by 1 · depth 16 - Degree of level N₀ℓ q over level N₀q under q↦ q^ℓ
ModularCurve.finrank_modularFunctionFieldFull_mul_eq_of_coe_eq_qExpand73 below · cited by 1 · depth 16 - Eichler–Shimura rank bound for congruence subgroups
ModularCurve.finrank_parabolicHoms_le_two_mul_finrank_cuspForm_of_isCongruenceSubgroup172 below · cited by 7 · depth 16 - Rank of the period lattice equals twice the genus
ModularCurve.finrank_periodLattice_eq_two_mul_genusFF1,396 below · cited by 3 · depth 16 - Geometric Frobenius fixes the place at infinity
ModularCurve.frobOnPlacesGeomLevel_charLGeomPlaceEquiv_placeInfty7 below · cited by 21 · depth 16 - Frobenius after Verschiebung is trivial on the Cartier dual
ModularCurve.frobenius_comp_verschiebung_eq_unit_counit_of_model_jZero_torsion11 below · cited by 1 · depth 16 - The genus formula vanishes at level 4
ModularCurve.genusFormula_four0 below · cited by 1 · depth 16 - The genus formula for X₀(N) vanishes at N=9
ModularCurve.genusFormula_nine0 below · cited by 1 · depth 16 - Geometric integrality of the two-chart model over ℤ[1/N]
ModularCurve.geometricallyIntegral_baseChangeToBase_twoChartIntegralModel_away853 below · cited by 3 · depth 16 - Curve package for the level-N modular function field
ModularCurve.hasCanonicalDivisor_and_dCoordGenerates_and_hasPrincipalDivisors_and_nontrivial_kaehler120 below · cited by 2 · depth 16 - q-expansion of pE₂(pτ)-E₂(τ)
ModularCurve.hasSum_eisensteinTwoCoeff_mul_cexp_pow0 below · cited by 1 · depth 16 - Nilpotence of Tₚ on a Hopf-algebra model of J₀(M)[p]
ModularCurve.heckeNilpotent_of_model_jZero_torsion_heckeNilpotent10 below · cited by 1 · depth 16 - Hecke operator T_q on J₀(p)(ℚ̄) realised by φ_η
ModularCurve.heckeOperatorBar_points_eq_comp_of_transform_rat359 below · cited by 1 · depth 16 - Valuation ring of the inertia field inside A
ModularCurve.inertiaField_comap_incl_and_surjective_and_isAlgClosed_residueField10 below · cited by 5 · depth 16 - Inertia acts cyclotomically on reducing T_λ-ordinary λ-power torsion
ModularCurve.inertia_smul_eq_nsmul_of_forall_exists_heckeOperatorBar_pow_apply_eq_of_reductionModL_eq_zero1,923 below · cited by 1 · depth 16 - Irreducibility of the level-ℓ modular equation at ̃ j_N
ModularCurve.irreducible_modularPolynomial_map_jqNModC_of_not_dvd122 below · cited by 1 · depth 16 - Boundedness at all cusps of p E₂(pz)-E₂(z)
ModularCurve.isBoundedAtImInfty_eisensteinTwoSlash_slash0 below · cited by 1 · depth 16 - The ℓ-degeneracy roof is a curve over K
ModularCurve.isCurveOver_charLDegeneracyRoof154 below · cited by 2 · depth 16 - The cusp ∞̄ of X₀(q) lies on the ∞-side
ModularCurve.isInftySide_cuspInftyBar0 below · cited by 14 · depth 16 - Integrality over the plane local ring at (a,a^q)
ModularCurve.isIntegral_modularLocalizedAtPoint_of_forall_ord_nonneg263 below · cited by 3 · depth 16 - Order conditions at affine places and cusps imply integrality
ModularCurve.isModPFormFn_of_forall_stackOrd_nonneg_of_forall_le_ord368 below · cited by 5 · depth 16 - The cusp ̄ 0 = w_q∞̄ lies on the zero side
ModularCurve.isZeroSide_cuspZeroBar78 below · cited by 10 · depth 16 - Order-two Hopf sheaves over ℤ: ℤ/2, μ₂, or H¹
ModularCurve.iso_restriction_or_natCard_fppfCohomology_of_sectionsEquiv_algHom_two47 below · cited by 1 · depth 16 - Normalisation of the j-line at level Nq: noetherian, normal, finite
ModularCurve.jIntegralClosure_isNoetherianRing_and_isIntegrallyClosed_level147 below · cited by 9 · depth 16 - j-invariant of the formal Tate curve over K((q))
ModularCurve.j_tateLaurent4 below · cited by 11 · depth 16 - j(q) lies in K(j(q^s), j(q^ℓ), j(q^{sℓ}))
ModularCurve.jqModC_mem_adjoin_jqNModC_of_prime_of_ne125 below · cited by 1 · depth 16 - Vanishing of leadₓᵃ of a trace at supersingular places
ModularCurve.lead_trace_eq_zero_of_forall_le_ord366 below · cited by 2 · depth 16 - Coefficientwise base change preserves linear independence of Laurent series
ModularCurve.linearIndependent_coeffMap1 below · cited by 2 · depth 16 - Coefficientwise base change preserves independence of pairs
ModularCurve.linearIndependent_map_prod_of_coe_eq_coeffMap1 below · cited by 6 · depth 16 - Integral weight-2m holomorphy gives membership in L(D)
ModularCurve.mem_riemannRochSpace_of_isModPFormFn1 below · cited by 2 · depth 16 - Riemann–Roch space of the weight divisor equals mod-p forms
ModularCurve.mem_riemannRochSpace_weightDivisor_iff_isModPFormFn372 below · cited by 3 · depth 16 - Extensionality for K-algebra maps out of the level-ℓ modular function field
ModularCurve.modularFunctionFieldC_algHom_ext0 below · cited by 6 · depth 16 - Collapse at level ℓ in characteristic ℓ
ModularCurve.modularFunctionFieldC_self_collapse_unconditional3 below · cited by 13 · depth 16 - Ogg's unit Δ(q)/Δ(qᵖ) at the two components
ModularCurve.modularUnitSeries_mem_valuationSubring_pair_of_not_dvd100 below · cited by 9 · depth 16 - Multiplier determined by periods: third-kind versus first-kind reciprocity
ModularCurve.multiplier_eq_exp_of_periodAlong_add_petersson_mem_periodLattice610 below · cited by 1 · depth 16 - Pole order of jmath̄ at a place is nonzero in K
ModularCurve.natAbs_ord_jGeomGen_cast_ne_zero_of_ord_neg126 below · cited by 2 · depth 16 - Component group at p has order num((p-1)/12)
ModularCurve.natCard_componentGroup_widthOfPlaces_eq_eisensteinNumerator422 below · cited by 1 · depth 16 - Degree bounds for the Kronecker congruence remainder
ModularCurve.natDegree_kroneckerRemainder_le75 below · cited by 2 · depth 16 - Order bound for β(d)h^m on the α-fibre of an index place
ModularCurve.neg_mul_poleOrder_add_one_le_ord_heckeBetaC_mul_pow366 below · cited by 4 · depth 16 - Floor bound for β(d)h^m along a supersingular fibre
ModularCurve.neg_mul_poleOrder_le_ord_heckeBetaC_mul_pow366 below · cited by 3 · depth 16 - Multiplicativity of the root count ν₃
ModularCurve.nuThree_mul_of_coprime0 below · cited by 1 · depth 16 - Multiplicativity of ν₂ at coprime arguments
ModularCurve.nuTwo_mul_of_coprime0 below · cited by 1 · depth 16 - Order of Δ/Δ_δ at poles of j
ModularCurve.ord_coeffEmb_modularUnitSeries_eq_sub_of_ord_jqModC_neg148 below · cited by 2 · depth 16 - ̃ j - a is a uniformiser when q ≤ 3
ModularCurve.ord_jGeomGen_sub_algebraMap_eq_one_of_evalAt_eq_of_le_three351 below · cited by 1 · depth 16 - The chosen uniformiser at a place has order one
ModularCurve.ord_unif0 below · cited by 11 · depth 16 - Segment periods under the anti-holomorphic involution τ↦-τ̄
ModularCurve.periodAlongOf_eq_neg_conj_periodAlongOf_J_smul0 below · cited by 5 · depth 16 - Period homomorphism of a weight-2 cusp form equals its segment period
ModularCurve.periodMapOf_apply_eq_periodOf3 below · cited by 20 · depth 16 - Periods of f∣₂σ and the raw diamond action
ModularCurve.periodMapOf_gammaH_eq_diamondRaw_of_coe_eq_slash4 below · cited by 9 · depth 16 - Period map intertwines U_q on forms with cohomological Hecke operator
ModularCurve.periodMapOf_gammaH_eq_heckeT_of_coe_eq_heckeU4 below · cited by 7 · depth 16 - Period map intertwines classical T_ℓ with cohomological T_ℓ
ModularCurve.periodMapOf_gammaH_eq_heckeT_of_coe_eq_heckeU_add_slash4 below · cited by 12 · depth 16 - Period maps of weight-2 cusp forms are parabolic
ModularCurve.periodMapOf_mem_parabolicHoms1 below · cited by 13 · depth 16 - Period map commutes with the degeneracy map V_d
ModularCurve.periodMap_rescaleLin_apply3 below · cited by 2 · depth 16 - Factorisation of Φ₃ modulo 3⁶
ModularCurve.phiThree_eq_mul_add_pow_six_mul0 below · cited by 2 · depth 16 - Explicit 2-adic factorisation identity for Φ₂
ModularCurve.phiTwo_eq_mul_add_pow_twelve_mul0 below · cited by 2 · depth 16 - The level-two modular equation Φ₂(j(q),j(q²))=0
ModularCurve.phiTwo_eval2_evalAtJ_jqN_two_eq_zero58 below · cited by 1 · depth 16 - Ramification and characteristic-q widths at a supersingular place
ModularCurve.placeRamificationJ_mul_jWidthChar_evalAt_jNGeomGen_eq_of_mem_ssPlaces439 below · cited by 1 · depth 16 - Ramification over the j- and j_N-lines versus automorphism widths
ModularCurve.placeRamificationJ_mul_jWidth_evalAt_jNGeomGen_eq422 below · cited by 2 · depth 16 - Level-one place specialization over the residue field of A
ModularCurve.placeSpecialization_exists_level_one_residueField228 below · cited by 5 · depth 16 - Width divisibility is unchanged by the shift k ↦ k+p+1
ModularCurve.placeWidth_dvd_div_two_iff_dvd_add_div_two_of_mem_ssPlaces378 below · cited by 1 · depth 16 - Igusa and Deligne–Rapoport point dictionaries agree through θ_ℚ
ModularCurve.pts_lift_comp_theta_fst_eq_pts_of_dRModelPackage_of_igusaModel209 below · cited by 1 · depth 16 - Frobenius identity for the q-expansion of j(q^N)
ModularCurve.qExpand_jqNModC_eq_pow_unconditional1 below · cited by 5 · depth 16 - q-expansion of the weight-2m trace Hecke operator
ModularCurve.qexpOfWeight_trace_heckeBetaC_mul_pow_eq_heckePS_of_eq_smul_map132 below · cited by 5 · depth 16 - Width transport along both degeneracy maps, characteristic ≥ 5
ModularCurve.ramificationIndexAlong_mul_placeWidth_eq_placeWidth_restrictAlong_of_five_le833 below · cited by 4 · depth 16 - Coprime integral coordinates on P¹(ℚ) are unique up to sign
ModularCurve.ratPoint_eq_ratPoint_iff_of_isCoprime0 below · cited by 1 · depth 16 - Vanishing supersingular leading coefficients versus stack order
ModularCurve.resFnFun_eq_zero_iff_forall_one_le_stackOrd364 below · cited by 1 · depth 16 - Ogg's unit reduces to the supersingular polynomial
ModularCurve.residue_coeffEmb_modularUnitSeries_eq_prod_ssJSet_of_regularProlongation102 below · cited by 1 · depth 16 - Fricke translation exchanges the two degeneracy restrictions
ModularCurve.restrictAlong_heckeAlphaBar_frickeInvolutionBar_smul81 below · cited by 14 · depth 16 - Necessity of the η-unit exponent when n(ℓ)=1
ModularCurve.sharpUnitNecessary_of_eisensteinNumerator_eq_one0 below · cited by 1 · depth 16 - Sharp-unit necessity for ℓ≡ 1,49 (mod 120)
ModularCurve.sharpUnitNecessary_of_mod_oneTwenty_eq_one_or_fortyNine17 below · cited by 1 · depth 16 - Sharp unit necessity at levels ℓ ≡ 13 (mod 60)
ModularCurve.sharpUnitNecessary_of_mod_sixty_eq_thirteen10 below · cited by 1 · depth 16 - Necessity of n∣ m at levels ℓ≡ 37(mod 60)
ModularCurve.sharpUnitNecessary_of_mod_sixty_eq_thirtySeven10 below · cited by 1 · depth 16 - Eta-quotient necessity for all levels ℓ≡ 11 (mod 12)
ModularCurve.sharpUnitNecessary_of_mod_twelve_eq_eleven10 below · cited by 1 · depth 16 - W-η necessity for levels ℓ ≡ 5 (mod 12)
ModularCurve.sharpUnitNecessary_of_mod_twelve_eq_five10 below · cited by 1 · depth 16 - Eta-unit necessity for all levels ℓ≡ 7(mod 12)
ModularCurve.sharpUnitNecessary_of_mod_twelve_eq_seven10 below · cited by 1 · depth 16 - Divisibility by the Eisenstein numerator from one Γ₀(ℓ) witness
ModularCurve.sharpUnitNecessary_of_witness8 below · cited by 6 · depth 16 - The sharp η-quotient series lies in the function field of X₀(ℓ)
ModularCurve.sharpUnitSeries_mem_modularFunctionField69 below · cited by 1 · depth 16 - Sharp eta quotient raised to gcd(ℓ-1,12) gives Δ/Δ_ℓ
ModularCurve.sharpUnitSeries_pow_sharpIndex0 below · cited by 5 · depth 16 - A j-fixing semilinear automorphism moves Pₐ to P_{τ(a)}
ModularCurve.smul_charLGeomPlaceOfPoint_of_smul_jqModC16 below · cited by 1 · depth 16 - Divisor of Ogg's modular unit at prime level
ModularCurve.smul_cuspidalDivisor_apply_eq_ord109 below · cited by 3 · depth 16 - Hasse-supersingular j-set as image of Deuring roots
ModularCurve.ssJSetHasse_eq_image_legendreJ_toFinset6 below · cited by 1 · depth 16 - Integrality bounds stack orders and orders at cusps
ModularCurve.stackOrd_nonneg_and_le_ord_of_isModPFormFn364 below · cited by 10 · depth 16 - Existence of the genus for X₀(N) over ℚ̄
ModularCurve.stichtenothGenusExists_modularFunctionFieldBar157 below · cited by 4 · depth 16 - Eichler–Deuring mass formula in Legendre–Deuring form
ModularCurve.sum_inv_jWidth_of_deuringPolynomial11 below · cited by 1 · depth 16 - Convergent q-expansion tends to its constant term at i∞
ModularCurve.tendsto_atImInfty_of_hasSum_qParam0 below · cited by 6 · depth 16 - Non-vanishing of θ(j(q^N))
ModularCurve.thetaL_jqNModC_ne_zero0 below · cited by 8 · depth 16 - Weight-2m floor at an affine geometric place
ModularCurve.weightFloor_eq_of_isAffineGeomPlace360 below · cited by 14 · depth 16 - Weil–Kähler agreement for the modular function field
ModularCurve.weilKaehlerAgree_modularFunctionFieldC115 below · cited by 2 · depth 16 - Branch swap with chain reversal preserves the resolved intersection table
ModularCurve.x0MqResolvedTable_inter_equiv_of_swap_of_rev0 below · cited by 1 · depth 16 - Invariance of the resolved intersection table under re-indexing
ModularCurve.x0MqResolvedTable_inter_equiv_of_width_eq0 below · cited by 1 · depth 16 - Integrality of the integral closure of k[jmath̄] over C
ModularCurve.algebra_isIntegral_integralClosure_adjoin_jGeomGen_of_exists_apply_eq2 below · cited by 2 · depth 17 - Arithmetic Galois transport of a slot chart at a place
ModularCurve.arithmeticGalois_smul_slot0 below · cited by 1 · depth 17 - Tate curve: c₄³ = j(q) Δ over any ring
ModularCurve.c4_pow_three_tateLaurent4 below · cited by 1 · depth 17 - c₄ of the Tate curve is E₄
ModularCurve.c4_tateLaurent0 below · cited by 3 · depth 17 - Riemann–Roch bound for mod-3 cusp forms of weight 2m
ModularCurve.card_le_dimFormulaCusp_of_isModPCuspFormFn_of_linearIndependent_of_char_three742 below · cited by 1 · depth 17 - Anharmonic orbit size times j-width equals 6
ModularCurve.card_orbit_mul_jWidth2 below · cited by 1 · depth 17 - Tameness of cusp pole orders on the ℓ-degeneracy roof
ModularCurve.cast_natAbs_ord_heckeAlphaC_ne_zero_and_heckeBetaC_of_ord_neg127 below · cited by 1 · depth 17 - Pole orders of jmath̄(qᵈ) at cusps are prime to p
ModularCurve.cast_natAbs_ord_qExpand_jqModC_ne_zero_of_ord_neg125 below · cited by 2 · depth 17 - Automorphisms with σ(j)=j(qᵖ) fix the j-chart, move the Gauss ring
ModularCurve.chartAlgFin_iff_and_comap_ne_and_aeval_mem_comap_of_algEquiv_map_j_eq_qExpand44 below · cited by 6 · depth 17 - Transvections generate SL₂(ℤ/Nℤ)
ModularCurve.closure_elemSet_eq_top1 below · cited by 2 · depth 17 - Compatibility of w_q with the degeneracy inclusion of j-expansion fields
ModularCurve.coe_atkinLehnerInvolutionFull_mul_eq_of_coe_eq74 below · cited by 2 · depth 17 - Partial Atkin–Lehner involution commutes with the ℓ-degeneracy map
ModularCurve.coe_atkinLehnerInvolutionFull_mul_eq_qExpand_of_coe_eq_qExpand74 below · cited by 1 · depth 17 - q-expansion of q j(q) through q¹²
ModularCurve.coeff_jNum_le_twelve0 below · cited by 1 · depth 17 - Nonvanishing of a determinant of values at rational places
ModularCurve.det_evalAt_ne_zero_of_span_inf_riemannRochSpace_eq_bot122 below · cited by 1 · depth 17 - Multiplicativity of the diamond operators on J_H
ModularCurve.diamondHBar_mul68 below · cited by 12 · depth 17 - E₄³-E₆²=1728 q η²⁴ in ℤ[[q]]
ModularCurve.eisenstein4_cube_sub_mk_sq3 below · cited by 9 · depth 17 - Eisenstein numerator divides m z₀ for eta-quotient roots
ModularCurve.eisensteinNumerator_dvd_mul_of_witness8 below · cited by 1 · depth 17 - Riemann–Roch bound for the cuspidal weight-2m floor divisor
ModularCurve.ell_le_dimFormulaCusp_of_forall_eq_weightFloor_sub733 below · cited by 2 · depth 17 - Poles of j on X₀(q)_ℚ̄ lie at the two cusps
ModularCurve.eq_cuspInftyBar_or_eq_cuspZeroBar_of_ord_jFun_neg76 below · cited by 7 · depth 17 - Uniqueness of the moduli place of a Γ₀(N)-class
ModularCurve.eq_of_isModuliPlaceOf409 below · cited by 17 · depth 17 - Effective divisor with ℓ(D)=1 is alone in its class
ModularCurve.eq_of_isPrincipal_sub_of_finrank_riemannRochSpace_eq_one0 below · cited by 1 · depth 17 - Vanishing in negative weight for mod-p modular functions
ModularCurve.eq_zero_of_forall_stackOrd_nonneg_of_forall_le_ord_of_neg424 below · cited by 1 · depth 17 - Vanishing of the second j-value at supersingular places for q<5
ModularCurve.evalAt_jNGeomGen_eq_zero_of_mem_ssPlaces_of_lt_five432 below · cited by 1 · depth 17 - Vanishing of (πᵃG)(x) iff stack order at least one
ModularCurve.evalAt_zpow_mul_eq_zero_iff_one_le_stackOrd4 below · cited by 2 · depth 17 - Non-vanishing of the reduced Kronecker remainder at supersingular j
ModularCurve.eval_kroneckerRemainder_ne_zero_of_mem_ssJSet188 below · cited by 21 · depth 17 - Supersingular moduli places on κ(j(q),j(q^N)): existence and uniqueness
ModularCurve.existsUnique_mem_ssPlaces_toValuationSubring_eq_comap_moduliPlace415 below · cited by 3 · depth 17 - A Hecke coordinate on inertia-displaced Eisenstein 2-torsion
ModularCurve.exists_addMonoidHom_eisensteinTorsionBar_inf_closure_inertia_smul_sub_heckeLatticeAlgebra_quotient_natCard_ker_le2,542 below · cited by 1 · depth 17 - Inertia-invariant classes come from inertia-stable divisors
ModularCurve.exists_arithmeticGalois_smul_eq_pic0Mk_eq178 below · cited by 2 · depth 17 - Roof reduction commutes place by place with both degeneracy maps
ModularCurve.exists_charLDegeneracyRoof_regularProlongation_heckeCompat_restrictAlong_eq_of_ne874 below · cited by 1 · depth 17 - Galois descent for stable subspaces of L((q))
ModularCurve.exists_coeffEmb_basis_of_forall_coeffMap_mem1 below · cited by 2 · depth 17 - Integral elements of ℚ((q)) that are quotients lie in ℤ((q))
ModularCurve.exists_coeffMap_eq_of_isIntegralElem_of_exists_eq_div0 below · cited by 2 · depth 17 - Order-one difference g-g^{q^2} at supersingular places
ModularCurve.exists_coeff_pow_eq_and_ord_sub_pow_sq_eq_one_of_mem_ssPlaces147 below · cited by 1 · depth 17 - Separating a supersingular place by an 𝔽_{q²}-rational affine function
ModularCurve.exists_coeff_pow_sq_eq_and_hasValue_zero_and_not_hasValue_zero_of_mem_ssPlaces_of_ne404 below · cited by 2 · depth 17 - Crossing presentation at a supersingular node of X₀(q)
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring468 below · cited by 10 · depth 17 - Unitary multipliers are exponentials of weight-2 periods
ModularCurve.exists_cuspForm_multiplier_eq_exp_of_norm_eq_one569 below · cited by 1 · depth 17 - Finite flat model of the Tₚ-bijective part of J₀(M)[p^k]
ModularCurve.exists_finiteFlat_model_jZero_torsion_heckeBijective_frobenius_verschiebung_reductionModL1,909 below · cited by 2 · depth 17 - Finite flat pⁿ-torsion of modular Jacobians away from the level
ModularCurve.exists_finiteFlat_prolongation_pi_torsion_pic0_qExpFunctionField_of_not_dvd1,613 below · cited by 2 · depth 17 - Supersingular places of the j-line versus supersingular j-invariants
ModularCurve.exists_finset_forall_mem_iff_mem_ssPlaces_equiv_evalAt_jGeomGen_eq7 below · cited by 2 · depth 17 - Integrality over A[j] descends to all large number fields
ModularCurve.exists_forall_mem_jIntegralClosure_of_integral_affineBaseFin116 below · cited by 2 · depth 17 - Adic Galois representation from a Hecke character on J₁(M)
ModularCurve.exists_galoisRepAdic_charpoly_frobenius_of_heckeDiamondChar1,377 below · cited by 1 · depth 17 - Eichler–Shimura representation attached to a Hecke character
ModularCurve.exists_galoisRepAdic_charpoly_frobenius_of_heckeDiamondChar_tateModule_quotient1,377 below · cited by 5 · depth 17 - Existence of an admissible equivariant primitive of a weight-2 cusp form
ModularCurve.exists_hasEquivariantPrimitiveOf1 below · cited by 28 · depth 17 - Equal branch values at degenerate supersingular nodes
ModularCurve.exists_hasValue_frobNodePair_of_isIntegral_modularLocalizedAtPoint_of_degenerate401 below · cited by 1 · depth 17 - Common branch value at a generic supersingular node
ModularCurve.exists_hasValue_frobNodePair_of_isIntegral_modularLocalizedAtPoint_of_ne_zero_of_ne_1728324 below · cited by 1 · depth 17 - Crossing lemma at a supersingular node of X₀(q) mod q
ModularCurve.exists_hasValue_frobNodePair_of_mem_ssJSet_of_pow_eq535 below · cited by 3 · depth 17 - Auxiliary places making K-E non-special on X₀(N)
ModularCurve.exists_injective_riemannRochSpace_canonicalDivisorOf_sub_eq_bot196 below · cited by 2 · depth 17 - Integral norm relation for g of Ogg's modular unit
ModularCurve.exists_int_poly_natDegree_aeval_jFull_eq_mul_aeval_modularUnitSeries270 below · cited by 4 · depth 17 - Mod p Eichler–Shimura: H¹ₚₐᵣ(Γ₀(N),𝒪) versus Hom(J₀(N)[p],k)
ModularCurve.exists_linearMap_H1_top_hom_torsionBy_jZero_heckeTL_eq_comp_of_mem_parabolicHoms736 below · cited by 1 · depth 17 - Pairwise disjoint base divisors off the cusp in L(2E-K)
ModularCurve.exists_mem_riemannRochSpace_sub_canonicalDivisorOf_forall_ne_not_and274 below · cited by 2 · depth 17 - Monic relation over ℚ[j] for the modular unit Δ(q)/Δ(qᵖ)
ModularCurve.exists_monic_rat_relation_int_coeff_modularUnit25 below · cited by 1 · depth 17 - Integrality at height-one primes with no pole centred at (a,a^q)
ModularCurve.exists_mul_eq_of_height_one_of_forall_pole_not_centred253 below · cited by 2 · depth 17 - Finite fppf H¹ over Specℤ for a two-point Hopf algebra
ModularCurve.exists_natCard_fppfCohomology_one_of_not_finite_of_sectionsEquiv_algHom_two45 below · cited by 1 · depth 17 - Cyclic generator for the reduced Eisenstein socle at q
ModularCurve.exists_nsmul_generator_heckeTorsion_span_sup_of_reductionModL_eisensteinMaximalIdeal_smul_eq_zero1,051 below · cited by 2 · depth 17 - Number field presentation of functions on X₀(q)_ℚ̄
ModularCurve.exists_numberField_presentation4 below · cited by 8 · depth 17 - Number field presentation of modular functions in j, j_N
ModularCurve.exists_numberField_presentation_of_neZero114 below · cited by 4 · depth 17 - Orbit map on X₀(N): j_N-values and dual ramification
ModularCurve.exists_orbitMap_cyclicAddSubgroup_places_evalAt_jqNModC_eq_and_ord_sub_eq_natCard397 below · cited by 2 · depth 17 - Base-point freeness of the embedding system on X₀(N)
ModularCurve.exists_ord_add_embDivisor_eq_zero254 below · cited by 16 · depth 17 - Model coordinates give uniformisers away from the cusp
ModularCurve.exists_ord_sub_evalAt_eq_one266 below · cited by 1 · depth 17 - Descent of rational dependence along a coefficient map
ModularCurve.exists_polynomial_mul_aeval_eq_aeval_of_coeffMap0 below · cited by 1 · depth 17 - Common width along both degeneracy legs of the ℓ-roof
ModularCurve.exists_ramificationIndexAlong_mul_eq_placeWidth_restrictAlong_heckeAlphaC_heckeBetaC859 below · cited by 1 · depth 17 - Fricke transform of a rational form on Γ_H(M)
ModularCurve.exists_slash_fricke_eq_sum_smul_of_ratCast_qExpansion34 below · cited by 5 · depth 17 - Forms with integral q-expansions span M_k(Γ_H(N))
ModularCurve.exists_sum_smul_eq_of_isIntegralQExp_gammaH103 below · cited by 5 · depth 17 - Forms with coefficients in K₀ are K₀-combinations of integral forms
ModularCurve.exists_sum_smul_eq_of_qExpansion_coeff_mem52 below · cited by 3 · depth 17 - Constant column sums of β_*α^* on supersingular places
ModularCurve.exists_sum_ssPlaces_correspondence_heckeAlphaC_heckeBetaC_single_eq_of_dvd336 below · cited by 2 · depth 17 - Supersingular places arise from enhanced supersingular curves
ModularCurve.exists_toValuationSubring_eq_comap_moduliPlace_of_mem_ssPlaces411 below · cited by 4 · depth 17 - Toric and finite parts of the torsion of J₀(Nq) at q
ModularCurve.exists_toricPart_finPart_torsion_jZero_of_not_dvd3,558 below · cited by 2 · depth 17 - The two valuations of ℚ(X₀(p)) above p
ModularCurve.exists_valuationSubring_pair_modularFunctionFieldFull_prime126 below · cited by 1 · depth 17 - Finiteness of the kernel of T_ℓ-(ℓ+1) on J₀(N)
ModularCurve.finite_setOf_heckeGen_sub_smul_eq_zero_of_not_dvd754 below · cited by 1 · depth 17 - Degree over ℚ̄(j) bounded by degree over k(j)
ModularCurve.finrank_gammaH_le_finrank_gammaH_residueField_of_not_dvd284 below · cited by 10 · depth 17 - Multiplicity one for J₀(M)[𝔪] when Tₚ ∈ 𝔪
ModularCurve.finrank_heckeTorsion_jZero_le_two_of_isAbsolutelyIrreducible_of_heckeGen_mem2,798 below · cited by 1 · depth 17 - Multiplicity one for J₀(M)[𝔪] in the ordinary case
ModularCurve.finrank_heckeTorsion_jZero_le_two_of_isAbsolutelyIrreducible_of_heckeGen_notMem2,391 below · cited by 1 · depth 17 - Rank of parabolic homomorphisms of Γ(N) bounded by 2dim S₂
ModularCurve.finrank_parabolicHoms_Gamma_le_two_mul_finrank_cuspForm164 below · cited by 1 · depth 17 - Descent of the parabolic rank bound along a normal subgroup
ModularCurve.finrank_parabolicHoms_le_two_mul_finrank_cuspForm_of_le_of_normal13 below · cited by 1 · depth 17 - Effective divisor with L(D-v)=0 has ℓ(D)=1
ModularCurve.finrank_riemannRochSpace_eq_one_of_sub_single_eq_bot214 below · cited by 2 · depth 17 - Regularity at all affine places equals integrality over k[̃ j,̃ j_N]
ModularCurve.forall_isAffineGeomPlace_mem_iff_isIntegral_adjoin79 below · cited by 6 · depth 17 - Fricke involution sends the second degeneracy leg to the first
ModularCurve.frickeInvolutionBar_comp_heckeBetaBar_one79 below · cited by 5 · depth 17 - Geometric Frobenius carries the place of (W,C) to that of its twist
ModularCurve.frobOnPlacesGeomLevel_toValuationSubring_eq_comap_moduliPlace_map_frobenius413 below · cited by 1 · depth 17 - Galois action commutes with Hecke and diamond operators on J_H
ModularCurve.galois_smul_genOpH_comm10 below · cited by 21 · depth 17 - The level-one modular function field has genus zero
ModularCurve.genusFF_modularFunctionFieldC_one_eq_zero52 below · cited by 4 · depth 17 - Genus of X_H(M) is unchanged at primes ℓ ∤ M
ModularCurve.genusFF_xHFunctionFieldC_eq_genusFF_xHFunctionFieldBar_of_not_dvd840 below · cited by 9 · depth 17 - q-expansion of the reciprocal eta quotient 1/G_ℓ
ModularCurve.hasSum_sharpUnitSeries_inv_qParam11 below · cited by 1 · depth 17 - q-expansion of the sharp eta quotient at ∞
ModularCurve.hasSum_sharpUnitSeries_qParam11 below · cited by 1 · depth 17 - q-expansion at the cusp 0 of the sharp eta quotient
ModularCurve.hasSum_smul_sharpUnitSeries_inv_qParam11 below · cited by 1 · depth 17 - q-expansion at the cusp 0 of the eta quotient unit
ModularCurve.hasSum_smul_sharpUnitSeries_qParam11 below · cited by 1 · depth 17 - Hasse invariant intertwines the two Hecke operators at ℓ
ModularCurve.hasse_smul_traceAlong_smul_pullbackAlong_smul_D_jGeomGen_eq155 below · cited by 1 · depth 17 - Hecke and diamond operators on J₁(M) pairwise commute
ModularCurve.heckeDiamondCommuteBar272 below · cited by 29 · depth 17 - The Hecke and diamond inputs for X₁(M) all hold
ModularCurve.heckeDiamondInputsAll66 below · cited by 34 · depth 17 - Commutativity of the Hecke operators on J_H(M)
ModularCurve.heckeOperatorHAlong_comm259 below · cited by 4 · depth 17 - Hecke operators on J_H(M) commute with diamond operators
ModularCurve.heckeOperatorHAlong_diamondHBar_comm42 below · cited by 6 · depth 17 - On ℓ-torsion, ̄ T_ℓ equals the Frobenius push-forward
ModularCurve.heckeOperatorModL_eq_frobeniusPushforwardModL_of_natCast_smul_eq_zero2 below · cited by 2 · depth 17 - Inertia field of a place over p: unramified DVR, residue surjectivity, place descent
ModularCurve.inertiaField_comap_isDVR_and_residue_surjective_and_place_fixed8 below · cited by 13 · depth 17 - Inertia acts cyclotomically on ordinary λ-torsion reducing to zero
ModularCurve.inertia_smul_eq_nsmul_of_forall_exists_heckeOperatorBar_pow_apply_eq_of_reductionModL_eq_zero_of_ne_two1,924 below · cited by 3 · depth 17 - Infinitely many places on X₀(N) over ℚ̄
ModularCurve.infinite_place_modularFunctionFieldBar227 below · cited by 2 · depth 17 - Integral closure of k[jmath̄] in the level-N fibre field
ModularCurve.isDedekindDomain_integralClosure_adjoin_jGeomGen_of_separable0 below · cited by 2 · depth 17 - Descent to K of integrality over K[jmatĥ⁻¹]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_inv_of_coeffMap_mul_thetaL_pow_eq_qExpansion91 below · cited by 2 · depth 17 - Descent of integrality over K[jmatĥ] along σ: K→ℂ
ModularCurve.isIntegral_adjoin_coeffEmb_jq_of_coeffMap_mul_thetaL_pow_eq_qExpansion84 below · cited by 2 · depth 17 - Reduction at ℓ∤ N preserves integrality over ℚ̄[j]
ModularCurve.isIntegral_adjoin_jqModC_coeffMap_residue_of_isIntegral_of_not_dvd748 below · cited by 2 · depth 17 - Reduction at ℓ∤ N preserves integrality over the j⁻¹-chart
ModularCurve.isIntegral_adjoin_jqModC_inv_coeffMap_residue_of_isIntegral_of_not_dvd748 below · cited by 2 · depth 17 - Descent of integrality from ℚ̄[j,j_N] to (A∩ K)[j]
ModularCurve.isIntegral_jRing_of_coeffMap_eq_of_isIntegral_adjoin_of_not_dvd148 below · cited by 1 · depth 17 - Mazur's cusp gives a level-p structure on Tate(qᵖ)
ModularCurve.isLevelPStructure_tateBase_cuspData_mazurCusp_of_five_le68 below · cited by 3 · depth 17 - Riemann–Roch space of the weight-2m floor divisor
ModularCurve.isModPFormFn_of_mem_riemannRochSpace3 below · cited by 4 · depth 17 - The mod-p form condition is K-linear
ModularCurve.isModPFormFn_zero_and_add_and_smul0 below · cited by 7 · depth 17 - Existence of a moduli place for every Γ₀(N)-moduli point
ModularCurve.isModuliPlaceOf_nonempty394 below · cited by 13 · depth 17 - Totalised q-expansion map on differentials satisfies its defining identities
ModularCurve.isQExpansionDiffAlong_qExpansionDiffAlong0 below · cited by 12 · depth 17 - Noetherian normality of the integral closure of A₀[j]
ModularCurve.jIntegralClosure_isNoetherian_and_isLocalization154 below · cited by 9 · depth 17 - Rank-one Kähler differentials of K(j(q),j(q^N))
ModularCurve.kaehlerRankOne_modularFunctionFieldC_of_isSeparable_jqNModC1 below · cited by 5 · depth 17 - Fibres of the Legendre j-map: the anharmonic orbit
ModularCurve.legendreJ_eq_legendreJ_iff2 below · cited by 1 · depth 17 - ℤ₍ₚ₎[1/j] is integrally closed with fraction field ℚ(j)
ModularCurve.mem_adjoin_jq_inv_of_isIntegral_and_exists_mul_eq4 below · cited by 1 · depth 17 - Descent of rational Laurent series from L· F₀
ModularCurve.mem_of_coeffEmb_mem_laurentBaseChange0 below · cited by 7 · depth 17 - Cuspidal dictionary: L(D_{2m}-cusps) versus mod p cusp forms
ModularCurve.mem_riemannRochSpace_iff_isModPCuspFormFn_of_forall_eq_weightFloor_sub372 below · cited by 3 · depth 17 - At level one the all-divisors modular function field is K(j(q))
ModularCurve.modularFunctionFieldFullC_one0 below · cited by 4 · depth 17 - Finite generation of the p-adic Tate module of J₁(M)
ModularCurve.moduleFinite_padicInt_tateModule_jOne300 below · cited by 26 · depth 17 - Fibres and ramification of j at moduli places of X₀(N)
ModularCurve.moduliPlace_orbitClauses394 below · cited by 11 · depth 17 - Degeneracy law for moduli places: level N down to M
ModularCurve.moduliPlace_restrictAlong_inclusion1 below · cited by 5 · depth 17 - Second degeneracy map on moduli places: (E,C)↦(E/⟨ Q⟩,φ C)
ModularCurve.moduliPlace_restrictAlong_qExpand_fullKernelQuotient91 below · cited by 5 · depth 17 - Second degeneracy map on moduli places via Vélu's odd-order model
ModularCurve.moduliPlace_restrictAlong_qExpand_veluQuotient92 below · cited by 4 · depth 17 - Fibre polynomials of monic bivariate integral polynomials are monic
ModularCurve.monic_fibrePoly0 below · cited by 3 · depth 17 - Order of the width-weighted component group at supersingular nodes
ModularCurve.natCard_componentGroup_placeWidth_nodePairsOfPlaces_eq_eisensteinNumerator421 below · cited by 1 · depth 17 - Counting q^k-torsion: #V_M=#R_M·#V_M⁰ for odd q≠ p
ModularCurve.natCard_heckeTorsion_span_sup_eq_natCard_heckeLatticeAlgebra_quotient_mul_natCard_inertia_smul_eq_nsmul_of_ne_two2,314 below · cited by 1 · depth 17 - Supersingular order bound for the Hecke difference on the roof
ModularCurve.neg_mul_add_one_le_ord_pow_mul_heckeBetaC_mul_pow_sub_of_mem_ssPlaces966 below · cited by 1 · depth 17 - Node units vanish only for constant data on F_N
ModularCurve.nodeUnit_eq_zero_iff_modularFunctionFieldC_of_perfectField124 below · cited by 2 · depth 17 - The cusp ̄ 0 of X₀(q) is not on the ∞-side
ModularCurve.not_isInftySide_cuspZeroBar78 below · cited by 4 · depth 17 - Integral even-weight forms divided by (θ j)^m lie in ℚ(j,j_N)
ModularCurve.ofPowerSeries_mul_thetaL_jq_zpow_neg_mem_modularFunctionField166 below · cited by 3 · depth 17 - Vanishing of Ω(D') at the edge weight 2m=p+1
ModularCurve.omegaSpace_eq_bot_of_two_mul_eq_add_one808 below · cited by 1 · depth 17 - Invariant differential of the Tate curve at the origin
ModularCurve.one_add_single_mul_derivative_tateOriginX0 below · cited by 1 · depth 17 - Order of d̄ j at affine places and tame cusps
ModularCurve.ordDifferential_D_jGeomGen_eq_of_not_dvd_of_cast_natAbs_ne_zero381 below · cited by 3 · depth 17 - Order of Ogg's unit Δ(q)/Δ(q^ℓ) at ∞̄
ModularCurve.ord_cuspInftyBar_coeffEmb_modularUnitSeries109 below · cited by 10 · depth 17 - Order of the modular unit at the cusp ̄ 0
ModularCurve.ord_cuspZeroBar_coeffEmb_modularUnitSeries109 below · cited by 10 · depth 17 - Poles of j(q) and j(q^ℓ) agree at every place
ModularCurve.ord_heckeAlphaC_jGeomGen_neg_iff_ord_heckeBetaC_jGeomGen_neg42 below · cited by 1 · depth 17 - Order of the Hecke multiplier at a tame place
ModularCurve.ord_heckeMultiplier_eq17 below · cited by 4 · depth 17 - Order of the Hecke multiplier at a pole of α^*j
ModularCurve.ord_heckeMultiplier_eq_of_ord_neg_of_eq_smul_map451 below · cited by 1 · depth 17 - Abel's theorem with Petersson correction on Γ₀(N)
ModularCurve.periodAlong_add_petersson_mem_periodLattice_of_multiplier_eq_exp35 below · cited by 1 · depth 17 - Segment period equals difference of primitive values
ModularCurve.periodOf_apply_eq_sub_of_hasEquivariantPrimitiveOf0 below · cited by 16 · depth 17 - Level-one places: j-ramification one and width jWidth(a)
ModularCurve.placeRamificationJ_charLGeomPlaceOfPoint_eq_one_and_placeWidth_eq_jWidth38 below · cited by 6 · depth 17 - Width invariance in characteristic p≥ 5: r· W(j_N)=r_N· W(j)
ModularCurve.placeRamificationJ_mul_jWidth_evalAt_jNGeomGen_eq_of_five_le825 below · cited by 2 · depth 17 - Frobenius invariance of the characteristic-q place width
ModularCurve.placeWidthChar_arithFrobC_smul0 below · cited by 2 · depth 17 - Width-weighted adjointness of the two degeneracy Hecke correspondences
ModularCurve.placeWidthChar_mul_correspondence_heckeBetaC_heckeAlphaC_single_apply_eq_of_prime_of_five_le493 below · cited by 2 · depth 17 - q-expansion of h π^*ω along σ
ModularCurve.qExpansionDiffAlong_smul_map0 below · cited by 4 · depth 17 - E₂ (θ j)^{-(p+1)/2} lies in the level-N function field
ModularCurve.qP_mul_thetaL_jqModC_zpow_mem_modularFunctionFieldC480 below · cited by 1 · depth 17 - Width transport along an embedding fixing q-expansions
ModularCurve.ramificationIndexAlong_mul_placeWidth_eq_placeWidth_restrictAlong_of_coe_eq92 below · cited by 3 · depth 17 - Cyclotomic q^k-torsion of J₀(N) reduces to zero above q
ModularCurve.reductionModL_eq_zero_of_forall_mem_inertiaSubgroupIn_smul_eq_nsmul_of_ne_two757 below · cited by 3 · depth 17 - Uniqueness of the regular prolongation reducing j and j_M
ModularCurve.regularProlongation_integers_eq_and_coe_residue_eq_of_residue_jq_jqN176 below · cited by 1 · depth 17 - Supersingularity along the two legs of the ℓ-roof
ModularCurve.restrictAlong_heckeAlphaC_mem_ssPlaces_iff_restrictAlong_heckeBetaC_mem_ssPlaces859 below · cited by 2 · depth 17 - Ring maps out of K(̃ j,̃ j_N) are determined by generators
ModularCurve.ringHom_ext_of_apply_jGeomGen_eq_of_apply_jNGeomGen_eq0 below · cited by 2 · depth 17 - Restriction of a slot place along ᾱ from level Nℓ
ModularCurve.slot_restrictAlong_heckeAlphaBar85 below · cited by 1 · depth 17 - Slot data under restriction along `heckeBetaBar`
ModularCurve.slot_restrictAlong_heckeBetaBar85 below · cited by 1 · depth 17 - Hecke stability of the toric m-torsion of J₀(p)
ModularCurve.smul_mem_jZeroToricTorsion237 below · cited by 1 · depth 17 - Supersingular Hecke entry at ℓ∣ N counts ℓ-isogenies
ModularCurve.ssHeckeMatrixC_apply_eq_natCard_subgroup_dualPair_of_dvd_of_five_le_of_moduliPlace961 below · cited by 1 · depth 17 - Supersingular Hecke matrix entries count ℓ-isogenies preserving Γ₀(N)-structure
ModularCurve.ssHeckeMatrixC_apply_eq_natCard_subgroup_dualPair_of_moduliPlace473 below · cited by 1 · depth 17 - Hasse-supersingular j-invariants are the j(λ) with H_q(λ)=0
ModularCurve.ssJSetHasse_eq_image_legendreJ5 below · cited by 4 · depth 17 - Stack order zero at supersingular places for ̃ P (thetajmath̄)^{-(p+1)/2}
ModularCurve.stackOrd_qP_mul_thetaL_jqModC_zpow_eq_zero_of_mem_ssPlaces497 below · cited by 1 · depth 17 - Existence of the genus for K(j(q),j(q^N)) over a perfect field
ModularCurve.stichtenothGenusExists_modularFunctionFieldC_of_perfectField150 below · cited by 2 · depth 17 - The Tate curve equation for the point at u=1+T
ModularCurve.tateOrigin_equation36 below · cited by 1 · depth 17 - (thetajmath̄)^{q-1} as a rational function of jmath̄
ModularCurve.thetaL_jqModC_pow_mul_prod_sq_eq179 below · cited by 5 · depth 17 - Dual compatibility of supersingular Hecke map with T_Ω
ModularCurve.theta_ssHeckeFun_eq_inv_smul_dualMap_of_forall_weilOfKaehler894 below · cited by 1 · depth 17 - Specialisation surjects onto the toric kernel at supersingular nodes
ModularCurve.toric_sp_surjective_of_jZeroSemistableSpecialization_ssPlaces3,333 below · cited by 1 · depth 17 - Twice the width of a moduli place counts level-preserving automorphisms
ModularCurve.two_mul_placeWidth_eq_natCard_rationalAut_map_eq_of_toValuationSubring_eq_comap_moduliPlace431 below · cited by 3 · depth 17 - Weil differentials bounded by D versus mod-p cusp functions
ModularCurve.weilOfKaehler_smul_D_jGeomGen_mem_omegaSpace_iff_isModPCuspFormFn505 below · cited by 1 · depth 17 - Supersingular widths give a cyclic combinatorial component group
ModularCurve.zmultiples_componentGroupProj_smul_coord_eq_top_of_width_eq_jWidth1 below · cited by 1 · depth 17 - At level one, the Atkin–Lehner involution at q is Fricke
ModularCurve.atkinLehnerInvolutionFull_one_eq_frickeInvolutionFull75 below · cited by 2 · depth 18 - The c₆ invariant of the formal Tate curve
ModularCurve.c6_tatePowerSeries0 below · cited by 7 · depth 18 - Dimension bound for mod p weight-2m modular functions
ModularCurve.card_le_dimFormula_of_isModPFormFn_of_linearIndependent735 below · cited by 2 · depth 18 - Ratio of differentials read off q-expansions
ModularCurve.coe_eq_thetaL_div_of_D_eq_smul0 below · cited by 2 · depth 18 - Coefficientwise reduction preserves the modular function field
ModularCurve.coeffMap_mem_modularFunctionFieldC79 below · cited by 1 · depth 18 - Atkin–Lehner toggle at q and Vélu quotients of moduli places
ModularCurve.congrEquiv_moduliPlace_eq_moduliPlace_fullKernelQuotient_of_atkinLehner136 below · cited by 1 · depth 18 - Hecke correspondences at two primes commute on divisors
ModularCurve.correspondence_heckeAlphaC_heckeBetaC_correspondence_heckeAlphaC_heckeBetaC_comm258 below · cited by 1 · depth 18 - Naturality of level-p cusp data under coefficient base change
ModularCurve.cuspData_map_coeffMap2 below · cited by 3 · depth 18 - Degeneracy maps commute with the ℓ-Hecke correspondence on divisors
ModularCurve.degeneracyPair_pushforwardAlong_correspondence_comm_of_ne_of_charP_of_isAlgClosed426 below · cited by 2 · depth 18 - Degeneracy identities for the level-prime Hecke correspondence
ModularCurve.degeneracyPair_pushforwardAlong_correspondence_levelPrime_identities421 below · cited by 1 · depth 18 - Degeneracy push-forwards commute with T_ℓ for ℓ∤ p
ModularCurve.degeneracyPushforwardPair_heckeOperatorBar_of_not_dvd211 below · cited by 7 · depth 18 - Canonical degree is 2 genusFormula(N)-2 in characteristic p≥ 5
ModularCurve.degree_canonicalDivisorOf_eq_two_mul_genusFormula_sub_two725 below · cited by 1 · depth 18 - Degree of the weight-2m floor divisor on X₀(N)
ModularCurve.degree_eq_of_forall_eq_weightFloor398 below · cited by 4 · depth 18 - Degree of the weight-2m floor divisor in characteristic 3
ModularCurve.degree_eq_of_forall_eq_weightFloor_of_char_three383 below · cited by 1 · depth 18 - Degree of the edge weight divisor at 2m = p+1
ModularCurve.degree_weightDivisor_sub_indexPlaces_eq_of_two_mul_eq_add_one472 below · cited by 1 · depth 18 - Diamond operators on J₁(M) commute
ModularCurve.diamondOneBar_comm0 below · cited by 1 · depth 18 - Invariance of the Eisenstein coefficients under n ↦ np
ModularCurve.eisensteinTwoCoeff_mul_level0 below · cited by 1 · depth 18 - Unique prime above a generic supersingular node over a number field
ModularCurve.eq_of_isPrime_of_liesOver_descendedNodeRing_of_ne_zero_of_ne_1728321 below · cited by 1 · depth 18 - Reduction kernel forces V=J₀(M)(ℚ̄)[𝔪]
ModularCurve.eq_top_of_sup_ker_reductionModL_eq_top_of_baseChange_equiv_of_isAbsolutelyIrreducible2,133 below · cited by 1 · depth 18 - Reduction of J_H is injective on prime-to-ℓ torsion
ModularCurve.eq_zero_of_reductionQExpModL_gammaH_eq_zero_of_nsmul_eq_zero858 below · cited by 4 · depth 18 - Cyclicity of Eisenstein classes on the characteristic-q fibre
ModularCurve.eq_zero_or_exists_eq_nsmul_of_heckePic0Fibre_eq_eisenstein_of_heckeOperatorModL_eq_of_smul_eq_neg316 below · cited by 1 · depth 18 - Weierstrass equation of the Tate curve with parameter qᵖ
ModularCurve.equation_tateBase_iff0 below · cited by 5 · depth 18 - The non-toric slot point lies on the base Tate curve
ModularCurve.equation_tateBase_nonToricPoint37 below · cited by 14 · depth 18 - Toric points of Tate(qᵖ) over a commutative ring
ModularCurve.equation_tateBase_tateToricPoint36 below · cited by 6 · depth 18 - Vanishing of preΨₚ at the Tate slot points, p≥ 5
ModularCurve.eval_prePsi_tateBase_nonToricPoint_eq_zero_of_five_le40 below · cited by 3 · depth 18 - Toric p-torsion points kill preΨₚ on Tate(qᵖ)
ModularCurve.eval_prePsi_tateBase_tateToricPoint_eq_zero_of_five_le38 below · cited by 3 · depth 18 - n-torsion of J₀(N) as Λ_N/nΛ_N, Hecke-equivariantly
ModularCurve.exists_addEquiv_torsionBy_jZero_periodLattice_quotient_heckeOperatorBar712 below · cited by 1 · depth 18 - Lifting unitary characters of Γ₀(N) to real additive characters
ModularCurve.exists_addMonoidHom_exp_eq_of_norm_eq_one_of_trace_sq_le_four565 below · cited by 1 · depth 18 - Hecke coordinate with bounded kernel on 2^m-torsion of multiplicative type
ModularCurve.exists_addMonoidHom_torsionBy_two_pow_inf_closure_inertia_smul_sub_heckeLatticeAlgebra_quotient_natCard_ker_le_of_multiplicativeTypeNat2,505 below · cited by 1 · depth 18 - Diamond automorphism lifts to the Hecke roof over Γ_H(M)
ModularCurve.exists_algEquiv_intertwinesAlong_diamondAutHBar31 below · cited by 1 · depth 18 - Automorphism of K(̄ j,̄ j_ℓ) swapping ̄ j and ̄ j_ℓ
ModularCurve.exists_algEquiv_swap_jqModC_jqNModC_of_finrank_eq_dedekindPsi42 below · cited by 1 · depth 18 - Schematic closure of a Galois-stable subspace of J₀(N)[𝔪]
ModularCurve.exists_bialgHom_surjective_model_submodule_heckeTorsion_jZero15 below · cited by 1 · depth 18 - Cyclotomic-equivariant nondegenerate pairing on a Galois plane in J₀(M)[𝔪]
ModularCurve.exists_bilinForm_heckeTorsion_jZero_nondegenerate_smul_eq_cyclotomic_of_baseChange_equiv1,040 below · cited by 1 · depth 18 - Twisted reciprocity: a chain bounding div F on X₀(N)
ModularCurve.exists_chain_periodAlong_add_petersson_eq_zero_of_multiplier_eq_exp33 below · cited by 1 · depth 18 - Eichler–Shimura relation on TₚJ_H(M) after diamond twisting
ModularCurve.exists_character_frobeniusQuadratic_diamondTwist_tateModule_jH1,016 below · cited by 1 · depth 18 - A mod p weight-(p+1) function from ℓ²E₂(q^ℓ)-ℓ E₂(q)
ModularCurve.exists_coe_eq_qExpand_qP_sub_mul_thetaL_zpow_and_one_le_stackOrd905 below · cited by 1 · depth 18 - Frobenius-fixed affine uniformiser at a supersingular place
ModularCurve.exists_coeff_pow_eq_and_ord_eq_one_of_mem_ssPlaces146 below · cited by 1 · depth 18 - Frobenius-fixed affine function separating a φ²-fixed place
ModularCurve.exists_coeff_pow_sq_eq_and_hasValue_zero_and_not_hasValue_zero_of_frobSq_fixed_of_isAffineGeomPlace_of_ne129 below · cited by 1 · depth 18 - Good constant reduction of the embedded modular curve, with charts
ModularCurve.exists_constantReduction_chartData_of_isEmbBasis749 below · cited by 2 · depth 18 - Crossing presentation of the node ring at j=0 or 1728
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring_of_eq_zero_or_eq_1728429 below · cited by 5 · depth 18 - Crossing presentation of the node ring at a width-one supersingular point
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring_of_ne_zero_of_ne_1728293 below · cited by 1 · depth 18 - Kronecker dictionary: Φ(H) carries j(q^N) to j(E/H)
ModularCurve.exists_equiv_algHom_modularFunctionFieldFullC_apply_jqNModC_eq_fullKernelQuotient_j286 below · cited by 5 · depth 18 - Degeneracy values j(qᵈ) read off j(E/H[d])
ModularCurve.exists_equiv_algHom_modularFunctionFieldFullC_forall_dvd_apply_jqNModC_eq_cyclicQuotientJ367 below · cited by 4 · depth 18 - Equivariant torsion reduction: j of the Vélu quotient and ramification
ModularCurve.exists_equivariant_torsion_reduction_ofJ_evalAt_fullKernelQuotient_j_ord_mul_natCard389 below · cited by 1 · depth 18 - Equivariant reduction of torsion for the generic curve over the j-line
ModularCurve.exists_equivariant_torsion_reduction_ofJ_forall_place_reduceHom45 below · cited by 7 · depth 18 - Descent of a modular function and a residue value to a number field
ModularCurve.exists_finiteDimensional_mem_fieldOver_and_redRestrict_eq5 below · cited by 10 · depth 18 - Local-local finite flat model of J₀(N)[𝔪] with Hecke action
ModularCurve.exists_finiteFlat_local_local_model_heckeTorsion_jZero_of_heckeGen_mem2,124 below · cited by 1 · depth 18 - Lifting integral mod-q modular functions to characteristic zero
ModularCurve.exists_full_lift_isIntegral_of_isIntegral744 below · cited by 1 · depth 18 - Adic Galois representation attached to a Hecke character of J₁(M)
ModularCurve.exists_galoisRepAdic_charpoly_frobenius_and_inertia_mul_eq_zero_and_hecke_frobenius_mul_inertia_eq_zero_of_heckeDiamondChar_of_dvd_of_not_sq_dvd_of_le_div5,207 below · cited by 2 · depth 18 - Riemann–Roch index identity for the modular function field of X₀(N)
ModularCurve.exists_genus_riemannIndex_modularFunctionFieldBar157 below · cited by 1 · depth 18 - Bounded Green kernel for chordal proximity at prime level
ModularCurve.exists_greenKernel_regularizedAt_of_prime_of_five_le1,663 below · cited by 1 · depth 18 - One point over supersingular nodes with j=0 or 1728
ModularCurve.exists_hasValue_frobNodePair_of_isIntegral_modularLocalizedAtPoint_of_eq_zero_or_eq_1728380 below · cited by 1 · depth 18 - Equal branch values at a supersingular node, q<5
ModularCurve.exists_hasValue_frobNodePair_of_isIntegral_modularLocalizedAtPoint_of_lt_five218 below · cited by 1 · depth 18 - Hecke–diamond ring of J₁(M) embeds into End_ℂS₂(Γ₁(M))
ModularCurve.exists_injective_ringHom_adjoin_heckeDiamondGenBar_cuspForm611 below · cited by 2 · depth 18 - Leading coefficient of the norm of g at Ogg's unit
ModularCurve.exists_leadingCoeff_eq_mul_pow_of_aeval_jFull_eq_norm_aeval_modularUnitSeries269 below · cited by 1 · depth 18 - Surjectivity of the Legendre j-invariant over algebraically closed fields
ModularCurve.exists_legendreJ_eq0 below · cited by 1 · depth 18 - Parabolic H¹ mod 𝔪 versus Hom(Λ_N,k), Hecke-equivariantly
ModularCurve.exists_linearMap_H1_top_periodLattice_hom_heckeTL_eq_comp_of_mem_parabolicHoms591 below · cited by 1 · depth 18 - Descent to 𝔽_q of Frobenius-fixed modular functions
ModularCurve.exists_mem_zmod_coeffMap_eq_of_coeff_pow_char_eq2 below · cited by 1 · depth 18 - Igusa test datum with rational N-torsion and injective reduction
ModularCurve.exists_moduliTestDatum_natCard_torsion_eq_sq394 below · cited by 1 · depth 18 - Gauss integrality as localisation of the Kronecker ring at level Nq
ModularCurve.exists_mul_coeffMap_eq_iff_coe_mem_modularLocalized_of_not_dvd126 below · cited by 1 · depth 18 - Vertical height-one primes of the j-integral closure
ModularCurve.exists_mul_eq_of_height_one_of_natCast_mem131 below · cited by 3 · depth 18 - Mazur's bound l₁ + a ≤ 1 on the fppf site
ModularCurve.exists_natCard_fppfH_one_of_not_finite_of_sectionsEquiv_algHom_two43 below · cited by 1 · depth 18 - Order of ι(jmath̄(qᵈ)) equals -(N/d)a'² for some a' ∣ d
ModularCurve.exists_order_algHom_qExpand_jqModC_eq_of_apply_jqModC_eq123 below · cited by 1 · depth 18 - Reduction of places of X₀(N) at ℓ∤ N
ModularCurve.exists_placeReductionModL_mapDomain_eq_ord_of_not_dvd738 below · cited by 2 · depth 18 - One witness package for the semistable specialisation of J₀(Nq)
ModularCurve.exists_placeSpecialization_prolongationTuple_width_comp_sp_gluedSpecialization_placeWidthChar3,321 below · cited by 2 · depth 18 - Places of X₀(q)_ℚ̄ centred at (a,a^q) cutting out 𝔭
ModularCurve.exists_place_centred_node_of_height_one_of_natCast_notMem208 below · cited by 8 · depth 18 - Gauss presentations of j(q) and j(qᵖ) over a DVR
ModularCurve.exists_powerSeries_coeffEmb_jq_mul_eq_and_div_eq_jqModC_and_qExpand5 below · cited by 23 · depth 18 - Galois action on the Fricke transform's q-expansion
ModularCurve.exists_qExpansion_slash_fricke_eq_and_conj_eq_slash_gamma024 below · cited by 3 · depth 18 - Gauss prolongation on the q-expansion modular function field
ModularCurve.exists_regularProlongation_laurentBaseChange_qExpFunctionFieldC1 below · cited by 32 · depth 18 - Places of X₀(N) accumulating at the cusp ∞
ModularCurve.exists_seq_place_tendsto_evalAt_cuspInftyBar226 below · cited by 3 · depth 18 - Poles of jmath̄ on the full level-N field are slot places
ModularCurve.exists_slot_of_ord_jqModC_neg123 below · cited by 1 · depth 18 - Smooth proper ℤ₍ₚ₎-model of X(Γ) away from the level
ModularCurve.exists_smoothProperModel_qExpFunctionField_genericFibre_galoisCompat_of_not_dvd938 below · cited by 1 · depth 18 - Annulus of modulus q² at the crossing j=1728
ModularCurve.exists_ssAnnulus_centred_ofNat1728_of_crossingPresentation_of_branchPrimes660 below · cited by 5 · depth 18 - Annulus of places centred at a width-one supersingular crossing
ModularCurve.exists_ssAnnulus_centred_of_widthOne805 below · cited by 5 · depth 18 - Width-three annulus at the supersingular crossing j=0
ModularCurve.exists_ssAnnulus_centred_zero_of_crossingPresentation_of_branchPrimes660 below · cited by 5 · depth 18 - Kronecker's congruence for j(q) and j(qᵖ)
ModularCurve.exists_sub_mul_sub_eq_natCast_mul_of_coe_eq_qExpand56 below · cited by 1 · depth 18 - Integral closure of K[̃ j] in F_N is Dedekind
ModularCurve.exists_subalgebra_isDedekindDomain_isFractionRing_mem_iff_isIntegral_jModElt49 below · cited by 1 · depth 18 - A copy of ρ̄ inside the 𝔪-torsion of J₀(M)
ModularCurve.exists_submodule_heckeTorsion_jZero_finrank_eq_two_baseChange_equiv_of_isAbsolutelyIrreducible1,298 below · cited by 2 · depth 18 - Kernel of reduction on 𝔪-torsion is a T/𝔪-subspace
ModularCurve.exists_submodule_heckeTorsion_jZero_mem_iff_reductionModL_eq_zero975 below · cited by 1 · depth 18 - Another Galois-stable copy of ρ̄ inside J₀(M)[𝔪]/V
ModularCurve.exists_submodule_quotient_heckeTorsion_jZero_linearEquiv_galoisStable_of_baseChange_equiv1,298 below · cited by 1 · depth 18 - Forms on Γ₁(N) are spanned by integral q-expansions
ModularCurve.exists_sum_smul_eq_of_isIntegralQExp_gamma150 below · cited by 3 · depth 18 - Transcendental generator and mod-ℓ reduction inputs for X_H(M)
ModularCurve.exists_transcendental_and_reductionInputsQExpModL_gammaH_of_not_dvd855 below · cited by 10 · depth 18 - Hecke endomorphisms of a finite flat model are bialgebra maps
ModularCurve.finiteFlatModel_comul_comp_heckeEndo10 below · cited by 3 · depth 18 - Elements of 𝔪 act trivially on a finite flat model
ModularCurve.finiteFlatModel_heckeEndo_eq_algebraMap_counit_of_mem9 below · cited by 2 · depth 18 - Hecke action on a finite flat model: unit, composition, convolution
ModularCurve.finiteFlatModel_heckeEndo_one_mul_add9 below · cited by 1 · depth 18 - Degree of k₀(̃ j) over k₀(r(̃ j)) equals deg r
ModularCurve.finrank_adjoin_aeval_jqModC3 below · cited by 2 · depth 18 - Rank at most one for 𝔪-torsion modulo the reduction kernel
ModularCurve.finrank_heckeTorsion_jZero_quotient_ker_reductionModL_le_one_of_heckeGen_notMem1,239 below · cited by 1 · depth 18 - Riemann–Roch for the function field of X₀(N) over ℚ̄
ModularCurve.finrank_riemannRochSpace_sub_finrank_canonicalDivisorOf_sub_eq185 below · cited by 2 · depth 18 - Ramification-weighted Hecke fibres count cyclic ℓ-overgroups with dual pairs
ModularCurve.finsum_ramificationIndexAlong_heckeAlphaC_eq_natCard_overgroup_dualPair_of_moduliPlace463 below · cited by 1 · depth 18 - Frobenius pull-back annihilates ℓ-torsion in Pic⁰ mod ℓ
ModularCurve.frobeniusPullbackModL_eq_zero_of_natCast_smul_eq_zero1 below · cited by 3 · depth 18 - Eichler–Shimura relation on the Tate module of J_H
ModularCurve.frobeniusQuadratic_tateModule_jH1,005 below · cited by 5 · depth 18 - Eichler–Shimura relation on the Tate module of J₁(M)
ModularCurve.frobeniusQuadratic_tateModule_jOne1,002 below · cited by 3 · depth 18 - Genus of the modular function field in characteristic p≥ 5
ModularCurve.genusFF_modularFunctionFieldFullC_eq_genusFormula711 below · cited by 4 · depth 18 - Genus invariance of q-expansion function fields under constant extension
ModularCurve.genusFF_qExpFunctionFieldC_eq_of_ringHom81 below · cited by 6 · depth 18 - Genus of X_H(M) does not drop modulo ℓ∤ M
ModularCurve.genusFF_xHFunctionFieldBar_le_genusFF_xHFunctionFieldC_of_not_dvd782 below · cited by 1 · depth 18 - Geometric base change of the level-one Atkin–Lehner involution is Fricke
ModularCurve.geomAut_atkinLehnerInvolutionFull_one_eq_frickeInvolutionBar76 below · cited by 15 · depth 18 - Lower-level torsion from a non-vanishing degeneracy push-forward
ModularCurve.hasLowerLevelTorsion_of_mem_heckeTorsion_of_degeneracyPushforwardPair_ne_zero212 below · cited by 1 · depth 18 - Hecke inputs at level N and index q, both primes
ModularCurve.heckeInputsFibre_of_prime95 below · cited by 5 · depth 18 - Hecke operators on J₁(M) commute
ModularCurve.heckeOperatorOneBar_comm259 below · cited by 1 · depth 18 - Hecke operators commute with diamond operators on J₁(M)
ModularCurve.heckeOperatorOneBar_comm_diamondOneBar42 below · cited by 1 · depth 18 - Uₚ=-wₚ on Pic⁰ of the special fibre
ModularCurve.heckePic0Fibre_eq_neg_fricke_smul_of_prime981 below · cited by 1 · depth 18 - Level-N₀ 𝔪-torsion from a p-old point when Uₚnotin𝔪
ModularCurve.heckeTorsion_ne_bot_of_mem_heckeTorsion_of_degeneracyPushforwardPair_ne_zero_of_not_mem252 below · cited by 1 · depth 18 - Integrality of Y²jmatĥ^{ m}(jmatĥ-1728)^m over ℂ[jmatĥ⁻¹]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_inv_of_mul_thetaL_pow_eq_qExpansion89 below · cited by 1 · depth 18 - Integrality of Y^{2N}jmatĥ^{ mN+1}(jmatĥ-1728)^{mN} over ℂ[jmatĥ⁻¹]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_inv_pow_of_cuspForm_mul_thetaL_pow_eq_qExpansion89 below · cited by 1 · depth 18 - Integrality of Y⁶jmatĥ^{4m}(jmatĥ-1728)^{3m} over ℂ[jmatĥ]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_of_mul_thetaL_pow_eq_qExpansion82 below · cited by 1 · depth 18 - j(q) is integral over K[j(qᵖ)]
ModularCurve.isIntegral_adjoin_jqNModC_jqModC40 below · cited by 1 · depth 18 - Coefficient Frobenius carries a moduli place to the Frobenius twist
ModularCurve.isModuliPlaceOf_map_frobenius_smul0 below · cited by 1 · depth 18 - The sharp eta quotient is monic of order -n(ℓ)
ModularCurve.isMonicOfOrder_sharpUnitSeries0 below · cited by 4 · depth 18 - Symmetry of the division-polynomial independence unit
ModularCurve.isUnit_indepElt_symm4 below · cited by 11 · depth 18 - Independence element of toric and slot points is a unit
ModularCurve.isUnit_indepElt_tateBase_tateToricPoint_nonToricPoint2 below · cited by 5 · depth 18 - j-generator of the level-N function field is the q-expansion j(q)
ModularCurve.jGeomGen_eq_mk_jqModC0 below · cited by 3 · depth 18 - Value of (jmath̄-j₀) θ f/(thetajmath̄ f) at an affine place
ModularCurve.jGeomGen_sub_mul_div_mem_and_evalAt_eq_of_coe_eq_thetaL_div376 below · cited by 1 · depth 18 - Frobenius and Uₚ on the toric part of J₀(N₀p)[pⁿ]
ModularCurve.jZeroNeronObjectAtP_smul_mem_toricPts_and_heckeGen_smul_eq_of_isFrobeniusAt_of_bridge1,998 below · cited by 1 · depth 18 - j-invariant of the Tate curve with parameter q^N
ModularCurve.j_map_qExpand_tateLaurent4 below · cited by 3 · depth 18 - Closed form for the Kronecker remainder Wronskian
ModularCurve.kroneckerRemainder_frobeniusGraph_ode184 below · cited by 1 · depth 18 - j(λ)=1728 exactly for λ∈{-1,2,1/2}
ModularCurve.legendreJ_eq_ofNat_iff0 below · cited by 1 · depth 18 - Vanishing of the Legendre j-invariant
ModularCurve.legendreJ_eq_zero_iff0 below · cited by 1 · depth 18 - Invariance of the Legendre j-map under t ↦ t⁻¹
ModularCurve.legendreJ_inv0 below · cited by 1 · depth 18 - Invariance of the Legendre j-map under t ↦ 1-t
ModularCurve.legendreJ_one_sub0 below · cited by 1 · depth 18 - Hecke faithfulness on the rational Tate module of J₁(M)
ModularCurve.linearIndependent_rationalHeckeRepOne_of_linearIndependent593 below · cited by 1 · depth 18 - Descent of ℂ-compositum: rational members lie in F_N
ModularCurve.mem_modularFunctionFieldFull_of_coeffEmb_mem_laurentBaseChange1 below · cited by 1 · depth 18 - The modular ring as the range of two-variable evaluation
ModularCurve.modularRing_eq_range_modularEval0 below · cited by 4 · depth 18 - Tₚ J_H(M) is finite free over ℤₚ
ModularCurve.moduleFinite_and_free_padicInt_tateModule_jH300 below · cited by 21 · depth 18 - m-torsion of the inertia-generated subgroup of J₀(Mq')
ModularCurve.natCard_closure_inertia_smul_sub_torsionBy_eq_pow_card_ssPlaces3,332 below · cited by 1 · depth 18 - Supersingular width component group: order num((q-1)/12) and cyclicity
ModularCurve.natCard_componentGroup_eq_and_isAddCyclic_of_width_eq_jWidth43 below · cited by 1 · depth 18 - Verschiebung cokernel bound for a local–local model of J₀(N)[𝔪]
ModularCurve.natCard_dieudonneModule_quot_range_verschiebung_le_of_local_local_model_heckeTorsion_jZero2,572 below · cited by 1 · depth 18 - Reduced Eisenstein torsion dominates the lattice Hecke quotient
ModularCurve.natCard_heckeLatticeAlgebra_quotient_le_natCard_image_reductionModL_heckeTorsion_span_sup2,221 below · cited by 1 · depth 18 - Reduced Eisenstein torsion bounded by lattice Hecke quotient
ModularCurve.natCard_image_reductionModL_heckeTorsion_span_sup_le_natCard_heckeLatticeAlgebra_quotient2,102 below · cited by 1 · depth 18 - Degree of the ℚ(j)-norm of g at Ogg's unit
ModularCurve.natDegree_eq_mul_of_aeval_jFull_eq_norm_aeval_modularUnitSeries222 below · cited by 3 · depth 18 - Cuspidal places of X₀(q) have no strict type
ModularCurve.not_isStrictType_of_isCuspidal8 below · cited by 9 · depth 18 - Copies of ρ̄ in J₀(M)[𝔪] are not killed mod p
ModularCurve.not_le_ker_reductionModL_of_baseChange_equiv_of_isAbsolutelyIrreducible_of_heckeGen_notMem2,126 below · cited by 1 · depth 18 - Placewise identity for ord_w(d̄ j) and weight floors
ModularCurve.ordDifferential_D_jGeomGen_sub_weightFloor_eq503 below · cited by 1 · depth 18 - Simple zero of jmath̃-jmath̃^{q^2} at jmath̃=a
ModularCurve.ord_charLGeomPlaceOfPoint_jqModC_sub_pow_sq_eq_one38 below · cited by 3 · depth 18 - Order of the Hecke multiplier at a pole of j
ModularCurve.ord_heckeMultiplier_eq_of_ord_neg102 below · cited by 1 · depth 18 - Order zero of the Hecke multiplier away from j=0,1728
ModularCurve.ord_heckeMultiplier_eq_zero_of_evalAt_ne380 below · cited by 1 · depth 18 - Order of vanishing of ̄ j at supersingular places in characteristic 3
ModularCurve.ord_jGeomGen_eq_three_or_eq_six_of_exists_prime_dvd_mod_three_eq_two_of_isAlgClosed385 below · cited by 6 · depth 18 - Ramification over j=0 and j=1728 divides 3 and 2
ModularCurve.ord_jqModC_dvd_three_and_ord_sub_dvd_two_of_charP354 below · cited by 3 · depth 18 - Inertia at q is unipotent of echelon two
ModularCurve.pic0_x1x0FunctionFieldC_smul_smul_sub_self_eq_of_mem_inertiaSubgroupIn2,760 below · cited by 1 · depth 18 - q-expansion of h dx along a field tower
ModularCurve.qExpansionDiffAlong_smul_map_D0 below · cited by 2 · depth 18 - Integral q-expansion of E₆
ModularCurve.qExpansion_E6_eq_map_mk0 below · cited by 10 · depth 18 - A characteristic-p identity for ̃ P (thetajmath̄)^{-(p+1)/2}
ModularCurve.qP_mul_thetaL_jqModC_zpow_mul_eq86 below · cited by 2 · depth 18 - Width transport along the degeneracy pair, characteristic ≥ 5
ModularCurve.ramificationIndexAlong_mul_placeWidth_eq_placeWidth_restrictAlong_degeneracyPair_of_five_le833 below · cited by 3 · depth 18 - Rank-two freeness of VₚJ₁(M) with nebentypus determinant
ModularCurve.rationalRankTwoNebentypus_family901 below · cited by 6 · depth 18 - Inertia acts trivially on the reduction map of J_H
ModularCurve.reductionQExpModL_gammaH_smul_eq_self_of_mem_inertiaSubgroupIn263 below · cited by 4 · depth 18 - Degree of the Hecke compositum over X_H(M) is multiplicative
ModularCurve.relfinrank_xHHeckeCompositum_eq_mul175 below · cited by 1 · depth 18 - Galois action on Tₚ J₁(M) commutes with Hecke operators
ModularCurve.rep_tateModule_jOne_comm11 below · cited by 6 · depth 18 - Inertia away from Mp acts trivially on Tₚ J₁(M)
ModularCurve.rep_tateModule_jOne_eq_self_of_mem_inertiaSubgroupIn969 below · cited by 3 · depth 18 - Supersingularity along both legs of the ℓ-roof when ℓ ∣ N
ModularCurve.restrictAlong_heckeAlphaC_mem_ssPlaces_iff_restrictAlong_heckeBetaC_mem_ssPlaces_of_dvd335 below · cited by 1 · depth 18 - Parabolic cohomology rank of Γ(N), denominator-free form
ModularCurve.six_mul_level_mul_finrank_parabolicHoms_Gamma_add_eq14 below · cited by 1 · depth 18 - Ogg's unit has order zero at affine places
ModularCurve.six_mul_ord_add_eq_of_coe_mul_thetaL_jqModC_eq_thetaL_jqNModC_of_isAffineGeomPlace813 below · cited by 1 · depth 18 - Level-one supersingular Hecke entry counts ℓ-isogeny kernels
ModularCurve.ssHeckeMatrixC_one_apply_eq_natCard_subgroup_dualPair487 below · cited by 1 · depth 18 - Row form of Hecke compatibility for the supersingular residue pairing
ModularCurve.ssResiduePairing_ssHeckeFun_eq_comp_traceAlong891 below · cited by 1 · depth 18 - Diamond sum on divisors as pull-back of push-forward
ModularCurve.sum_diamondAutBar_smul_eq_ncard_smul_pullbackAlong_pushforwardAlong43 below · cited by 2 · depth 18 - Chains with boundary vanishing mod Γ₀(N) have lattice periods
ModularCurve.sum_periodAlong_mem_periodLattice_of_boundary_eq_zero1 below · cited by 1 · depth 18 - Stability of K(jmath̄,jmath̄_N) under θ/thetajmath̄
ModularCurve.thetaL_div_thetaL_jqModC_mem_modularFunctionFieldC99 below · cited by 4 · depth 18 - Ramanujan's formula θ j·Δ=-E₄²E₆
ModularCurve.thetaL_jq_mul_deltaSeries80 below · cited by 5 · depth 18 - The identity (θ j)⁶ = j⁴(j-1728)³Δ
ModularCurve.thetaL_jq_pow_six83 below · cited by 5 · depth 18 - q d/dq commutes with q ↦ q^N up to N
ModularCurve.thetaL_qExpand0 below · cited by 4 · depth 18 - Toric points lie on the Tate curve over K((q))
ModularCurve.toricPoint_equation35 below · cited by 2 · depth 18 - Genus bound for dim_ℂ S₂(Γ(N))
ModularCurve.twelve_mul_add_mul_index_le_finrank_cuspForm_Gamma148 below · cited by 1 · depth 18 - Coprime widths make e(s₀) γ(s₀) generate the component group
ModularCurve.zmultiples_componentGroupProj_smul_coord_eq_top_of_pairwise_coprime0 below · cited by 1 · depth 18 - dj is non-zero in the level-N function field
ModularCurve.D_jqModC_ne_zero0 below · cited by 2 · depth 19 - Level one: ̃ j generates the modular function field
ModularCurve.adjoin_jqModC_eq_top0 below · cited by 1 · depth 19 - Bounding fibres of j over 0, 1728, ∞ in characteristic ℓ
ModularCurve.card_fibres_jqModC_qExpFunctionFieldC_gammaH_le_natCard_doubleCoset617 below · cited by 2 · depth 19 - Diamond correspondence on differentials computes ⟨ d⟩ on weight-2 forms
ModularCurve.coeffMap_diffQExp_correspondence_diamondAutBar_eq_qExpansion_diamondLinOne121 below · cited by 2 · depth 19 - Frobenius-fixed ℓ-torsion classes give 𝔽_ℓ-rational logarithmic differentials
ModularCurve.coeffMap_frobenius_inv_mul_thetaL_eq_of_frobeniusPushforwardModL_eq126 below · cited by 1 · depth 19 - Coefficientwise maps preserve constant-field extensions of a Laurent subfield
ModularCurve.coeffMap_mem_laurentBaseChange_of_ringHom165 below · cited by 7 · depth 19 - q-expansion of the Hecke correspondence T_ℓ on differentials
ModularCurve.coeff_diffQExp_correspondence_heckeBetaOneBar_heckeAlphaOneBar_of_not_dvd223 below · cited by 2 · depth 19 - Logarithmic q-derivative: no polar part, constant term the order
ModularCurve.coeff_inv_mul_thetaL_eq_zero_and_coeff_zero_eq_order75 below · cited by 2 · depth 19 - Cartier fixedness of the logarithmic q-derivative
ModularCurve.coeff_inv_mul_thetaL_mul_char_eq_pow76 below · cited by 4 · depth 19 - aₙₚ=aₙ for a Uₚ-fixed, Fricke-anti-invariant q-torsion class
ModularCurve.coeff_inv_mul_thetaL_mul_level_eq_of_heckePic0Fibre_self_eq_of_smul_eq_neg269 below · cited by 1 · depth 19 - q-expansion of T_ℓ on differentials of X₀(N)
ModularCurve.coeff_qExpansionDiffAlong_traceDiff_pullbackDiff_heckeBetaC213 below · cited by 2 · depth 19 - Coefficients of the universal Tate x-series after slot substitution
ModularCurve.coeff_slotSubst_tateUnivX0 below · cited by 31 · depth 19 - Coefficients of a slot substitution into the universal Tate Y-series
ModularCurve.coeff_slotSubst_tateUnivY0 below · cited by 19 · depth 19 - q-twisting by ζ shifts level-p cusp data
ModularCurve.cuspData_map_qTwist2 below · cited by 2 · depth 19 - Tate curve modulo order-d toric subgroup has j-invariant j(qᵈ)
ModularCurve.cyclicQuotientJ_tateLaurent_baseChange_eq_jqNModC_of_le_zmultiples131 below · cited by 4 · depth 19 - Riemann's inequality for the function field of X₀(q)_{ℚ̄}
ModularCurve.degree_add_one_sub_genusFF_le_finrank_riemannRochSpace_modularFunctionFieldBar185 below · cited by 3 · depth 19 - Determinant of inertia at p on 𝔪-torsion equals a
ModularCurve.det_eq_natCast_of_mem_inertiaSubgroupIn_of_baseChange_equiv_heckeTorsion_jZero1,163 below · cited by 2 · depth 19 - Diamond times Frobenius determinant equals ℓ on Tₚ J_H
ModularCurve.diamond_mul_coordDet_eq_of_basis_rationalTateModule_jH559 below · cited by 2 · depth 19 - Ramanujan's identity E₄ θΔ-3 θ E₄ Δ=E₆ Δ over ℚ
ModularCurve.eisenstein4_mul_thetaL_delta_sub_eq_eisenstein6_mul_delta78 below · cited by 1 · depth 19 - Places of the level-one ̄ j-line over an algebraically closed field
ModularCurve.eq_charLGeomPlaceOfPoint_or_eq_charLGeomPlaceEquiv_placeInfty5 below · cited by 16 · depth 19 - Primes over a supersingular node with j=0 or 1728
ModularCurve.eq_of_isPrime_of_liesOver_descendedNodeRing_of_eq_zero_or_eq_1728377 below · cited by 1 · depth 19 - Weight-zero cuspidal integrality forces vanishing
ModularCurve.eq_zero_of_isModPCuspFormFn_zero165 below · cited by 1 · depth 19 - Cuspidal weight-two mod p forms killed by θ vanish
ModularCurve.eq_zero_of_isModPFormFn_one_of_qexpOfWeight_eq_pow474 below · cited by 1 · depth 19 - Value at the q-adic place is the constant coefficient
ModularCurve.evalAt_qInftyPlaceBar_eq_coeff_zero5 below · cited by 5 · depth 19 - Vanishing of preΨₚ at the Tate slot abscissa
ModularCurve.eval_prePsi_tateBase_nonToricPoint_eq_zero48 below · cited by 3 · depth 19 - Kronecker congruence: f(q^q)-f(q)^q is uniquely q-divisible
ModularCurve.existsUnique_qExpand_sub_pow_eq_natCast_mul1 below · cited by 3 · depth 19 - Hecke coordinates mod 2^m on the inertia part of T₂J₀(p)
ModularCurve.exists_addMonoidHom_family_tateModule_inf_pi_closure_inertia_smul_sub_heckeLatticeAlgebra_quotient_natCard_ker_quotient_le2,503 below · cited by 1 · depth 19 - Norms to ℚ(j) are integer polynomials in j
ModularCurve.exists_aeval_jFull_eq_norm_of_mem_chartAlgFin119 below · cited by 2 · depth 19 - Diamond automorphism lifts along both degeneracy embeddings
ModularCurve.exists_algEquiv_intertwinesAlong_diamondAutBar31 below · cited by 3 · depth 19 - A Fricke involution exchanging j(q) and j(q^N)
ModularCurve.exists_algEquiv_swap_jqModC_jqNModC_modularFunctionFieldFullC121 below · cited by 8 · depth 19 - q-chart at the cusp: xᵃⁿ=q^{ord}· G(q)
ModularCurve.exists_analyticAt_realize_eq_qParam_zpow_mul16 below · cited by 2 · depth 19 - The period lattice of a congruence subgroup is a full lattice
ModularCurve.exists_basis_periodLatticeOf_linearIndependent_real_span_eq_top_of_isCongruenceSubgroup173 below · cited by 14 · depth 19 - Constant reduction realising the q-expansion reduction on Pic⁰
ModularCurve.exists_constantReduction_pic0Map_eq_reductionQExpModL15 below · cited by 4 · depth 19 - Inert quadratic descent of the node crossing presentation
ModularCurve.exists_crossingPresentation_modularLocalizedAtPoint_coeffSubring_of_inertQuadratic170 below · cited by 1 · depth 19 - Geometric generic fibre of the two-chart integral model
ModularCurve.exists_curveModel_genericFibre_twoChartIntegralModel_iso_and_galoisCompat5 below · cited by 1 · depth 19 - Two integral degeneracy embeddings of modular function fields
ModularCurve.exists_degeneracyPair_residueField132 below · cited by 1 · depth 19 - Cyclic N-subgroups versus embeddings of the modular function field
ModularCurve.exists_equiv_algHom_modularFunctionFieldFullC_apply_jqN_eq_cyclicQuotientJ304 below · cited by 1 · depth 19 - Supersingular places biject with supersingular j-invariants via evaluation
ModularCurve.exists_equiv_ssJSet_coe_eq_evalAt_jGeomGen_of_forall_mem_iff_mem_ssPlaces45 below · cited by 2 · depth 19 - Finite flat Hecke-equivariant model of J₀(N)[𝔪]
ModularCurve.exists_finiteFlat_model_heckeTorsion_jZero_of_not_dvd1,865 below · cited by 1 · depth 19 - Almost all places are integral for finitely many q-expansions
ModularCurve.exists_finset_forall_coeff_mem_valuationSubring0 below · cited by 1 · depth 19 - Coefficientwise reductions of independent modular functions at almost all primes
ModularCurve.exists_finset_linearIndependent_residue_coeff0 below · cited by 1 · depth 19 - Finiteness of zeros and poles modulo Γ₀(N)
ModularCurve.exists_finset_orbitReps_of_meromorphicOrderAt_ne_zero0 below · cited by 1 · depth 19 - Ultrametric regularised Green kernel at p for prime level N≥ 5
ModularCurve.exists_greenKernel_regularizedAt_ultrametric_of_prime_of_five_le1,662 below · cited by 1 · depth 19 - Parabolic H¹ with 𝒪-coefficients as dual of 𝒪⊗ Tₚ J_H
ModularCurve.exists_heckeEquivariant_parabolicHoms_to_dual_baseChange_tateModule_jH532 below · cited by 1 · depth 19 - Hecke-equivariant embedding of J_H(M) into the analytic Jacobian
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_jH_quotient_periodLatticeOf521 below · cited by 3 · depth 19 - Boston–Lenstra–Ribet: a second copy of ρ̄ in J₀(M)[𝔪]
ModularCurve.exists_injective_linearMap_range_inf_eq_bot_of_baseChange_equiv_of_ne_top1,163 below · cited by 1 · depth 19 - Level-p structures exist over algebraically closed fields
ModularCurve.exists_isLevelPStructure_of_isAlgClosed5 below · cited by 3 · depth 19 - Semistable specialisation with supersingular nodes, toric monodromy and Hecke transport
ModularCurve.exists_jZeroSemistableSpecialization_ssPlaces_monodromy_closure_surjective_heckeTransport_v23,331 below · cited by 1 · depth 19 - A Galois-stable copy of ρ inside J[𝔪]
ModularCurve.exists_linearBlrBlock_of_span_eq_top_of_frobeniusQuadratic_of_dense3 below · cited by 2 · depth 19 - Parabolic homomorphisms as the ℤ-dual of the period lattice
ModularCurve.exists_linearEquiv_parabolicHoms_dual_periodLattice_apply_period576 below · cited by 1 · depth 19 - Regular differentials on X₁(M) over ℚ̄ give S₂(Γ₁(M))
ModularCurve.exists_linearEquiv_tensor_regularDifferentials_x1FunctionFieldBar_cuspForm403 below · cited by 4 · depth 19 - Residue-pair independent integral families in finite-dimensional subspaces
ModularCurve.exists_linearIndependent_residuePair_of_finiteDimensional0 below · cited by 2 · depth 19 - Local-local ℤ₍ₚ₎-model of J₀(N)[𝔪], Hecke-free form
ModularCurve.exists_local_local_model_heckeTorsion_jZero_of_heckeGen_mem1,922 below · cited by 1 · depth 19 - Non-archimedean Łojasiewicz bound for a function on X₀(N)
ModularCurve.exists_log_absValue_evalAt_ge_of_forall_prox_le334 below · cited by 1 · depth 19 - Residual two-dimensional representation at a Hecke maximal ideal
ModularCurve.exists_matrixRep_trace_det_frobenius_of_heckeTorsion_ne_bot1,270 below · cited by 2 · depth 19 - Reciprocity law on X₀(N): divisor periods and Petersson integral
ModularCurve.exists_mem_periodLattice_sum_periodAlong_add_petersson_eq_of_multiplier_eq_exp31 below · cited by 1 · depth 19 - Degree 2g+1 reduced divisor separates rational points and tangents
ModularCurve.exists_mem_riemannRochSpace_mapDomain_embDivisor_sub_notMem237 below · cited by 1 · depth 19 - Finite meromorphic order from a non-zero limit at the cusp
ModularCurve.exists_meromorphicOrderAt_eq_coe_of_tendsto_atImInfty0 below · cited by 6 · depth 19 - Weight-two Eisenstein form on Γ₀(N) with prescribed q-coefficients
ModularCurve.exists_modularForm_qCoeff_eq_eisensteinTwoCoeff_of_neZero0 below · cited by 1 · depth 19 - Diamond operators cut out the Γ₁(M₀)∩Γ₀(t) q-expansion field
ModularCurve.exists_monoidHom_diamondAut_mem_x1x0FunctionFieldC_iff29 below · cited by 9 · depth 19 - Galois-equivariant pairing on stable Eisenstein torsion of J₀(p)
ModularCurve.exists_pairing_heckeTorsion_span_sup_galois_of_stable950 below · cited by 1 · depth 19 - Toric point of exact order M on the Tate curve
ModularCurve.exists_point_tateLaurent_nsmul_eq_toricPoint_of_isPrimitiveRoot6 below · cited by 8 · depth 19 - Kronecker congruence for the integral j-expansion: j(q^q)≡ j(q)^q (mod q)
ModularCurve.exists_qExpand_jqInt_sub_pow_eq_natCast_mul2 below · cited by 3 · depth 19 - Shimura reciprocity at S for level N modular functions
ModularCurve.exists_qExpansion_S_smul_eq_and_conj_eq_of_ratCast_qExpansion20 below · cited by 1 · depth 19 - Shimura reciprocity at the cusps, even weight, level Γ(N)
ModularCurve.exists_qExpansion_slash_coeff_eq_and_eq_apply_of_gamma_of_even17 below · cited by 3 · depth 19 - Crossing model for the completed node ring at j=0,1728
ModularCurve.exists_ringEquiv_adicCompletion_modularLocalizedAtPoint_uvCrossingModel_of_eq_zero_or_eq_1728422 below · cited by 1 · depth 19 - Algebraic places accumulate at a non-cuspidal place of X₀(N)
ModularCurve.exists_seq_place_tendsto_evalAt238 below · cited by 1 · depth 19 - Integral modular function with equal degrees in both characteristics
ModularCurve.exists_transcendental_finrank_adjoin_eq_xHFunctionFieldC_of_not_dvd285 below · cited by 2 · depth 19 - Two-divisibility of inertia displacements in Eisenstein torsion
ModularCurve.exists_two_nsmul_eq_of_mem_torsionBy_two_pow_inf_closure_inertia_smul_sub2,234 below · cited by 1 · depth 19 - Finite-dimensionality of Riemann–Roch spaces on X₀(q)_{ℚ̄}
ModularCurve.finiteDimensional_riemannRochSpace_modularFunctionFieldBar132 below · cited by 5 · depth 19 - Bound [k₀:𝔽ₚ] for 𝔪-torsion in regular differentials
ModularCurve.finrank_mTorsionDiffOf_le_finrank_of_adjoin_range_eq_top887 below · cited by 1 · depth 19 - Genus of X_H(M) unchanged at places above ℓ∤ M
ModularCurve.genusFF_gammaH_residueField_eq_of_not_dvd841 below · cited by 2 · depth 19 - Eta product: formal coefficients sum to prod(1-qⁿ⁺¹)ᵃ
ModularCurve.hasSum_coeff_etaProd_pow0 below · cited by 10 · depth 19 - Non-toric cusp points: x(t· w) on the line of x(w)
ModularCurve.inLine_cuspData_smul_of_five_le49 below · cited by 1 · depth 19 - Vanishing of the independence element and cyclic subgroup membership
ModularCurve.indepElt_eq_zero_iff_mem_zmultiples3 below · cited by 26 · depth 19 - Raw intersection rows for the resolved X₀(Mq) table
ModularCurve.intersectionAlpha_x0MqResolvedTable_eq_sum_x0MqAdj_and_sum_x0MqAdj_inl0 below · cited by 2 · depth 19 - Strict-transform rows of the resolved X₀(Mq) intersection table
ModularCurve.intersectionAlpha_x0MqResolvedTable_inl0 below · cited by 1 · depth 19 - Cyclicity of `componentGroup` for widths in {1,2,3}
ModularCurve.isAddCyclic_componentGroup_of_widths0 below · cited by 1 · depth 19 - Characteristic-ℓ chart ring of X₀(p) over the j-line
ModularCurve.isDedekindDomain_and_finite_and_isSeparable_chartRing_jqModC113 below · cited by 1 · depth 19 - Integrality over the node ring over A ∩ K
ModularCurve.isIntegral_modularLocalizedAtPoint_coeffSubring_of_forall_pole_not_centred263 below · cited by 4 · depth 19 - Normality of the node ring of X₀(q) at j∈{0,1728}
ModularCurve.isIntegrallyClosed_modularLocalizedAtPoint_coeffSubring_of_eq_zero_or_eq_1728431 below · cited by 4 · depth 19 - Normality of the q-node ring at a supersingular point, q<5
ModularCurve.isIntegrallyClosed_modularLocalizedAtPoint_coeffSubring_of_lt_five215 below · cited by 2 · depth 19 - Integral closedness at a generic supersingular node of X₀(q)
ModularCurve.isIntegrallyClosed_modularLocalizedAtPoint_coeffSubring_of_ne_zero_of_ne_1728321 below · cited by 6 · depth 19 - Cusp points are level-p structures on the Tate curve
ModularCurve.isLevelPStructure_cuspData81 below · cited by 1 · depth 19 - Frobenius and Uₚ on prime-to-p toric torsion
ModularCurve.jZeroNeronObjectAtP_smul_mem_toricPts_and_heckeGen_smul_eq_and_smul_heckeGen_eq_of_isFrobeniusAt_of_ne1,993 below · cited by 1 · depth 19 - Base-changed deck group: fixed field and degree formula
ModularCurve.laurentBaseChange_deck_galois_package0 below · cited by 3 · depth 19 - Kronecker congruence to second order in characteristic q
ModularCurve.laurentMap_evalAtJInt_kroneckerRemainder_eval_X_pow1 below · cited by 1 · depth 19 - Base change of the Tate curve along a coefficient map
ModularCurve.map_coeffMap_tateLaurent0 below · cited by 7 · depth 19 - Supersingularity of j with j^{q^2}=j under base change
ModularCurve.mem_ssJSet_algebraMap_of_pow_eq_of_ne_zero_of_ne_172815 below · cited by 3 · depth 19 - Divisor invariance and elliptic-point divisibility for multiplicative F
ModularCurve.meromorphicOrderAt_smul_eq_and_card_stabilizer_dvd_of_multiplier_eq_exp2 below · cited by 3 · depth 19 - Kronecker remainder equals q⁻¹(j_q-j^q)(j-j_q^{ q})
ModularCurve.modularEval_kroneckerRemainder0 below · cited by 14 · depth 19 - SL₂(ℤ)-invariance of the multiset over Γ₀(ℓ)-coset representatives
ModularCurve.multiset_map_cosetReps_smul1 below · cited by 1 · depth 19 - Order of the component group is the Eisenstein numerator
ModularCurve.natCard_componentGroup_eq_eisensteinNumerator6 below · cited by 1 · depth 19 - Lower bounds for the fibres of j over 0, 1728, ∞
ModularCurve.natCard_doubleCoset_le_card_fibres_of_finrank_eq_index92 below · cited by 3 · depth 19 - Eisenstein torsion of J₀(p) counted as a square
ModularCurve.natCard_heckeTorsion_span_sup_eq_sq_natCard_heckeLatticeAlgebra_quotient1,033 below · cited by 1 · depth 19 - Non-toric points satisfy the Tate curve equation over K((q))
ModularCurve.nonToricPoint_equation36 below · cited by 1 · depth 19 - Pinned multiplication formula for abscissae of Tate slot points
ModularCurve.nonToricPoint_fst_mul_psiSq_eq_phi49 below · cited by 1 · depth 19 - Rank two freeness of ℚₚ⊗ TₚJ_H(M) over the Hecke algebra
ModularCurve.nonempty_basis_fin_two_rationalTateModule_jH770 below · cited by 4 · depth 19 - The 0-side and ∞-side of a cuspidal place are disjoint
ModularCurve.not_isInftySide_of_isZeroSide157 below · cited by 4 · depth 19 - Order of jmath̃-c at the place jmath̃=a
ModularCurve.ord_charLGeomPlaceOfPoint_jqModC_sub_algebraMap38 below · cited by 54 · depth 19 - Order zero at an ordinary point for the ∞-branch reduction
ModularCurve.ord_charLGeomPlaceOfPoint_modularRedLocHom_eq_zero_of_not_mem_ssJSet499 below · cited by 4 · depth 19 - j(mathsf q)-a is a uniformiser on the ℓ-degeneracy roof
ModularCurve.ord_heckeAlphaC_jGeomGen_sub_algebraMap_eq_one360 below · cited by 1 · depth 19 - j(q^ℓ) - a' is a uniformiser at generic places of the roof
ModularCurve.ord_heckeBetaC_jGeomGen_sub_algebraMap_eq_one363 below · cited by 1 · depth 19 - Ramification of X₀(N) over the j-line, intrinsic form
ModularCurve.ord_mul_natCard_stabilizer_zmultiples_reduceHom_eq_ramificationIndexAlong_mul_natCard_stabilizer320 below · cited by 2 · depth 19 - Ramification over the j(q^N)-line for a good model
ModularCurve.ord_sub_mul_natCard_stabilizer_zmultiples_reduceHom_eq_ramificationIndexAlong_mul_natCard_stabilizer_fullKernelQuotient386 below · cited by 1 · depth 19 - Hecke and diamond stability of the Γ_H(M) period lattice
ModularCurve.periodLatticeOf_gammaH_heckeDiamondStable10 below · cited by 2 · depth 19 - Periods intertwine slashing by α with conjugation by α
ModularCurve.periodMapOf_gammaH_eq_comp_of_coe_eq_slash2 below · cited by 4 · depth 19 - Canonical identification J₁(M)≅ J_{Γ_bot}(M) over ℚ̄
ModularCurve.pic0Congr_jOne_jH_bot_compat0 below · cited by 1 · depth 19 - Point evaluation of a transported integral polynomial
ModularCurve.pointEval_kroneckerRemainder0 below · cited by 6 · depth 19 - Primitives of a finite flat model bounded by (S/pS)[𝔪]
ModularCurve.pow_finrank_primitives_baseChange_le_card_torsionBySet_intLattice_quotient2,563 below · cited by 1 · depth 19 - Fricke twist of δ lands in 𝔪-torsion differentials
ModularCurve.pullbackAlong_apply_mem_mTorsionDiffOf_of_mem_heckeTorsion_jZero_of_coe_eq_reductionModL1,021 below · cited by 1 · depth 19 - Fr^*Fr_*=ℓ on Pic⁰ in characteristic ℓ
ModularCurve.qExpFrobeniusPullbackModL_qExpFrobeniusPushforwardModL_of_transcendental42 below · cited by 5 · depth 19 - Generation of ℚ(j(qᵈ):d∣ M) by a coprime divisor family
ModularCurve.qExpand_jq_mem_adjoin_of_gcd_eq_one77 below · cited by 1 · depth 19 - q-expansion of an exact differential along σ
ModularCurve.qExpansionDiffAlong_D0 below · cited by 19 · depth 19 - Injectivity of the q-expansion of differentials on K(j,j_N)
ModularCurve.qExpansionDiffAlong_modularFunctionFieldC_injective_of_thetaL_ne_zero51 below · cited by 1 · depth 19 - Semilinearity of the q-expansion map on differentials
ModularCurve.qExpansionDiffAlong_smul0 below · cited by 18 · depth 19 - Deuring reduction data for q-expansion modular curves
ModularCurve.reductionInputsQExpModL_of_finrank_le_of_genusFF_eq101 below · cited by 1 · depth 19 - Reduction mod ℓ commutes with the Fricke involution
ModularCurve.reductionModL_frickeInvolutionBar_smul919 below · cited by 1 · depth 19 - Eichler–Shimura congruence on J₁(M) modulo ℓ
ModularCurve.reductionQExpModL_gamma1_heckeOperatorOneBar984 below · cited by 4 · depth 19 - Eichler–Shimura congruence for J_H(M) modulo ℓ
ModularCurve.reductionQExpModL_gammaH_heckeOperatorHAlong980 below · cited by 2 · depth 19 - Arithmetic Frobenius reduces to the Frobenius push-forward on J_H
ModularCurve.reductionQExpModL_gammaH_smul_of_isFrobeniusAt265 below · cited by 3 · depth 19 - Degree of the Hecke compositum over X₁(M) factorises
ModularCurve.relfinrank_x1HeckeCompositum_eq_mul175 below · cited by 1 · depth 19 - Inertia relation (⟨ u⟩σ-1)(σ-1)=0 on norm-free vectors
ModularCurve.rep_diamondGen_apply_inertia_sub_eq_of_nsmul_sub_sum_tateModule_jOne_of_dvd_of_not_sq_dvd_of_le_div5,079 below · cited by 1 · depth 19 - Frobenius and T_q on the norm-free part of TₚJ₁(M)
ModularCurve.rep_frobenius_rep_heckeGenOne_sub_smul_rep_diamondGen_rep_inertia_sub_eq_zero_normFreePartAt_tateModule_jOne_of_le_div5,079 below · cited by 1 · depth 19 - Closed form for cq^j/(1-cq^j)² in K((q))
ModularCurve.single_div_one_sub_sq0 below · cited by 2 · depth 19 - Closed form for w²/(1-w)³ at a monomial in K((q))
ModularCurve.single_sq_div_one_sub_cube0 below · cited by 2 · depth 19 - Manin-symbol bound for elliptic- and parabolic-killing characters of Γ₀(N)
ModularCurve.six_mul_card_add_le_index_of_linearIndependent_of_trace_sq_le_four0 below · cited by 1 · depth 19 - Injectivity of the slot substitution at p=2, j=1
ModularCurve.slotSubst_gen_injective0 below · cited by 2 · depth 19 - Residues of an embedding basis span the reduced Riemann–Roch space
ModularCurve.span_residue_eq_riemannRochSpace_of_isEmbBasis_of_isGood273 below · cited by 2 · depth 19 - Finiteness of the Hasse-supersingular j-invariants
ModularCurve.ssJSetHasse_finite6 below · cited by 1 · depth 19 - Kronecker congruence in subrings: (j^q-j_q)(j-j_q^q)∈ qR
ModularCurve.sub_mul_sub_mem_span_natCast_of_jqModC_mem_of_jqNModC_mem54 below · cited by 8 · depth 19 - Eichler–Deuring mass formula for supersingular j-invariants
ModularCurve.sum_inv_jWidth_eq_of_ssJSet33 below · cited by 3 · depth 19 - Hecke compatibility of the supersingular residue pairing
ModularCurve.sum_kaehlerResidueTerm_liftFun_ssHeckeFun_eq890 below · cited by 1 · depth 19 - Degree-zero divisor for a function with period multiplier
ModularCurve.sum_meromorphicOrderAt_div_card_stabilizer_eq_zero_of_multiplier_eq_exp10 below · cited by 2 · depth 19 - Reduction mod q is onto the q^k-torsion of J₀(N)
ModularCurve.surjOn_reductionModL_torsion_pow1,772 below · cited by 3 · depth 19 - Analytic Tate curve at q=tᵖ equals the formal Tate base
ModularCurve.tateCurve_curve_X_pow_eq_tateBase0 below · cited by 5 · depth 19 - Closed form of the Tate x-coordinate at a constant point
ModularCurve.tateCurve_pointX_C_eq_tateToricPoint_fst0 below · cited by 4 · depth 19 - Tate parametrisation at u=c t^j: closed form for X
ModularCurve.tateCurve_pointX_C_mul_X_pow_eq_nonToricPoint_fst2 below · cited by 2 · depth 19 - Universal Tate series satisfy the Tate Weierstrass equation
ModularCurve.tateUniv_equation35 below · cited by 5 · depth 19 - Limit at i∞ of a modular function regular there
ModularCurve.tendsto_realize_atImInfty_coeff_zero16 below · cited by 2 · depth 19 - Ramanujan's relation θ j· E₄=- j E₆ for formal q-series
ModularCurve.thetaL_jq_mul_eisenstein4_eq_neg_jq_mul_eisenstein681 below · cited by 5 · depth 19 - Dwork quotient satisfies the θ-identity mod q
ModularCurve.thetaL_laurentMap_dworkQuotient1 below · cited by 1 · depth 19 - Level-one identity for partial(θ j) in ℚ((q))
ModularCurve.thetaL_thetaL_jq_sub_smul_mul_jq_mul_jq_sub_eq83 below · cited by 1 · depth 19 - The q^{pM}-coefficient of a toric point's first coordinate
ModularCurve.toricPoint_fst_coeff_mul0 below · cited by 3 · depth 19 - Vanishing of q^m-coefficients of the toric x-coordinate for p ∤ m
ModularCurve.toricPoint_fst_coeff_of_not_dvd0 below · cited by 3 · depth 19 - Constant term of the toric point's x-coordinate
ModularCurve.toricPoint_fst_coeff_zero0 below · cited by 3 · depth 19 - The q^{pM}-coefficient of a toric point's second coordinate
ModularCurve.toricPoint_snd_coeff_mul_eq_sum_divisors0 below · cited by 3 · depth 19 - Toric point's second coordinate is supported on multiples of p
ModularCurve.toricPoint_snd_coeff_of_not_dvd0 below · cited by 3 · depth 19 - Constant term of the toric point's y-coordinate
ModularCurve.toricPoint_snd_coeff_zero0 below · cited by 3 · depth 19 - A formal Lambert series identity in K((q))
ModularCurve.tsum_lambertTerm_eq0 below · cited by 2 · depth 19 - Coefficientwise summation in K((q)) for growing q-orders
ModularCurve.tsum_of_coeff_lt_eq_zero0 below · cited by 2 · depth 19 - Riemann–Hurwitz inequality for j on X(Γ) over ℚ̄
ModularCurve.two_mul_genusFF_add_card_fibres_le_finrank_add_two_of_gamma1_le151 below · cited by 2 · depth 19 - Different bound at supersingular points in characteristics 2,3
ModularCurve.two_mul_index_le_sum_ordDiff_D_add_natCard_doubleCoset_of_lt_five612 below · cited by 1 · depth 19 - Triviality modulo M₀q/q gives membership in the kernel
ModularCurve.unitsMap_div_eq_one_imp_mem_unitsMap_ker_mul0 below · cited by 1 · depth 19 - Vélu quotient of the Tate curve over a non-toric slot set
ModularCurve.veluQuotient_nonToricSlotSet0 below · cited by 1 · depth 19 - q-expansion function field of Γ₁(M₀)∩Γ₀(q) as a Γ_H field
ModularCurve.x1x0FunctionFieldC_eq_xHFunctionFieldC_unitsMap_ker2 below · cited by 1 · depth 19 - Algebraic independence of the generic variable change of Tate(q)
ModularCurve.algebraicIndependent_variableChange_tateLaurent6 below · cited by 2 · depth 20 - Serre's δ intertwines ̄ T_q with tr_α∘β^*
ModularCurve.apply_eq_traceAlong_pullbackAlong_of_coe_eq_heckePic0Fibre175 below · cited by 1 · depth 20 - Serre's dlog on p-torsion classes yields regular differentials
ModularCurve.apply_mem_regularDifferentials_of_recipe5 below · cited by 1 · depth 20 - Diamond-invariant functions mod ℓ lie in level Γ₀(M)
ModularCurve.coe_mem_modularFunctionFieldFullC_of_forall_diamondPullbackModL_apply_eq201 below · cited by 3 · depth 20 - Coefficients of U_ℓ on differentials: aₙ↦ a_{ℓ n}
ModularCurve.coeff_qExpansionDiffAlong_traceDiff_pullbackDiff_heckeAlphaC_of_dvd215 below · cited by 1 · depth 20 - Smoothed fundamental function: smoothness and partition-of-unity properties
ModularCurve.contDiff_and_finsum_smoothedFundamental_eq_one0 below · cited by 8 · depth 20 - Cyclotomic determinant of the 𝔪-torsion representation of J₀(M)
ModularCurve.det_mTorsionGaloisRep_eq_natCast_of_multiplicityOneData1,191 below · cited by 1 · depth 20 - Injectivity of the q-expansion of differentials on ℚ̄·ℚ(X₁(M))
ModularCurve.diffQExp_x1FunctionFieldBar_injective5 below · cited by 1 · depth 20 - Cusp points lie on the Tate curve over R((q))
ModularCurve.equation_tateBase_cuspPoint40 below · cited by 1 · depth 20 - The function field of X₁(M) over ℚ̄ is essentially of finite type
ModularCurve.essFiniteType_x1FunctionFieldBar3 below · cited by 3 · depth 20 - Cusp points of the Tate curve are p-torsion, p ≥ 5
ModularCurve.eval_prePsi_tateBase_cuspPoint_eq_zero_of_five_le43 below · cited by 1 · depth 20 - Eichler–Shimura: parabolic characters as sums of periods
ModularCurve.existsUnique_periodMapOf_add_conj_periodMapOf_eq_of_mem_parabolicHoms173 below · cited by 1 · depth 20 - Galois-fixed adapted basis for the pole filtration at ∞̄
ModularCurve.exists_adapted_family_modularFunctionFieldBar125 below · cited by 1 · depth 20 - A 2-divisible multiplicative-type subgroup of dyadic Eisenstein torsion
ModularCurve.exists_addSubgroup_eisensteinTorsionBar_two_nsmul_surjOn_inertia_smul_eq_nat_smul_smul_sub_mem2,232 below · cited by 2 · depth 20 - Atkin–Lehner automorphism interchanging the two degeneracy embeddings
ModularCurve.exists_algEquiv_x1x0FunctionFieldC_atkinLehner30 below · cited by 1 · depth 20 - Hecke-equivariant Abel–Jacobi isomorphism for X_H(M) over ℂ
ModularCurve.exists_bijective_heckeEquivariant_addMonoidHom_pic0_complex_xH_quotient_periodLatticeOf490 below · cited by 2 · depth 20 - Chord relation for non-toric Tate slot points
ModularCurve.exists_chordSlope_nonToricPoint16 below · cited by 4 · depth 20 - Weight-2 cusp forms on Γ₁(M) as q-expansions of differentials
ModularCurve.exists_coeffMap_diffQExp_x1FunctionFieldBar_eq_qExpansion136 below · cited by 1 · depth 20 - Constant reduction of the modular function field at q ∤ M'
ModularCurve.exists_constantReduction_modularFunctionFieldBar_residue_eq_coeffMap861 below · cited by 9 · depth 20 - Regular differentials of ℚ̄· F(Γ₁(M)) are weight-two cusp forms
ModularCurve.exists_cuspForm_coeffMap_diffQExp_x1FunctionFieldBar_eq_qExpansion_of_mem_regularDifferentials243 below · cited by 1 · depth 20 - Level-one cusp and uniformisers with divisors [w]-[∞̄]
ModularCurve.exists_cusp_notMem_smul_eq_and_unif_divisor_laws_levelOne58 below · cited by 1 · depth 20 - Frobenius similitude pairing on the rational Tate module of J_H
ModularCurve.exists_diamondFrobeniusSimilitudePairing_rationalTateModule_jH364 below · cited by 1 · depth 20 - Finite flat Hopf model of J₀(N)[p^k] with Hecke action
ModularCurve.exists_finiteFlat_model_jZero_torsion_hecke_of_not_dvd1,863 below · cited by 2 · depth 20 - Parabolic cohomology of Γ_H(M) versus dual Tate module of J_H
ModularCurve.exists_heckeEquivariant_parabolicHoms_to_dual_tateModule_jH531 below · cited by 3 · depth 20 - Hecke-equivariant map from parabolic cohomology to the dual Tate module
ModularCurve.exists_heckeEquivariant_parabolicHoms_to_dual_tateModule_jH_charInvolution531 below · cited by 1 · depth 20 - Cusp places inject into double cosets, with trivial diamond stabilisers
ModularCurve.exists_injective_doubleCoset_forall_diamondPullbackModL_smul_place_eq_of_ord_neg316 below · cited by 1 · depth 20 - Elliptic places of X₀(M) mod ℓ and double cosets
ModularCurve.exists_injective_doubleCoset_forall_diamondPullbackModL_smul_place_eq_of_ord_pos606 below · cited by 1 · depth 20 - Base change of J_H(M) to ℂ: injectivity, torsion, Hecke
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_jH_pic0_complex97 below · cited by 3 · depth 20 - Untwisting a unitary multiplier to a Γ₀(N)-invariant function
ModularCurve.exists_invariant_untwist_of_multiplier_eq_exp5 below · cited by 1 · depth 20 - Galois model over K(j) with ramification dividing 6 in characteristic 3
ModularCurve.exists_isGalois_ord_dvd_six_qExpFunctionFieldC_gammaH_of_char_three472 below · cited by 3 · depth 20 - Galois model of X_H(M) over K(j) in characteristic 2
ModularCurve.exists_isGalois_ord_dvd_twelve_qExpFunctionFieldC_gammaH_of_char_two471 below · cited by 2 · depth 20 - Tate module of J_H(q²M') versus period lattice, with level automorphisms
ModularCurve.exists_linearEquiv_tateModule_jH_padicInt_tensor_periodLatticeOf_levelAut528 below · cited by 1 · depth 20 - q-expansion isomorphism k⊗_ℤS₂(Γ₀(N),ℤ)≅ H⁰(Ω¹)
ModularCurve.exists_linearEquiv_tensor_intLattice_regularDifferentials_qExpansionDiffAlong_eq881 below · cited by 1 · depth 20 - Rational weight-2m forms of level N: dimension lower bound
ModularCurve.exists_linearIndependent_isModPFormFn_rat_dimFormula_le_card767 below · cited by 1 · depth 20 - Winding pairing on X₀(N) lands in the period lattice
ModularCurve.exists_mem_periodLattice_tendsto_windingPairing_smoothedFundamental18 below · cited by 1 · depth 20 - Integrality criterion producing a weight-2m form on Γ₀(N)
ModularCurve.exists_modularForm_qExpansion_eq_mul_thetaL_pow_of_isIntegral95 below · cited by 1 · depth 20 - Bounded denominators for rational weight-2m modular functions
ModularCurve.exists_ne_zero_coeffMap_eq_mul_of_isModPFormFn4 below · cited by 1 · depth 20 - Generator and idempotent tower on the Eisenstein inertia Tate module
ModularCurve.exists_nsmul_generator_idempotent_tower_heckeAlg_tateModule_inf_pi_closure_inertia_smul_sub2,502 below · cited by 1 · depth 20 - A Fricke-twisted Weil pairing on J₀(N)[n]
ModularCurve.exists_pairing_nsmul_eq_zero_galois_hecke535 below · cited by 2 · depth 20 - Pinned specialisation family for the norm-free part at p ‖ M
ModularCurve.exists_qExpSemistableSpecializationPinnedV3_family_normFreePart_and_diamond_of_dvd_of_not_sq_dvd_of_le_div5,072 below · cited by 2 · depth 20 - Shimura reciprocity at the cusps of level N
ModularCurve.exists_qExpansion_comp_smul_coeff_eq_and_eq_apply_of_gamma_invariant16 below · cited by 2 · depth 20 - q-expansion field of X(Γ) embeds into level-M modular functions
ModularCurve.exists_ringHom_laurentBaseChange_qExpFunctionFieldC_levelN19 below · cited by 4 · depth 20 - Tangent doubling law at non-toric slot points
ModularCurve.exists_tangentSlope_nonToricPoint16 below · cited by 4 · depth 20 - Tangent slope at the non-toric Tate slot point, p=3
ModularCurve.exists_tangentSlope_nonToricPoint_of_eq_three10 below · cited by 1 · depth 20 - A p-torsion point independent of two given ones
ModularCurve.exists_torsionPt_indepElt_ne_zero4 below · cited by 2 · depth 20 - Uniform semistable covering of X₀(N) at every prime
ModularCurve.exists_uniform_dualGraphCovering_of_prime_of_five_le1,661 below · cited by 1 · depth 20 - Vélu μ₂-quotient of the Tate curve: E_{q^m}/⟨ T⟩≅ E_q^{2m}
ModularCurve.exists_variableChange_veluQuotient2_tateLaurent_eq_and_vcXInv_velu2X_toricPoint_eq_of_isPrimitiveRoot49 below · cited by 1 · depth 20 - Vélu quotient of the Tate curve by μ_ℓ
ModularCurve.exists_variableChange_veluQuotient_tateLaurent_eq_and_vcXInv_veluX_toricPoint_eq_of_isPrimitiveRoot53 below · cited by 1 · depth 20 - Degree of the q↦ q^ℓ degeneracy map on X₁(M)
ModularCurve.finrankAlong_heckeBetaOneBar218 below · cited by 15 · depth 20 - Degree q-1 over the subfield generated by the modular unit
ModularCurve.finrank_adjoin_coeffEmb_modularUnitSeries_inv216 below · cited by 6 · depth 20 - Degree of ℚ(j)(j(q^N)) over ℚ(j) equals ψ(N)
ModularCurve.finrank_adjoin_jqN_eq0 below · cited by 1 · depth 20 - Frobenius on places commutes with degeneracy restriction
ModularCurve.frobOnPlacesGeomLevel_restrictAlong_degeneracyPair0 below · cited by 1 · depth 20 - Frobenius pullback after pushforward on J₀(N) equals ℓ
ModularCurve.frobeniusPullbackModL_frobeniusPushforwardModL112 below · cited by 2 · depth 20 - Fr_*Fr^* = ℓ on J₀(N) in characteristic ℓ
ModularCurve.frobeniusPushforwardModL_frobeniusPullbackModL116 below · cited by 1 · depth 20 - Existence of canonical divisors on X₁(M) over ℚ̄
ModularCurve.hasCanonicalDivisor_x1FunctionFieldBar51 below · cited by 1 · depth 20 - The function field of X₁(M) over ℚ̄ is a curve
ModularCurve.isCurveOver_x1FunctionFieldBar44 below · cited by 24 · depth 20 - Polynomials in jmath̄ as separating elements of degree deg r·ψ(N)
ModularCurve.isSeparable_and_finrank_adjoin_aeval_jqModC_modularFunctionFieldFullC112 below · cited by 1 · depth 20 - Separability of ̄ j(q^N) over K(̄ j(q)) at invertible prime level
ModularCurve.isSeparable_jqNModC_of_modularPolynomialFamily7 below · cited by 1 · depth 20 - Unit independence element for two p-torsion cusp points
ModularCurve.isUnit_indepElt_tateBase_cuspPoint_of_five_le78 below · cited by 1 · depth 20 - ̄ j(q^ℓ) lies in K(̄ j(q)) in characteristic ℓ
ModularCurve.jqNModC_self_mem_adjoin_unconditional3 below · cited by 1 · depth 20 - Kirchhoff count equals the Eisenstein numerator
ModularCurve.kirchhoffCount_eq_eisensteinNumerator_of_massFormula0 below · cited by 1 · depth 20 - Linear independence of Laurent series under coefficient field extension
ModularCurve.linearIndependent_coeffMap_comp_of_linearIndependent0 below · cited by 7 · depth 20 - Differentials with cusp-form q-expansion are regular
ModularCurve.mem_regularDifferentials_x1FunctionFieldBar_of_coeffMap_diffQExp_eq_qExpansion237 below · cited by 1 · depth 20 - Hecke algebra modulo the annihilator of J₀(M) is ℤ-finite
ModularCurve.moduleFinite_int_heckeAlg_quotient_annihilator_jZero_of_neZero746 below · cited by 1 · depth 20 - Verschiebung cokernel of the Dieudonné module of J₀(N)[p] at 𝔪
ModularCurve.natCard_dieudonneModule_quot_range_verschiebung_sup_range_map_hecke_eq_card_torsionBySet_intLattice_quotient2,510 below · cited by 1 · depth 20 - Semistable specialisation datum for J₀(Nq) with Néron clauses
ModularCurve.nonempty_jZeroSemistableSpecialization_neronClauses_nodes_heckeTransport_inertiaNodeUnit3,330 below · cited by 1 · depth 20 - Igusa's supersingular ramification census for X_H(M) in characteristic three
ModularCurve.ord_census_qExpFunctionFieldC_gammaH_of_char_three581 below · cited by 1 · depth 20 - Census of the supersingular fibre of j in characteristic 2
ModularCurve.ord_census_qExpFunctionFieldC_gammaH_of_char_two582 below · cited by 1 · depth 20 - Reduction is regular at ordinary points jmath̃=a with a^{q^2}=a
ModularCurve.ord_charLGeomPlaceOfPoint_modularRedLocHom_eq_zero_of_not_mem_ssJSet_of_pow_sq_eq469 below · cited by 1 · depth 20 - Reduction is a unit at j=a when a^{q^2}≠ a
ModularCurve.ord_charLGeomPlaceOfPoint_modularRedLocHom_eq_zero_of_not_mem_ssJSet_of_pow_sq_ne494 below · cited by 1 · depth 20 - Ogg's modular unit is a unit away from the cusps
ModularCurve.ord_coeffEmb_modularUnitSeries_of_not_isCusp109 below · cited by 9 · depth 20 - Igusa ramification formula via equivariant torsion reduction
ModularCurve.ord_mul_natCard_stabilizer_eq_ramificationIndexAlong_mul_of_equivariant_torsion_reduction302 below · cited by 2 · depth 20 - Zeros of j-a are simple for a≠ 0,1728
ModularCurve.ord_sub_algebraMap_le_one_laurentBaseChange_qExpFunctionFieldC_of_ne_zero_of_ne_172824 below · cited by 2 · depth 20 - Base-point independence of the period of γ
ModularCurve.periodAlongOf_smul_sub_periodAlongOf_eq_periodOf3 below · cited by 4 · depth 20 - Uniqueness and Galois equivariance of the place transport 1· p → p
ModularCurve.placeEquiv_unique_and_arithmeticGalois_smul_of_forall_mem_iff0 below · cited by 1 · depth 20 - Principal divisors generated by A-primitive functions and diamond translates
ModularCurve.principal_le_closure_divisor_laurentIntegral_diamondAutHBar327 below · cited by 1 · depth 20 - Kronecker norm form of T_ℓ on X_H(M)
ModularCurve.qExpand_norm_heckeBetaHBar211 below · cited by 1 · depth 20 - Level-two relation between j(q²) and λ
ModularCurve.qExpand_two_jq_mul_lambdaModC_sq26 below · cited by 18 · depth 20 - Serre's dlog sends Frobenius to Uₚ on q-expansions
ModularCurve.qExpansionDiffAlong_apply_eq_heckeU_of_congr_coe_eq_frobeniusPushforwardModL147 below · cited by 1 · depth 20 - Cartier operator on q-expansions equals Uₚ twisted by σ⁻¹
ModularCurve.qExpansionDiffAlong_cartier_eq_coeffMap_frobeniusEquiv_symm_heckeU66 below · cited by 1 · depth 20 - Injectivity of the q-expansion map on differentials of K(jmath̄(q),jmath̄(q^N))
ModularCurve.qExpansionDiffAlong_modularFunctionFieldC_injective_of_thetaL_ne_zero_of_natCast_ne_zero47 below · cited by 3 · depth 20 - Degeneracy trace acts as formal T_q on q-expansions
ModularCurve.qExpansionDiffAlong_traceAlong_pullbackAlong_eq_heckeT231 below · cited by 1 · depth 20 - q-expansion of the trace down the degeneracy roof at level p
ModularCurve.qExpansionDiffAlong_traceDiff_pullbackDiff_heckeBetaC_self209 below · cited by 1 · depth 20 - q-expansion of the trace of a pulled-back differential
ModularCurve.qExpansionDiff_traceDiff_pullbackDiff_smul_D166 below · cited by 3 · depth 20 - Transformation of Siegel functions under SL₂(ℤ)
ModularCurve.siegelFun_specialLinearGroup_smul1 below · cited by 1 · depth 20 - Supersingular residue pairing against the degeneracy correspondence
ModularCurve.sum_kaehlerResidueTerm_eq_sum_kaehlerResidueTerm_traceAlong_of_ord_sub_traceFunAlong1 below · cited by 1 · depth 20 - Vanishing of ℓ-adic Tate sequences with trivial Igusa specialisation
ModularCurve.tateModule_eq_zero_of_forall_toPic0Pair_sp_eq_zero_of_ne_normFreePartAt_pinnedV3386 below · cited by 2 · depth 20 - θ = q d/dq against the substitution q ↦ q^N
ModularCurve.theta_qExpand73 below · cited by 2 · depth 20 - Toric Tate points add: P_c+P_d=P_{cd} over F((q))
ModularCurve.toricPoint_add_toricPoint_of_charZero5 below · cited by 4 · depth 20 - Transcendence of the λ-expansion over any commutative ring
ModularCurve.transcendental_lambdaModC0 below · cited by 15 · depth 20 - Segment periods generate the dual of S₂(Γ)
ModularCurve.addSubgroupClosure_range_periodAlongOf_eq_top7 below · cited by 2 · depth 21 - Riemann–Roch bound for mod-3 weight-2m functions on X₀(N)
ModularCurve.card_le_dimFormula_of_isModPFormFn_of_linearIndependent_of_charP_three772 below · cited by 1 · depth 21 - j(q) lies in the base change of ℚ(X₁(N))
ModularCurve.coeffEmb_jq_mem_laurentBaseChange_x1FunctionField3 below · cited by 18 · depth 21 - Non-vanishing of j(q) over a characteristic-zero field
ModularCurve.coeffEmb_jq_ne_zero0 below · cited by 9 · depth 21 - Ring homomorphisms intertwine the coefficient embeddings of ℚ((q))
ModularCurve.coeffMap_coeffEmb_of_ringHom165 below · cited by 8 · depth 21 - Frobenius push-forward p-th powers the q-expansion coefficients of δ
ModularCurve.coeff_qExpansionDiffAlong_apply_of_coe_eq_frobeniusPushforwardModL118 below · cited by 1 · depth 21 - Cartier laws force aₙ(Cω)ᵖ=aₙₚ(ω) on q-expansions
ModularCurve.coeff_qExpansionDiffAlong_cartier_pow65 below · cited by 2 · depth 21 - Determinant of Frobenius on J₀(M)[𝔪] equals ℓ
ModularCurve.detFrobeniusMod_jZero_of_multiplicityOneData1,168 below · cited by 1 · depth 21 - Growth of g(E₄²E₆)^m against hΔ^m at the cusps
ModularCurve.eventually_norm_slash_le_mul_of_isIntegral_adjoin_coeffEmb_jq_inv_sq15 below · cited by 1 · depth 21 - Transport of a p-torsion dlog datum along equal subfields
ModularCurve.exists_addMonoidHom_torsion_recipe_qExpansionDiffAlong_congr_eq2 below · cited by 1 · depth 21 - Mod ℓ cusp expansion of the function field of X_H(M)
ModularCurve.exists_algHom_qExpFunctionFieldC_gammaH_eq_slot_and_diamondPullbackModL_eq_qTwist116 below · cited by 1 · depth 21 - Hecke-equivariant Cartier self-duality of J₀(N)[p]
ModularCurve.exists_bialgEquiv_cartierDual_baseChange_model_jZero_torsion_comp_map_eq692 below · cited by 1 · depth 21 - Igusa function field over k(jmath̄): finite and separable
ModularCurve.exists_coe_eq_jqModC_and_transcendental_and_finiteDimensional_and_isSeparable_igusaFunctionFieldX1C226 below · cited by 12 · depth 21 - k(X₁(M)) is finite separable over k(jmath̄)
ModularCurve.exists_coe_eq_jqModC_and_transcendental_and_finiteDimensional_and_isSeparable_x1FunctionFieldC219 below · cited by 23 · depth 21 - Weight-two cusp forms with algebraic coefficients as differentials
ModularCurve.exists_coeffMap_qExpansionDiffAlong_laurentBaseChange_qExpFunctionFieldC_eq_qExpansion135 below · cited by 1 · depth 21 - Existence of a complex place dictionary for Γ
ModularCurve.exists_complexPlaceDictionaryOf6 below · cited by 4 · depth 21 - Cusp forms of weight 2m from integrality of X over ℂ[J] and ℂ[1/J]
ModularCurve.exists_cuspForm_qExpansion_eq_mul_thetaL_pow_of_isIntegral_qExpFunctionFieldC95 below · cited by 2 · depth 21 - Functions of level N are quotients of integral q-series
ModularCurve.exists_eq_coeffMap_div_coeffMap_of_mem_modularFunctionFieldFull0 below · cited by 1 · depth 21 - Parabolic integral logarithm of an invariant non-vanishing C¹ function
ModularCurve.exists_exp_eq_of_invariant_ne_zero_isParabolicHom1 below · cited by 2 · depth 21 - Fricke involution on J_H(M) over ℚ̄
ModularCurve.exists_frickeAlgEquiv_xHFunctionFieldBar79 below · cited by 2 · depth 21 - A Hecke generator up to bounded index for inertia displacements
ModularCurve.exists_generator_tateModule_inf_pi_closure_inertia_smul_sub_and_smul_eisensteinTorsionBar_eq_zero2,491 below · cited by 1 · depth 21 - A 2-adic Eisenstein idempotent tower acting on J₀(p)
ModularCurve.exists_heckeAlg_idempotent_tower_smul_eq_self_of_mem_eisensteinTorsionBar871 below · cited by 2 · depth 21 - Hecke-equivariant comparison Tₚ J_H(M) ≅ ℤₚ ⊗ Λ_{Γ_H(M)}
ModularCurve.exists_heckeEquivariant_linearEquiv_tateModule_jH_padicInt_tensor_periodLatticeOf528 below · cited by 5 · depth 21 - Tate module of J_H(M) versus the period lattice
ModularCurve.exists_heckeEquivariant_linearEquiv_tateModule_jH_padicInt_tensor_periodLatticeOf_charInvolution528 below · cited by 1 · depth 21 - Analytic uniformisation of J_H(q²M') with level automorphisms
ModularCurve.exists_injective_addMonoidHom_jH_quotient_periodLatticeOf_levelAut520 below · cited by 1 · depth 21 - Eichler–Shimura compatibility for the Hecke–diamond ring of J₁(M)
ModularCurve.exists_injective_ringHom_adjoin_heckeDiamondGenBar_cuspForm_qCoeff612 below · cited by 4 · depth 21 - Clearing denominators in integrality over ℚ[t]
ModularCurve.exists_int_ne_zero_isIntegral_adjoin_int_of_isIntegral_adjoin_rat0 below · cited by 1 · depth 21 - Invariant function with prescribed divisor and Abel–Jacobi limit
ModularCurve.exists_invariant_localModel_tendsto_integral_dbarLogDeriv_smoothedFundamental8 below · cited by 1 · depth 21 - A cyclotomic DVR inside a place above p
ModularCurve.exists_isCyclotomicExtension_isDiscreteValuationRing_isFractionRing_mem_valuationSubring_of_liesOverPrime0 below · cited by 1 · depth 21 - Finitely many Laurent elements defined over one finite Galois extension
ModularCurve.exists_isGalois_forall_coeffMap_eq_of_mem_laurentBaseChange166 below · cited by 1 · depth 21 - Regular functions as quotients of two forms of equal degree
ModularCurve.exists_isHomogeneous_mul_aeval_eq_aeval_and_secVal_ne_zero277 below · cited by 2 · depth 21 - Existence of a λ-modular polynomial with Kronecker's congruence
ModularCurve.exists_lambdaKroneckerCongruence202 below · cited by 8 · depth 21 - A uniform 2-adic exponent for torsion annihilators on J₀(p)
ModularCurve.exists_latticeRestrict_heckeEvalForms_mem_span_two_pow_of_forall_smul_eq_zero1,263 below · cited by 1 · depth 21 - Integral parabolic characters as the ℤ-dual of the period lattice
ModularCurve.exists_linearEquiv_parabolicHoms_dual_periodLatticeOf_apply_periodOf_of_isCongruenceSubgroup174 below · cited by 2 · depth 21 - Many independent weight-2m modular functions over ℚ̄
ModularCurve.exists_linearIndependent_isModPFormFn_algebraicClosure_dimFormula_le_card751 below · cited by 1 · depth 21 - Equivariant family with independent residue pairs on X₀(q)
ModularCurve.exists_linearIndependent_residuePair_forall_arithmeticGalois_smul_eq_of_finiteDimensional2 below · cited by 1 · depth 21 - Integral weight-two cusp forms as regular differentials over k
ModularCurve.exists_mem_regularDifferentials_qExpansionDiffAlong_eq_of_forall_qCoeff_eq_intCast803 below · cited by 1 · depth 21 - Galois-equivariant specialisation of X₁(M) functions at (E,P)
ModularCurve.exists_natural_algHom_qExpFunctionFieldC_gamma1_of_transcendental_j444 below · cited by 2 · depth 21 - Moduli description of X_H(M) points at transcendental j
ModularCurve.exists_natural_diamond_algHom_qExpFunctionFieldC_gammaH_of_transcendental_j549 below · cited by 3 · depth 21 - Places over j₀ on X_H(M) as ± H-orbits of M-torsion
ModularCurve.exists_orbitMap_torsionOrbit_places_qExpFunctionFieldC_gammaH576 below · cited by 3 · depth 21 - Ordinary Frobenius line in the Tate module of J₁(M)
ModularCurve.exists_ordLine_frobenius_quadratic_mem_tateModule_jOne_quotient_of_isUnit_of_not_dvd2,268 below · cited by 1 · depth 21 - Ordinary line at p‖M in a p-new quotient of TₚJ₁(M)
ModularCurve.exists_ordLine_frobenius_sub_smul_mem_tateModule_jOne_quotient_of_diamond_eq_one_of_forall_linearMap_eq_zero_of_dvd_of_not_sq_dvd3,409 below · cited by 1 · depth 21 - Rational basis for a Galois-stable space of q-expansions
ModularCurve.exists_rational_basis_isModPFormFn_of_forall_coeffMap_mem4 below · cited by 1 · depth 21 - Cusp at infinity as an A-section of the two-chart model
ModularCurve.exists_ringHom_section_comp_iotaInf_modelTo_of_coe_eq_coeffEmb_jq2 below · cited by 1 · depth 21 - Poles of j on the level-N modular function field are slot expansions
ModularCurve.exists_slot_algHom_modularFunctionFieldFullC_of_ord_neg123 below · cited by 1 · depth 21 - Trace census by ℓ+1 embeddings at level Nℓ
ModularCurve.exists_traceCensus150 below · cited by 1 · depth 21 - Uniform dual-graph covering at a prime dividing prime level
ModularCurve.exists_uniform_dualGraphCovering_of_dvd_of_not_sq_dvd_of_prime_of_five_le1,410 below · cited by 1 · depth 21 - Uniform semistable covering at a prime not dividing the level
ModularCurve.exists_uniform_dualGraphCovering_of_not_dvd777 below · cited by 1 · depth 21 - Finiteness of level Nℓ over level N after base change
ModularCurve.finiteDimensional_extendScalars_full_prime142 below · cited by 1 · depth 21 - Degree of the first degeneracy embedding over X₁(M)
ModularCurve.finrankAlong_heckeAlphaOneBar216 below · cited by 7 · depth 21 - Deuring's inequality for the q-expansion function field over K(jmath̄)
ModularCurve.finrank_adjoin_jqModC_qExpFunctionFieldC_le_index_of_isAlgClosed119 below · cited by 5 · depth 21 - Bijectivity of the Frobenius push-forward on J₀(N) in characteristic ℓ
ModularCurve.frobeniusPushforwardModL_bijective113 below · cited by 1 · depth 21 - Genus identity for X₁(Mp), Igusa curve and supersingular points
ModularCurve.genusFF_laurentBaseChange_gamma1_mul_add_one_eq_two_mul_genusFF_igusaFunctionFieldX1C_add_natCard1,101 below · cited by 5 · depth 21 - Principal divisors on the ℚ̄-function field of X_H(M)
ModularCurve.hasPrincipalDivisors_xHFunctionFieldBar46 below · cited by 34 · depth 21 - Stokes with side pairing on a Γ-fundamental set
ModularCurve.integral_dbar_mul_cuspForm_gammaFundamentalSet_eq_sidePairing3 below · cited by 2 · depth 21 - Integrality over ℤ[j(q)] gives integrality over ℤ((q))
ModularCurve.isIntegralElem_coeffMap_of_isIntegral_adjoin_jq0 below · cited by 1 · depth 21 - Integrality of Y⁶jmatĥ⁴(jmatĥ-1728)³ over ℂ[jmatĥ]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_of_mul_thetaL_eq_qExpansion_of_finiteIndex82 below · cited by 1 · depth 21 - Ratios of forms are integral over ℂ[j]
ModularCurve.isIntegral_adjoin_jqModC_qExpansion_div_of_forall_isBoundedUnder13 below · cited by 1 · depth 21 - Integrality of x⁶j⁴(j-1728)³ when x dj is regular
ModularCurve.isIntegral_and_isIntegral_of_smul_D_mem_regularDifferentials_qExpFunctionFieldC157 below · cited by 3 · depth 21 - Normality of the λ-node ring at supersingular points
ModularCurve.isIntegrallyClosed_lambdaLocalizedAtPoint_coeffSubring340 below · cited by 2 · depth 21 - Places of the level-one modular function field are rational
ModularCurve.isRational_place_modularFunctionFieldC_one39 below · cited by 27 · depth 21 - Separability of level-M Laurent field over L(j)
ModularCurve.isSeparable_adjoin_jq_extendScalars_full72 below · cited by 1 · depth 21 - Unit independence element at two non-toric cusp slots
ModularCurve.isUnit_indepElt_tateBase_cuspPoint_slot_slot_of_five_le49 below · cited by 1 · depth 21 - Collapse of K(̄ j(q),̄ j(q^{Nℓ})) under separability
ModularCurve.modularFunctionFieldC_mul_eq_of_isSeparable3 below · cited by 1 · depth 21 - Inertia at 2 is of multiplicative type on the reduction kernel
ModularCurve.multiplicativeTypeNat_inf_ker_reductionModL_eisensteinTorsionBar1,972 below · cited by 1 · depth 21 - Hecke torsion in the cotangent space of J₀(N)[p]
ModularCurve.natCard_iInf_ker_mapCotangent_baseChange_model_jZero_torsion_eq_card_torsionBySet_intLattice_quotient2,309 below · cited by 1 · depth 21 - Existence of an integral weight-one form on Γ₁(M)
ModularCurve.nonempty_integralWeightOneForm5 below · cited by 53 · depth 21 - The norm-free endomorphism satisfies N∘ N=|Δ| N
ModularCurve.normFreeEnd_normFreeEnd_eq_card_nsmul2 below · cited by 3 · depth 21 - Cuspidality on the j_q-side excludes strict type
ModularCurve.not_isStrictType_of_isCuspidalSnd10 below · cited by 6 · depth 21 - Lower bound ord_w y ≥ 1-ord_w j at cusps
ModularCurve.one_sub_ord_le_ord_of_coeffMap_mul_thetaL_eq_qExpansion_of_gamma_le90 below · cited by 1 · depth 21 - No cancellation of leading terms under independent constants
ModularCurve.order_sum_algebraMap_mul_coeffMap0 below · cited by 1 · depth 21 - Self-adjointness of the norm-free endomorphism under the Weil pairing
ModularCurve.pair_normFreeEnd_eq_pair_normFreeEnd69 below · cited by 1 · depth 21 - Igusa's q-expansion theorem for Γ₀(M) in characteristic p ∤ M
ModularCurve.qExpFunctionFieldC_gamma0_eq_modularFunctionFieldC_of_not_dvd185 below · cited by 13 · depth 21 - An eta-product identity for E₄ over ℤ((q))
ModularCurve.qExpand_two_eisenstein4_mul_etaProd_pow_eight23 below · cited by 1 · depth 21 - Jacobi's quartic identity as an eta-product identity
ModularCurve.qExpand_two_etaProd_pow_twentyfour20 below · cited by 3 · depth 21 - q-expansion differential along an inclusion agrees with `diffQExp`
ModularCurve.qExpansionDiffAlong_val_eq_diffQExp75 below · cited by 8 · depth 21 - A j-fixing semilinear automorphism fixes the place at infinity
ModularCurve.smul_charLGeomPlaceEquiv_placeInfty_of_smul_jqModC16 below · cited by 1 · depth 21 - Tate ordinate at a constant point over F((t))
ModularCurve.tateCurve_pointY_C_eq_tateToricPoint_snd0 below · cited by 2 · depth 21 - θ = q d/dq commutes with the twist q ↦ uq
ModularCurve.theta_qTwist73 below · cited by 1 · depth 21 - Uniqueness of a valuation prolongation to the compositum with unramified constants
ModularCurve.valuationSubring_eq_of_comap_eq_of_forall_exists_coeffMap_eq0 below · cited by 1 · depth 21 - Universal order-2 Vélu step on the Tate curve
ModularCurve.variableChange_veluQuotient2_tateLaurent_cyclotomicUniv_eq_and_vcXInvR_velu2XR_tateToricPoint_eq48 below · cited by 1 · depth 21 - Universal Vélu quotient of the Tate curve by μ_ℓ
ModularCurve.variableChange_veluQuotient_tateLaurent_cyclotomicUniv_eq_and_vcXInvR_veluXR_tateToricPoint_eq52 below · cited by 1 · depth 21 - q-expansion function field of X₁(Mp) is Igusa in characteristic p
ModularCurve.x1FunctionFieldC_mul_eq_igusaFunctionFieldX1C1,182 below · cited by 2 · depth 21 - Abel's theorem on ΓbackslashH^*: necessity
ModularCurve.abelJacobi_mem_periodLatticeOf_of_meromorphicOrderAt_eq_card_stabilizer22 below · cited by 1 · depth 22 - Base change of the Igusa function field of X₁(M)
ModularCurve.adjoin_image_coeffMap_igusaFunctionFieldX1C_eq0 below · cited by 8 · depth 22 - Integrality over ℂ[̄ j] bounds analytic orders on H
ModularCurve.analyticOrderAt_le_of_isIntegral_adjoin_jqModC_pow14 below · cited by 2 · depth 22 - Uniform proximity–parameter comparison on a doubly attached annulus
ModularCurve.annulusComparison_of_attached_at_both_ends_of_adaptedFamily83 below · cited by 1 · depth 22 - Annulus proximity comparison at a wide node above p ∥ N
ModularCurve.annulusComparison_of_attached_at_both_ends_of_certifiedFamily85 below · cited by 1 · depth 22 - The diamond kernel at p ∥ M has p-1 representatives
ModularCurve.card_normFreeRepsAt_eq_sub_one0 below · cited by 1 · depth 22 - U_ℓ on q-expansions of differentials of X₁(M)
ModularCurve.coeff_diffQExp_correspondence_heckeBetaOneBar_heckeAlphaOneBar_of_dvd219 below · cited by 1 · depth 22 - q-expansion of dy/y has ℓ-th power coefficients
ModularCurve.coeff_qExpansionDiffAlong_dlog_of_frobeniusModL_eq_pow3 below · cited by 1 · depth 22 - Cartier operator on q-expansions: aₙ(Cω)ᵖ=aₙₚ(ω)
ModularCurve.coeff_qExpansionDiffAlong_kw_cart_C_pow0 below · cited by 3 · depth 22 - Degree of the weight-2m floor divisor in characteristic 3
ModularCurve.degree_eq_of_forall_eq_weightFloor_of_charP_three430 below · cited by 1 · depth 22 - Diamond operator on J₁(M) is trivial for d not coprime to M
ModularCurve.diamondOneBar_eq_id_of_not_coprime0 below · cited by 4 · depth 22 - Multiplicativity of the diamond operators on J₁(M)
ModularCurve.diamondOneBar_mul_of_coprime0 below · cited by 3 · depth 22 - Riemann–Roch count for the weight-2m floor divisor
ModularCurve.ell_eq_dimFormula_of_forall_eq_weightFloor441 below · cited by 1 · depth 22 - Uniqueness of the q-expansion differential map along σ
ModularCurve.eq_qExpansionDiffAlong_of_isQExpansionDiffAlong0 below · cited by 3 · depth 22 - Kronecker remainder non-vanishing at supersingular λ-values
ModularCurve.eval_lambdaKroneckerRemainder_ne_zero184 below · cited by 2 · depth 22 - Cusp decay of g(E₄²E₆)^m against hΔ^m under integrality over ℂ[1/j]
ModularCurve.eventually_norm_slash_le_of_isIntegral_adjoin_jqModC_inv_pow15 below · cited by 1 · depth 22 - Uniqueness of the q-th Kronecker remainder for Ψ_q
ModularCurve.existsUnique_lambdaKroneckerRemainder0 below · cited by 2 · depth 22 - Fricke pull-back on ℚ̄· F(Γ_H(M)) and its Galois twist
ModularCurve.exists_algEquiv_xHFunctionFieldBar_slash_fricke_and_galois_smul35 below · cited by 8 · depth 22 - Embedding of the Γ₁(M) q-expansion field fixed by ± P₀
ModularCurve.exists_algHom_qExpFunctionFieldC_gamma1_comp_eq_of_map_eq_or_eq_neg441 below · cited by 1 · depth 22 - Mod ℓ q-expansion principle at an arbitrary cusp of X_H(M)
ModularCurve.exists_algHom_qExpFunctionFieldC_gammaH_coe_eq_div_of_map_eq_smul_qExpansion_slash106 below · cited by 3 · depth 22 - Equivariant Abel–Jacobi bijection for X_H(M) over ℂ
ModularCurve.exists_bijective_heckeEquivariant_addMonoidHom_pic0_complex_xH_quotient_periodLatticeOf_slash490 below · cited by 1 · depth 22 - Fricke-twisted p-adic Weil pairing on TₚJ₁(M)⊗ K
ModularCurve.exists_bilinForm_tateModule_jOne_hecke_selfAdjoint_rep_eq_cyclotomicCharacter_mul_of_forall_pow_eq_one608 below · cited by 1 · depth 22 - Integrality of q-expansions on both charts after base change
ModularCurve.exists_coeffMap_eq_coe_of_mem_chartAlg_twoChartModel_laurentBaseChange1 below · cited by 3 · depth 22 - Models of X₁(M) over ℚ̄ and k, with place reduction
ModularCurve.exists_curveModels_twoChartIntegralModel_x1FunctionField_chartCentre_isLaurentPlaceReduction_of_ringHom945 below · cited by 1 · depth 22 - Existence of the weight-2m floor divisor
ModularCurve.exists_divisor_forall_eq_weightFloor113 below · cited by 1 · depth 22 - Level-Γ_H(M) structures as K(j)-embeddings of the modular function field
ModularCurve.exists_equiv_algHom_qExpFunctionFieldC_gammaH_of_transcendental_j549 below · cited by 1 · depth 22 - Gauss valuation subring of the q-expansion field of X₁(N)
ModularCurve.exists_gaussValuationSubring_laurentBaseChange_x1FunctionField0 below · cited by 14 · depth 22 - Rational cyclicity of the inertia-displacement Tate module at 2
ModularCurve.exists_generator_tateModule_closure_inertia_smul_sub_adjoin_tateHeckeRep2,460 below · cited by 1 · depth 22 - Uniformisation of J_H(M) with Hecke and conjugation compatibility
ModularCurve.exists_injective_heckeEquivariant_addMonoidHom_jH_quotient_periodLatticeOf_complexConjugation520 below · cited by 1 · depth 22 - Invariant function with prescribed degree-zero Γ₀(N)-invariant divisor
ModularCurve.exists_invariant_localModel_dbarLogDeriv_eq_sum_finsum_translate4 below · cited by 1 · depth 22 - Cotangent space of a p-torsion model of J₀(N) versus S₂(Γ₀(N),ℤ)⊗ k
ModularCurve.exists_linearEquiv_baseChange_cotangent_model_jZero_torsion_tensor_intLattice_comp_mapCotangent_eq2,306 below · cited by 1 · depth 22 - Regular differentials and q-expansions under constant field change
ModularCurve.exists_mem_regularDifferentials_qExpansionDiffAlong_eq_coeffMap_of_mem_regularDifferentials173 below · cited by 1 · depth 22 - Integral weight-two cusp forms give regular differentials mod p
ModularCurve.exists_mem_regularDifferentials_residueField_qExpansionDiffAlong_eq_of_forall_qCoeff_eq_intCast801 below · cited by 1 · depth 22 - Elements of the base-changed q-expansion field are ratios of forms
ModularCurve.exists_modularForm_mul_qExpansion_eq_of_mem_laurentBaseChange_qExpFunctionFieldC1 below · cited by 11 · depth 22 - Elements of ℂ· F(Γ) are ratios of modular forms
ModularCurve.exists_modularForm_mul_qExpansion_eq_of_mem_laurentBaseChange_qExpFunctionFieldC_of_T_mem0 below · cited by 6 · depth 22 - Diamond-equivariant moduli embeddings for X₁(M) at transcendental j
ModularCurve.exists_natural_diamond_algHom_qExpFunctionFieldC_gammaH_bot_of_transcendental_j444 below · cited by 2 · depth 22 - Compatible family of Weil pairings on J₀(N)[ℓ^k]
ModularCurve.exists_pairing_family_pow_nsmul_eq_zero_galois_hecke_compat536 below · cited by 2 · depth 22 - Frobenius versus Uₚ on diamond-fixed TₚJ₁(M), p‖ M
ModularCurve.exists_pow_smul_diamond_frobenius_sub_hecke_mem_span_degeneracy_inertiaAugmentation_diamondFixed_tateModule_jOne_of_dvd_of_not_sq_dvd3,323 below · cited by 1 · depth 22 - GAGA for X_H(M): invariant meromorphic functions are algebraic
ModularCurve.exists_realizeOf_eventuallyEq_of_meromorphic_gammaH55 below · cited by 1 · depth 22 - Gauss reduction of X₀(N) above the prime level N
ModularCurve.exists_regularProlongation_modularFunctionFieldBar_self123 below · cited by 1 · depth 22 - Constant-term section at ∞ on the j⁻¹-chart
ModularCurve.exists_ringHom_chartAlgInf_algebraMap_eq_coeff_zero_of_coe_eq_coeffEmb_jq0 below · cited by 6 · depth 22 - Separable annihilator for good Hecke and diamond operators on J₁(M)
ModularCurve.exists_separable_aeval_smul_eq_zero_jOne_of_mem_adjoin_good901 below · cited by 1 · depth 22 - Eisenstein torsion points lift to the 2-adic Tate module
ModularCurve.exists_tateModule_apply_eq_of_mem_eisensteinTorsionBar_two777 below · cited by 1 · depth 22 - Inertia-fixed Tate vector with independent Hecke orbit
ModularCurve.exists_tateModule_inertia_fixed_linearIndependent_heckeLatticeAlgebra_orbit816 below · cited by 1 · depth 22 - Cuspidal units have nonzero limits at every cusp
ModularCurve.exists_tendsto_realizeOf_smul_of_forall_ord_eq_zero17 below · cited by 1 · depth 22 - Toric and old lattices in TₚJ₁(M) for p ∥ M
ModularCurve.exists_toricLattice_oldLattice_diamondNorm_tateModule_jOne_of_dvd_of_not_sq_dvd3,069 below · cited by 1 · depth 22 - Uniform adapted bases with bounded chordal distortion at p∤ N
ModularCurve.exists_uniform_adapted_basis247 below · cited by 1 · depth 22 - Uniform certified multiplicative covering of the level-N modular curve
ModularCurve.exists_uniform_multCovering_with_certifiedFamily_of_prime_of_five_le1,396 below · cited by 1 · depth 22 - Uniform p-window for an embedding basis of X₀(N)
ModularCurve.exists_uniform_window_smul_mem_integers274 below · cited by 1 · depth 22 - Vélu 2-quotient of the Tate curve of q^m is that of q^{2m}
ModularCurve.exists_variableChange_veluQuotient2_toricPoint_neg_one_tateLaurent_map_qExpand_eq_map_qExpand_mul_two37 below · cited by 1 · depth 22 - Vélu quotient of the Tate curve by toric ℓ-torsion
ModularCurve.exists_variableChange_veluQuotient_toricPoint_tateLaurent_map_qExpand_eq_map_qExpand_mul39 below · cited by 1 · depth 22 - Finiteness of ℚ̄-function field of X₁(M) over that of X_H(M)
ModularCurve.finiteAlong_inclusion_xHFunctionFieldBar_x1FunctionFieldBar3 below · cited by 6 · depth 22 - Connected part at an ordinary prime spans at most a line
ModularCurve.finrank_map_reductionKernelSpan_tateModule_jOne_le_one_of_isUnit2,266 below · cited by 1 · depth 22 - The Igusa function field has degree p-1 over X₁(M)
ModularCurve.finrank_x1FunctionFieldC_igusaFunctionFieldX1C_eq_sub_one1,039 below · cited by 4 · depth 22 - Norm along the Frobenius of the modular function field is the ℓ-th power
ModularCurve.frobeniusModL_norm_eq_pow73 below · cited by 1 · depth 22 - Genus inequality for X₁(Mp) via Igusa curves
ModularCurve.genusFF_laurentBaseChange_gamma1_mul_add_one_le_two_mul_genusFF_igusaFunctionFieldX1C_add_natCard1,069 below · cited by 2 · depth 22 - Genus of the level-N modular function field over ℚ̄
ModularCurve.genusFF_modularFunctionFieldFullC_eq_genusFormula_algebraicClosure730 below · cited by 2 · depth 22 - Genus of X₁(M) unchanged in characteristic p ∤ M
ModularCurve.genusFF_x1FunctionFieldC_eq_genusFF_laurentBaseChange_gamma1_of_isAlgClosed846 below · cited by 3 · depth 22 - q-expansion of E₄ converges to E₄(τ)
ModularCurve.hasSum_coeff_eisenstein4_qParam1 below · cited by 6 · depth 22 - Kummer relation, finiteness and separability of the Igusa field
ModularCurve.hasseRootFn_pow_mem_and_finite_and_isSeparable_igusaFunctionFieldX1C10 below · cited by 7 · depth 22 - Λ-eigenspace meets the span of (̂ t-Λ(t))-images trivially
ModularCurve.iInf_ker_tateHeckeRepOne_baseChange_sub_inf_span_eq_bot_of_separable_of_good0 below · cited by 1 · depth 22 - Nonzero simultaneous Hecke eigenspace in K⊗ Tₚ J
ModularCurve.iInf_ker_tateHeckeRepOne_baseChange_sub_ne_bot0 below · cited by 1 · depth 22 - K·ℚ(X₁(M)) is a curve over K
ModularCurve.isCurveOver_laurentBaseChange_qExpFunctionFieldC_gamma144 below · cited by 5 · depth 22 - The q-expansion function field of X(Γ) is a curve over K
ModularCurve.isCurveOver_qExpFunctionFieldC_of_isAlgClosed51 below · cited by 61 · depth 22 - Integrality of Y^{2M}j^{M+1}(j-1728)^M over ℂ[1/j]
ModularCurve.isIntegral_adjoin_coeffEmb_jq_inv_of_mul_thetaL_eq_qExpansion_of_gamma_le81 below · cited by 1 · depth 22 - Integrality over ℂ[j] of q-expansion quotients of modular forms
ModularCurve.isIntegral_adjoin_jqModC_qExpansion_div_of_forall_isBoundedUnder_of_finiteIndex13 below · cited by 1 · depth 22 - Normality of the plane model of X₀(q) at (a,a^q)
ModularCurve.isIntegrallyClosed_modularLocalizedAtPoint_coeffSubring_of_pow_sq_ne175 below · cited by 1 · depth 22 - Hasse root is a Kummer generator of exponent p-1
ModularCurve.isKummerGenerator_hasseRootFn_x1FunctionFieldC225 below · cited by 3 · depth 22 - (varpi) prime and 𝔪=(varpi,j-x) at a smooth point
ModularCurve.isPrime_span_uniformizer_and_maximalIdeal_modularLocalizedAtPoint_eq_of_pow_sq_ne172 below · cited by 2 · depth 22 - Properness of the two-chart integral j-model over A
ModularCurve.isProper_toBase_twoChartIntegralModel_of_eq_laurentBaseChange8 below · cited by 12 · depth 22 - Separability of the X₁(M) q-expansion field over K(j)
ModularCurve.isSeparable_adjoin_qExpFunctionFieldC_gamma110 below · cited by 4 · depth 22 - Kronecker congruence for the λ modular equation
ModularCurve.kroneckerCongruence_lambda179 below · cited by 2 · depth 22 - Kronecker remainder of the λ-modular equation, evaluated
ModularCurve.lambdaEval_kroneckerRemainder0 below · cited by 2 · depth 22 - Base change of a subfield of ℚ((q)) as an L-span
ModularCurve.mem_laurentBaseChange_iff_exists_eq_sum_smul_coeffEmb0 below · cited by 4 · depth 22 - Holomorphic weight-2m mod 3 forms lie in L(D)
ModularCurve.mem_riemannRochSpace_of_isModPFormFn_of_charP_three387 below · cited by 1 · depth 22 - Level-two modular polynomial as minimal polynomial of λ(q^q)
ModularCurve.minpoly_lambdaNModC_eq173 below · cited by 4 · depth 22 - Existence of an integral λ-modular polynomial for odd q
ModularCurve.nonempty_lambdaModularPolynomialData197 below · cited by 1 · depth 22 - Modular functions with nonzero cusp limits are units at cuspidal places
ModularCurve.ord_eq_zero_of_not_mem_of_realizeOf_tendsto30 below · cited by 2 · depth 22 - Ramification over j=0 and j=1728 divides 3 and 2
ModularCurve.ord_jqModC_dvd_three_and_ord_sub_dvd_two_algebraicClosure315 below · cited by 1 · depth 22 - Ramification bounds for j on X(Γ) over ℚ̄
ModularCurve.ord_le_three_and_ord_sub_le_two_and_ord_sub_le_one_laurentBaseChange_qExpFunctionFieldC_algebraicClosure99 below · cited by 7 · depth 22 - Igusa cover of X₁(M) unramified off supersingular places
ModularCurve.ramificationIndex_igusaFunctionFieldX1C_eq_one_of_not_evalAt_mem_ssJSet1,057 below · cited by 1 · depth 22 - Independence of the realisation from the chosen presentation
ModularCurve.realizeOf_eq_div0 below · cited by 24 · depth 22 - Coefficientwise reduction of p(j(q),j(q^N))
ModularCurve.redRes_modularEval0 below · cited by 1 · depth 22 - Inertia at ℓ∤ N acts trivially on prime-to-ℓ torsion of J₀(N)
ModularCurve.smul_eq_self_of_mem_inertiaSubgroupIn_of_nsmul_eq_zero_of_not_dvd979 below · cited by 1 · depth 22 - Lower bound for supersingular places of X₁(M) in characteristic p
ModularCurve.sub_one_mul_index_gamma1_le_twelve_mul_natCard_evalAt_mem_ssJSet_x1FunctionFieldC1,034 below · cited by 3 · depth 22 - Diamond norm annihilates the norm-free endomorphism on J₁(M)
ModularCurve.sum_diamondOneBar_normFreeEnd_eq_zero1 below · cited by 1 · depth 22 - Hecke operators killing inertia displacements kill the Eisenstein Tate module
ModularCurve.tateHeckeRep_eq_zero_of_forall_closure_inertia_smul_sub_eq_zero1,167 below · cited by 1 · depth 22 - Smoothed unfolding of a weight-two pairing against F
ModularCurve.tendsto_integral_mul_smoothedFundamental_mul_finsum_translate1 below · cited by 2 · depth 22 - Genus formula for X₁(M) with M ≥ 5
ModularCurve.twelve_mul_genusFF_laurentBaseChange_gamma1_add_six_mul_natCard_doubleCoset_eq_index_add_twelve413 below · cited by 4 · depth 22 - At most (p-1)μ/12 supersingular places on X₁(M)
ModularCurve.twelve_mul_natCard_evalAt_mem_ssJSet_le_sub_one_mul_index_gamma1_x1FunctionFieldC728 below · cited by 1 · depth 22 - Vélu's μ₂-isogeny sends toric point c to c²
ModularCurve.vcXInv_velu2X_and_vcYInv_velu2Y_toricPoint_tateLaurent_map_qExpand_eq_toricPoint_sq13 below · cited by 1 · depth 22 - Vélu's μ_ℓ-isogeny sends toric point c to c^ℓ
ModularCurve.vcXInv_veluX_and_vcYInv_veluY_toricPoint_tateLaurent_map_qExpand_eq_toricPoint_pow10 below · cited by 1 · depth 22 - Inverting F and t leaves the Abel fibre sum unchanged
ModularCurve.abelFibreSumOf_inv0 below · cited by 1 · depth 23 - λ-series generates the level-4 modular function field
ModularCurve.adjoin_lambdaModC_eq_laurentBaseChange_modularFunctionFieldFull_four170 below · cited by 4 · depth 23 - Adjoining λ and λ(X^q) gives level 4q
ModularCurve.adjoin_lambdaModC_lambdaNModC_eq_laurentBaseChange_modularFunctionFieldFull171 below · cited by 2 · depth 23 - Integrality descent for polynomials in the λ-series
ModularCurve.aeval_lambdaModC_intCoeffs_descent0 below · cited by 1 · depth 23 - Base-changed diamond automorphisms fix the Γ₀(N)-level functions
ModularCurve.algEquiv_apply_eq_self_of_coe_mem_laurentBaseChange_qExpFunctionFieldC_gamma0_of_coe_eq_baseChangeAut_diamondAut31 below · cited by 2 · depth 23 - Tate-curve toric coordinates lie in the Γ₁(M) q-expansion field
ModularCurve.c4_mul_toricPoint_fst_div_c6_mem_qExpFunctionFieldC_gamma18 below · cited by 1 · depth 23 - Stabilisers in Γ versus in ±Γ
ModularCurve.card_stabilizer_sup_zpowers_negOne_mul_card_inf_eq_two_mul_card_stabilizer0 below · cited by 1 · depth 23 - Diamond automorphisms of X_H(M) and X₁(M) agree under ι
ModularCurve.coe_diamondAutHBar_eq_diamondAutBar_of_coe_eq67 below · cited by 2 · depth 23 - Coefficientwise reduction commutes with evaluating at λ-expansions
ModularCurve.coeffRed_lambdaEval0 below · cited by 1 · depth 23 - Supersingular Legendre parameter is a root of the Deuring polynomial
ModularCurve.deuringPolynomial_eval_eq_zero_of_exists_mem_ssJSet16 below · cited by 1 · depth 23 - Independence of powers of E₄E₆/Δ mod q over X₁(M)
ModularCurve.eq_zero_of_sum_mul_eisensteinRatio_pow_eq_zero_of_mem_x1FunctionFieldC1,033 below · cited by 3 · depth 23 - Kronecker remainder is nonzero at supersingular λ-points
ModularCurve.eval_lambdaKroneckerRemainder_ne_zero_of_deuringPolynomial169 below · cited by 1 · depth 23 - Local constancy of the Abel fibre sum modulo periods
ModularCurve.eventually_abelFibreSumOf_sub_mem_periodLatticeOf20 below · cited by 1 · depth 23 - Uniqueness of finite flat Hopf models of J₀(N)[p] with Hecke action
ModularCurve.exists_algEquiv_finiteFlat_model_jZero_torsion_hecke_of_ne_two_of_neZero91 below · cited by 1 · depth 23 - Coefficientwise extension of automorphisms of L·ℚ(X₁(N)) to ℚ̄
ModularCurve.exists_algEquiv_x1FunctionFieldBar_coe_eq_coeffMap_of_algEquiv_laurentBaseChange2 below · cited by 1 · depth 23 - Generic evaluation of the Γ₁(M) function field with torsion point
ModularCurve.exists_algHom_qExpFunctionFieldC_gammaH_bot_comp_eq_iff_and_comp_diamondPullbackModL_of_transcendental_j441 below · cited by 1 · depth 23 - Slot embeddings realised by q-expansions at a cusp
ModularCurve.exists_algHom_slot_mul_qExpansion_slash_eq82 below · cited by 1 · depth 23 - Degeneracy embedding of X_H(M) function fields at p ‖ M
ModularCurve.exists_algHom_xHFunctionFieldBar_div_coe_eq_and_isIntegral_and_finrankAlong_eq207 below · cited by 4 · depth 23 - Hecke-self-adjoint perfect pairing on the Tate module of J₁(M)
ModularCurve.exists_bilinForm_tateModule_jOne_hecke_selfAdjoint_reductionKernelSpan_orthogonal_le616 below · cited by 1 · depth 23 - Twisted pairing on Tₚ J₁(M) and orthogonality at p
ModularCurve.exists_bilinForm_tateModule_jOne_hecke_selfAdjoint_rep_diamond_cyclotomic_toricOrthogonal_mem_span_degeneracy_inertiaAugmentation_of_dvd_of_not_sq_dvd3,319 below · cited by 1 · depth 23 - Hecke-self-adjoint Galois pairing on the Tate module of J₀(p)
ModularCurve.exists_bilinForm_tateModule_nondegenerate_hecke_galois539 below · cited by 1 · depth 23 - Good reduction of the two-chart integral model at p ∤ M
ModularCurve.exists_curveModel_specialFibreIso_twoChartIntegralModel_qExpFunctionFieldC_chartCentre_of_not_dvd901 below · cited by 5 · depth 23 - Fibres of the two-chart model of X₁(M), with q-expansion readings
ModularCurve.exists_curveModels_fibres_twoChartIntegralModel_x1FunctionField_readCharts_and_iso_pullback_of_ringHom925 below · cited by 1 · depth 23 - Decomposition characters on the multiplicative submodule of TₚJ_H(M)
ModularCurve.exists_decompositionCharacters_multiplicativeSubmodule_cornerSubmodule_tateModule_jH_of_ordinary3,753 below · cited by 2 · depth 23 - Fricke involution package for J₁(M) over ℚ̄
ModularCurve.exists_frickeAlgEquiv_x1FunctionFieldBar_galois_smul83 below · cited by 1 · depth 23 - Gauss fraction form inside the q-expansion function field
ModularCurve.exists_gaussFracForm_mem_laurentBaseChange_qExpFunctionFieldC3 below · cited by 8 · depth 23 - Rank-one multiplicative submodule at an ordinary non-Eisenstein corner
ModularCurve.exists_generator_multiplicativeSubmodule_cornerSubmodule_tateModule_jH_of_ordinary_of_not_isEisenstein_of_mem_infSubgroup4,863 below · cited by 1 · depth 23 - Geometric generic fibre of the two-chart integral model
ModularCurve.exists_genericFibreIso_ofGenerator_twoChartIntegralModel_chartCentre_and_galoisCompat5 below · cited by 2 · depth 23 - p-torsion Hopf algebra of a group-law model of J₀(N)
ModularCurve.exists_hopfAlgebra_torsion_model_jZero_points_hecke_of_relativeGroupLaw3 below · cited by 1 · depth 23 - Deuring reduction of places preserves chart centres when p ∤ M
ModularCurve.exists_isPlaceReductionQExpModL_chartCentre_of_not_dvd854 below · cited by 4 · depth 23 - Rational q-adic Tate module of J₀(p) versus period lattice
ModularCurve.exists_linearEquiv_rationalTateModule_tensor_periodLattice744 below · cited by 1 · depth 23 - Eichler–Shimura for J₁(M): Tate charpoly is Q²
ModularCurve.exists_map_eq_charpoly_heckeTLinOne_and_charpoly_tateHeckeRepOne_jOne_eq_map_sq601 below · cited by 2 · depth 23 - Inertia at 2 moves every nonzero Hecke image in T_P
ModularCurve.exists_mem_inertiaSubgroupIn_tateModule_rep_ne_of_adjoin_tateHeckeRep_apply_ne_zero1,166 below · cited by 1 · depth 23 - q-expansion principle for constant reductions of X₀(N)
ModularCurve.exists_mem_integers_residue_eq_coeffMap_of_isPlaceReductionModL229 below · cited by 1 · depth 23 - Gauss residues descend from Γ₀(qN) to Γ₀(N)
ModularCurve.exists_mem_qExpFunctionFieldC_gamma0_and_eq_qExpand_of_qExpand_mul_coeffMap_eq133 below · cited by 4 · depth 23 - Meromorphic Γ-invariant functions are quotients of modular forms
ModularCurve.exists_modularForm_eventuallyEq_div_of_meromorphic_of_finiteIndex0 below · cited by 1 · depth 23 - Unit-root factor of Tₚ on TₚJ₁(M) and p-rank
ModularCurve.exists_monic_unitRoot_mul_aeval_tateModule_jOne_eq_zero_pow_finrank_ker_eq_card_torsion_sq1,317 below · cited by 1 · depth 23 - X_H function field as diamond-fixed subfield of X₁
ModularCurve.exists_monoidHom_diamondAut_mem_xHFunctionField_iff29 below · cited by 4 · depth 23 - Coefficients in one number field with bounded denominators
ModularCurve.exists_numberField_isIntegral_mul_coeff_of_forall_ord_nonneg185 below · cited by 1 · depth 23 - Tangent space of the relative Jacobian of X₀(N) at p
ModularCurve.exists_pts_relJacobian_jZero_level_dualNumber_kernel_equiv_addMonoidHom_intLattice_latticeHeckeFamily_integral_of_representsRelSubPic_of_ratCurveModel_of_not_dvd1,563 below · cited by 1 · depth 23 - Gauss valuation ring: coefficientwise reduction to κ((q))
ModularCurve.exists_ringHom_laurentSeries_residueField_of_gaussValuationSubring0 below · cited by 3 · depth 23 - Hasse invariant in the j-coordinate, characteristic p≥ 5
ModularCurve.exists_separable_thetaL_jqModC_pow_mul_aeval_eq185 below · cited by 5 · depth 23 - Primitive P-integral normalisation of a bounded q-expansion
ModularCurve.exists_smul_coe_eq_coeffMap_and_residue_ne_zero_of_smul_coe_eq_coeffMap_xHFunctionFieldBar1 below · cited by 12 · depth 23 - Cusp limits transfer from F to the ratio g/h
ModularCurve.exists_tendsto_div_smul_of_eventuallyEq_realizeOf_of_tendsto1 below · cited by 1 · depth 23 - Trace and determinant of Galois on a corner of Tₚ(J_H)
ModularCurve.exists_trace_det_latticeMatrix_tateGaloisRep_cornerSubmodule_tateModule_jH1,379 below · cited by 1 · depth 23 - 2-divisibility of the Eisenstein P-power torsion at 2
ModularCurve.exists_two_nsmul_eq_of_mem_eisensteinTorsionBar_two776 below · cited by 1 · depth 23 - Uniform p-power window for model bases at q-criterion charts
ModularCurve.exists_uniform_window_smul_mem_integers_of_qCoeff_criterion123 below · cited by 1 · depth 23 - Gauss valuation ring of L(j) inside the X₁(N) function field
ModularCurve.exists_valuationSubring_adjoin_isDiscreteValuationRing_mem_iff_of_laurentBaseChange_x1FunctionField2 below · cited by 2 · depth 23 - Vélu quotient of the Tate curve by its toric p-slots
ModularCurve.exists_variableChange_veluQuotient_toricSlotSet0 below · cited by 1 · depth 23 - Finiteness of the n-torsion of J₁(M) over ℚ̄
ModularCurve.finite_torsion_jOne528 below · cited by 2 · depth 23 - Degree of the j-cover of X₁(M) equals [SL₂(ℤ):±Γ₁(M)]
ModularCurve.finrank_adjoin_jqModC_laurentBaseChange_qExpFunctionFieldC_gamma1_eq_index224 below · cited by 10 · depth 23 - Degree q+1 of λ(qτ) over L(λ), λ the X₀(4) Hauptmodul
ModularCurve.finrank_adjoin_lambdaModC_adjoin_lambdaNModC172 below · cited by 4 · depth 23 - Igusa's good reduction of X₁(M) in degree form
ModularCurve.finrank_adjoin_x1FunctionFieldC_eq_index_gamma1_sup_of_isAlgClosed296 below · cited by 6 · depth 23 - Frobenius identity for the λ-series in characteristic ℓ
ModularCurve.frobenius_identity_lambda1 below · cited by 2 · depth 23 - Order of the Hasse ratio at supersingular places is prime to p-1
ModularCurve.gcd_sub_one_natAbs_ord_eq_one_of_evalAt_mem_ssJSet_of_coe_eq_hasseRootFn_pow1,036 below · cited by 2 · depth 23 - The genus formula for X₀(N) takes natural number values
ModularCurve.genusFormula_isNat1 below · cited by 1 · depth 23 - Hasse root function outside κ(X₁(M))_q in characteristic three
ModularCurve.hasseRootFn_notMem_x1FunctionFieldC_charThree964 below · cited by 3 · depth 23 - Integral q-expansions of the λ modular equation coefficients
ModularCurve.intCoeffs_minpoly_lambdaNModC_coeff180 below · cited by 1 · depth 23 - The Hasse root function generates a Kummer extension of exponent p-1
ModularCurve.isKummerGenerator_hasseRootFn1 below · cited by 4 · depth 23 - Hasse root function is a Kummer generator of degree p-1
ModularCurve.isKummerGenerator_hasseRootFn_and_relfinrank_igusaFunctionFieldX1C1,032 below · cited by 5 · depth 23 - At p=2 the Hasse root lies in k(X₁(M))
ModularCurve.isKummerGenerator_one_hasseRootFn_of_charP_two221 below · cited by 1 · depth 23 - Hasse root as exponent-two Kummer generator in characteristic 3
ModularCurve.isKummerGenerator_two_hasseRootFn_of_charP_three0 below · cited by 3 · depth 23 - Places of the q-expansion function field of X₁(M) are rational
ModularCurve.isRational_place_x1FunctionFieldC_of_isAlgClosed259 below · cited by 7 · depth 23 - The rational ℓ-adic Hecke algebra on J₀(p) is reduced
ModularCurve.isReduced_rationalHeckeAlgebra891 below · cited by 2 · depth 23 - Igusa function field splits Xᵖ⁻¹-b over the X₁(M) field
ModularCurve.isSplittingField_igusaFunctionFieldX1C_X_pow_sub_C0 below · cited by 1 · depth 23 - Supersingularity transfers along the mathsf q ↦ mathsf q^{q^e} substitution
ModularCurve.isSupersingularPlace_of_forall_mem_iff_of_coe_eq_qExpand6 below · cited by 5 · depth 23 - Level-four Hauptmodul relation between j and λ/16
ModularCurve.jq_mul_lambdaModC_mul_one_sub_pow_four27 below · cited by 5 · depth 23 - The Hauptmodul λ lies in the level-four function field
ModularCurve.lambdaModC_mem_modularFunctionFieldFull_four171 below · cited by 8 · depth 23 - Membership in F ≤ K((q)) descends along coefficientwise embeddings
ModularCurve.mem_of_coeffMap_mem_adjoin_image_of_ringHom0 below · cited by 1 · depth 23 - Supersingular j-invariants are the roots of X^{e₄}(X-1728)^{e₆}S
ModularCurve.mem_ssJSet_iff_eval_eq_zero_of_thetaL_pow_mul_aeval_eq184 below · cited by 4 · depth 23 - Minimal polynomial of λ(q^q) has coefficients in ℚ[λ]
ModularCurve.minpoly_lambdaNModC_coeff_mem_adjoin194 below · cited by 1 · depth 23 - Poles of j on X₁(M) counted by cusps
ModularCurve.natCard_place_ord_neg_laurentBaseChange_gamma1_eq_natCard_doubleCoset303 below · cited by 2 · depth 23 - n-torsion of Pic⁰ of X_H(M) has order n^{2g}
ModularCurve.natCard_torsion_pic0_xHFunctionFieldBar_eq_pow_two_mul_genusFF852 below · cited by 7 · depth 23 - Places of j-order exactly 3 and 6 in characteristic 3
ModularCurve.ncard_setOf_ord_jGeomGen_eq_three_and_eq_six_of_exists_prime_dvd_mod_three_eq_two402 below · cited by 1 · depth 23 - Order of j at places above j=0 divides 12/(p-1)
ModularCurve.ord_dvd_twelve_div_sub_one_of_ord_pos_x1FunctionFieldC_of_lt_five474 below · cited by 2 · depth 23 - Orders of j and j-1728 on X₁(M) for M≥ 4
ModularCurve.ord_eq_three_of_ord_pos_and_ord_sub_eq_two_laurentBaseChange_gamma1294 below · cited by 6 · depth 23 - Ramification index of X₁(M) over the j-line in characteristic p
ModularCurve.ord_sub_algebraMap_eq_jWidthChar_of_place_x1FunctionFieldC628 below · cited by 2 · depth 23 - Ramification of X₁(M) over the j-line at finite j₀
ModularCurve.ord_sub_algebraMap_eq_jWidth_of_place_x1FunctionFieldC966 below · cited by 5 · depth 23 - Reduction kernel in TₚJ₁(M): corank and a Tₚ-polynomial
ModularCurve.pow_finrank_sub_finrank_reductionKernelSpan_tateModule_jOne_eq_card_torsion_and_exists_monic_aeval_mem1,844 below · cited by 1 · depth 23 - Hasse-radicand identity in the function field of X₁(M)_κ
ModularCurve.pow_twelve_mul_pow_sub_one_eq_of_coe_eq_hasseRootFn_pow85 below · cited by 2 · depth 23 - Pull-back to X₁(M) intertwines U_q for q ∣ M
ModularCurve.pullbackAlongHom_heckeOperatorHAlong_eq_heckeOperatorOneBar_pullbackAlongHom271 below · cited by 3 · depth 23 - Base change of the q-expansion function field along k ⊆ K
ModularCurve.qExpFunctionFieldC_eq_adjoin_image_coeffMap_qExpFunctionFieldC0 below · cited by 16 · depth 23 - Linear independence of powers of j over q-th power Laurent series
ModularCurve.qExpand_linearIndependent_pow2 below · cited by 5 · depth 23 - Stability of the q-expansion function field under q↦ q^q in characteristic q
ModularCurve.qExpand_mem_qExpFunctionFieldC_of_charP1 below · cited by 5 · depth 23 - Half-period shift of the Legendre λ-series
ModularCurve.qTwist_neg_one_lambdaModC_mul27 below · cited by 1 · depth 23 - Kernel of q-expansion reduction at p∤ M is Hecke-stable
ModularCurve.reductionQExpModL_gamma1_heckeAlgOne_smul_eq_zero992 below · cited by 2 · depth 23 - Degree p of κ(̄ j) over κ(̄ j^{ p})
ModularCurve.relfinrank_adjoin_jqModC_pow_adjoin_jqModC_eq1 below · cited by 3 · depth 23 - Reduction of a regular differential x dj on X₀(N) modulo p∤ N
ModularCurve.smul_D_jqModC_mem_regularDifferentials_residueField_of_smul_D_mem_regularDifferentialsBar774 below · cited by 1 · depth 23 - Eichler–Deuring mass formula for supersingular places at level N
ModularCurve.sum_inv_placeWidth_eq_eichlerMass_of_ssPlaces412 below · cited by 2 · depth 23 - Distribution relation for Tate's X-series under μ_ℓ
ModularCurve.sum_range_toricPoint_fst_sub_sum_Ico_eq_mul_toricPoint_pow_fst_add_C0 below · cited by 1 · depth 23 - Distribution relation for Tate's Y-series under μ_ℓ
ModularCurve.sum_range_toricPoint_snd_sub_sum_Ico_eq_mul_toricPoint_pow_snd_add0 below · cited by 1 · depth 23 - Galois and Hecke actions on T_q J₀(N) commute
ModularCurve.tateModule_rep_comp_tateHeckeRep_comm237 below · cited by 2 · depth 23 - Toric point at level ap is the qᵃ-expansion
ModularCurve.toricPoint_level_mul0 below · cited by 18 · depth 23 - Divisibility by 12 of ordₓ(̄ f¹²/Δ̄) at affine places
ModularCurve.twelve_dvd_ord_of_coe_eq_div_of_ord_nonneg_x1FunctionFieldC963 below · cited by 4 · depth 23 - Twelve divides ordₓ T-ordₓ J at cusps
ModularCurve.twelve_dvd_ord_sub_ord_of_coe_eq_div_of_ord_neg_x1FunctionFieldC963 below · cited by 1 · depth 23 - Fibre identity for j on X(Γ) over ℚ̄
ModularCurve.two_mul_genusFF_add_card_fibres_eq_finrank_add_two_of_gamma1_le269 below · cited by 3 · depth 23 - Vélu maps at toric points as μ_ℓ-orbit sums
ModularCurve.veluX_and_veluY_tateLaurent_toricPoint_eq_sum_range_sub_sum_Ico6 below · cited by 1 · depth 23 - q-expansion field of Γ_H(M)∩Γ₀(Mℓ) equals that of Γ_{H'}(Mℓ)
ModularCurve.xHTopFunctionFieldC_mul_eq_xHFunctionField_comap_unitsMap1 below · cited by 1 · depth 23 - The c₄ invariant of the formal Tate curve is E₄
ModularCurve.c4_tatePowerSeries0 below · cited by 3 · depth 24 - Characteristic-ℓ fibre counts over j=0,1728,∞ for Γ₁(M)
ModularCurve.card_fibres_jqModC_x1FunctionFieldC_le_natCard_doubleCoset_gamma1618 below · cited by 1 · depth 24 - Norm along the q↦ q^ℓ degeneracy as a product of twists
ModularCurve.coe_heckeBetaOneBar_norm_eq_prod_qTwist_of_finrankAlong_eq0 below · cited by 4 · depth 24 - Uniform p-adic normalisation of a rational Laurent q-expansion
ModularCurve.coeffEmb_smul_coeff_mem_and_not_mem_nonunits_of_le_padicValRat6 below · cited by 27 · depth 24 - Inertia acts trivially on a pinned constant reduction
ModularCurve.constantReduction_residue_arithmeticGalois_smul_eq189 below · cited by 1 · depth 24 - Places of the level-N modular function field are rational
ModularCurve.deg_eq_one_modularFunctionFieldC0 below · cited by 1 · depth 24 - Eta-product identity for E₄ in ℤ((q))
ModularCurve.eisenstein4_mul_etaProd_identity24 below · cited by 1 · depth 24 - Adapted basis of the p-ordinary corner of Tₚ J_H
ModularCurve.exists_adaptedLatticeBasis_inertiaEigenspace_cornerSubmodule_tateModule_jH_of_ordinary3,684 below · cited by 1 · depth 24 - An involution with μ ↦ 1/16-μ at level 4q
ModularCurve.exists_algEquiv_full_four_mul_lambdaModC_eq_sixteenth_sub185 below · cited by 1 · depth 24 - Involution of the level-4q function field scaling level-4 q-expansions
ModularCurve.exists_algEquiv_full_four_mul_restrict_eq_qExpand176 below · cited by 2 · depth 24 - Reduction mod P intertwines ⟨ d⟩ with a residual automorphism
ModularCurve.exists_algEquiv_reductionQExpModL_gamma1_diamondOneBar_eq_smul545 below · cited by 1 · depth 24 - Chart data from a unit family with polynomial residues
ModularCurve.exists_chartData_of_lineResidues57 below · cited by 4 · depth 24 - Reduced integral forms: poles of ̄ g¹²/Δ̄^k bounded by kord J
ModularCurve.exists_coe_eq_div_pow_and_mul_min_ord_le_ord_x1FunctionFieldC102 below · cited by 4 · depth 24 - Constant extension of the q-expansion function field model
ModularCurve.exists_curveModel_qExpFunctionFieldC_iso_pullback_germ_eq_coeffMap_of_ringHom74 below · cited by 1 · depth 24 - Fricke-twisted Weil pairing on the rational Tate module of J_H
ModularCurve.exists_diamondCyclotomicSimilitudePairing_rationalTateModule_jH364 below · cited by 4 · depth 24 - Coefficients of ℚ̄-modular functions lie in a number field
ModularCurve.exists_finiteDimensional_forall_coeff_mem0 below · cited by 2 · depth 24 - Bounded denominators for functions with poles only at ∞̄
ModularCurve.exists_forall_padicValRat_pow_mul_coeff_nonneg_of_forall_ord_nonneg125 below · cited by 2 · depth 24 - Multiplicity one mod 𝔪 for the ordinary multiplicative part
ModularCurve.exists_forall_sub_smul_mem_maximalIdeal_smul_multiplicativeSubmodule_tateModule_jH_of_ordinary_of_not_isEisenstein_of_mem_infSubgroup4,783 below · cited by 1 · depth 24 - Compatible Fricke involutions on X₁(M), X_H(M) and X_{H'}(M/p)
ModularCurve.exists_frickeAlgEquiv_triple_x1_xH_galois_smul_and_apply_inclusion_eq_and_forall_apply_degeneracy_eq83 below · cited by 1 · depth 24 - Fricke involution package for J₁(M) over ℚ̄
ModularCurve.exists_frickeAlgEquiv_x1FunctionFieldBar87 below · cited by 1 · depth 24 - Fricke involution on J_H(M) over ℚ̄
ModularCurve.exists_frickeAlgEquiv_xHFunctionFieldBar_galois_smul78 below · cited by 6 · depth 24 - Integral q-expansions of division-value forms and their toric values
ModularCurve.exists_gamma1_isIntegralQExp_fourier_and_toricPoint_eq_sum4 below · cited by 2 · depth 24 - Hecke endomorphism T_q of the relative Jacobian, with moduli description
ModularCurve.exists_heckeEndomorphism_relJacobian_moduli_of_ratCurveModel591 below · cited by 1 · depth 24 - Integral weight-k forms on Γ₁(M) with independent reductions
ModularCurve.exists_isIntegralQExp_linearIndependent_intSeriesC_gamma1_of_le542 below · cited by 3 · depth 24 - Existence of the p-adic content of a rational Laurent series
ModularCurve.exists_isLeast_padicValRat_coeff_of_mul_coeffEmb_coeff_mem6 below · cited by 26 · depth 24 - Global 1-forms of the ℤ₍ₚ₎-model versus p-integral cusp forms
ModularCurve.exists_linearEquiv_kaehlerH0_baseChange_intLattice_of_ratCurveModel_of_cuspSection_compat_of_neZero872 below · cited by 1 · depth 24 - Integral form ratios are quotients of shifted integral q-expansions
ModularCurve.exists_mem_qExpFunctionFieldC_single_mul_intSeriesC_mul_eq_of_mem_intFormRatiosC2 below · cited by 2 · depth 24 - Order homomorphism taking the Hasse root function to value coprime to p-1
ModularCurve.exists_monoidHom_units_x1FunctionFieldC_coprime_of_coe_eq_hasseRootFn_pow1,030 below · cited by 1 · depth 24 - Bounded denominators for the Fricke transform on X₀(p)
ModularCurve.exists_mul_coeff_frickeInvolutionBar_mem_of_mem_riemannRochSpace_cuspInftyBar184 below · cited by 26 · depth 24 - Degeneracy pair [[1,F^*],[F^*,⟨ d⟩]] is an ℓ-power-torsion isogeny
ModularCurve.exists_nsmul_eq_zero_and_exists_eq_frobeniusDegeneracyPair_torsion_qExpFunctionFieldC_of_ne1,430 below · cited by 1 · depth 24 - Nonvanishing integral forms at places above j=0 and 1728
ModularCurve.exists_ord_eq_zero_of_ord_pos_x1FunctionFieldC963 below · cited by 1 · depth 24 - Places descend along constant field extension of q-expansion fields
ModularCurve.exists_place_algebraicClosure_ord_comp_eq_of_laurentBaseChange47 below · cited by 3 · depth 24 - M-torsion of Tate(q^M) and the inertia transvection
ModularCurve.exists_point_tateBase_qTwist_eq_add_of_isPrimitiveRoot47 below · cited by 1 · depth 24 - Denominators of the minimal polynomial of λ(q^q) over ℚ(λ)
ModularCurve.exists_pow_mul_minpoly_lambdaNModC_coeff_mem_adjoin66 below · cited by 1 · depth 24 - Fricke-orthogonal diamond-fixed vectors are p-old modulo inertia coboundaries
ModularCurve.exists_pow_smul_mem_span_degeneracy_inertiaAugmentation_of_forall_weilPairing_fricke_eq_zero_diamondFixed_tateModule_jOne_of_dvd_of_not_sq_dvd3,305 below · cited by 1 · depth 24 - Clearing simple poles on the j-line gives a polynomial
ModularCurve.exists_prod_mul_eq_aeval_of_forall_ord_nonneg_of_forall_neg_one_le_ord58 below · cited by 8 · depth 24 - Clearing prescribed poles on the j-line by polynomials
ModularCurve.exists_prod_pow_mul_eq_aeval_of_forall_ord_nonneg_of_forall_neg_le_ord58 below · cited by 2 · depth 24 - Division-value functions on the Tate curve and diamond permutation
ModularCurve.exists_qExpFunctionFieldC_gammaH_bot_coe_eq_toricPoint_pow_and_diamondPullbackModL_apply_eq8 below · cited by 1 · depth 24 - Embedding basis of X₀(p) with rational q-expansions
ModularCurve.exists_ratFamily_isEmbBasis141 below · cited by 2 · depth 24 - Gauss prolongation to the q-expansion function field of level Γ
ModularCurve.exists_regularProlongation_laurentBaseChange_qExpFunctionFieldC_residue_div1 below · cited by 8 · depth 24 - Chart rings of a mathbf Z₍ₚ₎-model embed into ̄ F_N
ModularCurve.exists_ringHom_cover_modularFunctionFieldBar_of_ratCurveModel_of_neZero2 below · cited by 3 · depth 24 - Embedding the ℚ̄-rational q-expansion field of X(Γ) into ℂ(X(M))
ModularCurve.exists_ringHom_laurentBaseChange_qExpFunctionFieldC_levelN_qExpansion19 below · cited by 4 · depth 24 - Cusp ∞ extends to a section of a proper ℤ_{(q)}-model
ModularCurve.exists_schemeHomOver_placeOfPoint_eq_cuspInftyFull_of_isProper_of_ratCurveModel10 below · cited by 1 · depth 24 - Inertia differences on J_H reduce to node units
ModularCurve.exists_schemeHomOver_pts_smul_sub_eq_and_ptsSp_symm_mem_range_nodeUnit_of_mem_inertia_jHNeronObjectAtP2,785 below · cited by 4 · depth 24 - Bounded denominators with attained Gauss norm at ∞̄
ModularCurve.exists_smul_forall_coeff_mem_and_exists_not_mem_nonunits118 below · cited by 4 · depth 24 - Wide supersingular annuli of X₀(p) attached to both charts
ModularCurve.exists_ssAnnulus_oppAnnulus_isAttached_of_chartSpec_of_eq_zero_or_eq_ofNat1728_levelOne795 below · cited by 4 · depth 24 - Cusp-regular Gauss-integral functions as A-combinations of rational ones
ModularCurve.exists_sum_smul_coeffEmb_of_mem_integers_of_cuspRegular267 below · cited by 8 · depth 24 - Atkin–Lehner translate spans ℚ(ζₚ)-combinations of integral forms
ModularCurve.exists_sum_smul_eq_smul_atkinLehnerSlash_gamma1_mul78 below · cited by 4 · depth 24 - Frobenius at p acts as Uₚ⟨ d⟩ on ordinary corner
ModularCurve.exists_tateGaloisRep_frobenius_sub_U_mul_diamond_smul_eq_cyclotomicCharacter_smul_of_isFrobeniusAt_cornerSubmodule_tateModule_jH_of_ordinary3,500 below · cited by 1 · depth 24 - Surjectivity of the level maps of T_ℓ J₀(p)
ModularCurve.exists_tateModule_apply_eq_of_pow_smul_eq_zero414 below · cited by 1 · depth 24 - Uniform p-adic window for a finite family of modular functions
ModularCurve.exists_uniform_window_smul_mem_integers_of_qCoeff_criterion_of_ne_zero184 below · cited by 1 · depth 24 - Finiteness of the first degeneracy extension along α₁
ModularCurve.finiteAlong_heckeAlphaOneBar_of_neZero2 below · cited by 2 · depth 24 - Finiteness of the β₁ degeneracy extension at level Γ₁(N)
ModularCurve.finiteAlong_heckeBetaOneBar_of_heckeBetaOneDefined2 below · cited by 3 · depth 24 - Finitely many Γ-orbits of zeros and poles of F-t
ModularCurve.finite_image_orbitRel_meromorphicOrderAt_sub_ne_zero_of_finiteIndex0 below · cited by 1 · depth 24 - Degree of the base-changed X₀(N) q-expansion field over K(j)
ModularCurve.finrank_adjoin_jqModC_laurentBaseChange_qExpFunctionFieldC_gamma0_eq_index189 below · cited by 11 · depth 24 - Degree over K(j) of the Γ_H(M) q-expansion field equals [SL₂(ℤ):±Γ_H(M)]
ModularCurve.finrank_adjoin_jqModC_laurentBaseChange_qExpFunctionFieldC_gammaH_eq_index224 below · cited by 3 · depth 24 - Degree over k(t) does not drop under coefficient extension
ModularCurve.finrank_adjoin_simple_le_finrank_adjoin_simple_of_coe_eq_coeffMap2 below · cited by 1 · depth 24 - Odd order of the Hasse square at supersingular places, p=3
ModularCurve.gcd_two_natAbs_ord_eq_one_of_evalAt_mem_ssJSet_three_of_coe_eq_hasseRootFn_sq976 below · cited by 1 · depth 24 - Genus of the q-expansion function field under algebraically closed constant extension
ModularCurve.genusFF_qExpFunctionFieldC_eq_genusFF_qExpFunctionFieldC_of_isAlgClosed61 below · cited by 2 · depth 24 - Attachment of the opposite supersingular annulus, level 1· p
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec_levelOne702 below · cited by 3 · depth 24 - Attachment of the supersingular annulus at level 1· p
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec_levelOne702 below · cited by 3 · depth 24 - Compositum L· F₀ in L((q)) as Frac(L⊗_ℚF₀)
ModularCurve.isFractionRing_tensorProduct_laurentBaseChange0 below · cited by 3 · depth 24 - Wronskian closed form for the λ-line Kronecker remainder
ModularCurve.lambdaKroneckerRemainder_frobeniusGraph_ode167 below · cited by 1 · depth 24 - Level-Nt roof field generated by X₁(Nt) and q↦ q^t
ModularCurve.laurentBaseChange_x1FunctionField_sup_adjoin_qExpand_x1x0FunctionFieldC175 below · cited by 1 · depth 24 - Compositum of the function fields of X₁(Nt) and X(Γ₁(N)∩Γ₀(Nq))
ModularCurve.laurentBaseChange_x1FunctionField_sup_x1x0FunctionFieldC177 below · cited by 1 · depth 24 - Base-changed compositum for Γ₁(M) and Γ_H(M)∩Γ₀(Mq)
ModularCurve.laurentBaseChange_x1FunctionField_sup_xHTopFunctionFieldC177 below · cited by 2 · depth 24 - Base change preserves linear independence of Laurent series
ModularCurve.linearIndependent_coeffMap_algebraMap0 below · cited by 3 · depth 24 - Level-two modular relation splits into q+1 conjugates
ModularCurve.map_eq_phiProd_lambda_of_eval_qExpand_eq_zero0 below · cited by 2 · depth 24 - Cancelling powers of λ inside ℚ[λ]
ModularCurve.mem_adjoin_lambdaModC_of_pow_mul_mem0 below · cited by 1 · depth 24 - q-expansion principle at ∞: Gauss non-units have constant term in mathfrak m_A
ModularCurve.mem_maximalIdeal_apply_of_coe_mem_nonunits_gauss_of_mem_chartAlgInf_laurentBaseChange2 below · cited by 4 · depth 24 - Supersingularity of j from that of j^{p^e}
ModularCurve.mem_ssJSet_of_pow_mem_ssJSet0 below · cited by 6 · depth 24 - Integrality of j(q) over ℤ[j(q^ℓ)] at a place
ModularCurve.mem_toValuationSubring_of_coe_eq_jqModC_of_qExpand_mem80 below · cited by 3 · depth 24 - The q-expansion field of X₁(M) is closed under p-th roots
ModularCurve.mem_x1FunctionFieldC_of_pow_mem_x1FunctionFieldC220 below · cited by 1 · depth 24 - Integrality of the minimal polynomial of λ(q^q) over ℚ(λ)
ModularCurve.minpoly_lambdaNModC_coeff_coeff_eq_zero_of_neg180 below · cited by 1 · depth 24 - Ogg's modular unit lies in ℚ(j(q),j(q^N))
ModularCurve.modularUnitSeries_mem_modularFunctionField_all128 below · cited by 1 · depth 24 - Places above j=0,1728,∞ counted by double cosets
ModularCurve.natCard_fibres_jqModC_eq_natCard_doubleCoset_of_finrank_eq_index201 below · cited by 2 · depth 24 - p-torsion of J₁(M) in characteristic p counted by Tₚ
ModularCurve.natCard_torsion_jOneC_eq_pow_natDegree_sub_natTrailingDegree_of_map_eq_charpoly_heckeTLinOne974 below · cited by 1 · depth 24 - Inertia at p ∥ M commutes with a Fricke-type automorphism
ModularCurve.ofAlgAut_smul_galois_smul_eq_of_mem_inertiaSubgroupIn_of_frickeGaloisTwist4 below · cited by 4 · depth 24 - j = 1728 is supersingular in characteristic 11
ModularCurve.ofNat_1728_mem_ssJSet_eleven16 below · cited by 2 · depth 24 - The coordinate jmath̃ has order -1 at the place at infinity
ModularCurve.ord_charLGeomPlaceEquiv_placeInfty_jqModC11 below · cited by 11 · depth 24 - Order of P(̃ j) at a point of the j-line is the root multiplicity
ModularCurve.ord_charLGeomPlaceOfPoint_aeval_jqModC_eq_rootMultiplicity39 below · cited by 4 · depth 24 - No elliptic points on X₁(M), M ≥ 4, over ℚ̄
ModularCurve.ord_eq_three_of_ord_pos_and_ord_sub_eq_two_laurentBaseChange_gamma1_algebraicClosure292 below · cited by 1 · depth 24 - Segment period equals difference of a primitive
ModularCurve.periodAlongOf_apply_eq_sub_of_hasDerivAt0 below · cited by 3 · depth 24 - Supersingular places in characteristic 3 have width 1
ModularCurve.placeWidthChar_eq_one_of_mem_ssPlaces_of_eq_three_of_dvd401 below · cited by 3 · depth 24 - Width one at supersingular places in characteristic two
ModularCurve.placeWidthChar_eq_one_of_mem_ssPlaces_of_eq_two_of_dvd402 below · cited by 3 · depth 24 - Fricke-transported Hauptmodul relation between j(X⁴) and μ
ModularCurve.qExpand_four_jq_mul_one_sub_mul_lambdaModC_pow_four27 below · cited by 1 · depth 24 - Inertia at 2 cannot fix a Hecke eigenplane of J₀(p)
ModularCurve.rationalTateModule_false_of_inertia_fixed_eigenplane928 below · cited by 1 · depth 24 - Realisation of jmatĥ equals E₄³/Δ near every point
ModularCurve.realize_coeffEmb_jq_eventuallyEq19 below · cited by 1 · depth 24 - Hecke operators preserve the kernel of reduction at P
ModularCurve.reductionQExpModL_gamma1_heckeOperatorOneBar_eq_zero_of_ne964 below · cited by 1 · depth 24 - Fricke laws on the Tate module of J₁(M)
ModularCurve.rep_tateModule_jOne_frickeAlgEquiv_transpose_diamond_galois_inertia_of_laws1 below · cited by 1 · depth 24 - Hecke adjunction for the integral Serre pairing, sectional charts
ModularCurve.serrePairingInt_deformationClass_heckeGen_eq_of_isCompletionAlong_of_res_eq_heckeDiffBar365 below · cited by 1 · depth 24 - Characteristic-3 supersingular places of X₀(M'): 6|W|=ψ(M')
ModularCurve.six_mul_card_eq_dedekindPsi_of_ssPlaces_of_eq_three_of_dvd358 below · cited by 2 · depth 24 - Order of the Hasse radicand at supersingular places is ≡ 1 mod (p-1)
ModularCurve.sub_one_dvd_ord_sub_one_of_coe_eq_hasseRootFn_pow_of_eval_eq_zero970 below · cited by 1 · depth 24 - Surjectivity of reduction on ℓ^k-torsion of J_H(M)
ModularCurve.surjOn_reductionQExpModL_gammaH_torsion_pow1,728 below · cited by 1 · depth 24 - Inertia acts by the cyclotomic character on an ordinary corner
ModularCurve.tateGaloisRep_smul_sub_eq_cyclotomicCharacter_smul_of_mem_inertiaSubgroupIn_cornerSubmodule_tateModule_jH_of_ordinary3,405 below · cited by 5 · depth 24 - Divisibility of kcdotordₓ J by 3 and of kcdotordₓ(J-1728) by 2
ModularCurve.three_dvd_mul_ord_and_two_dvd_mul_ord_sub_of_ord_eq_zero_x1FunctionFieldC2 below · cited by 1 · depth 24 - Characteristic 2 supersingular places: 12 |W| = ψ(M')
ModularCurve.twelve_mul_card_eq_dedekindPsi_of_ssPlaces_of_eq_two_of_dvd358 below · cited by 1 · depth 24 - Genus–cusp–elliptic-point count for X₁(M) over κ
ModularCurve.two_mul_genusFF_x1FunctionFieldC_add_natCard_doubleCoset_eq_index_add_two873 below · cited by 4 · depth 24 - ℓ-power torsion of Pic⁰ of the q-expansion function field
ModularCurve.abelJacobiCard_genusFF_qExpFunctionFieldC949 below · cited by 1 · depth 25 - Diamond automorphisms with d≡ 1 (mod M) fix the floor field
ModularCurve.algEquiv_apply_eq_self_of_coe_mem_laurentBaseChange_x1x0FunctionFieldC_of_coe_eq_baseChangeAut_diamondAut31 below · cited by 2 · depth 25 - Width-two and width-three places counted by ν₂ and ν₃
ModularCurve.card_eq_nuTwo_and_card_eq_nuThree_of_forall_mem_iff_placeWidth_eq395 below · cited by 1 · depth 25 - Hecke correspondence on differentials matches the Hecke operator on q-expansions
ModularCurve.coeffMap_diffQExpBar_heckeDiffBar_eq_qExpansion_latticeRestrictHom_heckeProj_heckeGen162 below · cited by 1 · depth 25 - Inertia acts trivially on residues of the constant reduction (q=3)
ModularCurve.constantReduction_residue_arithmeticGalois_smul_eq_of_eq_three189 below · cited by 1 · depth 25 - Inertia preserves residues of a constant reduction (q=2)
ModularCurve.constantReduction_residue_arithmeticGalois_smul_eq_of_eq_two189 below · cited by 1 · depth 25 - The cusp at infinity of the full modular function field has degree one
ModularCurve.deg_cuspInftyFull5 below · cited by 1 · depth 25 - H_q(16λ)² (θλ)^{q-1}=(λ(1-16λ))^{q-1} in characteristic q
ModularCurve.deuringPolynomial_sq_mul_thetaL_lambda_pow163 below · cited by 1 · depth 25 - Diamond tokens are multiplicative and trivial on H'
ModularCurve.diamondActionModL_gammaLift_mul_and_eq_one_of_mem_and_ofAlgAut_smul4 below · cited by 10 · depth 25 - Diamond operators ⟨ d⟩ with d∈ H act trivially on J_H
ModularCurve.diamondHBar_apply_eq_self_of_mem0 below · cited by 6 · depth 25 - Frobenius on the multiplicative part of an ordinary factor
ModularCurve.exists_U_mul_diamond_smul_tateGaloisRep_frobenius_eq_cyclotomicCharacter_smul_of_forall_inertia_cornerSubmodule_tateModule_jH_of_ordinary3,453 below · cited by 1 · depth 25 - Good-reduction abelian-scheme model of J_H(M) at ℓ∤ M
ModularCurve.exists_abelianSchemePropertyBundle_model_jH1,690 below · cited by 1 · depth 25 - Base change to ℚ̄ of the two chart rings
ModularCurve.exists_algEquiv_tensor_chartAlgFin_chartRing_and_chartAlgInf_laurentBaseChange_twoChartModel_of_coe_eq_coeffEmb1 below · cited by 1 · depth 25 - Degeneracy inclusion of ℚ̄-function fields of X_H
ModularCurve.exists_algHom_xHFunctionFieldBar_div_infSubgroup_isIntegral_and_coe_eq4 below · cited by 2 · depth 25 - Functions on the j-line with one pole and prescribed values
ModularCurve.exists_eq_algebraMap_add_prod_mul_aeval_of_forall_ord_nonneg_of_hasValue52 below · cited by 3 · depth 25 - Unitary multiplier and nonzero cusp limits force constancy
ModularCurve.exists_eq_const_of_norm_multiplier_eq_one_of_finiteIndex0 below · cited by 1 · depth 25 - Degeneracy roof at the generic fibre: function-field Hecke correspondence
ModularCurve.exists_functionField_degeneracyRoof_kaehlerToFunctionField_eq_correspondence_of_res_eq_heckeDiffBar208 below · cited by 1 · depth 25 - Partial Atkin–Lehner automorphism w_Q for squarefree Q coprime to N
ModularCurve.exists_isAtkinLehnerAutFull_of_squarefree_of_coprime74 below · cited by 1 · depth 25 - Exponential decay at cusps of cusp-regular differentials
ModularCurve.exists_isBigO_slash_realizeOf_mul_deriv_realizeOf_of_forall_ordDifferential_nonneg15 below · cited by 1 · depth 25 - Integral independent family on Γ₁(M) of dimension-formula size
ModularCurve.exists_isIntegralQExp_linearIndependent_complex_gamma1_dimFormula_le_card540 below · cited by 1 · depth 25 - Rank of an integral family survives reduction on Γ₁(M)
ModularCurve.exists_isIntegralQExp_linearIndependent_intSeriesC_of_linearIndependent0 below · cited by 1 · depth 25 - Integral weight-two cusp forms as relative differentials on the model
ModularCurve.exists_kaehlerH0_coeffMap_diffQExpBar_eq_qExpansion_of_mem_intLattice_of_ratCurveModel_of_cuspSection_compat_of_neZero456 below · cited by 1 · depth 25 - Integral weight-two cusp forms and regular differentials on X₁(M)_k
ModularCurve.exists_linearEquiv_tensor_regularDifferentials_x1FunctionFieldC_qExpansionDiffAlong_eq_and_injective943 below · cited by 1 · depth 25 - Ordinary multiplicative submodule dual to mod-p two-cusp eigenspace
ModularCurve.exists_linearMap_bijOn_semilinearMaps_multiplicativeSubmodule_tateModule_jH_twoCuspEigenspace_of_ordinary_of_mem_infSubgroup4,749 below · cited by 1 · depth 25 - Functions on X₀(N) as quotients with poles only at ∞̄
ModularCurve.exists_mem_riemannRochSpace_mul_eq_of_ne_zero168 below · cited by 1 · depth 25 - Atkin–Lehner translate at p lies in M_k(Γ₁(Mp))
ModularCurve.exists_modularForm_coe_eq_atkinLehnerSlash_gamma1_mul0 below · cited by 7 · depth 25 - Weil bound for Frobenius on prime-to-p torsion of Pic⁰
ModularCurve.exists_monic_aeval_qExpFrobeniusPushforwardModL_torsion_eq_zero_norm_root_eq_sqrt1,209 below · cited by 1 · depth 25 - p-divisible finite part of the Néron object for J_H(M)
ModularCurve.exists_pDivisibleGroup_points_eq_finPts_raynaudExtension_closedImmersion_jHNeronObjectAtP_of_representsRelSubPic3,046 below · cited by 5 · depth 25 - Affine supersingular place where a weight-one form is non-zero
ModularCurve.exists_place_x1FunctionFieldC_ord_aeval_pos_of_integralWeightOneForm967 below · cited by 1 · depth 25 - Integrality of q-expansions of global 1-forms on a ℤ₍ₚ₎-model
ModularCurve.exists_powerSeries_diffQExpBar_eq_ofPowerSeries_map_of_kaehlerH0_of_ratCurveModel_of_cuspSection_compat_of_neZero349 below · cited by 1 · depth 25 - Substitution q↦ q^N preserves integrality and vanishing
ModularCurve.exists_qExpand_ofPowerSeries_map_eq_ofPowerSeries_map0 below · cited by 1 · depth 25 - Width-one supersingular annulus on X₀(q), at level q
ModularCurve.exists_ssAnnulus_centred_of_widthOne_level806 below · cited by 2 · depth 25 - Wide supersingular annuli of X₀(p) attached to both charts
ModularCurve.exists_ssAnnulus_oppAnnulus_isAttached_of_chartSpec_of_eq_zero_or_eq_ofNat1728796 below · cited by 2 · depth 25 - Uniform p-power window for scaling a finite family in L(n∞̄)
ModularCurve.exists_uniform_window_smul_mem_integers_of_qCoeff_criterion_of_mem_riemannRochSpace123 below · cited by 1 · depth 25 - Finiteness of the δ'(F^⋆)²-fixed locus on Pic⁰
ModularCurve.finite_setOf_diamondInv_frobeniusInvSmul_sq_eq_self1,242 below · cited by 1 · depth 25 - Finiteness of the fixed locus of ⟨ ̄ p⟩ ∘ F²
ModularCurve.finite_setOf_diamond_qExpFrobeniusPushforwardModL_sq_eq_self363 below · cited by 1 · depth 25 - Reduction at p ∤ Mℓ of the T_ℓ degeneracy maps
ModularCurve.finrankAlong_inclusion_qExpFunctionFieldC_residueField_eq_of_not_dvd296 below · cited by 1 · depth 25 - Igusa degree equality for Γ₁(L)≤Γ≤Γ₀(L)
ModularCurve.finrank_adjoin_jqModC_qExpFunctionFieldC_eq_index_of_gamma1_le_of_le_gamma0551 below · cited by 11 · depth 25 - Supersingularity of a depends only on kerφ
ModularCurve.forall_apply_mem_ssJSet_of_ker_eq_of_apply_mem_ssJSet26 below · cited by 14 · depth 25 - Pivot families force integral coordinates under constant reduction
ModularCurve.forall_mem_valuationSubring_of_sum_mul_coeffMap_mem_integers_of_pivot0 below · cited by 1 · depth 25 - Crossing parameter at supersingular j∈{0,1728}: Gauss unit, order one
ModularCurve.gaussUnit_frickeInvolutionBar_and_ord_eq_one_of_crossingPresentation_of_eq_zero_or_eq_ofNat1728282 below · cited by 2 · depth 25 - Fricke automorphism transposes Hecke operators and inverts diamonds
ModularCurve.heckePic0HBarTranspose_smul_diamondHBar_smul_smul_of_qExpansion_slash_fricke78 below · cited by 1 · depth 25 - Attachment of the opposite supersingular annulus to the ∞-chart
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec701 below · cited by 1 · depth 25 - Attachment of the opposite supersingular annulus, level 1· p
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec_levelOne_univ702 below · cited by 2 · depth 25 - Attachment of the wide supersingular annulus to a component chart
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec_of_paramGauss_of_eq_zero_or_eq_ofNat1728_levelOne757 below · cited by 3 · depth 25 - Attachment of the supersingular annulus at the zero chart
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec701 below · cited by 1 · depth 25 - Supersingular annulus attached to the zero chart, level 1· p
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec_levelOne_univ702 below · cited by 2 · depth 25 - Attachment of the wide supersingular annulus to the zero chart
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec_of_paramGauss_of_eq_zero_or_eq_ofNat1728_levelOne757 below · cited by 3 · depth 25 - Finiteness and flatness of [ℓ^k] on an abelian ℤ_{(ℓ)}-scheme
ModularCurve.isFinite_and_flat_schemeNsmul_pow_of_jHC_points141 below · cited by 2 · depth 25 - Integrality of g¹²/Δ^k over ℤ[1/M][j] and of g¹²/E₄^{3k} over ℤ[1/M][j⁻¹]
ModularCurve.isIntegralElem_div_delta_pow_and_div_eisenstein4_pow_of_isIntegralQExp_gamma1101 below · cited by 1 · depth 25 - Place-wise order of f₁⁻⁽ᵖ⁻¹⁾ via the supersingular polynomial
ModularCurve.jWidth_mul_ord_eq_ord_aeval_of_coe_eq_hasseRootFn_pow85 below · cited by 1 · depth 25 - Global differentials injected into Ω_̄ F_N/ℚ̄
ModularCurve.kaehlerH0_res_injective_of_injective_chartMap_of_neZero1 below · cited by 1 · depth 25 - Kronecker remainder along the Frobenius graph for λ
ModularCurve.laurentMap_evalAtLambdaInt_kroneckerRemainder_eval_X_pow0 below · cited by 1 · depth 25 - Vanishing of the Fricke-transformed reduction on the 0-line
ModularCurve.modularRedLocHom_frickeInvolutionBar_eq_zero_of_hasValue_zero_of_forall_ord_nonneg553 below · cited by 1 · depth 25 - p-torsion of J₁(M) counted by Cartier-fixed differentials
ModularCurve.natCard_torsion_jOneC_eq_natCard_regularDifferentials_x1FunctionFieldC_coeff_mul_eq_pow174 below · cited by 1 · depth 25 - Petersson functional in the period lattice of Γ_H(M)
ModularCurve.petersson_mem_periodLatticeOf_iff_re_periodOf_int_gammaH187 below · cited by 1 · depth 25 - Diamond automorphisms commute with Frobenius on places
ModularCurve.qExpFrobeniusPlaceModL_ofAlgAut_diamondActionModL_smul260 below · cited by 38 · depth 25 - Frobenius pushforward commutes with diamond operators on Pic⁰
ModularCurve.qExpFrobeniusPushforwardModL_ofAlgAut_diamondActionModL_smul260 below · cited by 4 · depth 25 - Formal q-expansion identity for 16E₄(q⁴) and Euler products
ModularCurve.qExpand_four_eisenstein4_mul_etaProd_identity24 below · cited by 1 · depth 25 - Coefficients of (f∣_kγ)(pτ) lie in ℚ(ζₚ)
ModularCurve.qExpansion_coeff_atkinLehnerSlash_mem_adjoin_exp_gamma1_mul41 below · cited by 2 · depth 25 - Inertia twist q ↦ ζ q on non-toric Tate points
ModularCurve.qTwist_nonToricPoint_of_pow_eq_one2 below · cited by 1 · depth 25 - Cyclic degree p-1 Galois extension of modular function fields
ModularCurve.relfinrank_eq_sub_one_and_isGalois_and_isCyclic_x1FunctionField_mul_x1x0235 below · cited by 5 · depth 25 - Global 1-forms restrict to regular differentials on ℚ̄(X₀(N))
ModularCurve.res_mem_regularDifferentialsBar_of_chartMap_of_neZero170 below · cited by 1 · depth 25 - Reduction of a cusp-regular integral function stays regular
ModularCurve.residue_mem_valuationSubring_of_cuspRegular_of_residue_jq_mem255 below · cited by 1 · depth 25 - Slope law of the opposite supersingular annulus, level 1· p
ModularCurve.slopeLaw_oppAnnulus_inftyChart_of_chartSpec_levelOne700 below · cited by 2 · depth 25 - Slope law on the zero chart, level 1· p
ModularCurve.slopeLaw_ssAnnulus_zeroChart_of_chartSpec_levelOne700 below · cited by 2 · depth 25 - A θ-identity for the Dwork quotient of λ
ModularCurve.thetaL_laurentMap_lambdaDworkQuotient1 below · cited by 1 · depth 25 - Additivity of the Tate parametrisation at parameters c q^j
ModularCurve.toricPoint_add_nonToricPoint_of_charZero45 below · cited by 10 · depth 25 - Discriminant of the formal Tate curve is q η²⁴
ModularCurve.Delta_tatePowerSeries4 below · cited by 2 · depth 26 - Atkin–Lehner map inverts Ogg's unit up to p¹²
ModularCurve.algEquiv_apply_eq_pow_twelve_mul_inv_of_coe_eq_coeffEmb_modularUnitSeries_of_qExpand_of_arithmeticGalois_comm340 below · cited by 5 · depth 26 - Atkin–Lehner slash at p conjugates diamonds on Γ₁(Mp)
ModularCurve.atkinLehnerSlashFun_slash_eq_slash_atkinLehnerSlashFun_of_upperLeft_gamma1_mul0 below · cited by 1 · depth 26 - Coefficientwise compatibility of base-changed q-expansion automorphisms
ModularCurve.coe_baseChangeAut_eq_coeffMap_coe_baseChangeAut_of_coe_eq_coeffMap0 below · cited by 1 · depth 26 - The q-adic place of a rational Laurent subfield has degree 1
ModularCurve.deg_qInftyPlaceRat4 below · cited by 1 · depth 26 - Level-two Deuring polynomial identity in characteristic q
ModularCurve.delta_pow_mul_deuringPolynomial_lambda_pow_twelve84 below · cited by 1 · depth 26 - Diamond automorphisms of ℚ(X₁(N)) form a (ℤ/N)^×-action
ModularCurve.diamondAut_congr_and_mul_and_one_and_inv_and_diamondAutBar31 below · cited by 1 · depth 26 - Points dictionary for the finite part at p of J_H(M)
ModularCurve.exists_addMonoidHom_points_finitePart_eq_finPts_jHNeronObjectAtP_of_closedImmersion4 below · cited by 2 · depth 26 - Smooth chart algebras commute with base change ℤ₍ₚ₎→ A
ModularCurve.exists_algEquiv_tensorProduct_chartAlg_laurentBaseChange_of_smooth4 below · cited by 1 · depth 26 - Galois conjugation of Atkin–Lehner expansions over an abstract cyclotomic field
ModularCurve.exists_apply_eq_qExpansion_coeff_atkinLehnerSlash_and_slash_mul_eq_apply_aut_gamma1_mul26 below · cited by 1 · depth 26 - Ordinary idempotent on a p-divisible subgroup of J_H
ModularCurve.exists_bialgHom_family_idempotent_inverse_U_of_cornerIdempotent_tateModule_jH293 below · cited by 1 · depth 26 - Gauss-integral witnesses may be taken modular on X_H(M)
ModularCurve.exists_coeffMap_mem_xHFunctionFieldBar_mul_eq_of_mul_coeffMap_eq4 below · cited by 7 · depth 26 - Riesz representation for the weight-2 Petersson pairing on Γ_H(M)
ModularCurve.exists_cuspForm_petersson_eq_gammaH6 below · cited by 1 · depth 26 - p-saturation of global differentials via q-expansions
ModularCurve.exists_eq_smul_of_diffQExpBar_eq_ofPowerSeries_smul_of_kaehlerH0_of_ratCurveModel_of_cuspSection_compat_of_neZero366 below · cited by 1 · depth 26 - Fibres of the two-chart model of X_Γ at p∤ M
ModularCurve.exists_fibreCurveModels_twoChartIntegralModel_qExpFunctionFieldC_isPlaceReductionQExpModL_of_not_dvd911 below · cited by 1 · depth 26 - Degeneracy roof at q over the generic fibre
ModularCurve.exists_functionField_degeneracyRoof_lift_of_ratCurveModel4 below · cited by 1 · depth 26 - Ogg's unit Δ(q)/Δ(qᵖ): integrality and level Γ_H(M)
ModularCurve.exists_int_coeffMap_eq_modularUnitSeries_and_mem_qExpFunctionFieldC_gammaH_of_dvd39 below · cited by 8 · depth 26 - Finite order of a diamond operator on Pic⁰ mod p
ModularCurve.exists_iterate_diamond_eq_self_pic0_fbar4 below · cited by 2 · depth 26 - Generic restriction of a global 1-form factors through the cusp stalk
ModularCurve.exists_kaehlerDifferential_stalk_and_ringHom_res_eq_mapOfRingHom_cuspSection_of_ratCurveModel_compat_of_neZero2 below · cited by 2 · depth 26 - Weight-two cusp forms and periods are insensitive to -1
ModularCurve.exists_linearEquiv_cuspForm_sup_zpowers_neg_one_dualMap_periodOf1 below · cited by 2 · depth 26 - Mod-p two-cusp forms dual to the multiplicative part
ModularCurve.exists_linearMap_injective_range_eq_dual_multiplicativeSubmodule_tateModule_jH_twoCuspForms_of_ordinary_of_mem_infSubgroup4,747 below · cited by 1 · depth 26 - Tate module comparison for the finite part of J_H
ModularCurve.exists_linearMap_tateModule_points_finitePart_injective_range_galois_jHNeronObjectAtP0 below · cited by 1 · depth 26 - Integral weight-two cusp forms as regular differentials
ModularCurve.exists_mem_regularDifferentials_qExpFunctionFieldC_qExpansionDiffAlong_eq_of_forall_qCoeff_eq_intCast940 below · cited by 2 · depth 26 - Base change of ℚ-automorphisms to L-automorphisms of L· F₀
ModularCurve.exists_monoidHom_algEquiv_laurentBaseChange_apply_coeffEmb1 below · cited by 2 · depth 26 - Finite faithful action on the X₁(M)∩ X₀(p) function field with fixed field that of X₀(Mp)
ModularCurve.exists_mulSemiringAction_faithful_fixed_iff_x1x0FunctionFieldC_gamma0240 below · cited by 3 · depth 26 - Simple zero of p/(jₚ-jᵖ) at a supersingular value
ModularCurve.exists_natCast_mul_inv_nodeCoord_mem_integers_ord_residue_eq_one_inftyChart196 below · cited by 1 · depth 26 - Simple zero of jₚ - jᵖ on the zero-cusp chart
ModularCurve.exists_nodeCoord_mem_integers_ord_residue_eq_one_zeroChart0 below · cited by 1 · depth 26 - A p-divisible group inside the Néron object of J_H(M)
ModularCurve.exists_pDivisibleGroup_closedImmersion_finitePart_jHNeronObjectAtP_of_representsRelSubPic2,270 below · cited by 2 · depth 26 - Raynaud quotient of the finite part by its toric subgroup
ModularCurve.exists_pDivisibleGroup_raynaudQuotient_toricPts_finitePart_jHNeronObjectAtP_of_closedImmersion2,394 below · cited by 1 · depth 26 - Level-one weight 12m forms: q-expansion equals P(j)Δ^m
ModularCurve.exists_polynomial_ofPowerSeries_qExpansion_eq_aeval_jqModC_mul_of_levelOne2 below · cited by 2 · depth 26 - p-power multiple of an integral weight-2 cusp form as a differential
ModularCurve.exists_pow_smul_kaehlerH0_coeffMap_diffQExpBar_eq_qExpansion_of_mem_intLattice_of_ratCurveModel_of_cuspSection_compat_of_neZero322 below · cited by 1 · depth 26 - Integral q-expansions of germs at the cusp of a ℤ₍ₚ₎-model
ModularCurve.exists_powerSeries_map_eq_ffEquiv_symm_stalkMap_stalkSpecializes_cuspSection_of_ratCurveModel_compat_of_neZero346 below · cited by 2 · depth 26 - Function field of the geometric generic fibre is ℚ̄F_N
ModularCurve.exists_ringEquiv_functionField_pullback_comp_baseToFunctionField_eq_and_germToFunctionField_eq_chartMap_of_neZero2 below · cited by 2 · depth 26 - Hecke algebra of TₚJ_H acting on reduced two-cusp forms
ModularCurve.exists_ringHom_moduleEnd_twoCuspForms_apply_eq_twoCuspGenMod529 below · cited by 2 · depth 26 - Gauss normalisation of q-expansions with poles among those of j
ModularCurve.exists_smul_forall_coeff_mem_and_exists_not_mem_nonunits_of_forall_ord_neg118 below · cited by 1 · depth 26 - Two-step special-fibre tower of the Raynaud quotient with descended Uₚ
ModularCurve.exists_twoStepTower_raynaudQuotient_descent_finPts_jHNeronObjectAtP_of_finPtsWitness_of_bridgePins2,702 below · cited by 1 · depth 26 - Finiteness of Frobenius-fixed divisor classes on ̄ F in characteristic p
ModularCurve.finite_fixedPoints_iterate_qExpFrobeniusPushforwardModL147 below · cited by 3 · depth 26 - Degree p+1 along the q-expansion-preserving map of X_H function fields
ModularCurve.finrankAlong_eq_add_one_of_coe_eq_xHFunctionFieldBar246 below · cited by 9 · depth 26 - Degree of jmath̄(q^{Mℓ}) over the Igusa function field
ModularCurve.finrank_adjoin_jqNModC_igusaFunctionFieldX1C_eq1,069 below · cited by 1 · depth 26 - Gauss's lemma for the j-expansion and its order
ModularCurve.forall_coeff_mem_of_forall_coeff_aeval_jqModC_mem0 below · cited by 2 · depth 26 - Inertia displacements reduce to one on the Raynaud quotient
ModularCurve.forall_raynaudQuotient_point_reducesToOne_of_eq_smul_sub_of_mem_inertia_finitePart_jHNeronObjectAtP2,791 below · cited by 1 · depth 26 - Residue package for the two legs of the T_q degeneracy roof
ModularCurve.functionField_residuePackage_degeneracyRoof_of_finiteAlong85 below · cited by 1 · depth 26 - Deligne–Rapoport genus identity for X_H(M) when p ‖ M
ModularCurve.genusFF_xHFunctionFieldBar_add_one_eq_two_mul_genusFF_residueField_add_natCard_ssNodePairsQExp1,405 below · cited by 10 · depth 26 - Reduced modular unit vanishes at supersingular places
ModularCurve.hasValue_zero_of_mem_ssPlacesQExp_of_coe_eq_coeffMap_modularUnitSeries263 below · cited by 3 · depth 26 - Attaching a wide supersingular annulus to the ∞-chart
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec_of_paramGauss_of_eq_zero_or_eq_ofNat1728_levelOne_univ757 below · cited by 1 · depth 26 - Attachment of the opposite supersingular annulus to the ∞-chart
ModularCurve.isAttached_oppAnnulus_inftyChart_of_chartSpec_univ701 below · cited by 1 · depth 26 - Attachment of a wide supersingular annulus to the zero chart
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec_of_paramGauss_of_eq_zero_or_eq_ofNat1728_levelOne_univ757 below · cited by 1 · depth 26 - Attachment of the supersingular annulus to the zero chart
ModularCurve.isAttached_ssAnnulus_zeroChart_of_chartSpec_univ701 below · cited by 1 · depth 26 - Geometric generic fibre of a ℤ₍ₚ₎-model of X₀(N) is integral
ModularCurve.isIntegral_pullback_and_nonempty_of_chartMap_of_neZero7 below · cited by 2 · depth 26 - Torsion of Pic⁰ over an algebraic closure of mathbb Fₚ
ModularCurve.isOfFinAddOrder_pic0_fbar_of_forall_pow_eq_self277 below · cited by 1 · depth 26 - Generic-fibre degeneracy roof for Hecke action on differentials
ModularCurve.kaehlerToFunctionField_eq_correspondence_degeneracyRoof_of_res_eq_heckeDiffBar161 below · cited by 1 · depth 26 - Kronecker's congruence read in the Gauss valuation ring
ModularCurve.map_j_sub_pow_mem_nonunits_gauss_of_coe_map_eq_qExpand6 below · cited by 3 · depth 26 - Width-N q-expansion as q ↦ q^N substitution
ModularCurve.ofPowerSeries_qExpansion_natCast_eq_qExpand_of_one_mem_strictPeriods0 below · cited by 7 · depth 26 - Reduction of Δ(q)/Δ(qᵖ) is a unit at ordinary affine places
ModularCurve.ord_eq_zero_of_not_mem_ssPlacesQExp_of_hasValue_of_coe_eq_coeffMap_modularUnitSeries262 below · cited by 3 · depth 26 - Orders under the q↦ qᵖ Frobenius on places
ModularCurve.ord_qExpFrobeniusPlaceModL_eq_ord_of_qExpFrobeniusModL_eq_pow3 below · cited by 1 · depth 26 - Petersson functional lies in the period lattice iff periods have integral real part
ModularCurve.petersson_mem_periodLatticeOf_iff_re_periodOf_int184 below · cited by 2 · depth 26 - Uniqueness of a reduction map on places of X_H(M)
ModularCurve.placeReduction_unique_of_forall_mapDomain_eq_ord_gammaH_of_not_dvd955 below · cited by 2 · depth 26 - Coefficients of (f|_kγ)(pτ) lie in ℚ(ζₚ), k even
ModularCurve.qExpansion_coeff_atkinLehnerSlash_mem_adjoin_exp_gamma1_mul_of_even23 below · cited by 3 · depth 26 - Odd weight: ℚ(ζₚ)-rationality of f∣_kγ at pτ
ModularCurve.qExpansion_coeff_atkinLehnerSlash_mem_adjoin_exp_gamma1_mul_of_odd40 below · cited by 1 · depth 26 - Saturated Tate image with free cokernel of toric rank
ModularCurve.range_saturated_and_nonempty_coker_linearEquiv_tateModule_points_finitePart_jHNeronObjectAtP11 below · cited by 1 · depth 26 - Degree p-1 of the diamond cover X₁(Mp)→ X_{Γ_1(M)∩Γ_0(p)}
ModularCurve.relfinrank_eq_sub_one_x1FunctionField_mul_x1x0232 below · cited by 2 · depth 26 - Slope law for the opposite supersingular annulus
ModularCurve.slopeLaw_oppAnnulus_inftyChart_of_chartSpec699 below · cited by 2 · depth 26 - Slope law on the supersingular annulus, 0-component chart
ModularCurve.slopeLaw_ssAnnulus_zeroChart_of_chartSpec699 below · cited by 2 · depth 26 - Supersingular j-invariants base-change along algebraically closed extensions
ModularCurve.ssJSet_eq_image_algebraMap_of_isAlgClosed19 below · cited by 18 · depth 26 - Tate ordinate at u = c t^j equals the non-toric slot point
ModularCurve.tateCurve_pointY_C_mul_X_pow_eq_nonToricPoint_snd2 below · cited by 1 · depth 26 - θ(jmath̄) Δ = -E₄²E₆ over any field
ModularCurve.thetaL_jqModC_mul_intSeriesC_X_mul_dedekindEtaUnit81 below · cited by 2 · depth 26 - Jacobi's discriminant formula for the level-two parameter μ
ModularCurve.thetaL_lambdaModC_pow_six106 below · cited by 1 · depth 26 - Mod ℓ collapse of Γ₀(ℓ^r)-level q-expansion function fields
ModularCurve.xHTopFunctionFieldC_residueField_mul_pow_eq_xHFunctionFieldC_of_not_dvd293 below · cited by 1 · depth 26 - The q-adic place of a subfield of ℚ((q)) has degree 1
ModularCurve.deg_of_toValuationSubring_eq_qIntegersBar3 below · cited by 1 · depth 27 - Diamond operators fix the mod p reduction of Δ(q)/Δ(qᵖ)
ModularCurve.diamondActionModL_apply_eq_self_of_coe_eq_coeffMap_modularUnitSeries19 below · cited by 1 · depth 27 - Diamonds preserve supersingular places; Frobenius squares to ⟨ e⟩
ModularCurve.diamondActionModL_smul_mem_ssPlacesQExp_iff_and_qExpFrobeniusPlaceModL_qExpFrobeniusPlaceModL_eq_smul1,237 below · cited by 12 · depth 27 - Diamond automorphisms fix the base-changed function field of X₀(N)
ModularCurve.diamondAutBar_apply_coeffEmb_modularFunctionFieldFull_eq216 below · cited by 1 · depth 27 - Atkin–Lehner automorphism acting by q ↦ q^t
ModularCurve.exists_algEquiv_laurentBaseChange_xHTopFunctionFieldC_coe_eq_qExpand_of_coprime30 below · cited by 1 · depth 27 - Integral q-parameter at the cusp of a ℤ₍ₚ₎-model
ModularCurve.exists_algHom_retraction_param_stalk_cuspSection_ffEquiv_symm_eq_ofPowerSeries_isUnit_coeff_one_of_ratCurveModel_compat_of_neZero342 below · cited by 2 · depth 27 - Diamond operator ⟨ d⟩ as an automorphism of the finite part
ModularCurve.exists_bialgEquiv_family_diamond_finPts_jHNeronObjectAtP_of_finPtsWitness79 below · cited by 1 · depth 27 - Pinned idempotent pieces of B mod p identified with A
ModularCurve.exists_bialgHom_baseChange_levelTorsion_raynaudQuotient_image_eq_idempotent_finPts_jHNeronObjectAtP_of_finPtsWitness2,599 below · cited by 1 · depth 27 - Residues of the Gauss prolongation as ratios of chart residues
ModularCurve.exists_chartAlg_residue_mul_eq_residue_of_coe_eq_coeffMap_of_residue_surjective10 below · cited by 3 · depth 27 - Function-field witnesses for integral quotients in ℚ̄((q))
ModularCurve.exists_coeffMap_mem_xHFunctionFieldBar_mul_eq_of_mul_coeffMap_eq_all3 below · cited by 2 · depth 27 - Automorphism fixing the X₀(Mp) q-expansions is a diamond
ModularCurve.exists_coprime_forall_eq_diamondAutBar_of_forall_apply_coeffEmb_modularFunctionFieldFull_eq_x1FunctionFieldBar217 below · cited by 1 · depth 27 - Petersson pairing represents every functional on S₂(Γ)
ModularCurve.exists_cuspForm_petersson_eq_of_finiteIndex6 below · cited by 1 · depth 27 - Descent of Uₚ and ⟨ d⟩ to torus and Raynaud quotients
ModularCurve.exists_descent_torusQuotient_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness_of_injective14 below · cited by 1 · depth 27 - Germs at the cusp with p-divisible q-expansion are p-divisible
ModularCurve.exists_eq_germ_mul_of_ffEquiv_symm_stalkMap_stalkSpecializes_eq_ofPowerSeries_smul_cuspSection_of_ratCurveModel_compat_of_neZero3 below · cited by 1 · depth 27 - Component projectors on the mod-p Raynaud quotient
ModularCurve.exists_idempotent_pair_baseChange_raynaudQuotient_projector_components_finPts_jHNeronObjectAtP_of_finPtsWitness_of_bridgePins2,654 below · cited by 1 · depth 27 - Periods as a parabolic homomorphism Γ → S₂(Γ)^∨
ModularCurve.exists_isParabolicHom_apply_eq_periodOf3 below · cited by 2 · depth 27 - Manin: every period via an integer parabolic homomorphism
ModularCurve.exists_isParabolicHom_sum_intCast_mul_edgeIntegral_eq_periodOf4 below · cited by 1 · depth 27 - Frobenius-periodicity of all divisor classes of ̄ F'
ModularCurve.exists_iterate_qExpFrobeniusPushforwardModL_eq_self_of_forall_pow_eq_self187 below · cited by 1 · depth 27 - Ordinary duality: multiplicative part of TₚJ_H and polar differentials
ModularCurve.exists_linearMap_injective_range_eq_dual_multiplicativeSubmodule_tateModule_jH_ssPolarDifferentials_of_ordinary_of_mem_infSubgroup4,616 below · cited by 1 · depth 27 - Integral parabolic homomorphisms give period-lattice edge-integral functionals
ModularCurve.exists_mem_periodLatticeOf_eq_sum_intCast_mul_edgeIntegral_of_isParabolicHom3 below · cited by 2 · depth 27 - Reduction of regular differentials to the residue field of a place over p
ModularCurve.exists_mem_regularDifferentials_qExpFunctionFieldC_residueField_of_mem_regularDifferentials888 below · cited by 1 · depth 27 - Descent of the modular equation Φ_ℓ to a q-series subfield
ModularCurve.exists_monic_map_eq_prod_X_sub_C_qTwist_qExpand_jqModC_mul_X_sub_C77 below · cited by 2 · depth 27 - Faithful diamond action on the q-expansion field of X_H(M)
ModularCurve.exists_monoidHom_gamma0_algEquiv_qExpFunctionFieldC_gammaH_apply_eq_one_iff_of_charZero35 below · cited by 4 · depth 27 - Integrality of p/(jₚ-jᵖ) and a simple zero at supersingular a
ModularCurve.exists_natCast_mul_inv_nodeCoord_mem_integers_ord_residue_eq_one_inftyChart_univ196 below · cited by 1 · depth 27 - Node coordinate jₚ-jᵖ has a simple zero
ModularCurve.exists_nodeCoord_mem_integers_ord_residue_eq_one_zeroChart_univ0 below · cited by 1 · depth 27 - Existence of an odd-weight integral form with ℚ(ζₚ)-coefficients
ModularCurve.exists_odd_isIntegralQExp_qExpansion_atkinLehnerSlash_coeff_mem_adjoin_exp39 below · cited by 1 · depth 27 - Uniformiser in K₁ at non-cuspidal places of X₁(Mp)
ModularCurve.exists_ord_eq_one_of_place_x1FunctionField_mul_of_ord_nonneg316 below · cited by 1 · depth 27 - Split Raynaud quotient of the finite part by toric points
ModularCurve.exists_pDivisibleGroup_raynaudQuotient_toricPts_and_exists_retraction_finitePart_jHNeronObjectAtP_of_closedImmersion2,393 below · cited by 2 · depth 27 - Integral q-expansions at the cusp ∞ from a retraction
ModularCurve.exists_powerSeries_map_eq_ffEquiv_symm_of_placeOfPoint_eq_cuspInftyFull4 below · cited by 1 · depth 27 - Level drop at p ∥ M for q-expansion fields in characteristic p
ModularCurve.exists_qExpFunctionFieldC_infSubgroup_coe_eq_of_charP131 below · cited by 6 · depth 27 - Galois conjugation of Atkin–Lehner q-expansions as a diamond operator
ModularCurve.exists_qExpansion_coeff_atkinLehnerSlash_eq_and_slash_mul_eq_apply_gamma1_mul25 below · cited by 1 · depth 27 - Inertia displacements of p-power torsion reduce to the identity
ModularCurve.exists_schemeHomOver_pts_smul_sub_eq_and_resPt_eq_one_of_mem_inertia_jHNeronObjectAtP2,789 below · cited by 2 · depth 27 - Base change of a subfield of ℚ((q)) is its L-span
ModularCurve.exists_sum_single_mul_coeffEmb_of_mem_laurentBaseChange0 below · cited by 3 · depth 27 - Torus quotient of the finite part: multiplicative tower, Raynaud-exact
ModularCurve.exists_torusQuotient_multiplicative_exact_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness65 below · cited by 1 · depth 27 - Gauss ring restricted to L(j) is a discrete valuation ring
ModularCurve.exists_valuationSubring_adjoin_isDiscreteValuationRing_mem_iff_of_laurentBaseChange_qExpFunctionFieldC2 below · cited by 8 · depth 27 - Equal degrees of the two degeneracy maps X₁(N)leftleftarrows X₁(Np)
ModularCurve.finrankAlong_x1LevelInclBar_eq_finrankAlong_x1LevelSubstBar253 below · cited by 1 · depth 27 - Degree of F(x) from a twist-orbit factorisation
ModularCurve.finrank_adjoin_eq_of_map_eq_prod_X_sub_C_qTwist_mul_X_sub_C0 below · cited by 1 · depth 27 - Degree ℓ+1 for a twist-fixed Laurent series
ModularCurve.finrank_adjoin_eq_succ_of_map_eq_prod_X_sub_C_qTwist_mul_X_sub_C_of_not_mem0 below · cited by 1 · depth 27 - Degree ℓ over the Igusa field when ℓ ∣ M
ModularCurve.finrank_adjoin_jqNModC_mul_igusaFunctionFieldX1C_eq_of_dvd187 below · cited by 1 · depth 27 - Degrees over κ((q)) are preserved by q ↦ qⁿ
ModularCurve.finrank_adjoin_qExpand_eq_finrank_adjoin_of_coe_eq_image0 below · cited by 1 · depth 27 - Real rank bound: dim_ℝ H¹ₚₐᵣ(Γ,ℝ) ≤ 2dim S₂(Γ)
ModularCurve.finrank_real_parabolicHoms_le_two_mul_finrank_cuspForm_of_isCongruenceSubgroup173 below · cited by 2 · depth 27 - Base change commutes with fixed fields of q-expansion fields
ModularCurve.forall_algEquiv_laurentBaseChange_apply_eq_iff_mem_laurentBaseChange_of_fixedField0 below · cited by 1 · depth 27 - Integrality and residue of γ on Pl⊗ B
ModularCurve.forall_mem_integers_and_coe_residue_eq_of_tmul_of_forall_coe_eq_coeffMap0 below · cited by 1 · depth 27 - Injectivity of 𝔽ₚ⊗ψᵥ for the Raynaud quotient
ModularCurve.injective_tensorProduct_map_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness_of_isDiscreteValuationRing2,579 below · cited by 3 · depth 27 - Riemann bilinear relation on X_Γ via twisted edge integrals
ModularCurve.integral_petersson_gammaFundamentalSet_eq_sum_conj_periodOf_mul_edgeIntegral6 below · cited by 1 · depth 27 - The q-expansion field over a residue field is a curve
ModularCurve.isCurveOver_and_exists_finset_adjoin_eq_top_qExpFunctionFieldC_residueField49 below · cited by 8 · depth 27 - ̄ j(q^{Mℓ}) avoids the Igusa function field
ModularCurve.jqNModC_mul_not_mem_igusaFunctionFieldX1C_of_not_dvd243 below · cited by 1 · depth 27 - ̄ j(q^ℓ) lies outside the Igusa function field
ModularCurve.jqNModC_not_mem_igusaFunctionFieldX1C_of_not_dvd1,057 below · cited by 1 · depth 27 - Base-changed X₁ function fields: compositum and equal relative degrees
ModularCurve.laurentBaseChange_x1FunctionField_sup_levelRaise_eq_and_relfinrank_eq242 below · cited by 1 · depth 27 - Base change to ℚ̄ of the Hecke correspondence on differentials
ModularCurve.map_differentialCorrespondence_eq_heckeDiffBar_map1 below · cited by 1 · depth 27 - Mutual integrality of j(q) and j(q^N) on both charts
ModularCurve.mem_chartAlgFin_and_forall_mem_chartAlgInf_exists_mul_mem_of_coe_eq_coeffEmb_jq_qExpand_of_one_lt85 below · cited by 26 · depth 27 - Regular differentials spanned by restrictions of global 1-forms
ModularCurve.mem_span_range_res_of_mem_regularDifferentialsBar_of_chartMap_of_neZero172 below · cited by 1 · depth 27 - Unitary multiplier equals exponential of real period
ModularCurve.multiplier_eq_exp_of_periodAlongOf_add_petersson_mem_periodLatticeOf219 below · cited by 1 · depth 27 - q ↦ qᵖ raises the Γ₁(N)∩Γ₀(Nt) level by p
ModularCurve.qExpand_mem_x1x0FunctionFieldC_mul_of_mem2 below · cited by 1 · depth 27 - The two q-expansion maps on differentials agree
ModularCurve.qExpansionDiffAlong_val_eq_diffQExpBar75 below · cited by 1 · depth 27 - Coefficients of Γ₁(Mp)-invariant functions at γ∞ lie in ℚ(ζₚ)
ModularCurve.qExpansion_coeff_comp_smul_mem_adjoin_exp_of_gamma1_mul17 below · cited by 2 · depth 27 - Unramifiedness of X₁(N)→ X₁(M) away from the cusps
ModularCurve.ramificationIndexAlong_inclusion_eq_one_of_ord_nonneg_laurentBaseChange_x1FunctionField_of_dvd315 below · cited by 2 · depth 27 - Relative degree is preserved under extension of constants
ModularCurve.relfinrank_adjoin_image_coeffMap_eq_relfinrank_of_le0 below · cited by 2 · depth 27 - Slope law on the infinity chart for the opposite annulus
ModularCurve.slopeLaw_oppAnnulus_inftyChart_of_chartSpec_univ699 below · cited by 1 · depth 27 - Slope law on the supersingular annulus at the 0-chart
ModularCurve.slopeLaw_ssAnnulus_zeroChart_of_chartSpec_univ699 below · cited by 1 · depth 27 - Weight-two cusp forms give regular differentials x dj
ModularCurve.smul_D_mem_regularDifferentials_qExpFunctionFieldC_algebraicClosure_of_mul_thetaL_jqModC_eq411 below · cited by 2 · depth 27 - Joint Tₚ–⟨ d⟩ eigenvalues avoid a²=(p+1)²e
ModularCurve.sq_ne_add_one_sq_mul_of_joint_eigenvector_tateGenOpH_T_dia606 below · cited by 1 · depth 27 - Vanishing twisted edge sum: the (2,0) bilinear relation on X_Γ
ModularCurve.sum_periodOf_mul_edgeIntegral_eq_zero5 below · cited by 1 · depth 27 - Realisation laws for the field of automorphic functions
ModularCurve.automorphicField_realize_laws0 below · cited by 3 · depth 28 - Gauss point lies on the pole chart of the integral model
ModularCurve.chartAlgInf_subset_and_exists_ideal_gaussCentre_twoChartIntegralModel_qExpFunctionFieldC8 below · cited by 3 · depth 28 - Vanishing constant term of q-expansions at the cusp ∞
ModularCurve.coeff_zero_ffEquiv_symm_eq_zero_of_mem_maximalIdeal_of_placeOfPoint_eq_cuspInftyFull0 below · cited by 2 · depth 28 - Diamond operators on mod p differentials match ⟨ d⟩ on q-expansions
ModularCurve.diffQExp_diamondDiffModLH_eq_intSeriesC_of_diffQExp_eq_of_mem_twoCuspIntegralSet1,253 below · cited by 3 · depth 28 - U_q on differentials mod p matches U_q on q-expansions
ModularCurve.diffQExp_heckeDiffModLH_eq_intSeriesC_of_diffQExp_eq_of_mem_twoCuspIntegralSet_of_dvd588 below · cited by 1 · depth 28 - T_ℓ on mod p differentials matches T_ℓ on cusp forms
ModularCurve.diffQExp_heckeDiffModLH_eq_intSeriesC_of_diffQExp_eq_of_mem_twoCuspIntegralSet_of_not_dvd1,290 below · cited by 1 · depth 28 - Injectivity of the q-expansion map on differentials
ModularCurve.diffQExp_qExpFunctionFieldC_injective51 below · cited by 29 · depth 28 - Rigidity of embeddings of modular function fields fixing j
ModularCurve.exists_algEquiv_modularFunctionFieldC_apply_jGeomGen_eq_comp115 below · cited by 1 · depth 28 - Cusp-0 expansion of the level-M Igusa function field
ModularCurve.exists_algHom_igusaFunctionFieldX1C_apply_eq_jqNModC_and_apply_eq_jqModC114 below · cited by 1 · depth 28 - Finite part at p descends to the decomposition ring
ModularCurve.exists_bialgEquiv_baseChange_decompositionRing_finitePart_jHNeronObjectAtP0 below · cited by 1 · depth 28 - Special fibre of the Raynaud quotient as a tensor square
ModularCurve.exists_bialgEquiv_baseChange_raynaudQuotient_tensorProduct_levelTorsion_finPts_jHNeronObjectAtP_of_finPtsWitness1,625 below · cited by 2 · depth 28 - Unitary multipliers as exponentials of real cusp-form periods
ModularCurve.exists_cuspForm_multiplier_eq_exp_periodOf_of_norm_eq_one177 below · cited by 1 · depth 28 - Descent of Uₚ and a diamond to the Raynaud quotient
ModularCurve.exists_descent_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness12 below · cited by 1 · depth 28 - Descent of Uₚ and ⟨ d⟩ to the torus quotient
ModularCurve.exists_descent_torusQuotient_of_descent_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness0 below · cited by 1 · depth 28 - Divisor of a weight-one form on X₁(M), M≥ 5
ModularCurve.exists_divisor_two_mul_eq_ord_add_weightFloor_one_laurentBaseChange_gamma1380 below · cited by 1 · depth 28 - Component projectors on the special fibre of the Raynaud quotient
ModularCurve.exists_idempotent_pair_baseChange_raynaudQuotient_projector_components_finPts_jHNeronObjectAtP_of_bialgEquiv_of_hecke_descent0 below · cited by 1 · depth 28 - Constant field extension of places of the q-expansion curve
ModularCurve.exists_injective_place_extension_ssPlacesQExp_qExpFrobeniusPlaceModL_of_isAlgClosed34 below · cited by 2 · depth 28 - Frobenius push-forward on differentials of X_{H'}(N) in characteristic p
ModularCurve.exists_isFrobPushDiff_qExpFunctionFieldC_gammaH131 below · cited by 6 · depth 28 - Invertibility of j at generic-fibre points over the cusp
ModularCurve.exists_isUnit_stalk_ffEquiv_symm_stalkMap_genericPoint_eq_jq_of_specializes_cuspSection_of_ratCurveModel_compat_of_neZero288 below · cited by 1 · depth 28 - Cusp parameter has q-expansion 1/j up to a unit
ModularCurve.exists_isUnit_stalk_ffEquiv_symm_stalkMap_mul_stalkSpecializes_eq_jq_inv_cuspSection_of_ratCurveModel_compat_of_neZero8 below · cited by 1 · depth 28 - Places of q-expansion function fields are Frobenius-periodic
ModularCurve.exists_iterate_qExpArithFrobC_smul_place_eq_self_of_forall_pow_eq_self180 below · cited by 1 · depth 28 - Two-cusp forms mod p as regular differentials on the special fibre
ModularCurve.exists_linearEquiv_intTwoCuspForms_twoCompRegularDifferentials1,750 below · cited by 3 · depth 28 - Dual of P⁰ embeds in supersingular-polar differentials
ModularCurve.exists_linearMap_injective_range_eq_dual_multiplicativeSubmodule_ssPolarDifferentials_of_jHNeronObjectAtP_of_twoCompRegularDifferentials_of_ordinary_torusCoords_of_mem_infSubgroup4,608 below · cited by 1 · depth 28 - Constant-field change for polar differentials on X_{H'}(M/p)
ModularCurve.exists_linearMap_injective_tensorProduct_kaehler_map_ssPolarDifferentials_eq163 below · cited by 2 · depth 28 - Residual surjectivity of K in the Gauss ring over A
ModularCurve.exists_map_mem_and_sub_mem_nonunits_gauss_of_coe_eq_coeffMap_of_residue_surjective2 below · cited by 4 · depth 28 - Weight-k form from a square on Γ₁(M)
ModularCurve.exists_modularForm_gamma1_qExpansion_eq_mul_pow_of_qExpansion_eq_sq3 below · cited by 1 · depth 28 - Weight-2m criterion on Γ₁(M): X·(vartheta j)^m is a q-expansion
ModularCurve.exists_modularForm_gamma1_qExpansion_eq_mul_thetaL_pow_of_isIntegral95 below · cited by 2 · depth 28 - Prescribed zeros of modular forms for cusp-free discrete Γ
ModularCurve.exists_modularForm_ne_zero_le_meromorphicOrderAt_of_discreteTopology7 below · cited by 1 · depth 28 - Modular forms separate points and give local parameters
ModularCurve.exists_modularForm_separate_and_localParameter_of_discreteTopology6 below · cited by 2 · depth 28 - Integral monic polynomial annihilating Tₚ on Tₚ J_H(N)
ModularCurve.exists_monic_aeval_tateGenOpH_T_eq_zero_forall_norm_root_lt542 below · cited by 1 · depth 28 - Modular invariant j as a unit along the special fibre
ModularCurve.exists_notMem_span_germ_and_ffEquiv_symm_stalkMap_stalkSpecializes_eq_jq_mul_cuspSection_of_ratCurveModel_compat_of_neZero323 below · cited by 1 · depth 28 - Odd-weight integral form with ℚ(ζₚ)-rational transform, M ≤ 2
ModularCurve.exists_odd_isIntegralQExp_qExpansion_atkinLehnerSlash_coeff_mem_adjoin_exp_of_le_two35 below · cited by 1 · depth 28 - Odd-weight integral form on Γ₁(Mp) with cyclotomic dilated q-coefficients
ModularCurve.exists_odd_isIntegralQExp_qExpansion_atkinLehnerSlash_coeff_mem_adjoin_exp_of_three_le26 below · cited by 1 · depth 28 - A uniformiser from X₀(Mp) at tame j-finite places
ModularCurve.exists_ord_eq_one_of_place_x1x0FunctionFieldC_gamma0_of_ord_nonneg_of_tame287 below · cited by 1 · depth 28 - Toric quotient of the finite part over the decomposition ring
ModularCurve.exists_pDivisibleGroup_toricQuotient_decompositionRing_finitePart_jHNeronObjectAtP38 below · cited by 1 · depth 28 - Points of H as places of the automorphic function field
ModularCurve.exists_placeDictionary_automorphicField_of_discreteTopology4 below · cited by 1 · depth 28 - Diamond operators have finite order on the Tate module
ModularCurve.exists_pos_tateGenOpH_dia_pow_eq_one109 below · cited by 1 · depth 28 - Every place of a cocompact automorphic function field is a point
ModularCurve.exists_pt_eq_of_isCompact58 below · cited by 1 · depth 28 - Galois conjugation of period-p expansions via a diamond operator
ModularCurve.exists_qExpansion_comp_smul_coeff_eq_and_comp_mul_smul_coeff_eq_apply_of_gamma1_mul19 below · cited by 1 · depth 28 - Every Γ-invariant meromorphic function on H is a ratio of forms
ModularCurve.exists_realize_eventuallyEq_of_isCompact1 below · cited by 1 · depth 28 - Good reduction of ℚ̄· F(Γ) at a place above p∤ M
ModularCurve.exists_regularProlongation_placeMap_qExpFunctionFieldC_of_not_dvd855 below · cited by 1 · depth 28 - Transport of the q-expansion fields of X₀(M), X₁(M) along τ
ModularCurve.exists_ringHom_modularFunctionFieldC_x1FunctionFieldC_coe_eq_coeffMap_of_forall_exists_pow_eq0 below · cited by 1 · depth 28 - Elements of L· F₀ in L((q)) as fractions of finite L-combinations
ModularCurve.exists_sum_smul_coeffEmb_mul_eq_of_mem_laurentBaseChange0 below · cited by 2 · depth 28 - Torus quotient tower over 𝔽ₚ of the finite part
ModularCurve.exists_torusQuotient_exact_raynaudQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness37 below · cited by 1 · depth 28 - Verschiebung isomorphisms on the torus quotient of the special fibre
ModularCurve.exists_verschiebung_bialgEquiv_torusQuotient_finPts_jHNeronObjectAtP_of_finPtsWitness41 below · cited by 1 · depth 28 - Finiteness along the X₁(N')/X(Γ₁(N)∩Γ₀(Nt)) level inclusion
ModularCurve.finiteAlong_x1x0LevelInclBar2 below · cited by 2 · depth 28 - Riemann–Roch spaces of K·ℚ(X₁(M)) are finite-dimensional
ModularCurve.finiteDimensional_riemannRochSpace_laurentBaseChange_qExpFunctionFieldC_gamma151 below · cited by 2 · depth 28 - Unique place of the Igusa field above a supersingular valuation
ModularCurve.forall_valuationSubring_igusaFunctionFieldX1C_comap_eq_imp_eq_and_exists_of_mem_ssJSet1,052 below · cited by 1 · depth 28 - Hecke generators on differentials commute with base change k→ K
ModularCurve.genDiffModL_comp_eq_comp_baseChange_of_forall_apply_tmul350 below · cited by 2 · depth 28 - Raynaud quotient has twice the height of A
ModularCurve.height_raynaudQuotient_eq_two_mul_height_levelTorsion_finPts_jHNeronObjectAtP_of_finPtsWitness2,258 below · cited by 2 · depth 28 - Automorphic function field of a cocompact Fuchsian group is a curve
ModularCurve.isCurveOver_automorphicField_of_isCompact45 below · cited by 2 · depth 28 - Integrality shape for the weight-m floor Riemann–Roch space
ModularCurve.isIntegral_and_isIntegral_of_mem_riemannRochSpace_weightFloor3 below · cited by 2 · depth 28 - Integrality of the level inclusion x1x0LevelInclBar
ModularCurve.isIntegral_x1x0LevelInclBar3 below · cited by 1 · depth 28 - Diamond operators make the Igusa cover of X₁(M) Galois
ModularCurve.lift_fixedField_range_act_eq_x1FunctionFieldC_and_isGalois_igusaFunctionFieldX1C1,040 below · cited by 1 · depth 28 - Reduced diamond action commutes with extension of constants
ModularCurve.map_diamondActionModL_eq_diamondActionModL_map_of_coe_eq_coeffMap290 below · cited by 1 · depth 28 - Gauss valuation ring as a localisation of the pole chart
ModularCurve.mem_gaussValuationSubring_iff_exists_chartAlgInf_mul_eq_of_not_mem_gaussCentre134 below · cited by 2 · depth 28 - Gauss rings are compatible with coefficientwise extension of scalars
ModularCurve.mem_iff_map_mem_and_mem_nonunits_iff_of_gaussPresentation_of_coe_eq_coeffMap0 below · cited by 11 · depth 28 - κ is algebraically closed in the q-expansion function field
ModularCurve.mem_range_algebraMap_of_isAlgebraic_qExpFunctionFieldC1 below · cited by 5 · depth 28 - Supersingularity is invariant along Φ_ℓ (ℓ≠ p)
ModularCurve.mem_ssJSet_iff_of_isRoot_map_modularPolynomialData317 below · cited by 8 · depth 28 - Inertia at a j-integral place of κ(X₁(M)) has order at most 3
ModularCurve.natCard_smul_valuationSubring_eq_and_forall_sub_mem_nonunits_le_three_of_ringEquiv_x1FunctionFieldC974 below · cited by 1 · depth 28 - Existence of a diamond datum on the Igusa function field of X₁(M)
ModularCurve.nonempty_igusaDiamondDataX1C1,039 below · cited by 1 · depth 28 - Germs at the cusp ∞ have pole-free q-expansions
ModularCurve.order_ffEquiv_symm_nonneg_of_placeOfPoint_eq_cuspInftyFull0 below · cited by 3 · depth 28 - Abel-type theorem: periods plus Petersson term lie in Λ_Γ
ModularCurve.periodAlongOf_add_petersson_mem_periodLatticeOf_of_multiplier_eq_exp37 below · cited by 1 · depth 28 - Width equals n for a supersingular place with ord∘ι = ncdotord_w
ModularCurve.placeWidthChar_eq_of_mem_ssPlaces_of_ord_comp_eq_mul_ord629 below · cited by 1 · depth 28 - Degeneracy image g(qᵈ) lies in the X_H(N) function field over L
ModularCurve.qExpand_coeffEmb_mem_laurentBaseChange_xHFunctionField_of_mem_modularFunctionFieldFull187 below · cited by 65 · depth 28 - j(q^ℓ) in the q-expansion field of Γ_H(N)∩Γ₀(Nℓ)
ModularCurve.qExpand_jqModC_mem_qExpFunctionFieldC_gammaH_inf_gamma0_mul5 below · cited by 2 · depth 28 - j(qᵈ) not in the q-expansion function field of Γ_{H'}(N)
ModularCurve.qExpand_jqModC_not_mem_qExpFunctionFieldC_gammaH_of_not_dvd123 below · cited by 6 · depth 28 - Unramified places with j finite in the X₁ tower over ℚ̄
ModularCurve.ramificationIndexAlong_inclusion_eq_one_of_ord_nonneg_laurentBaseChange_x1FunctionField_of_dvd_algebraicClosure307 below · cited by 1 · depth 28 - Level raising by ℓ ∤ N has degree ℓ²-1
ModularCurve.relfinrank_x1FunctionField_mul_eq_sq_sub_one_of_prime_of_not_dvd228 below · cited by 1 · depth 28 - Genus of X₁(M) over an algebraically closed field of characteristic 0
ModularCurve.twelve_mul_genusFF_laurentBaseChange_gamma1_add_six_mul_natCard_doubleCoset_eq_index_add_twelve_of_isAlgClosed436 below · cited by 2 · depth 28 - Twice the characteristic width of a moduli place counts level-preserving automorphisms
ModularCurve.two_mul_placeWidthChar_eq_natCard_rationalAut_map_eq_of_toValuationSubring_eq_comap_moduliPlace432 below · cited by 3 · depth 28 - Compositum of q-expansion fields of X₁(Ma) and X₁(Mb)
ModularCurve.x1FunctionField_mul_sup_x1FunctionField_mul_eq_of_coprime231 below · cited by 1 · depth 28 - Coefficientwise σ-fixed Laurent elements are fixed by arithmeticGalois(σ)
ModularCurve.arithmeticGalois_smul_eq_self_of_forall_coeff_eq0 below · cited by 3 · depth 29 - Inertia-cyclotomic vectors are isotropic for a twisted pairing
ModularCurve.bilinForm_apply_eq_zero_of_inertia_cyclotomic4 below · cited by 1 · depth 29 - Hecke-self-adjoint pairings stay non-degenerate on idempotent corners
ModularCurve.bilinForm_nondegenerate_on_map_proj_cornerSubmodule_tateModule_jH_of_selfAdjoint378 below · cited by 2 · depth 29 - Integral j-chart ring lies in valuation subrings containing A[j]
ModularCurve.coe_mem_valuationSubring_of_forall_aeval_mem_chartAlgFin0 below · cited by 12 · depth 29 - q-expansion of U_ℓ on differentials for ℓ ∣ N
ModularCurve.coeff_diffQExp_heckeDiffModLH_of_dvd581 below · cited by 2 · depth 29 - q-expansion of T_ℓ on differentials in characteristic p
ModularCurve.coeff_diffQExp_heckeDiffModLH_of_not_dvd_of_charP619 below · cited by 1 · depth 29 - Supersingular places under constant field extension
ModularCurve.comap_ne_top_and_mem_ssPlacesQExp_of_mem_and_mem_ssPlacesQExp_of_comap_eq3 below · cited by 4 · depth 29 - Base change of diamond operators on differentials of X_{H'}(M/p)
ModularCurve.diamondDiffModLH_comp_eq_comp_baseChange_of_forall_apply_tmul104 below · cited by 1 · depth 29 - Uniqueness of the place over the cusp ∞ after base change
ModularCurve.eq_cuspInftyBar_of_comap_toSubring_eq_cuspInftyFull0 below · cited by 1 · depth 29 - Joint injectivity of two reduction maps on mod-p two-cusp forms
ModularCurve.eq_zero_of_isInfReductionMap_apply_eq_zero_of_apply_eq_zero_alSlash1,341 below · cited by 2 · depth 29 - Even parity of ord_P v+ord_P y at cusp places
ModularCurve.even_ord_add_ord_of_not_mem_toValuationSubring_laurentBaseChange_gamma1364 below · cited by 1 · depth 29 - Interior parity of ord_P(v) plus the weight-floor term
ModularCurve.even_ord_add_weightFloor_of_mem_toValuationSubring_laurentBaseChange_gamma1324 below · cited by 1 · depth 29 - Integrality over ℂ[j⁻¹] bounds growth at every cusp
ModularCurve.eventually_norm_slash_le_mul_of_isIntegral_adjoin_jqModC_inv_sq15 below · cited by 1 · depth 29 - Unitary characters trivial on tr²≤ 4 have real logarithms
ModularCurve.exists_addMonoidHom_exp_eq_of_norm_eq_one_of_trace_sq_le_four_of_finiteIndex3 below · cited by 1 · depth 29 - Hecke-equivariant dlog from J_H[p] to supersingular polar differentials
ModularCurve.exists_addMonoidHom_torsion_ssPolarDifferentials_dlog_of_ordinary_of_mem_infSubgroup4,602 below · cited by 1 · depth 29 - Atkin–Lehner-type operator induces an automorphism of the q-expansion field
ModularCurve.exists_algEquiv_laurentBaseChange_qExpFunctionFieldC_coeffMap_apply_eq_of_slash_heckeDiagMatrix7 below · cited by 2 · depth 29 - Two spellings of the q-expansion field of X₁(M)
ModularCurve.exists_algEquiv_x1FunctionFieldC_qExpFunctionFieldC_gammaH_bot_coe_eq0 below · cited by 1 · depth 29 - Expansion at the cusp 0 for X_H(M) in characteristic ℓ
ModularCurve.exists_algHom_qExpFunctionFieldC_gammaH_cuspZero_apply_eq_and_apply_div_pow_eq111 below · cited by 1 · depth 29 - Component maps of the Raynaud quotient on the special fibre
ModularCurve.exists_bialgHom_levelTorsion_raynaudQuotient_baseChange_spec_comp_eq_of_finPtsWitness51 below · cited by 1 · depth 29 - Twisted Weil pairing as a bilinear form on J_H(M)[p]
ModularCurve.exists_bilinForm_torsion_jH_nondegenerate_genOpH_selfAdjoint_galois515 below · cited by 2 · depth 29 - Chain bounding div F with periods cancelling a Petersson integral
ModularCurve.exists_chain_periodAlongOf_add_petersson_eq_zero_of_multiplier_eq_exp35 below · cited by 1 · depth 29 - Discrete valuation subring of a place capturing S and k₀
ModularCurve.exists_coeffRing_forall_exists_mul_eq_and_forall_mem_range_residue1 below · cited by 1 · depth 29 - Automorphisms fixing the X₀(M) function field are diamonds
ModularCurve.exists_eq_diamondPullbackModL_of_forall_coe_mem_gamma0_apply_eq0 below · cited by 1 · depth 29 - Non-units of the Gauss valuation ring are divisible by varpi
ModularCurve.exists_eq_mul_of_mem_nonunits_of_forall_mem_iff_gaussPresentation0 below · cited by 7 · depth 29 - Inertia-stable divisor representing an inertia-fixed class of J_H
ModularCurve.exists_inertiaStable_degZero_pic0Mk_eq_xH33 below · cited by 1 · depth 29 - Vertical order of j at the cusp section's special point
ModularCurve.exists_int_notMem_span_germ_and_ffEquiv_symm_stalkMap_stalkSpecializes_eq_jq_mul_zpow_mul_cuspSection_of_ratCurveModel_compat_of_neZero4 below · cited by 1 · depth 29 - Igusa radicand at a supersingular place: exponent coprime to p-1
ModularCurve.exists_irreducible_isCoprime_eq_mul_pow_of_coe_eq_hasseRootFn_pow_of_mem_ssJSet1,045 below · cited by 1 · depth 29 - Valuation-ring lift of a ℚ̄-point along a specialisation
ModularCurve.exists_liesOverPrime_schemeHomOver_comp_eq_base_closedPoint_eq_of_specializes0 below · cited by 1 · depth 29 - Base change injectivity for differentials of q-expansion function fields
ModularCurve.exists_linearMap_injective_tensorProduct_kaehler_qExpFunctionFieldC_apply_tmul52 below · cited by 1 · depth 29 - Inertia-fixed place on an annulus over a fixed value
ModularCurve.exists_mem_dom_forall_inertia_smul_eq_and_evalAt_param_eq7 below · cited by 2 · depth 29 - Rationality of quotients of modular forms on Γ_H(N)
ModularCurve.exists_mem_laurentBaseChange_coeffMap_mul_qExpansion_eq_of_forall_coeff_mem_range53 below · cited by 37 · depth 29 - Extending supersingular polar differentials across the glued two-component curve
ModularCurve.exists_mem_twoCompRegularDifferentials_of_mem_ssPolarDifferentials107 below · cited by 2 · depth 29 - Linear bounds on vanishing orders and dim M_k(Γ) for cocompact Γ
ModularCurve.exists_meromorphicOrderAt_le_and_finrank_modularForm_le_of_isCompact0 below · cited by 1 · depth 29 - Peaked Poincaré series with geometric tail bound
ModularCurve.exists_modularForm_peak_sub_le_of_discreteTopology1 below · cited by 1 · depth 29 - Igusa: k(X₁(M))_q/k(X₀(M))_q is Galois in characteristic ℓ∤ M
ModularCurve.exists_mulSemiringAction_faithful_fixed_iff_card_eq_index_qExpFunctionFieldC_gamma1_gamma0_charP292 below · cited by 1 · depth 29 - Odd-weight integral form with ℚ(ζ₃)-rational twist, 3 ∣ d
ModularCurve.exists_odd_isIntegralQExp_qExpansion_atkinLehnerSlash_coeff_mem_adjoin_exp_of_le_two_three3 below · cited by 1 · depth 29 - Diamond-equivariant modular description of a fibre of j
ModularCurve.exists_orbitMap_torsionOrbit_places_qExpFunctionFieldC_gammaH_smul_eq576 below · cited by 1 · depth 29 - Every place descends to a rational place over some 𝔽_{p^m}
ModularCurve.exists_place_qExpFunctionFieldC_galoisField_toValuationSubring_eq_comap_and_deg_eq_one139 below · cited by 1 · depth 29 - Forms on Γ₁(Mp)∩Γ₀(Mpℓ) spanned over K₀ by integral forms
ModularCurve.exists_sum_smul_eq_of_qExpansion_coeff_mem_x1x0_gamma0_mul104 below · cited by 1 · depth 29 - Integrality of inertia-fixed positions in a p-adic annulus
ModularCurve.exists_valuation_evalAt_param_eq_valuation_pow_of_forall_inertia_smul_eq15 below · cited by 1 · depth 29 - Toric quotient tower is of multiplicative type
ModularCurve.exists_verschiebung_bialgEquiv_and_reducesToOne_and_tower_toricClosure_finitePart_jHNeronObjectAtP23 below · cited by 1 · depth 29 - No pole of j along the special fibre at the cusp
ModularCurve.false_of_ffEquiv_symm_stalkMap_stalkSpecializes_eq_jq_mul_pow_mul_cuspSection_of_ratCurveModel_compat_of_neZero309 below · cited by 1 · depth 29 - No zero of j along the special fibre
ModularCurve.false_of_pow_mul_ffEquiv_symm_stalkMap_stalkSpecializes_eq_jq_mul_cuspSection_of_ratCurveModel_compat_of_neZero319 below · cited by 1 · depth 29 - Equal dimensions for mod p cusp forms and glued differentials
ModularCurve.finrank_tensorProduct_intTwoCuspForms_eq_finrank_twoCompRegularDifferentials1,749 below · cited by 1 · depth 29 - Frobenius push-forward of differentials commutes with base change
ModularCurve.frobPushDiffModL_comp_eq_comp_baseChange_of_forall_apply_tmul52 below · cited by 1 · depth 29 - The degeneracy map q ↦ q^ℓ on q-expansion function fields
ModularCurve.heckeBetaModLHDefined2 below · cited by 10 · depth 29 - Base change compatibility of Hecke operators on modular differentials
ModularCurve.heckeDiffModLH_comp_eq_comp_baseChange_of_forall_apply_tmul_of_prime248 below · cited by 1 · depth 29 - Frobenius push-forward divides supersingular pole orders by p
ModularCurve.isRegularAt_and_exists_eq_smul_dCoord_uniformizer_pow_mul_mem_of_isFrobPushDiff152 below · cited by 1 · depth 29 - Coefficient extension preserves linear independence of Laurent series
ModularCurve.linearIndependent_coeffEmb_of_linearIndependent0 below · cited by 4 · depth 29 - Uniqueness of the Gauss point at level M'
ModularCurve.mem_iff_mem_constantReduction_integers_of_jq_mem_residuallyTranscendental114 below · cited by 3 · depth 29 - Image of P⁰ in J_H[p]: cyclotomic inertia part, finite
ModularCurve.mem_map_proj_multiplicativeSubmodule_iff_inertia_cyclotomic_and_map_proj_le_finPts_of_ordinary13 below · cited by 1 · depth 29 - Non-positive weight forms for cocompact discrete Γ
ModularCurve.modularForm_eq_const_and_eq_zero_of_isCompact0 below · cited by 1 · depth 29 - Half-rank of the inertia-cyclotomic part of the ordinary corner
ModularCurve.ncard_inertiaCyclotomic_sq_eq_ncard_map_proj_cornerSubmodule_tateModule_jH_of_ordinary3,439 below · cited by 2 · depth 29 - Level-one torus quotient is a form of 𝔽ₚ[(ℤ/p)^t]
ModularCurve.nonempty_bialgEquiv_baseChange_residueField_torusQuotient_one_addMonoidAlgebra_of_finPtsWitness22 below · cited by 1 · depth 29 - j(qᵈ) not in the mod-ℓ level-Γ₁(N) expansion field
ModularCurve.qExpand_jqModC_not_mem_qExpFunctionFieldC_gammaH_bot_of_charP114 below · cited by 1 · depth 29 - j(qᵈ) lies outside the Γ_H(N,bot) q-expansion field
ModularCurve.qExpand_jqModC_not_mem_qExpFunctionFieldC_gammaH_bot_of_charZero9 below · cited by 1 · depth 29 - Tame j-finite places of X(Γ₁(M)∩Γ₀(p)) unramified over X₀(Mp)
ModularCurve.ramificationIndexAlong_inclusion_eq_one_of_ord_nonneg_laurentBaseChange_x1x0_gamma0_of_tame286 below · cited by 1 · depth 29 - Degree of the Γ₁(M)∩Γ₀(p) function field over Γ₀(Mp)
ModularCurve.relfinrank_laurentBaseChange_gamma0_mul_x1x0FunctionFieldC_eq_index239 below · cited by 2 · depth 29 - Chains closed modulo Γ have periods in the period lattice
ModularCurve.sum_periodAlongOf_mem_periodLatticeOf_of_boundary_eq_zero2 below · cited by 1 · depth 29 - Surjective product map and rank identity for the Raynaud quotient
ModularCurve.surjective_productMap_and_finrank_eq_levelTorsion_raynaudQuotient_baseChange_of_finPtsWitness1,608 below · cited by 1 · depth 29 - Moduli place width counts automorphisms of (W,C), all characteristics
ModularCurve.two_mul_placeWidthChar_eq_natCard_rationalAut_map_eq_of_toValuationSubring_eq_comap_moduliPlace_of_prime434 below · cited by 4 · depth 29 - Unique branch over the j-line for Γ₀(M') when q ∤ M'
ModularCurve.valuationSubring_unique_laurentBaseChange_gamma0_of_not_dvd943 below · cited by 12 · depth 29 - Cusp-0 expansion sends ̄ j(qᵈ) to ̄ j(q^N)
ModularCurve.apply_eq_qExpand_jqModC_of_coe_eq_qExpand_jqModC_of_cuspExpansion_S6 below · cited by 1 · depth 30 - Integral level-M' functions lie in the j-finite chart algebra
ModularCurve.coeffEmb_mem_chartAlgFin_of_forall_coeff_of_isIntegral_adjoin_jq822 below · cited by 7 · depth 30 - q-expansion law for an abstract Cartier operator
ModularCurve.coeff_qExpansionDiffAlong_pow_eq_coeff_mul_of_cartierLaws9 below · cited by 2 · depth 30 - Trace along q↦ q^ℓ on q-expansions when ℓ∣ N
ModularCurve.coeff_trace_along_heckeBetaModLH_of_dvd580 below · cited by 1 · depth 30 - Trace along q↦ q^ℓ of q-expansions, ℓ∤ N
ModularCurve.coeff_trace_along_heckeBetaModLH_of_not_dvd580 below · cited by 1 · depth 30 - Traces of places on 𝔽_{p^m}-forms of q-expansion function fields
ModularCurve.deg_eq_one_of_trace_qExpFunctionFieldC_galoisField_of_deg_dvd3 below · cited by 1 · depth 30 - q-expansion of Kähler differentials commutes with coefficient extension
ModularCurve.diffQExp_map_eq_coeffMap_diffQExp0 below · cited by 1 · depth 30 - Equality y(qᵈ)=jmath̄(q^N) forces d∣ N
ModularCurve.dvd_of_qExpand_eq_qExpand_jqModC0 below · cited by 1 · depth 30 - Cusp with ord j(q^N)=Nord j is ∞
ModularCurve.eq_cuspInftyBar_of_isCusp_of_ord_jqN_eq_mul_ord_jq_of_neZero153 below · cited by 1 · depth 30 - Atkin–Lehner-twisted dlog on J_H(M)[p] into supersingular differentials
ModularCurve.exists_addMonoidHom_torsion_ssPolarDifferentials_dlog_finPts_of_abelJacobiPin_tauFree_raynaud_bridgePins_export_of_algEquiv3,097 below · cited by 3 · depth 30 - Atkin–Lehner automorphism W in positive characteristic
ModularCurve.exists_algEquiv_qExpFunctionFieldC_heckeBetaModLH_eq_heckeAlphaModLH_and_eq_diamondActionModL_of_charP591 below · cited by 2 · depth 30 - Base change of smooth chart algebras to integrally closed domains
ModularCurve.exists_algEquiv_tensorProduct_chartAlg_laurentBaseChange_of_smooth_of_isIntegrallyClosed10 below · cited by 3 · depth 30 - Level-Nd expansions at the cusp 0 lie in K((qᵈ))
ModularCurve.exists_apply_eq_qExpand_of_coe_mem_qExpFunctionFieldC_gammaH_bot_of_cuspExpansion_S98 below · cited by 1 · depth 30 - Open stabilisers of places under constant-field base change
ModularCurve.exists_finiteDimensional_forall_mem_fixingSubgroup_smul_eq_place_laurentBaseChange26 below · cited by 1 · depth 30 - Finitely many Γ-orbits of zeros and poles of a multiplier-automorphic function
ModularCurve.exists_finset_orbitReps_of_meromorphicOrderAt_ne_zero_of_finiteIndex0 below · cited by 1 · depth 30 - Frobenius-semilinear transport of differentials preserving pole orders
ModularCurve.exists_frobeniusSemilinear_transport_kaehler_poleOrder_qExpFunctionFieldC62 below · cited by 1 · depth 30 - Toric Hopf quotient of Gᵥ: free of rank p^{vt}
ModularCurve.exists_hopfCokernel_free_finrank_eq_pow_of_finPtsWitness39 below · cited by 1 · depth 30 - Divisor periods plus Petersson pairing lie in the period lattice
ModularCurve.exists_mem_periodLatticeOf_sum_periodAlongOf_add_petersson_eq_of_multiplier_eq_exp33 below · cited by 1 · depth 30 - Poincaré series of weight k≥ 4 for cusp-free discrete groups
ModularCurve.exists_modularForm_eq_tsum_of_discreteTopology0 below · cited by 1 · depth 30 - A Γ₀(M)-action on the q-expansion function field of X_H
ModularCurve.exists_monoidHom_gamma0_algEquiv_qExpFunctionFieldC_gammaH104 below · cited by 3 · depth 30 - Bounded denominators at q for cusp-regular modular functions
ModularCurve.exists_ne_zero_forall_algebraMap_mul_coeff_mem_of_cuspRegular127 below · cited by 5 · depth 30 - Double-annihilator property of the twisted pairing on J_H(M)[n]
ModularCurve.exists_perfectPairing_nsmul_eq_zero_galois_heckeH_diamondH_forall_addSubgroup_eq_biannihilator514 below · cited by 1 · depth 30 - Atkin–Lehner slash at p as ℚ(ζₚ)-combination of integral forms
ModularCurve.exists_sum_smul_eq_smul_atkinLehnerSlash_x1x0_gamma0127 below · cited by 2 · depth 30 - Cusp places of X(Γ): width, ord_P y = -h, and limits
ModularCurve.exists_tendsto_realizeOf_mul_exp_of_not_mem_toValuationSubring348 below · cited by 1 · depth 30 - Finitely many zeros and poles of ̄ j on the special fibre
ModularCurve.false_of_infinite_setOf_ord_pointEquivPlace_jqModC_ne_zero_cuspSection_of_ratCurveModel_compat_of_neZero113 below · cited by 2 · depth 30 - Finiteness and separability along the degeneracy maps α,β
ModularCurve.finiteAlong_and_separableAlong_heckeAlphaModLH_heckeBetaModLH205 below · cited by 8 · depth 30 - Rosenlicht count for the two-component curve at level Γ_{H'}(N)
ModularCurve.finiteDimensional_and_finrank_twoCompRegularDifferentials_add_one180 below · cited by 1 · depth 30 - Finite-dimensionality of supersingular polar differentials on X_{H'}(N)
ModularCurve.finiteDimensional_ssPolarDifferentials100 below · cited by 2 · depth 30 - Deligne–Rapoport genus identity at level Γ_H(M), p ‖ M
ModularCurve.genusFF_xHFunctionFieldBar_add_one_eq_two_mul_genusFF_add_natCard_ssNodePairsQExp1,413 below · cited by 2 · depth 30 - Genus identity for X_H(M) at p ‖ M, arbitrary κ
ModularCurve.genusFF_xHFunctionFieldBar_add_one_eq_two_mul_genusFF_add_natCard_ssNodePairsQExp_univ1,412 below · cited by 2 · depth 30 - Genus of X_H(M) at most dim_ℂ S₂(Γ_H(M))
ModularCurve.genusFF_xHFunctionFieldBar_le_finrank_cuspForm_gammaH_two288 below · cited by 1 · depth 30 - Frobenius push-forward preserves ss-polar differentials and residues
ModularCurve.hasSimpleResidue_qExpFrobeniusPlaceModL_of_isFrobPushDiff152 below · cited by 2 · depth 30 - An open set containing infinitely many κ-points of the special fibre
ModularCurve.infinite_setOf_base_closedPoint_mem_of_fromSpecStalk_span_germ_mem_cuspSection_of_ratCurveModel_compat_of_neZero47 below · cited by 2 · depth 30 - Nonvanishing mod p of q-expansions at cusps γ∞
ModularCurve.intSeriesC_ne_zero_of_coe_eq_slash_of_mem_Gamma0_of_level_mul1,240 below · cited by 2 · depth 30 - Base change of Laurent subfields along a field homomorphism
ModularCurve.laurentBaseChange_eq_adjoin_image_coeffMap_and_exists_ringHom166 below · cited by 6 · depth 30 - Base change of the Γ₀(M') q-expansion field lies in X_H(N)
ModularCurve.laurentBaseChange_qExpFunctionFieldC_gamma0_le_laurentBaseChange_xHFunctionField187 below · cited by 14 · depth 30 - Integral ratios in the q-expansion field lie in intFormRatiosC
ModularCurve.mem_intFormRatiosC_of_coe_eq_intSeriesC_div0 below · cited by 1 · depth 30 - Regularity of differentials with integral cuspidal q-expansion
ModularCurve.mem_regularDifferentials_of_diffQExp_eq_intSeriesC_of_isIntegralQExp_cuspForm_gammaH_of_not_dvd942 below · cited by 1 · depth 30 - Kernel of an Atkin–Lehner-pinned reduction map, mod p
ModularCurve.mem_span_tmul_intTwoCuspReduce_of_apply_eq_zero_of_diffQExp_apply_eq_intSeriesC_alSlash_diamondLinH8 below · cited by 1 · depth 30 - Invariance of orders and stabiliser divisibility for multiplicative functions
ModularCurve.meromorphicOrderAt_smul_eq_and_card_stabilizer_dvd_of_multiplier_eq_exp_periodOf3 below · cited by 2 · depth 30 - Minimal polynomial of j(qᵖ)/jᵖ over the lower level field
ModularCurve.minpoly_div_pow_eq_and_natDegree_minpoly_eq_finrank_of_monic_of_coe_eq_xHFunctionFieldBar277 below · cited by 1 · depth 30 - A moduli place determines its Γ₀(N)-class uniquely
ModularCurve.moduliPoint_eq_of_isModuliPlaceOf_of_isModuliPlaceOf411 below · cited by 4 · depth 30 - Pole of ̄ j at the reduction of an A-point
ModularCurve.ord_apply_pointEquivPlace_jqModC_neg_of_stalkClosedPointTo_mem_maximalIdeal_of_ffEquiv_symm_stalkMap_eq_jq_inv_cuspSection_of_ratCurveModel_compat_of_neZero277 below · cited by 1 · depth 30 - An A-point with j∈mathfrak m_A reduces to a zero of ̄ j
ModularCurve.ord_apply_pointEquivPlace_jqModC_pos_of_stalkClosedPointTo_mem_maximalIdeal_of_ffEquiv_symm_stalkMap_eq_jq_cuspSection_of_ratCurveModel_compat_of_neZero287 below · cited by 1 · depth 30 - Ordinary corner count against supersingular polar differentials
ModularCurve.pow_finrank_range_corner_ssPolarDifferentials_mul_ncard_reducesToOne_eq_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins4_of_algEquiv3,519 below · cited by 1 · depth 30 - j(q^{N'}) as a ratio of integral forms on Γ_H(N)
ModularCurve.qExpand_jqModC_mem_intFormRatiosC_gammaH5 below · cited by 7 · depth 30 - Algebraicity of q-expansion coefficients of F∣_kγ
ModularCurve.qExpansion_slash_coeff_mem_range_of_isIntegralQExp98 below · cited by 1 · depth 30 - Unramified at tame j-finite places over X₀(Mp)
ModularCurve.ramificationIndexAlong_inclusion_eq_one_of_ord_nonneg_laurentBaseChange_x1x0_gamma0_of_tame_algebraicClosure278 below · cited by 1 · depth 30 - Ordinary corner of p-torsion spans inside τ(e) Ω
ModularCurve.span_image_corner_le_range_of_addMonoidHom_torsion_ssPolarDifferentials0 below · cited by 1 · depth 30 - Descended abelian quotients kill the Hopf cokernel of ψᵥ
ModularCurve.specMap_cokernel_comp_levelBaseChange_comp_abq_eq_one_of_finPtsWitness16 below · cited by 1 · depth 30 - Degree zero for a Γ-multiplicative meromorphic function
ModularCurve.sum_meromorphicOrderAt_div_card_stabilizer_eq_zero_of_multiplier_eq_exp_periodOf11 below · cited by 2 · depth 30 - Integrality of q-coefficients for cusp-regular modular functions
ModularCurve.algebraMap_coeff_mem_of_mem_integers_of_cuspRegular135 below · cited by 11 · depth 31 - Geometric function field identification is base change of the rational one
ModularCurve.coe_ffEquiv_symm_stalkMap_eq_coeffEmb_ffEquiv_symm_of_galoisCompat_of_placeCompat44 below · cited by 2 · depth 31 - Riemann's inequality for X_H(M) over ℚ̄
ModularCurve.degree_add_one_sub_genusFF_le_finrank_riemannRochSpace_xHFunctionFieldBar203 below · cited by 5 · depth 31 - Diamond pullback preserves supersingular-polar differentials and permutes residues
ModularCurve.diamondDiffModLH_mem_ssPolarDifferentials_and_residue_eq_residue_inv_smul1,244 below · cited by 3 · depth 31 - Transposed Hecke correspondence sends dlog f to dlog N_α(β f)
ModularCurve.differentialCorrespondence_heckeAlphaModLH_heckeBetaModLH_inv_smul_D3 below · cited by 1 · depth 31 - A cusp of type N is the cusp at infinity
ModularCurve.eq_cuspInftyBar_of_mul_ord_eq152 below · cited by 1 · depth 31 - Reduced Atkin–Lehner automorphism w_ℓ in characteristic p
ModularCurve.exists_algEquiv_qExpFunctionFieldC_floor_and_diamondActionModL_and_heckeBetaModLH_eq_heckeAlphaModLH_of_charP590 below · cited by 1 · depth 31 - Cusp places of ℂ·ℚ(X(Γ)) are exhausted by Pl
ModularCurve.exists_apply_eq_of_forall_ord_eq_zero_tendsto_realizeOf35 below · cited by 2 · depth 31 - Frobenius-semilinear transport of differentials on q-expansion function fields
ModularCurve.exists_frobeniusSemilinear_transport_kaehler_qExpFunctionFieldC62 below · cited by 1 · depth 31 - Invariance of the in-line polynomial under P↦[a]P
ModularCurve.exists_inLineMulPoly_eq_C_mul_of_toPoint_eq_zsmul_of_eval_prePsi_eq_zero6 below · cited by 2 · depth 31 - Untwisting a unitary multiplier into a Γ-invariant C¹ function
ModularCurve.exists_invariant_untwist_of_multiplier_eq_exp_periodOf5 below · cited by 1 · depth 31 - A rational Γ₀(p^k)-structure on the Tate curve
ModularCurve.exists_isGamma0PowAt_tateBase_and_map_coeffMap_eq_prod_X_sub_C_toricPoint22 below · cited by 6 · depth 31 - Atkin–Lehner twist intertwining transposed Hecke action on supersingular polar differentials
ModularCurve.exists_linearEquiv_ssPolarDifferentials_twist_transposeHecke_genDiffModL_of_isInfReductionMap_of_mem_infSubgroup1,849 below · cited by 1 · depth 31 - Winding pairing against dlogΦ modulo the period lattice
ModularCurve.exists_mem_periodLatticeOf_tendsto_windingPairing_smoothedFundamental20 below · cited by 1 · depth 31 - Atkin–Lehner operator preserves forms on Γ₁(M)∩Γ₀(p)
ModularCurve.exists_modularForm_coe_eq_atkinLehnerSlash_x1x0_gamma00 below · cited by 1 · depth 31 - Diamond automorphisms preserve Gauss p-integrality at level M
ModularCurve.exists_mul_ofPowerSeries_eq_of_diamondAutHBar_apply_eq_coeffEmb_of_level_mul1,239 below · cited by 1 · depth 31 - Fricke-twisted Weil pairing on J_H(M)[n]
ModularCurve.exists_pairing_nsmul_eq_zero_galois_heckeH_diamondH476 below · cited by 1 · depth 31 - Reduced p-th root functions detect the finite part of J_H(M)[p]
ModularCurve.exists_reducedRootFunction_torsion_mem_finPts_iff_forall_dvd_ord_of_abelJacobiPin_tauFree_of_algEquiv2,608 below · cited by 1 · depth 31 - Coefficientwise automorphisms descend along a normal constant subfield
ModularCurve.exists_ringEquiv_restrict_coeffMap_laurentBaseChange_of_normal0 below · cited by 2 · depth 31 - Cuspidal place and q_N-expansion at σ∞ on X(Γ)
ModularCurve.exists_ringHom_place_order_eq_mul_ord_of_qExpansion_slash18 below · cited by 2 · depth 31 - Forms with K₀-rational q-expansions on Γ₁(M)∩Γ₀(p)
ModularCurve.exists_sum_smul_eq_of_qExpansion_coeff_mem_x1x0_gamma0104 below · cited by 1 · depth 31 - Finiteness and separability along both degeneracy maps
ModularCurve.finiteAlong_and_separableAlong_heckeAlphaModLH_heckeBetaModLH_of_natCast_ne_zero201 below · cited by 4 · depth 31 - Dimension count for two-component supersingular glued differentials
ModularCurve.finiteDimensional_and_finrank_twoCompRegularDifferentials_add_one_eq_two_mul_finrank_regularDifferentials_add_natCard110 below · cited by 1 · depth 31 - Regular differentials of the modular function field have dimension the genus
ModularCurve.finite_and_finrank_regularDifferentials_qExpFunctionFieldC_eq_genusFF_of_isAlgClosed121 below · cited by 1 · depth 31 - Degree of the degeneracy map q↦ q^ℓ on X_{H'}(N)
ModularCurve.finrankAlong_heckeBetaModLH575 below · cited by 2 · depth 31 - Transfer of the valuation condition from j to j(qτ)
ModularCurve.forall_aeval_mem_iff_forall_aeval_qExpand_mem_of_valuationSubring41 below · cited by 3 · depth 31 - Frobenius-twisted Gauss ring is a branch ring over j
ModularCurve.forall_algebraMap_mem_comap_and_forall_aeval_mem_comap_gauss_of_coe_map_eq_qExpand45 below · cited by 1 · depth 31 - Uₚ on mod-p differentials sends dlog f to dlog σ f
ModularCurve.genDiffModL_U_self_inv_smul_D_of_coe_eq_coeffMap_frobenius1 below · cited by 1 · depth 31 - Diamond operator on differentials sends dlog to dlog of the pullback
ModularCurve.genDiffModL_dia_inv_smul_D0 below · cited by 1 · depth 31 - Hecke correspondence preserves supersingular-polar differentials, transforming residues
ModularCurve.heckeDiffModLH_mem_ssPolarDifferentials_and_residue_eq_sum_fiberAlong_of_prime418 below · cited by 3 · depth 31 - Vanishing of dlogΨ on ordinary corner finite-part classes
ModularCurve.inv_smul_D_reducedRootFunction_eq_zero_iff_exists_point_reducesToOne_of_mem_corner_of_mem_finPts_tauFree_raynaud_bridgePins1,448 below · cited by 1 · depth 31 - Cusp-regular level-M' functions are integral over ℚ[j]
ModularCurve.isIntegral_adjoin_jq_of_cuspRegular190 below · cited by 3 · depth 31 - Integrality of the modular function field over K(j(qᵖ))
ModularCurve.isIntegral_inclusion_adjoin_jqNModC41 below · cited by 1 · depth 31 - Inverse-twisted cusp data gives a level-p structure on Tate(qⁿ)
ModularCurve.isLevelPStructure_tateBase_cuspData_neg_of_dvd74 below · cited by 5 · depth 31 - Supersingular node pairs are stable under diamond operators
ModularCurve.isNodeStable_ofAlgAut_diamondActionModL_of_forall_mem_iff_mem_ssNodePairsQExp_of_not_dvd1,239 below · cited by 1 · depth 31 - Transport of the in-line divisibility under a Weierstrass variable change
ModularCurve.kernelVariableChangeDeg_dvd_inLineMulPoly_variableChange2 below · cited by 12 · depth 31 - Coefficient extension preserves linear independence of Laurent series
ModularCurve.linearIndependent_coeffMap_of_linearIndependent0 below · cited by 4 · depth 31 - Residue transport along Frobenius for q-decimating differential operators
ModularCurve.mem_ssPolarDifferentials_and_residue_qExpFrobeniusPlaceModL_eq_of_isFrobPushDiff161 below · cited by 3 · depth 31 - Supersingular place count is invariant under algebraically closed constant extension
ModularCurve.natCard_ssPlacesQExp_eq_natCard_ssPlacesQExp_of_isAlgClosed34 below · cited by 3 · depth 31 - Order of j at a cusp place equals minus its width
ModularCurve.ord_eq_neg_width_of_order_eq_mul_ord_of_qExpansion_slash347 below · cited by 1 · depth 31 - No elliptic points: ord_P j=3, ord_P(j-1728)=2 on X₀(N)
ModularCurve.ord_eq_three_and_ord_sub_eq_two_of_ord_pos_laurentBaseChange_gamma0_of_no_elliptic265 below · cited by 1 · depth 31 - Regular-differential half of the ordinary corner count at p
ModularCurve.pow_finrank_map_corner_regularDifferentials_mul_ncard_reducesToOne_eq_ncard_finPts_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins4_of_algEquiv3,513 below · cited by 2 · depth 31 - Ordinary corner: supersingular residues versus finite p-torsion
ModularCurve.pow_finrank_map_residue_range_corner_mul_ncard_finPts_eq_natCard_corner_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins4_of_algEquiv3,518 below · cited by 1 · depth 31 - Reduced root function of T_ℓ x and U_q x as a norm
ModularCurve.reducedRootFunction_genOpH_T_eq_smul_pow_mul_norm_heckeBetaModLH_of_abelJacobiPin_tauFree_of_algEquiv676 below · cited by 1 · depth 31 - Frobenius twist of the reduced root function under Uₚ
ModularCurve.reducedRootFunction_genOpH_U_self_eq_smul_pow_mul_of_coe_eq_coeffMap_frobenius_of_abelJacobiPin_tauFree_of_mem_infSubgroup_of_algEquiv446 below · cited by 1 · depth 31 - Reduced root function under the diamond operator ⟨ e⟩
ModularCurve.reducedRootFunction_genOpH_dia_eq_smul_pow_mul_diamondActionModL_of_abelJacobiPin_tauFree484 below · cited by 1 · depth 31 - j transcendental and K/L(j) finite separable at level (N,H)
ModularCurve.transcendental_and_finiteDimensional_and_isSeparable_adjoin_of_coe_eq_coeffEmb_jq_of_eq_laurentBaseChange_xHFunctionField_of_charZero114 below · cited by 12 · depth 31 - Base change of models reads stalk maps coefficientwise on q-expansions
ModularCurve.coe_ffEquiv_symm_algebraMap_stalkMap_fst_eq_coeffEmb_ffEquiv_symm_of_baseChange41 below · cited by 1 · depth 32 - Coefficient base change intertwines reduced diamond actions on q-expansions
ModularCurve.coeffMap_coe_apply_eq_coe_apply_coeffMap_of_isDiamondPullbackModL103 below · cited by 1 · depth 32 - Coefficient base change commutes with the degeneracy norm
ModularCurve.coeffMap_coe_norm_along_heckeAlphaModLH_eq_coe_norm_along_heckeAlphaModLH_coeffMap1 below · cited by 1 · depth 32 - Coefficients of 2yᵥ+xᵥ at Tate cusp points
ModularCurve.coeff_two_mul_cuspPoint_snd_add_fst1 below · cited by 5 · depth 32 - Constant term of (xᵥ+1/12)² at Tate cusps
ModularCurve.coeff_zero_cuspPoint_fst_add_inv_twelve_sq1 below · cited by 7 · depth 32 - Constant term of 2yᵥ + xᵥ at Tate cusp points
ModularCurve.coeff_zero_two_mul_cuspPoint_snd_add_fst2 below · cited by 5 · depth 32 - Places, L(D) and Riemann's equality for ̄ F_Γ over ̄ K
ModularCurve.deg_eq_one_and_finiteDimensional_lSpace_and_ell_eq_qExpFunctionFieldC_of_isAlgClosed122 below · cited by 5 · depth 32 - Descent of a Hecke idempotent to J_H[p]
ModularCurve.exists_addMonoidHom_torsion_proj_smul_eq_of_isIdempotentElem_tateModule_jH263 below · cited by 4 · depth 32 - Atkin–Lehner laws ascend along an extension of constants
ModularCurve.exists_algEquiv_laws_of_algEquiv_laws_of_algebra_of_charP108 below · cited by 1 · depth 32 - A root function for D pushes to one for Uₚ[D]
ModularCurve.exists_coe_eq_correspondence_and_mk_eq_genOpH_U_mk_and_smul_norm_ne_zero_and_forall_mul_smul_eq_ord0 below · cited by 1 · depth 32 - Root functions transport along the Hecke correspondence at ℓ
ModularCurve.exists_coe_eq_correspondence_and_mk_eq_heckeOperatorHAlong_mk_and_smul_norm_ne_zero_and_forall_mul_smul_eq_ord0 below · cited by 1 · depth 32 - Logarithmic supersingular polar differentials lie in the image of Theta₀
ModularCurve.exists_dlogReducedRoot_eq_of_eq_inv_smul_d_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins42,411 below · cited by 1 · depth 32 - Choice slack for p-th root witnesses of divisor classes
ModularCurve.exists_eq_add_ord_and_eq_algebraMap_mul_mul_smul_pow_of_pic0Mk_eq56 below · cited by 3 · depth 32 - Gauss reduction onto the mod-p q-expansion fields
ModularCurve.exists_gaussReduction_pair_surjective_ker_heckeAlpha_heckeBeta_diamondLift_of_liesOverPrime_xHTop564 below · cited by 1 · depth 32 - Gauss valuation ring on ℚ̄· F(Γ) at a place above p
ModularCurve.exists_gaussValuationSubring_laurentBaseChange_qExpFunctionFieldC_of_liesOverPrime2 below · cited by 2 · depth 32 - Invariant function with prescribed divisor pairing to Abel–Jacobi sums
ModularCurve.exists_invariant_localModel_tendsto_integral_dbarLogDeriv_smoothedFundamental_periodAlongOf10 below · cited by 1 · depth 32 - Mod-p Atkin–Lehner twist of supersingular-polar differentials
ModularCurve.exists_linearEquiv_ssPolarDifferentials_atkinLehnerPinAlong_and_mem_regularDifferentials_iff1,785 below · cited by 1 · depth 32 - Regular logarithmic supersingular-polar differentials lie in the image of Theta₀
ModularCurve.exists_mem_finPts_and_dlogReducedRoot_eq_of_mem_regularDifferentials_of_eq_inv_smul_d_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins42,365 below · cited by 2 · depth 32 - Weight-two form with Tate abscissa q-expansion X(cqᵇ,q^N)
ModularCurve.exists_modularForm_qExpansion_coeff_eq_coeff_slotSubst_tateUnivX3 below · cited by 2 · depth 32 - Toric Tate abscissa as weight-two form on Γ₁(N)∩Γ₀(N²)
ModularCurve.exists_modularForm_qExpansion_coeff_eq_coeff_tateToricPoint3 below · cited by 2 · depth 32 - Monic relation of degree ≤ p+1 for j over K(j(qᵖ))
ModularCurve.exists_monic_natDegree_le_aeval_jqModC_eq_zero0 below · cited by 1 · depth 32 - Integral n-th power adjustment of modular functions over ℚ̄
ModularCurve.exists_mul_pow_coe_eq_coeffMap_and_coeffMap_residue_ne_zero_laurentBaseChange_qExpFunctionFieldC2 below · cited by 1 · depth 32 - Atkin–Lehner pin commutes with the Hecke norm up to a scalar
ModularCurve.exists_ofAlgAut_smul_norm_heckeBetaHBar_inv_smul_eq_algebraMap_mul_norm_heckeBetaHBar_of_ne333 below · cited by 1 · depth 32 - Reduction of divisors of X_H(M) at p ‖ M
ModularCurve.exists_regularProlongation_placeMap_xHFunctionFieldBar_of_dvd_of_not_sq_dvd1,260 below · cited by 2 · depth 32 - Reduction of ℤ̄ to a characteristic-p algebraically closed field
ModularCurve.exists_ringHom_integralClosure_int_complex_apply_natCast_eq_zero0 below · cited by 2 · depth 32 - Reduction of a finite p-torsion class: pE is div(Ψ x)
ModularCurve.exists_section_toPic0Pair_reduction_eq_mk_and_mul_eq_ord_reducedRootFunction_of_mem_finPts_tauFree1,168 below · cited by 1 · depth 32 - Degree ℓ+1 of the ℓ-degeneracy inclusion when p ∥ M
ModularCurve.finrankAlong_heckeAlphaModLH_eq_and_relfinrank_eq_add_one_of_charP_of_dvd591 below · cited by 1 · depth 32 - Degree ℓ for the ℓ-degeneracy extension when ℓ∣ M/p
ModularCurve.finrankAlong_heckeAlphaModLH_eq_and_relfinrank_eq_of_charP_of_dvd_div591 below · cited by 1 · depth 32 - Exchanging j and j(q^ℓ) preserves the Gauss valuation ring
ModularCurve.forall_apply_mem_gaussValuationSubring_iff_of_apply_j_eq_of_liesOverPrime_xHTop294 below · cited by 1 · depth 32 - Vanishing of dlog(e_K g) versus triviality of [E]
ModularCurve.inv_smul_D_eq_zero_iff_mk_eq_zero_of_coe_eq_coeffMap_of_forall_mul_eq_ord75 below · cited by 1 · depth 32 - Tate curve: toric generators give a Γ₀(p^k)-structure
ModularCurve.isGamma0PowAt_tateBase_prod_X_sub_C_toricPoint_fst19 below · cited by 1 · depth 32 - Mazur's cusp is a level-p structure on Tate(qᵖ)
ModularCurve.isLevelPStructure_tateBase_cuspData_mazurCusp73 below · cited by 2 · depth 32 - Maximality at the cusp of the j⁻¹-chart
ModularCurve.isMaximal_of_jInvChartInf_mem_of_forall_mem_nonunits_gauss_chartAlgInf0 below · cited by 3 · depth 32 - Unramifiedness of A[j₀]→ B₀ at horizontal primes avoiding 0,1728
ModularCurve.isUnramifiedAt_polynomial_chartAlgFin_gamma0_of_height_eq_one_of_jChartFin_not_mem116 below · cited by 1 · depth 32 - Unramifiedness at vertical primes over the j-line for X₀(M')
ModularCurve.isUnramifiedAt_polynomial_chartAlgFin_gamma0_of_height_eq_one_of_mem_of_not_dvd899 below · cited by 1 · depth 32 - Cartier-surjective subspaces consist of logarithmic regular differentials
ModularCurve.le_span_setOf_logarithmic_of_forall_mem_regularDifferentials_of_frobPushDiffModL_surjOn182 below · cited by 1 · depth 32 - Uₚ-surjective polar differentials lie in the logarithmic span
ModularCurve.le_span_setOf_logarithmic_of_frobPushDiffModL_surjOn197 below · cited by 1 · depth 32 - Glued differentials vanishing on one component are regular
ModularCurve.mem_regularDifferentials_of_mem_twoCompRegularDifferentials_of_fst_eq_zero58 below · cited by 1 · depth 32 - Order of J_H(M)[p] equals p^{h+toricRank}
ModularCurve.natCard_torsion_eq_pow_height_add_toricRank_of_abelJacobiPin_tauFree2,285 below · cited by 3 · depth 32 - Lower bound -h ≤ ord_P(j) at a cusp place
ModularCurve.neg_width_le_ord_of_order_eq_mul_ord_of_qExpansion_slash7 below · cited by 1 · depth 32 - Orders of j and j-1728 on X₀(N) without elliptic points
ModularCurve.ord_eq_three_and_ord_sub_eq_two_of_ord_pos_laurentBaseChange_gamma0_of_no_elliptic_algebraicClosure263 below · cited by 1 · depth 32 - Weil pairing on J_H(M)[p]: bilinearity, perfectness, equivariance
ModularCurve.perfectPairing_nsmul_eq_zero_galois_heckeH_diamondH_forall_addSubgroup_eq_biannihilator_toric_orthogonal_fin_of_abelJacobiPin_of_divisorialWeilPairingData_of_degeneracyData3,127 below · cited by 2 · depth 32 - Decimation of a logarithmic vartheta-derivative commutes with Frobenius
ModularCurve.qDecimate_inv_mul_qEuler_eq_inv_mul_qEuler_coeffMap_frobenius0 below · cited by 1 · depth 32 - Diamond operators commute with the arithmetic Frobenius
ModularCurve.qExpArithFrobC_smul_diamondActionModL_eq_diamondActionModL_qExpArithFrobC_smul290 below · cited by 1 · depth 32 - Level drop at p ‖ M for q-expansion fields over Γ₀(ℓ)
ModularCurve.qExpFunctionFieldC_gammaH_inf_gamma0_mul_eq_infSubgroup_inf_gamma0_mul_of_charP133 below · cited by 1 · depth 32 - Degree ℓ+1 of the first degeneracy inclusion of q-expansion fields
ModularCurve.relfinrank_qExpFunctionFieldC_gammaH_gammaH_inf_gamma0_mul_eq_add_one554 below · cited by 3 · depth 32 - Degree ℓ for Γ_{H'}(N)∩Γ₀(Nℓ) when ℓ∣ N
ModularCurve.relfinrank_qExpFunctionFieldC_gammaH_gammaH_inf_gamma0_mul_eq_of_dvd554 below · cited by 2 · depth 32 - Degeneracy maps preserve and detect supersingular places
ModularCurve.restrictAlong_heckeBetaModLH_mem_ssPlacesQExp_iff_and_restrictAlong_heckeAlphaModLH_mem_ssPlacesQExp_iff_of_prime256 below · cited by 1 · depth 32 - Atkin–Lehner-pinned differentials span the supersingular-polar space
ModularCurve.span_ssPolarDifferentials_atkinLehnerPinned_eq_top363 below · cited by 2 · depth 32 - Toric points of Tate(q^M) add by multiplying parameters
ModularCurve.toricPoint_add_toricPoint_tateBase_of_charZero7 below · cited by 12 · depth 32 - A pinned twist intertwines the transposed ℓ-correspondence with T_ℓ
ModularCurve.twist_correspondence_heckeT_eq_genDiffModL_T_of_atkinLehnerPinAlong1,432 below · cited by 1 · depth 32 - Pinned twist carries the q-correspondence to U_q, q≠ p
ModularCurve.twist_correspondence_heckeU_eq_genDiffModL_U_of_atkinLehnerPinAlong_of_ne914 below · cited by 1 · depth 32 - Pinned Atkin–Lehner twist commutes with Uₚ
ModularCurve.twist_genDiffModL_U_self_eq_of_atkinLehnerPinAlong_of_mem_infSubgroup217 below · cited by 1 · depth 32 - Pinned twist sends ⟨ d⁻¹⟩ to ⟨ d⟩
ModularCurve.twist_genDiffModL_dia_inv_eq_genDiffModL_dia_of_atkinLehnerPinAlong1,401 below · cited by 2 · depth 32 - Laws of the Atkin–Lehner operator θ=θₚ⁻¹w_M
ModularCurve.atkinLehner_complement_laws_of_fricke_of_atkinLehner_p365 below · cited by 2 · depth 33 - Pinned Atkin–Lehner operators on supersingular-polar differentials are bijective
ModularCurve.bijective_of_atkinLehnerPinAlong1,404 below · cited by 2 · depth 33 - Base change of the q-expansion function field along ι
ModularCurve.coeffMap_mem_qExpFunctionFieldC_and_eq_adjoin_image_coeffMap0 below · cited by 2 · depth 33 - Mod-p reduction of a p-fold twisted product law
ModularCurve.coeffMap_residue_eq_C_mul_coeffMap_frobenius_of_qExpand_eq_C_mul_prod_qTwist0 below · cited by 1 · depth 33 - 2Y+X=DX for the universal Tate coordinates
ModularCurve.coeff_two_mul_tateUnivY_add_tateUnivX0 below · cited by 1 · depth 33 - φ-linearity of the q-expansion pin of ρ^∞
ModularCurve.diffQExp_sum_smul_apply_tmul_intTwoCuspReduce_eq_ofPowerSeries_map_of_isInfReductionMap0 below · cited by 4 · depth 33 - Tate-curve divisibility of the level kernel into `inLineMulPoly`
ModularCurve.dvd_inLineMulPoly_of_map_eq_variableChange_tateBase_tateToricPoint_of_map_eq_kernelVariableChangeDeg19 below · cited by 3 · depth 33 - Toric p-torsion points on Tate(qᵖ) kill ψₚ
ModularCurve.eval_prePsi_tateBase_tateToricPoint_eq_zero37 below · cited by 1 · depth 33 - Galois descent of automorphisms along a constant-field extension
ModularCurve.exists_algEquiv_laurentBaseChange_coeffMap_eq_of_arithmeticGalois_comm1 below · cited by 2 · depth 33 - Descent of p-divisible divisors along constant-field extensions
ModularCurve.exists_eq_mul_pow_mul_of_coe_eq_coeffMap_of_forall_dvd_ord176 below · cited by 1 · depth 33 - Bounded denominators on X_H(M) from A-integral j-values
ModularCurve.exists_forall_coeff_smul_mem_of_forall_ord_neg_xH54 below · cited by 1 · depth 33 - Gauss presentations are invariant under q ↦ q^N
ModularCurve.exists_gaussPresentation_qExpand_iff0 below · cited by 1 · depth 33 - Γ-invariant function with prescribed divisor and periodised partial̄-kernel
ModularCurve.exists_invariant_localModel_dbarLogDeriv_eq_sum_finsum_translate_of_finiteIndex4 below · cited by 1 · depth 33 - Galois-fixed basis of a stable Riemann–Roch space on X_H(M)
ModularCurve.exists_linearIndependent_riemannRochSpace_forall_arithmeticGalois_smul_eq_xH2 below · cited by 1 · depth 33 - Integral descent of Γ₀(M')-kernel tuples, componentwise
ModularCurve.exists_map_eq_and_isGamma0PowAt_tuple_of_isGamma0PowAt_map1 below · cited by 4 · depth 33 - Level-ℓ structures over K descend to the valuation ring
ModularCurve.exists_map_eq_and_isLevelPStructure_of_isLevelPStructure_map13 below · cited by 3 · depth 33 - Finite-part lifting for the reduced root function Ψ
ModularCurve.exists_mem_finPts_and_reducedRoot_eq_mul_pow_mul_of_coe_eq_coeffMap_of_forall_dvd_ord_of_abelJacobiPin_of_representsRelSubPicLevel_raynaud_bridgePins_noKTransport_bridgePins42,332 below · cited by 1 · depth 33 - qᵖ-expansion of the Atkin–Lehner conjugate of a Uₚ-norm
ModularCurve.exists_qExpand_coe_smul_norm_heckeBetaHBar_inv_smul_eq_C_mul_prod_qTwist330 below · cited by 1 · depth 33 - Completed stalk of two-chart model versus 𝔭-adic moduli completion
ModularCurve.exists_ringEquiv_adicCompletion_stalk_adicCompletion_comap_of_ker_classify_le_pow11 below · cited by 2 · depth 33 - Supersingularity of the universal j-invariant at 𝔭
ModularCurve.forall_map_j0_mem_ssJSet_of_ker_eq_comap_of_jOf_eq_jqNModC397 below · cited by 2 · depth 33 - Fricke laws on Pic⁰ of X_H(M), pinned by slash and Galois clauses
ModularCurve.frickeAlgEquiv_pic0_laws_of_slash_fricke_of_galois_smul78 below · cited by 2 · depth 33 - Toric ℓ-torsion points give Γ₁(ℓ)-points on the Tate curve
ModularCurve.isGamma1Point_tateBase_tateToricPoint_of_isPrimitiveRoot17 below · cited by 3 · depth 33 - Uniformiser generates a prime ideal of the j-chart algebra
ModularCurve.isPrime_span_algebraMap_chartAlgFin_xHFunctionField_top_of_natCast_ne_zero879 below · cited by 3 · depth 33 - Regularity of the two-chart model at the cusp ∞
ModularCurve.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_laurentBaseChange_cuspInfty223 below · cited by 3 · depth 33 - Residue separability over the j-line for X₀(M') when q ∤ M'
ModularCurve.isSeparable_residueField_polynomial_chartAlgFin_gamma0_of_height_eq_one_of_mem_of_not_dvd896 below · cited by 1 · depth 33 - Pinned Atkin–Lehner map preserves and reflects regular differentials
ModularCurve.mem_regularDifferentials_iff_of_atkinLehnerPinAlong1,784 below · cited by 1 · depth 33 - Index of the finite part in J_H(M)[p] for p ‖ M
ModularCurve.natCard_torsion_eq_pow_card_ssPlacesQExp_sub_one_mul_natCard_finPts_of_abelJacobiPin_tauFree10 below · cited by 1 · depth 33 - Model-fixing coordinate changes versus rational automorphisms preserving ⟨ g⟩
ModularCurve.natCard_variableChange_smul_eq_and_kernelVariableChangeDeg_eq_eq_natCard_rationalAut_map_zmultiples_eq22 below · cited by 1 · depth 33 - Invariance of automorphism counts under a field isomorphism
ModularCurve.natCard_variableChange_smul_eq_and_kernelVariableChangeDeg_eq_eq_of_ringEquiv0 below · cited by 1 · depth 33 - Fricke-twisted Weil pairing on J_H(M)[n]: nine properties
ModularCurve.pairing_nsmul_eq_zero_galois_heckeH_diamondH_biannihilator_of_divisorialWeilPairingData_frickeAlgEquiv513 below · cited by 3 · depth 33 - Unramifiedness of X₀(M')→ X(1) away from j=0,1728
ModularCurve.ramificationIndexAlong_val_adjoin_eq_one_of_ord_eq_zero_of_ord_sub_eq_zero_laurentBaseChange_gamma0109 below · cited by 1 · depth 33 - Torsion basis at the Tate cusp pair of level q
ModularCurve.torsion_basis_of_map_eq_variableChange_tateBase_cuspData86 below · cited by 2 · depth 33 - Atkin–Lehner pin preserves regularity of supersingular-polar differentials
ModularCurve.coe_map_mem_regularDifferentials_of_atkinLehnerPinAlong1,778 below · cited by 1 · depth 34 - Divisor-sum coefficients of (xᵥ+1/12)² at Tate cusp points
ModularCurve.coeff_cuspPoint_fst_add_inv_twelve_sq39 below · cited by 2 · depth 34 - Order of the slot-substituted universal Tate x-series
ModularCurve.coeff_slotSubst_tateUnivX_eq_zero_of_lt_min_and_coeff_eq_and_order_eq_min1 below · cited by 3 · depth 34 - Twisting the Tate slot parameter by ζ modulo ζ-1
ModularCurve.coeff_slotSubst_tateUnivX_mul_sub_coeff_mem_span_sub_one1 below · cited by 1 · depth 34 - Scaling j by q² moves the Gauss valuation ring
ModularCurve.comap_gauss_ne_of_coe_apply_eq_qExpand_sq_jqModC0 below · cited by 1 · depth 34 - Triviality of ⟨ d⟩ on X_H(M) for d ∈ ± H
ModularCurve.diamondAutHBar_eq_refl_of_mem_or_neg_mem0 below · cited by 1 · depth 34 - Normalisation of the x-chart commutes with base change to A
ModularCurve.exists_algEquiv_tensorProduct_chartAlg_laurentBaseChange_of_smooth_genericFibre_of_isReduced_specialFibre14 below · cited by 2 · depth 34 - Vanishing residue forces divisibility by q for cusp-regular modular functions
ModularCurve.exists_eq_natCast_mul_of_residue_eq_zero_of_mem_integers_of_cuspRegular136 below · cited by 2 · depth 34 - Descent of a Γ₀(p^k) kernel polynomial to an integrally closed domain
ModularCurve.exists_map_eq_and_isGamma0PowAt_of_isGamma0PowAt_map0 below · cited by 1 · depth 34 - Non-toric division values as weight-two forms on Γ₁(n)∩Γ₀(Nn)
ModularCurve.exists_modularForm_gamma1_inf_gamma0_mul_qExpansion_coeff_eq_coeff_slotSubst_tateUnivX3 below · cited by 2 · depth 34 - Toric abscissa on Tate(q^N) as a weight-two form
ModularCurve.exists_modularForm_gamma1_inf_gamma0_mul_qExpansion_coeff_eq_coeff_tateToricPoint3 below · cited by 2 · depth 34 - Completed stalk of the two-chart model as 𝔭-adic completion
ModularCurve.exists_ringEquiv_adicCompletion_stalk_adicCompletion_comap_of_ker_classify_le_pow_of_isPrimitiveRoot_mul_of_dvd12 below · cited by 2 · depth 34 - Reduced q-expansions embed the residue field at a vertical prime
ModularCurve.exists_ringHom_residueField_laurentSeries_injective_mem_range_chartAlgFin_gamma0_of_height_eq_one_of_not_dvd890 below · cited by 1 · depth 34 - Residue degree bound ≤ ψ(M') at height-one primes of X₀(M')
ModularCurve.finite_and_finrank_residueField_le_dedekindPsi_polynomial_chartAlgFin_gamma0_of_height_eq_one_of_not_dvd128 below · cited by 1 · depth 34 - Degree of X(Γ_H(M)∩Γ₀(Mt))→ X_H(M) is positive and bounded by the index
ModularCurve.finrankAlong_heckeAlphaHBar_pos_and_le_relIndex202 below · cited by 1 · depth 34 - Kernel tuple fixed iff the cyclic subgroup is preserved
ModularCurve.forall_kernelVariableChangeDeg_eq_iff_image_equivOfVariableChangeEq_zmultiples_eq17 below · cited by 1 · depth 34 - Supersingularity of the universal j-invariant at 𝔭
ModularCurve.forall_map_j0_mem_ssJSet_of_ker_eq_comap_of_jOf_eq_jqNModC_of_isPrimitiveRoot_mul_of_dvd79 below · cited by 2 · depth 34 - μ_M gives a cyclic M-kernel polynomial on Tate(qⁿ)
ModularCurve.isCyclicKernel_tateBase_prod_X_sub_C_toricPoint_fst15 below · cited by 1 · depth 34 - Cusp of Tate(qⁿ) gives a level-ℓ structure when ℓ ∣ n
ModularCurve.isLevelPStructure_tateBase_cuspData_of_dvd74 below · cited by 1 · depth 34 - j(q^N) lies in the j-finite chart algebra
ModularCurve.jqNModC_mem_chartAlgFin_of_mem86 below · cited by 7 · depth 34 - Inverting the Tate parameter negates the point
ModularCurve.nonToricPoint_inv_tsub0 below · cited by 2 · depth 34 - Cusp place on the ∞-component is unramified over X₀(M')
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul602 below · cited by 1 · depth 34 - Toric Tate cusp points: (xᵥ+1/12)² coefficients
ModularCurve.coeff_cuspPoint_fst_add_inv_twelve_sq_of_eq_zero37 below · cited by 1 · depth 35 - Non-toric Tate cusp: coefficients of (xᵥ+tfrac112)²
ModularCurve.coeff_cuspPoint_fst_add_inv_twelve_sq_of_ne_zero37 below · cited by 1 · depth 35 - Closed product formula for ψ(N)
ModularCurve.dedekindPsi_eq_prod_primeFactors2 below · cited by 3 · depth 35 - Mod-p Fricke involutions exchanging the two degeneracy maps
ModularCurve.exists_algEquiv_pair_qExpFunctionFieldC_intertwines_heckeAlphaModLH_heckeBetaModLH_and_reduction_slash_fricke584 below · cited by 1 · depth 35 - Atkin–Lehner automorphism of the function field at cofactor M/p
ModularCurve.exists_algEquiv_xHFunctionFieldBar_slash_atkinLehnerCofactor42 below · cited by 1 · depth 35 - Γ₁(ℓ)-points of the generic fibre descend to R₀
ModularCurve.exists_map_eq_and_isGamma1Point_of_isGamma1Point_map1 below · cited by 3 · depth 35 - Weight-two toric division values on Γ₁(n)∩Γ₀(Nn)
ModularCurve.exists_modularForm_gamma1_inf_gamma0_mul_qExpansion_eq_tateToricPoint_fst_and_slash_conjElemN_eq3 below · cited by 1 · depth 35 - Weight-three toric family on Γ₁(n)∩Γ₀(Nn)
ModularCurve.exists_modularForm_gamma1_inf_gamma0_mul_weight_three_qExpansion_eq_tateToricPoint_and_slash_conjElemN_eq5 below · cited by 1 · depth 35 - Automorphisms sending j to j(q^N) preserve ∞-integrality mod q
ModularCurve.exists_mul_coeffMap_eq_iff_of_algEquiv_apply_jq_eq_jqN_of_not_dvd353 below · cited by 1 · depth 35 - Integral (ζ'-1)-divisible difference of Tate toric x-series
ModularCurve.exists_powerSeries_forall_coeff_mem_span_sub_one_ofPowerSeries_eq_tateToricPoint_mul_sub_tateToricPoint0 below · cited by 1 · depth 35 - Toric minus non-toric Tate abscissa: integral, nonzero reduction
ModularCurve.exists_powerSeries_map_residue_ne_zero_ofPowerSeries_eq_tateToricPoint_sub_nonToricPoint2 below · cited by 1 · depth 35 - An integral q-series with non-zero reduction: toric minus non-toric abscissa
ModularCurve.exists_powerSeries_map_residue_ne_zero_ofPowerSeries_eq_tateToricPoint_sub_nonToricPoint_units2 below · cited by 1 · depth 35 - Atkin–Lehner transform at the cofactor of a rational even-weight form
ModularCurve.exists_slash_atkinLehnerCofactor_eq_sum_smul_of_ratCast_qExpansion_of_even41 below · cited by 2 · depth 35 - Degree of X_H(M) over the j-line over any ℚ-algebra field
ModularCurve.finrank_adjoin_jqModC_laurentBaseChange_qExpFunctionFieldC_gammaH_eq_index_of_algebraRat227 below · cited by 6 · depth 35 - Equal degrees of K over L(j(q)) and L(j(q^N))
ModularCurve.finrank_adjoin_jqNModC_eq_finrank_adjoin_jqModC_of_squarefree122 below · cited by 2 · depth 35 - Slot substitution commutes with base change
ModularCurve.map_slotSubst0 below · cited by 1 · depth 35 - Regularity of ω₁ versus (ω₁,0) being glued-regular
ModularCurve.mem_regularDifferentials_iff_mem_twoCompRegularDifferentials_prod_zero58 below · cited by 1 · depth 35 - Counting Γ₀-power cyclic kernel tuples equals ψ(M')
ModularCurve.natCard_isGamma0PowAt_tuple_eq_prod_of_isAlgClosed32 below · cited by 3 · depth 35 - Counting Katz level-ℓ structures on an elliptic curve
ModularCurve.natCard_levelPData_isLevelPStructure_eq_natCard_GL_of_isAlgClosed6 below · cited by 2 · depth 35 - Cusps on the Gauss branch: ord_w j(q²τ)=q² ord_w j
ModularCurve.ord_jqN_sq_eq_sq_mul_ord_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul81 below · cited by 1 · depth 35 - q-expansion function field depends only on ±Γ
ModularCurve.qExpFunctionFieldC_eq_of_le_of_forall_mem_or_neg_mem0 below · cited by 2 · depth 35 - Level drop from Γ₀(N) to Γ₀(N/p) in characteristic p
ModularCurve.qExpFunctionFieldC_gamma0_eq_qExpFunctionFieldC_gamma0_div_of_sq_dvd6 below · cited by 2 · depth 35 - The substitution q ↦ q^N preserves W₀ and its maximal ideal
ModularCurve.qExpand_mem_and_mem_nonunits_of_forall_mem_iff_exists_powerSeries0 below · cited by 2 · depth 35 - Unramified cusp place above X₀(M') at level q²M', q=3
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul_of_eq_three602 below · cited by 1 · depth 35 - Unramifiedness over X₀(M') at an ∞-type cusp place, q=2
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul_of_eq_two602 below · cited by 1 · depth 35 - Cusps of ∞-type on X₀(q²M') are unramified over X₀(M')
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jqN_sq_eq_sq_mul_ord_gamma0_sq_mul597 below · cited by 1 · depth 35 - Cusps of X_H(N) unramified over X₀(N)
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_neg_laurentBaseChange_gammaH_algebraicClosure267 below · cited by 3 · depth 35 - Fricke-type automorphism sends j(q) to j(qⁿ)
ModularCurve.coe_apply_eq_qExpand_jqModC_of_forall_coeffMap_mul_qExpansion_slash_fricke_eq14 below · cited by 1 · depth 36 - Integrality of the toric Tate point's q-coefficients
ModularCurve.coeff_tateToricPoint_mem_of_mem0 below · cited by 3 · depth 36 - Toric plus non-toric Tate point sums to a non-toric point
ModularCurve.eq_variableChange_nonToricPoint_pow_of_toPoint_add_toPoint_nonToricPoint_one_eq_some55 below · cited by 1 · depth 36 - Toric points on the Tate curve add: U^q+U^ℓ=U^{q+ℓ}
ModularCurve.eq_variableChange_tateToricPoint_pow_add_of_toPoint_add_toPoint_eq_some55 below · cited by 1 · depth 36 - Doubling the toric point: 2R reads as U^{2q}
ModularCurve.eq_variableChange_tateToricPoint_pow_two_mul_of_two_smul_toPoint_eq_some55 below · cited by 1 · depth 36 - Doubling the toric point U^q on a transformed Tate curve
ModularCurve.eq_variableChange_tateToricPoint_pow_two_mul_of_two_smul_toPoint_eq_some_self55 below · cited by 1 · depth 36 - Base change κ⊆ K of a Fricke automorphism pair
ModularCurve.exists_algEquiv_pair_qExpFunctionFieldC_laws_of_algEquiv_pair_laws_of_algebra_of_charP0 below · cited by 1 · depth 36 - Fricke involutions intertwining the degeneracy maps at level Mℓ
ModularCurve.exists_algEquiv_pair_xHFunctionFieldBar_slash_fricke_intertwines_heckeAlphaHBar_heckeBetaHBar77 below · cited by 1 · depth 36 - Cyclic subgroups of order p^k match Γ₀-kernel polynomials
ModularCurve.exists_equiv_addSubgroup_isAddCyclic_isGamma0PowAt_of_isAlgClosed22 below · cited by 1 · depth 36 - Gauss reduction at a place over p intertwining α and β
ModularCurve.exists_gaussReduction_pair_surjective_ker_heckeAlpha_heckeBeta_of_liesOverPrime_xHTop564 below · cited by 1 · depth 36 - Coefficients of j(mathsf q^{qn})-j(mathsf qⁿ)^q lie in the maximal ideal
ModularCurve.exists_mem_maximalIdeal_eq_coeff_jqNModC_mul_sub_pow3 below · cited by 2 · depth 36 - K₀(J₂)=K₀[J₂]: generation of the Γ₀(q²M') q-expansion field
ModularCurve.exists_polynomial_eval2_inclusion_eq_of_laurentBaseChange_gamma0_sq_mul594 below · cited by 1 · depth 36 - Integrality of x_T(c)-x_T(c²) with unit constant term
ModularCurve.exists_powerSeries_map_residue_ne_zero_ofPowerSeries_eq_tateToricPoint_sub_tateToricPoint_sq0 below · cited by 1 · depth 36 - Two regular prolongations above q∤ M, and no third
ModularCurve.exists_regularProlongation_pair_xHTopFunctionFieldC_eq_or_eq_of_not_dvd350 below · cited by 1 · depth 36 - ℚ̄-q-expansion field of Γ inside the level-M function field
ModularCurve.exists_ringHom_laurentBaseChange_qExpFunctionFieldC_levelN_qExpansion_forall_le19 below · cited by 1 · depth 36 - Gauss valuation ring inside L((q)) over a DVR subring
ModularCurve.exists_subfield_valuationSubring_laurentSeries_gauss_of_isDiscreteValuationRing0 below · cited by 2 · depth 36 - Tate curve over the Gauss ring: smooth model and toric point
ModularCurve.exists_tateBase_eq_map_and_tateToricPoint_mem_nonunits1 below · cited by 2 · depth 36 - Stability of the Gauss valuation ring under jleftrightarrow j(q^m)
ModularCurve.forall_apply_mem_gaussValuationSubring_iff_of_apply_jqModC_eq_qExpand_of_liesOverPrime294 below · cited by 1 · depth 36 - Integrality of g¹²/Δ^k over ℤ[j] and ℤ[j⁻¹]
ModularCurve.isIntegralElem_div_delta_pow_and_div_eisenstein4_pow_of_forall_qExpansion_slash_isIntegral5 below · cited by 1 · depth 36 - Irreducibility of the π₀-fibre of the j-finite chart
ModularCurve.le_of_mem_minimalPrimes_span_of_isPrime_chartAlgFin_xHFunctionField_of_not_dvd319 below · cited by 1 · depth 36 - Gauss-branch cusps: ord_w J₂ = q²ord_w j for q=3
ModularCurve.ord_jqN_sq_eq_sq_mul_ord_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul_of_eq_three81 below · cited by 1 · depth 36 - Gauss-branch cusp: ord_w J₂ = q²ord_w j for q=2
ModularCurve.ord_jqN_sq_eq_sq_mul_ord_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_gamma0_sq_mul_of_eq_two81 below · cited by 1 · depth 36 - Unramified ∞-type cusps of X₀(q²M') over X₀(M'), q=3
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jqN_sq_eq_sq_mul_ord_gamma0_sq_mul_of_eq_three597 below · cited by 1 · depth 36 - Unramifiedness of ∞-type cusps of X₀(4M') over X₀(M')
ModularCurve.ramificationIndexAlong_inclusion_gamma0_eq_one_of_ord_jqN_sq_eq_sq_mul_ord_gamma0_sq_mul_of_eq_two597 below · cited by 1 · depth 36 - Coefficientwise Weierstrass ODE for the universal Tate abscissa
ModularCurve.sub_one_mul_coeff_tateUnivX_eq36 below · cited by 2 · depth 36 - Transported Tate cusp pair as a q-torsion basis
ModularCurve.torsion_basis_of_map_eq_variableChange_tateBase_cuspData_of_mul_eq86 below · cited by 2 · depth 36 - Fricke involution acts on Δ through diag(p,1)
ModularCurve.discriminant_slash_fricke_eq_discriminant_slash_heckeDiagMatrix0 below · cited by 1 · depth 37 - Galois-invariant elements of ℤ[ζ_q] are rational integers
ModularCurve.exists_algebraMap_int_eq_of_mem_zetaSubring_of_forall_algEquiv_apply_eq0 below · cited by 1 · depth 37 - K₀[j(q²τ)] = K for X₀(q²M'), case q=3
ModularCurve.exists_polynomial_eval2_inclusion_eq_of_laurentBaseChange_gamma0_sq_mul_of_eq_three594 below · cited by 1 · depth 37 - At q=2, K(Γ₀(q²M'))=K₀[j(q²τ)]
ModularCurve.exists_polynomial_eval2_inclusion_eq_of_laurentBaseChange_gamma0_sq_mul_of_eq_two594 below · cited by 1 · depth 37 - Ogg's unit reduces to the supersingular polynomial in ̄ j
ModularCurve.exists_residue_eq_prod_ssJSet_of_coe_eq_coeffEmb_modularUnitSeries108 below · cited by 1 · depth 37 - A Gauss-type unramified valuation on the X_H(N) function field
ModularCurve.exists_valuationSubring_gaussType_unramified_laurentBaseChange_xHFunctionField3 below · cited by 1 · depth 37 - Generation of the level p² function field by j and j_{p²}
ModularCurve.functionFieldGeneration_sq65 below · cited by 3 · depth 37 - Igusa's degree bound over the residue field of A∩ k₀
ModularCurve.index_gammaH_le_finrank_adjoin_jqModC_qExpFunctionFieldC_residueField_comap218 below · cited by 2 · depth 37 - Frobenius on q-expansions: j(qⁿ)^q = j(q^{qn})
ModularCurve.jqNModC_pow_eq_jqNModC_mul_of_charP2 below · cited by 1 · depth 37 - Exactly ℓ-1 Γ₁(ℓ)-points linked to a Γ₀(M')-tuple
ModularCurve.natCard_levelPData_isGamma1Point_and_isGamma1Link_eq_of_isAlgClosed18 below · cited by 1 · depth 37 - q-expansion field of Γ_H(N)∩Γ₀(Nℓ) generated by j(q^ℓ)
ModularCurve.qExpFunctionFieldC_gammaH_sup_adjoin_qExpand_jqModC_eq_qExpFunctionFieldC_gammaH_inf_gamma0_of_not_dvd573 below · cited by 3 · depth 37 - Tate data at level Np as the q↦ q^N image of level p
ModularCurve.tateBase_mul_eq_map_qExpand_and_tateToricPoint_eq_and_nonToricPoint_eq0 below · cited by 6 · depth 37 - Uniqueness of Gauss-type valuations on the j-line of X_H(N)
ModularCurve.valuationSubring_eq_of_gaussType_laurentBaseChange_xHFunctionField_of_not_dvd308 below · cited by 1 · depth 37 - Mod-q collapse of level Γ_H(M)∩Γ₀(Mq) to level Γ_H(M)
ModularCurve.xHTopFunctionFieldC_residueField_mul_eq_xHFunctionFieldC_of_not_dvd346 below · cited by 1 · depth 37 - j-invariant of cyclic quotients of a base-changed Tate curve
ModularCurve.cyclicQuotientJ_smul_tateBase_baseChange_zmultiples_eq_algebraMap_jqNModC132 below · cited by 1 · depth 38 - Tate-curve point of order M' cut out by kernel polynomials
ModularCurve.exists_point_smul_tateBase_baseChange_cutOut_muTuple_of_isPrimitiveRoot9 below · cited by 1 · depth 38 - Multiples bℓ^{k-1}G are linked Γ₁(ℓ)-points
ModularCurve.isGamma1Point_and_dvd_inLineMulPoly_of_toPoint_eq_mul_pow_smul_of_isRoot15 below · cited by 1 · depth 38 - j(qᵖ) lies in ℚ(j(q),j(q^{p^2}))
ModularCurve.jqN_mem_modularFunctionField_sq46 below · cited by 1 · depth 38 - Relative degree at most ℓ+1 for Γ_H(N)∩Γ₀(Nℓ)
ModularCurve.relfinrank_qExpFunctionFieldC_gammaH_gammaH_inf_gamma0_mul_le_add_one_and_pos557 below · cited by 1 · depth 38 - Degree of j(q^ℓ) over a field containing j(q) is 1 or ℓ+1
ModularCurve.relfinrank_sup_adjoin_qExpand_jqModC_eq_one_or_eq_add_one91 below · cited by 1 · depth 38 - Coefficientwise ring maps carry Tate(q^N) to Tate(q^N)
ModularCurve.tateBase_map_coeffMap0 below · cited by 2 · depth 38 - Quotient j-invariant of the width-w Tate curve
ModularCurve.cyclicQuotientJ_smul_tateBase_baseChange_zmultiples_eq_algebraMap_jqNModC_width132 below · cited by 1 · depth 40 - Order-M' toric point on a twisted Tate curve of width w
ModularCurve.exists_point_smul_tateBase_baseChange_cutOut_muTuple_of_isPrimitiveRoot_width9 below · cited by 1 · depth 40
ModularCurve.B3 10
- Monodromy orbits upstairs match automorphism orbits downstairs
ModularCurve.B3.b3_specialisationEquivariance9 below · cited by 1 · depth 12 - Reduction is an isomorphism on p-torsion
ModularCurve.B3.exists_torsionBy_reduction_addEquiv5 below · cited by 0 · depth 12 - Special fibre of the good model is ofJ j₀ up to coordinate change
ModularCurve.B3.exists_variableChange_specialFibre_goodModel9 below · cited by 0 · depth 12 - Integral model at j₀=1728 with unit discriminant
ModularCurve.B3.goodModel_1728_spec0 below · cited by 0 · depth 12 - Good model at generic j₀: integrality, unit Δ, special fibre
ModularCurve.B3.goodModel_generic_spec0 below · cited by 0 · depth 12 - Integral model at j₀=0: unit discriminant, elliptic special fibre
ModularCurve.B3.goodModel_zero_spec0 below · cited by 0 · depth 12 - Unit discriminant gives an elliptic special fibre
ModularCurve.B3.isElliptic_specialFibre9 below · cited by 0 · depth 12 - Ellipticity of the special fibre of the good model
ModularCurve.B3.isElliptic_specialFibre_goodModel0 below · cited by 0 · depth 12 - The near-curve is the explicit model at j₀+s
ModularCurve.B3.nearCurve_eq_ofJNe0Or17280 below · cited by 0 · depth 12 - Level-N specialisation is monodromy-to-automorphism equivariant
ModularCurve.B3.specialisationEquivariance_level12 below · cited by 1 · depth 15
ModularCurve.ChainDirichlet 1
- Simultaneous chain Dirichlet problems after a depth-one correction
ModularCurve.ChainDirichlet.exists_depthOne_correction_dirichlet0 below · cited by 2 · depth 16
ModularCurve.CharPModel 88
- Good-reduction specialisation datum for J₀(N), N prime
ModularCurve.CharPModel.FibreModel.exists_jZeroGoodReductionSpecialization_sp_eq_spPic0_of_prime1,099 below · cited by 1 · depth 9 - Finite flat model of Eisenstein quotient torsion along `spPic0`
ModularCurve.CharPModel.FibreModel.exists_le_finiteFlat_model_eisensteinQuotient_torsion_spPic0_of_ne_two2,013 below · cited by 1 · depth 9 - Packaging the fibre-model specialisation as a place-specialisation packet
ModularCurve.CharPModel.FibreModel.exists_placeSpecialization_spPic0_eq_of_prime998 below · cited by 3 · depth 9 - Divisor specialisation preserves degree zero and principality
ModularCurve.CharPModel.FibreModel.spDiv_preservesPrincipal_of_reductionInputs313 below · cited by 2 · depth 9 - Normal fibre model with cusp chart for X₀(p)
ModularCurve.CharPModel.exists_fibreModel_cuspChart_integrallyClosed_of_prime743 below · cited by 1 · depth 9 - Specialisation on J₀(N) is injective on prime-to-ℓ torsion
ModularCurve.CharPModel.FibreModel.eq_zero_of_spPic0_eq_zero_of_prime_pow_smul_eq_zero_residueField1,002 below · cited by 1 · depth 10 - Hecke descent along the fibre specialisation, Eichler–Shimura at ℓ
ModularCurve.CharPModel.FibreModel.exists_heckeDescentFamily_spPic0_and_match_of_prime1,032 below · cited by 1 · depth 10 - Specialisation pushforward computes the divisor of the reduced q-expansion
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_eq_ord_coeffMap292 below · cited by 4 · depth 10 - Reduction mod ℓ on J₀(N) equals the constructed specialisation
ModularCurve.CharPModel.FibreModel.reductionModL_eq_pic0Congr_spPic0812 below · cited by 2 · depth 10 - Specialisation on Pic⁰ is induced by pushforward of divisors
ModularCurve.CharPModel.FibreModel.spPic0_compat0 below · cited by 4 · depth 10 - Finite-chart value dictionary for j under place specialization
ModularCurve.CharPModel.FibreModel.spPlace_d0_j110 below · cited by 16 · depth 10 - Specialisation of a place preserves vanishing of j(q^N)-a
ModularCurve.CharPModel.FibreModel.spPlace_d0_jN110 below · cited by 7 · depth 10 - Poles of jmath̃_N at places where jmath̄_N has no value in A
ModularCurve.CharPModel.FibreModel.spPlace_d0_jN_pole113 below · cited by 6 · depth 10 - Places giving jmath̄ no A-value specialise to j-poles
ModularCurve.CharPModel.FibreModel.spPlace_d0_j_pole113 below · cited by 8 · depth 10 - Eichler–Shimura relation at specialised places of X₀(N)
ModularCurve.CharPModel.FibreModel.spPlace_d1_of_cuspChart257 below · cited by 2 · depth 10 - Unramified Frobenius lift for the specialisation map, clause d2
ModularCurve.CharPModel.FibreModel.spPlace_d2972 below · cited by 2 · depth 10 - Specialisation carries arithmetic Frobenius to geometric Frobenius
ModularCurve.CharPModel.FibreModel.spPlace_d6_frobenius_of_cuspChart263 below · cited by 2 · depth 10 - Inertia acts trivially on specialised places
ModularCurve.CharPModel.FibreModel.spPlace_d6_inertia245 below · cited by 2 · depth 10 - Cusp dictionary at the j-pole for `spPlace`
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictInfty0 below · cited by 2 · depth 10 - Cusp dictionary for j/j_N^N at spPlace, prime level
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero_of_prime150 below · cited by 1 · depth 10 - Surjectivity of the place specialisation map of a fibre model
ModularCurve.CharPModel.FibreModel.spPlace_surjective111 below · cited by 7 · depth 10 - Normal fibre model of X₀(N) with cusp chart
ModularCurve.CharPModel.exists_fibreModel_cuspChart_integrallyClosed79 below · cited by 4 · depth 10 - Igusa lifting on both j-charts
ModularCurve.CharPModel.exists_integral_lift_jChart_and_jInvChart733 below · cited by 8 · depth 10 - Reduction on the finite chart is coefficientwise
ModularCurve.CharPModel.FibreModel.coe_piFin_eq_coeffRed116 below · cited by 16 · depth 11 - q-expansion principle for the pole chart of a fibre model
ModularCurve.CharPModel.FibreModel.coe_piInf_eq_coeffRed_of_cuspChart96 below · cited by 9 · depth 11 - Specialisation of J₀(N) intertwines T_q with the special-fibre operator
ModularCurve.CharPModel.FibreModel.heckePic0Fibre_spPic0_eq_spPic0_heckeGen_smul963 below · cited by 1 · depth 11 - Eichler–Shimura congruence on all divisors, squarefree level
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_heckeDivBar970 below · cited by 1 · depth 11 - Reduction of places equals the fibre-model specialisation map
ModularCurve.CharPModel.FibreModel.placeReductionModL_eq_spPlace810 below · cited by 3 · depth 11 - Places of the reduced modular field agreeing on the finite chart
ModularCurve.CharPModel.FibreModel.place_eq_of_forall_finChart_mem_nonunits_iff4 below · cited by 13 · depth 11 - Extensionality of places on the pole chart of a fibre model
ModularCurve.CharPModel.FibreModel.place_eq_of_forall_infChart_mem_nonunits_iff1 below · cited by 12 · depth 11 - Specialisation on Pic⁰ computes by divisor pushforward
ModularCurve.CharPModel.FibreModel.spPic0_apply0 below · cited by 1 · depth 11 - Cusp-zero dictionary, t-small branch, prime level
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero_of_t_small_of_prime145 below · cited by 1 · depth 11 - Chart dichotomy for places of the modular function field
ModularCurve.CharPModel.chart_dichotomy_jBar112 below · cited by 5 · depth 11 - Integral kernel elements at the j-chart are constant multiples
ModularCurve.CharPModel.exists_eq_const_mul_of_modularRedLocHom_eq_zero76 below · cited by 4 · depth 11 - Kernel elements integral at the inverted j-chart are constant multiples
ModularCurve.CharPModel.exists_eq_const_mul_of_modularRedLocHom_eq_zero_inv77 below · cited by 4 · depth 11 - Integrality of y x^{-m} over the inverted base
ModularCurve.CharPModel.exists_monic_eval2_inv_mul_inv_pow_eq_zero0 below · cited by 4 · depth 11 - Transported specialisation is a reduction of places mod ℓ
ModularCurve.CharPModel.FibreModel.isPlaceReductionModL_congr_spPlace299 below · cited by 3 · depth 12 - Eichler–Shimura relation on specialised principal divisors
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_heckeDivBar_of_mem_principal788 below · cited by 1 · depth 12 - A fibre model forces the residue map to be surjective
ModularCurve.CharPModel.FibreModel.red_surjective0 below · cited by 3 · depth 12 - Factoring out a constant from the kernel of a reduction
ModularCurve.CharPModel.exists_eq_const_mul_of_redHom_eq_zero0 below · cited by 2 · depth 12 - Existence and uniqueness of the value homomorphism at a place
ModularCurve.CharPModel.exists_unique_valueHom0 below · cited by 1 · depth 12 - Kernel of a reduction map equals an extended prime
ModularCurve.CharPModel.ker_eq_map_of_hasGoingDown0 below · cited by 2 · depth 12 - Fibre-model specialisation carries divisors to divisors of reductions
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_eq_ord_coeffMap_of_surjective293 below · cited by 6 · depth 13 - Eichler–Shimura relation on principal divisors, with cusp chart
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_heckeDivBar_of_mem_principal_of_cuspChart312 below · cited by 1 · depth 13 - Divisor specialisation on X₀(N) preserves degree zero and principality
ModularCurve.CharPModel.FibreModel.spDiv_preservesPrincipal_of_not_dvd296 below · cited by 2 · depth 13 - Eichler–Shimura relation for the specialisation map on places
ModularCurve.CharPModel.FibreModel.spPlace_d1_of_cuspChart_of_level257 below · cited by 2 · depth 13 - Unique unramified β-lift above a singular point, level prime to ℓ
ModularCurve.CharPModel.FibreModel.spPlace_d2_of_derivative_evalEval_eq_zero_of_level975 below · cited by 2 · depth 13 - Unique unramified β-lift above a smooth point of the reduced model
ModularCurve.CharPModel.FibreModel.spPlace_d2_of_derivative_evalEval_ne_zero_of_level228 below · cited by 2 · depth 13 - Unique unramified crossing place above a Frobenius-moved pole
ModularCurve.CharPModel.FibreModel.spPlace_d2_of_pole_of_cuspChart_of_level833 below · cited by 2 · depth 13 - Specialisation transports arithmetic Frobenius to geometric Frobenius
ModularCurve.CharPModel.FibreModel.spPlace_d6_frobenius_of_cuspChart_of_level263 below · cited by 2 · depth 13 - Inertia acts trivially on specialised places of X₀(N)
ModularCurve.CharPModel.FibreModel.spPlace_d6_inertia_of_level245 below · cited by 2 · depth 13 - Cusp dictionary in the chart j_N/j^N at j-poles
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictInfty_of_level0 below · cited by 2 · depth 13 - Cusp dictionary at the j_N-pole in the chart j/j_N^N
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero_of_level301 below · cited by 2 · depth 13 - Geometric Frobenius fixes the poles of ̃ j
ModularCurve.CharPModel.frobOnPlacesGeomLevel_eq_self_of_ord_jqModC_neg143 below · cited by 2 · depth 13 - Eichler–Shimura relation for sp_* at level N prime to ℓ
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_heckeDivBar_of_cuspChart_of_level831 below · cited by 2 · depth 14 - Finite chart ring localises to 𝒪ᵥ where jmath̃ is regular
ModularCurve.CharPModel.FibreModel.piFin_range_localizes_of_jqModC_mem93 below · cited by 8 · depth 14 - Pole chart of a fibre model localizes at non-affine places
ModularCurve.CharPModel.FibreModel.piInf_range_localizes_of_not_affine93 below · cited by 3 · depth 14 - Type-one β-ramification sum equals one at nodal points
ModularCurve.CharPModel.FibreModel.spPlace_d2_sum_ramification_typeOne_eq_one_of_level973 below · cited by 1 · depth 14 - Reduction relation at j-integral functions pins down spPlace P
ModularCurve.CharPModel.FibreModel.spPlace_eq_of_forall_residue_sub_mem_nonunits182 below · cited by 2 · depth 14 - Pole-chart place is pinned by the reduction relation
ModularCurve.CharPModel.FibreModel.spPlace_eq_of_forall_residue_sub_mem_nonunits_jInv182 below · cited by 2 · depth 14 - Fibre model with cusp chart for X₀(N) at ℓ∤ N
ModularCurve.CharPModel.exists_fibreModel_cuspChart_of_not_dvd743 below · cited by 5 · depth 14 - Integrality over A[jmath̄] from Gauss-integrality and pole-freeness
ModularCurve.CharPModel.exists_monic_eval2_affineBaseFin_eq_zero_of_mem_modularLocalized_of_forall_mem_of_jBar_mem138 below · cited by 6 · depth 14 - Pole-chart integrality over A[1/j] at level N
ModularCurve.CharPModel.exists_monic_eval2_affineBaseInf_eq_zero_of_mem_modularLocalized_of_forall_inv_jBar_mem138 below · cited by 2 · depth 14 - Place specialisation from a fibre model at arbitrary level
ModularCurve.CharPModel.exists_placeSpecialization_of_fibreModel_of_level1,056 below · cited by 2 · depth 14 - Finiteness of the level-N modular function field over ℚ̄(jmath̄)
ModularCurve.CharPModel.finiteDimensional_adjoin_jBar110 below · cited by 7 · depth 14 - Finite dimensionality of C_k(N) over k(jmath̃)
ModularCurve.CharPModel.finiteDimensional_adjoin_jLine110 below · cited by 3 · depth 14 - Places of the modular function field determined by smooth coordinates
ModularCurve.CharPModel.place_eq_of_ord_pos_of_derivative_evalEval_ne_zero1 below · cited by 4 · depth 14 - Uniqueness of places over a point smooth in the first variable
ModularCurve.CharPModel.place_eq_of_ord_pos_of_derivative_swapBivar_evalEval_ne_zero0 below · cited by 1 · depth 14 - Eichler–Shimura relation on principal divisors, level prime to ℓ
ModularCurve.CharPModel.FibreModel.mapDomain_spPlace_heckeDivBar_of_mem_principal_of_cuspChart_of_level312 below · cited by 1 · depth 15 - Centre-pinned specialisation of places on the finite j-chart
ModularCurve.CharPModel.FibreModel.placeFullC_eq_congr_spPlace_of_finChart_centrePin186 below · cited by 1 · depth 15 - Centre-pinned specialisation of places on the pole chart at a cusp
ModularCurve.CharPModel.FibreModel.placeFullC_eq_congr_spPlace_of_infChart_centrePin_of_mem_maximalIdeal184 below · cited by 1 · depth 15 - Specialisation of the Hecke correspondence at finite-centre places
ModularCurve.CharPModel.FibreModel.spDiv_heckeDivBar_eq_heckeFibreGeomLevel_of_finiteCentre_of_level972 below · cited by 1 · depth 15 - A fibre model with cusp chart for X₀(N)
ModularCurve.CharPModel.exists_fibreModel_cuspChart79 below · cited by 1 · depth 15 - Place specialisation at non-squarefree level prime to ℓ
ModularCurve.CharPModel.exists_placeSpecialization_of_fibreModel_of_level_of_not_squarefree1,021 below · cited by 1 · depth 15 - Existence of a place specialization at squarefree level
ModularCurve.CharPModel.exists_placeSpecialization_of_fibreModel_of_squarefree1,021 below · cited by 1 · depth 15 - Plane curve local ring is a DVR when partial_Y P≠ 0
ModularCurve.CharPModel.isDiscreteValuationRing_localizationAtPrime_of_derivative_evalEval_ne_zero0 below · cited by 1 · depth 15 - Poles of ̄ j on X₀(N) have order dividing N
ModularCurve.CharPModel.ord_jBar_dvd_of_ord_jBar_neg153 below · cited by 1 · depth 15 - First coordinate is a uniformiser where partial_YΦ̄≠ 0
ModularCurve.CharPModel.ord_sub_eq_one_of_derivative_evalEval_ne_zero0 below · cited by 1 · depth 15 - A fibre model with cusp chart yields a place-specialization packet
ModularCurve.CharPModel.FibreModel.exists_placeSpecialization_spPic0_eq998 below · cited by 1 · depth 16 - Integrally closed fibre model with cusp chart and lifts
ModularCurve.CharPModel.exists_fibreModel_cuspChart_integrallyClosed_of_lift79 below · cited by 1 · depth 16 - A charted fibre model realising a place specialization at level N>1
ModularCurve.CharPModel.exists_fibreModel_cuspChart_placeSpecialization_sp_eq_spPlace_of_one_lt1,020 below · cited by 3 · depth 16 - Degeneracy images of fibre-model elements lie in j-integral closure
ModularCurve.CharPModel.FibreModel.exists_forall_le_coe_heckeAlphaBar_mem_jIntegralClosure_and_coe_heckeBetaBar_mem115 below · cited by 2 · depth 17 - Cusp dictionary in the chart j/j_N^N at poles of j_N
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero303 below · cited by 1 · depth 17 - Cusp zero-chart specialisation dictionary under a small opposite coordinate
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero_of_t_small302 below · cited by 1 · depth 18 - Cusp zero-chart dictionary under a small opposite coordinate
ModularCurve.CharPModel.FibreModel.spPlace_d7_dictZero_of_t_small_of_level301 below · cited by 1 · depth 19 - Fibre-model independence of the specialisation of places
ModularCurve.CharPModel.FibreModel.spPlace_eq_of_surjective749 below · cited by 1 · depth 21 - Fibre-model independence of the specialisation of places
ModularCurve.CharPModel.FibreModel.spPlace_eq745 below · cited by 1 · depth 22 - Relative degree of the level-N function field over k(̃ j_N)
ModularCurve.CharPModel.relfinrank_adjoin_jqNModC_eq_dedekindPsi_of_evalSymm4 below · cited by 2 · depth 27
ModularCurve.CharPReduction 7
- Integral closedness of the localised modular subring
ModularCurve.CharPReduction.mem_modularLocalized_of_eval2_monic4 below · cited by 9 · depth 11 - Reduced localised modular Laurent series lie in k(̃ j(q),̃ j(q^N))
ModularCurve.CharPReduction.modularRedLocHom_mem0 below · cited by 36 · depth 11 - Localised reduction agrees with coefficientwise reduction
ModularCurve.CharPReduction.modularRedLocHom_eq_coeffRed0 below · cited by 30 · depth 12 - Surjectivity of the localised modular reduction onto the fibre field
ModularCurve.CharPReduction.exists_modularRedLocHom_eq0 below · cited by 3 · depth 13 - Valuation dichotomy in the localised modular presentation ring
ModularCurve.CharPReduction.mem_or_inv_mem_modularLocalized4 below · cited by 7 · depth 14 - Integrality over A[j] forces q-expansion coefficients in A
ModularCurve.CharPReduction.exists_coeffMap_eq_of_mem_modularLocalized_of_monic1 below · cited by 1 · depth 16 - Scalar normalisation making a modular function a unit of the localised reduction
ModularCurve.CharPReduction.exists_smul_mem_modularLocalized_and_modularRedLocHom_ne_zero_of_charP125 below · cited by 2 · depth 21
ModularCurve.CompEq 1
- Degree ψ(N) of the level-N function field over ℚ̄(j)
ModularCurve.CompEq.finrank_adjoin_jBar_eq_dedekindPsi112 below · cited by 17 · depth 11
ModularCurve.ComplexPlaceDictionary 21
- Abel's theorem for X₀(N): principal divisors have trivial Abel–Jacobi image
ModularCurve.ComplexPlaceDictionary.abelJacobi_mem_periodLattice_of_isPrincipal181 below · cited by 2 · depth 10 - Hecke compatibility of the Abel–Jacobi period map
ModularCurve.ComplexPlaceDictionary.exists_mapDomain_eq_heckeDivBar_abelJacobi_sub_mem_periodLattice221 below · cited by 2 · depth 10 - Every place where j is regular comes from H
ModularCurve.ComplexPlaceDictionary.exists_pt_eq_of_mem153 below · cited by 8 · depth 10 - Abel's theorem for X₀(N): principality from the period condition
ModularCurve.ComplexPlaceDictionary.isPrincipal_of_abelJacobi_mem_periodLattice651 below · cited by 2 · depth 10 - Abel's theorem for X₀(N): necessity, dictionary form
ModularCurve.ComplexPlaceDictionary.abelJacobi_mem_periodLattice_of_meromorphicOrderAt_eq177 below · cited by 1 · depth 11 - Abel's theorem for X₀(N): analytic sufficiency half
ModularCurve.ComplexPlaceDictionary.exists_meromorphic_meromorphicOrderAt_eq_of_abelJacobi_mem_periodLattice643 below · cited by 1 · depth 11 - Hecke divisor correspondence on a single point of X₀(N)
ModularCurve.ComplexPlaceDictionary.heckeDivBar_single_pt192 below · cited by 1 · depth 11 - Multiplier of a multiplicative meromorphic function is a period exponential
ModularCurve.ComplexPlaceDictionary.exists_cuspForm_mul_exp_period_eq_one_of_abelJacobi_mem_periodLattice642 below · cited by 1 · depth 12 - Weight-two functions with prescribed integral residues on X₀(N)
ModularCurve.ComplexPlaceDictionary.exists_slashInvariant_residue_eq_of_degree_eq_zero256 below · cited by 2 · depth 12 - Places of a dictionary coincide iff Γ₀(N)-equivalent
ModularCurve.ComplexPlaceDictionary.pt_eq_pt_iff153 below · cited by 5 · depth 12 - Degeneracy inclusion restricts the place of τ to the place of τ
ModularCurve.ComplexPlaceDictionary.restrictAlong_pt_heckeAlphaBar16 below · cited by 1 · depth 12 - Degeneracy map q↦ q^ℓ sends the place of τ to that of ℓτ
ModularCurve.ComplexPlaceDictionary.restrictAlong_pt_heckeBetaBar16 below · cited by 1 · depth 12 - Doubled ramification equals stabiliser order in Γ₀(N)
ModularCurve.ComplexPlaceDictionary.two_mul_ramification_eq_card_stabilizer19 below · cited by 5 · depth 12 - Third-kind differential on X₀(N) with two prescribed poles
ModularCurve.ComplexPlaceDictionary.exists_slashInvariant_residue_ne_zero_of_pt_ne231 below · cited by 1 · depth 13 - Triviality of unitary multipliers with Abel–Jacobi class a period
ModularCurve.ComplexPlaceDictionary.multiplier_eq_one_of_norm_eq_one_of_abelJacobi_mem_periodLattice641 below · cited by 1 · depth 13 - Residue sum zero for Γ₀(N)-invariant weight-two functions
ModularCurve.ComplexPlaceDictionary.sum_residue_eq_zero_of_slashInvariant167 below · cited by 1 · depth 13 - Unitary multiplier function for a degree-zero divisor on X₀(N)
ModularCurve.ComplexPlaceDictionary.exists_norm_multiplier_eq_one_and_abelJacobi_add_petersson_mem_periodLattice639 below · cited by 1 · depth 14 - Residue of a dx pulled back to the upper half-plane
ModularCurve.ComplexPlaceDictionary.exists_realize_mul_deriv_realize_eq_div_add16 below · cited by 1 · depth 14 - Unitary multiplier as exponential of a cusp-form period
ModularCurve.ComplexPlaceDictionary.multiplier_eq_exp_of_abelJacobi_add_petersson_eq_zero623 below · cited by 1 · depth 15 - Analytic extension across τ of a function regular at P_τ
ModularCurve.ComplexPlaceDictionary.exists_analyticAt_eventuallyEq_realize20 below · cited by 5 · depth 19 - Regularity of jmatĥ at the interior complex places
ModularCurve.ComplexPlaceDictionary.coeffEmb_jq_mem_pt21 below · cited by 1 · depth 23
ModularCurve.ComplexPlaceDictionaryOf 24
- Abel's theorem for X_H(M): principal divisors give periods
ModularCurve.ComplexPlaceDictionaryOf.abelJacobi_mem_periodLatticeOf_gammaH_of_isPrincipal282 below · cited by 2 · depth 21 - Coefficientwise conjugation sends pt(τ) to pt(-τ̄)
ModularCurve.ComplexPlaceDictionaryOf.arithmeticGalois_complexConjAlgEquiv_smul_pt2 below · cited by 2 · depth 21 - Divisor-level Hecke compatibility of the Abel–Jacobi map for X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.exists_mapDomain_eq_heckeDivHBar_abelJacobi_sub_mem_periodLatticeOf287 below · cited by 2 · depth 21 - Every place where j is regular is a point place
ModularCurve.ComplexPlaceDictionaryOf.exists_pt_eq_of_mem128 below · cited by 6 · depth 21 - Abel's theorem for X_H(M): sufficiency
ModularCurve.ComplexPlaceDictionaryOf.isPrincipal_of_abelJacobi_mem_periodLatticeOf_gammaH451 below · cited by 2 · depth 21 - Pull-back along γ sends pt(τ) to pt(γ⁻¹τ)
ModularCurve.ComplexPlaceDictionaryOf.ofAlgAut_smul_pt_eq_pt_inv_smul2 below · cited by 1 · depth 21 - Analytic sufficiency in Abel's theorem for X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.exists_meromorphic_meromorphicOrderAt_eq_of_abelJacobi_mem_periodLatticeOf_gammaH438 below · cited by 1 · depth 22 - Hecke correspondence on a point divisor of X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.heckeDivHBar_single_pt266 below · cited by 1 · depth 22 - Places of X_H(M) separate Γ_H(M)-orbits on H
ModularCurve.ComplexPlaceDictionaryOf.pt_eq_pt_iff_gammaH71 below · cited by 5 · depth 22 - Ramification of the X_H(M) dictionary equals stabiliser order
ModularCurve.ComplexPlaceDictionaryOf.two_mul_ramification_eq_card_stabilizer_gammaH256 below · cited by 3 · depth 22 - Abel's theorem for X_H(M): multiplier is a period exponential
ModularCurve.ComplexPlaceDictionaryOf.exists_cuspForm_mul_exp_periodOf_eq_one_of_abelJacobi_mem_periodLatticeOf_gammaH437 below · cited by 1 · depth 23 - Weight-two forms with prescribed residue divisor on X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.exists_slashInvariant_residue_eq_of_degree_eq_zero_gammaH305 below · cited by 2 · depth 23 - Pull-back along α sends pt(τ) to pt(α⁻¹τ)
ModularCurve.ComplexPlaceDictionaryOf.ofAlgAut_smul_pt_eq_pt_inv_smul_of_qExpansion_slash2 below · cited by 1 · depth 23 - Restriction of places along a q-expansion field inclusion
ModularCurve.ComplexPlaceDictionaryOf.restrictAlong_pt_eq_pt_of_le2 below · cited by 1 · depth 23 - Degeneracy map q↦ q^ℓ on complex place dictionaries
ModularCurve.ComplexPlaceDictionaryOf.restrictAlong_pt_qExpand1 below · cited by 1 · depth 23 - Ramification of a complex place dictionary equals half the ±Γ-stabiliser order
ModularCurve.ComplexPlaceDictionaryOf.two_mul_ramification_eq_card_stabilizer187 below · cited by 2 · depth 23 - Stabiliser order in ±Γ divides twice the ramification
ModularCurve.ComplexPlaceDictionaryOf.card_stabilizer_dvd_two_mul_ramification2 below · cited by 1 · depth 24 - Weight-two form with simple poles at two prescribed points
ModularCurve.ComplexPlaceDictionaryOf.exists_slashInvariant_residue_ne_zero_of_pt_ne_gammaH188 below · cited by 1 · depth 24 - Unitary multiplier is trivial when Abel–Jacobi class is a period
ModularCurve.ComplexPlaceDictionaryOf.multiplier_eq_one_of_norm_eq_one_of_abelJacobi_mem_periodLatticeOf_gammaH436 below · cited by 1 · depth 24 - Residue theorem for weight-two forms on X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.sum_residue_eq_zero_of_slashInvariant_gammaH265 below · cited by 1 · depth 24 - Abel's theorem for X_H(M): unitary multiplier and period reciprocity
ModularCurve.ComplexPlaceDictionaryOf.exists_norm_multiplier_eq_one_and_abelJacobi_add_petersson_mem_periodLatticeOf_gammaH433 below · cited by 1 · depth 25 - Polar part at τ of a realised differential a dx
ModularCurve.ComplexPlaceDictionaryOf.exists_realizeOf_mul_deriv_realizeOf_eq_div_add_gammaH3 below · cited by 1 · depth 25 - Unitary multiplier as exponential of a period on X_H(M)
ModularCurve.ComplexPlaceDictionaryOf.multiplier_eq_exp_of_abelJacobi_add_petersson_eq_zero_gammaH419 below · cited by 1 · depth 26 - Ramification index one for Γ₁(M), M ≥ 4
ModularCurve.ComplexPlaceDictionaryOf.ramification_eq_one_gamma1296 below · cited by 1 · depth 30
ModularCurve.ComponentChart 2
- Fricke image of a pole-free unit reduces to a nonzero constant
ModularCurve.ComponentChart.exists_residue_frickeInvolutionBar_eq_algebraMap_of_forall_pole_eq_cuspInftyBar561 below · cited by 24 · depth 24 - Vanishing of the Fricke transform at a supersingular node
ModularCurve.ComponentChart.residue_frickeInvolutionBar_eq_zero_of_hasValue_zero_of_forall_pole_eq_cuspInftyBar562 below · cited by 24 · depth 24
ModularCurve.CupPairing 9
- Antisymmetrised cup product of rational characters has a primitive
ModularCurve.CupPairing.exists_isPrimitive1 below · cited by 4 · depth 20 - Unimodularity of the integral cup pairing on parabolic homomorphisms
ModularCurve.CupPairing.exists_perfectPairing_intCast_eq_pair8 below · cited by 1 · depth 20 - The cusp-sum formula for `pair` holds for every primitive
ModularCurve.CupPairing.pair_eq_cuspSum_div_of_isPrimitive0 below · cited by 5 · depth 20 - Harmonic functions on SL₂(ℤ)/Γ as coboundaries of parabolic characters
ModularCurve.CupPairing.exists_isParabolicHom_eq_sub_of_forall_finsum_eq_zero4 below · cited by 1 · depth 21 - Projection formula for corestriction and the cup pairing
ModularCurve.CupPairing.mult_mul_pair_coresAdd_eq4 below · cited by 2 · depth 21 - Cup pairing of parabolic characters as a quotient sum
ModularCurve.CupPairing.mult_mul_pair_eq_neg_finsum4 below · cited by 1 · depth 21 - Invariance of the cup pairing under conjugation by GL₂⁺(ℝ)
ModularCurve.CupPairing.pair_comp_eq_of_conjRel8 below · cited by 1 · depth 21 - Invariance of cusp sums under conjugation by GL₂⁺(ℝ)
ModularCurve.CupPairing.cuspSum_comp_eq_of_conjRel1 below · cited by 1 · depth 22 - Cusp sums over Γ'≤Γ split along cusps of Γ
ModularCurve.CupPairing.cuspSum_eq_sum_finsum_of_le0 below · cited by 1 · depth 22
ModularCurve.CuspSpace 7
- Number of cusps of Γ₀(N) equals sum_{d∣ N}φ(gcd(d,N/d))
ModularCurve.CuspSpace.card_cuspSpace_eq_cuspCount4 below · cited by 3 · depth 13 - Classification of the cusps of Γ₀(N)
ModularCurve.CuspSpace.classification3 below · cited by 1 · depth 14 - Normal form for a cusp of Γ₀(N)
ModularCurve.CuspSpace.exists_normalForm0 below · cited by 2 · depth 15 - Cusp normal form criterion for Γ₀(N)
ModularCurve.CuspSpace.normalFormCriterion1 below · cited by 2 · depth 15 - Conjugates of T^m in Γ₀(N) and cusp width
ModularCurve.CuspSpace.conj_T_zpow_mem_Gamma0_iff0 below · cited by 2 · depth 23 - Cusp widths of Γ₀(N) sum to ψ(N)
ModularCurve.CuspSpace.sum_cuspWidth_eq_dedekindPsi8 below · cited by 1 · depth 23 - Fibre over a cusp has cardinality its width
ModularCurve.CuspSpace.card_fromCoset_fiber1 below · cited by 1 · depth 24
ModularCurve.DRLevel 28
- Special fibre at q: two components with section and w_q-translate
ModularCurve.DRLevel.exists_comp_pair_fibre877 below · cited by 1 · depth 13 - Geometric fibres of Igusa's X₀(N₀) as curve models
ModularCurve.DRLevel.exists_curveModel_iso_fibre0_chartPin897 below · cited by 1 · depth 13 - Cusps, Atkin–Lehner involution, degeneracy map and smooth locus
ModularCurve.DRLevel.exists_cusps_involution_forgetful_smoothLocus917 below · cited by 1 · depth 13 - Nodes of the special fibre match supersingular places and their Frobenius twists
ModularCurve.DRLevel.exists_nodeEquiv_placeOfPoint_eq1,081 below · cited by 1 · depth 13 - Reduced intersection of the two fibre components at q
ModularCurve.DRLevel.isReduced_pullback_comp918 below · cited by 1 · depth 13 - Frobenius on places via the second component mod q
ModularCurve.DRLevel.placeOfPoint_comp_one_fibreMap0_eq_arithFrobC_smul835 below · cited by 1 · depth 13 - Reduction of the cusps ∞ and 0 onto the two components
ModularCurve.DRLevel.range_sectionFibre_cusps_subset_range_comp_of_jointlySurjective883 below · cited by 1 · depth 13 - Base change of a two-component fibre description along κ₀→κ
ModularCurve.DRLevel.exists_comp_pair_fibre_of_ringHom0 below · cited by 1 · depth 14 - Fibre of X₀(N₀q) at q: two closed-immersed copies
ModularCurve.DRLevel.exists_comp_pair_fibre_residueField871 below · cited by 1 · depth 14 - Base change of the level-N₀ special-fibre dictionary
ModularCurve.DRLevel.exists_curveModel_iso_fibre0_chartPin_of_ringHom882 below · cited by 1 · depth 14 - Special fibre of X₀(N₀) over a place above q
ModularCurve.DRLevel.exists_curveModel_iso_fibre0_chartPin_residueField807 below · cited by 1 · depth 14 - Points of mathbb Z_{(q)} factor through residue fields of places of ℚ̄
ModularCurve.DRLevel.exists_place_residueField_ringHom_comp_eq5 below · cited by 8 · depth 14 - Chart retraction attached to a section of π mod q
ModularCurve.DRLevel.exists_retraction_chart_comp_zero_eq0 below · cited by 5 · depth 14 - Crossings of the mod q fibre avoid the cusps
ModularCurve.DRLevel.fst_pullback_comp_mem_range_iotaFin915 below · cited by 2 · depth 14 - Chart points of the special fibre give rational affine places
ModularCurve.DRLevel.isAffineGeomPlace_and_evalAt_jGeomGen_eq_of_chartPin83 below · cited by 1 · depth 14 - Regularity of j forces a point into the finite chart
ModularCurve.DRLevel.mem_range_iotaFin_of_isAffineGeomPlace_placeOfPoint0 below · cited by 1 · depth 14 - Reduction of the cusp ∞ meets only the first component
ModularCurve.DRLevel.range_sectionFibre_epsInf_subset_range_of_comp_fibreMap0_eq_id880 below · cited by 1 · depth 14 - Density of the j-finite chart in a fibre
ModularCurve.DRLevel.dense_range_chart_fibre140 below · cited by 1 · depth 15 - Closed-immersion section of π on the fibre at q from a chart retraction
ModularCurve.DRLevel.exists_isClosedImmersion_comp_fibreMap0_eq_id_of_retraction844 below · cited by 1 · depth 15 - Distinct minimal primes over q for the two fibre components
ModularCurve.DRLevel.exists_minimalPrimes_chartAlgInf_map_le_of_mem_range_comp860 below · cited by 1 · depth 15 - Two components cover the mod-q fibre and differ
ModularCurve.DRLevel.forall_mem_range_or_mem_range_and_range_ne_of_minimalPrimes_eq0 below · cited by 1 · depth 15 - Generic fibre of the Igusa model of level Mq is Dedekind
ModularCurve.DRLevel.isIntegral_and_isLocallyNoetherian_and_forall_stalk_pullback_toBase_specMap_rat864 below · cited by 2 · depth 15 - The Igusa fibre has no isolated points
ModularCurve.DRLevel.not_isOpen_singleton_fibre139 below · cited by 1 · depth 16 - Frobenius twisting of κ-points translates places by arithmetic Frobenius
ModularCurve.DRLevel.pointEquivPlace_comp_inv_of_fst_eq_frobenius_comp_eq_arithFrobC_smul84 below · cited by 1 · depth 16 - Chart-pinned models of the special fibre agree generically
ModularCurve.DRLevel.fromSpecStalk_comp_eq_of_chartPin82 below · cited by 1 · depth 17 - Base-changed points of the level model lie over V(q)
ModularCurve.DRLevel.bcMap_apply_notMem_preimage_basicOpen0 below · cited by 7 · depth 18 - Image of a residue-field section is a closed point
ModularCurve.DRLevel.isClosed_singleton_bcMap_residue_apply0 below · cited by 1 · depth 18 - Residue field of a k_O-rational point of the closed fibre
ModularCurve.DRLevel.isIso_residueFieldMap_snd_bcMap_residue_apply0 below · cited by 1 · depth 19
ModularCurve.DRModel 26
- The cusp ∞ reduces onto the W₀-component mod p
ModularCurve.DRModel.dvd_coeffZero_of_mem_nonunits_and_exists_not_dvd_of_prime139 below · cited by 5 · depth 15 - Two branch valuation rings of ℚ(X₀(p)) above p
ModularCurve.DRModel.exists_chartAlgFin_valuationSubring_pair_levelP130 below · cited by 17 · depth 15 - Geometric fibre at p: two rational components, supersingular intersection
ModularCurve.DRModel.exists_curveModel_ratFunc_closedImmersion_pair_pFibre_and_range_sectionFibre_subset_of_residue_generators383 below · cited by 1 · depth 15 - Involution over ℤ of the two-chart model of X₀(p)
ModularCurve.DRModel.exists_iso_comp_toBase_eq_and_hom_comp_hom_eq_id_and_exists_algHom_comp_hom_eq208 below · cited by 1 · depth 15 - Reducedness of the p-fibre of the Deligne–Rapoport model
ModularCurve.DRModel.isReduced_pFibre133 below · cited by 1 · depth 15 - Characteristic-p fibres of the DR model are reduced
ModularCurve.DRModel.isReduced_pullback_toBase_of_charP184 below · cited by 8 · depth 15 - Pole-chart residue X⁻¹ and separation of the two cusps
ModularCurve.DRModel.exists_chartAlgInf_residue_eq_inv_and_cusps_separate_of_valuationSubring_pair162 below · cited by 4 · depth 16 - Both branches of X₀(p) mod p are affine lines
ModularCurve.DRModel.exists_ringEquiv_quotient_chartAlgFin_polynomial_of_valuationSubring_pair62 below · cited by 5 · depth 16 - Fibre at p of each Deligne–Rapoport chart: reduced, two components
ModularCurve.DRModel.isReduced_quotient_and_ncard_minimalPrimes_span_natCast_chartAlg_int132 below · cited by 11 · depth 16 - Ogg's unit on the ∞-component is the supersingular polynomial
ModularCurve.DRModel.map_ringEquiv_quotient_chartAlgFin_modularUnit_eq_prod_ssJSet260 below · cited by 3 · depth 16 - Minimal primes over p are the two branch centres
ModularCurve.DRModel.mem_minimalPrimes_chartAlgFin_iff_of_valuationSubring_pair123 below · cited by 6 · depth 16 - Minimal primes of p in the pole chart ring are branch centres
ModularCurve.DRModel.mem_minimalPrimes_chartAlgInf_iff_of_valuationSubring_pair123 below · cited by 5 · depth 16 - Distinct minimal primes over q and 1/j generate the unit ideal
ModularCurve.DRModel.sup_sup_span_jInvChartInf_eq_top_of_mem_minimalPrimes231 below · cited by 1 · depth 16 - Branches above p: Gauss ring and Atkin–Lehner conjugate
ModularCurve.DRModel.valuationSubring_pair_eq_gauss_and_exists_algEquiv_swap127 below · cited by 7 · depth 16 - Pole-chart lift of 1/̄ jₚ on the W₁ branch
ModularCurve.DRModel.exists_chartAlgInf_mul_sub_one_mem_nonunits_of_valuationSubring_pair86 below · cited by 1 · depth 17 - Special fibre of X₀(p) on the finite chart
ModularCurve.DRModel.exists_minimalPrimes_pair_and_ringEquiv_quotient_polynomial143 below · cited by 5 · depth 17 - Maximal ideals of the pole chart containing p and 1/j
ModularCurve.DRModel.forall_mem_iff_dvd_or_forall_mem_of_isMaximal_of_jInvChartInf_mem_of_prime144 below · cited by 1 · depth 17 - Integrality of the X₀(p) two-chart model over an unramified DVR
ModularCurve.DRModel.isIntegral_pullback_toBase859 below · cited by 11 · depth 17 - Kronecker congruence pins j ≡ jₚ^{ p} on the second branch
ModularCurve.DRModel.jFull_sub_pow_mem_nonunits_of_valuationSubring_pair58 below · cited by 1 · depth 17 - Base change from characteristic p lands in the p-fibre
ModularCurve.DRModel.baseChangeMap_apply_notMem_preimage_basicOpen0 below · cited by 18 · depth 18 - Fibre at p of the Deligne–Rapoport model: two lines
ModularCurve.DRModel.exists_curveModel_closedImmersion_pair_pFibre_cover_levelSet_singleton232 below · cited by 1 · depth 18 - Rationality of a closed-fibre section gives a closed point
ModularCurve.DRModel.isClosed_singleton_baseChangeMap_residue_apply122 below · cited by 1 · depth 18 - Two points off the finite chart in the geometric p-fibre
ModularCurve.DRModel.exists_ne_and_notMem_chartFin_pFibre216 below · cited by 4 · depth 19 - Residue field map is an isomorphism at a rational point
ModularCurve.DRModel.isIso_residueFieldMap_snd_baseChangeMap_residue_apply0 below · cited by 1 · depth 19 - Characteristic-p fibres of the Deligne–Rapoport model are reducible
ModularCurve.DRModel.not_irreducibleSpace_pullback_toBase_of_charP205 below · cited by 4 · depth 19 - An involution of the two-chart model X₀(p) over ℤ
ModularCurve.DRModel.exists_iso_and_algHom_chartAlgFin_comp_eq_and_involutive208 below · cited by 1 · depth 20
ModularCurve.DRModelPackage 90
- Degree-zero cohomological flatness of the integral model at level p
ModularCurve.DRModelPackage.bijective_algebraMap_sections_baseChange921 below · cited by 1 · depth 15 - Local pools of disjoint étale multisections near the cusp
ModularCurve.DRModelPackage.exists_locallySplitPools_of_five_le1,048 below · cited by 1 · depth 15 - Residue-field points above p killed by [m], p∤ m
ModularCurve.DRModelPackage.exists_schemeNsmul_eq_one_residueField_point435 below · cited by 1 · depth 15 - Two-line degeneration of a non-smooth Deligne–Rapoport fibre
ModularCurve.DRModelPackage.exists_twoLineDegeneration_of_not_smooth962 below · cited by 1 · depth 15 - No p-power torsion in the Pic⁰-cut on characteristic p fibres
ModularCurve.DRModelPackage.forall_fibre_pow_torsionFree_algEquivZeroGroupCut432 below · cited by 1 · depth 15 - Geometric fibre at p of the Deligne–Rapoport model is non-smooth
ModularCurve.DRModelPackage.not_smooth_pullback_snd_toBase_of_charP148 below · cited by 3 · depth 15 - Finite-map datum of degree ≥ 1 away from p
ModularCurve.DRModelPackage.exists_finiteMapData_baseChange_away_one_le_m1,108 below · cited by 1 · depth 16 - Node-ratio embedding of the Pic⁰ cut at p
ModularCurve.DRModelPackage.exists_injective_monoidHom_algEquivZeroGroupCut_pFibre430 below · cited by 2 · depth 16 - Locally split pools at primes 𝔭⊆(ℓ), ℓ≠ p
ModularCurve.DRModelPackage.exists_locallySplitPools_of_le_span_of_ne846 below · cited by 1 · depth 16 - Locally split pools at primes above p
ModularCurve.DRModelPackage.exists_locallySplitPools_of_le_span_prime487 below · cited by 1 · depth 16 - Trivial component classes extend to A-points of Pic⁰
ModularCurve.DRModelPackage.exists_schemeHomOver_of_comp_eq_zero_of_abelJacobiPin_of_surjective2,060 below · cited by 1 · depth 16 - Two-affine cover of the bad fibre avoiding crossing points
ModularCurve.DRModelPackage.exists_twoAffineOpenCover_compl_eq_pair_compInf_compZero292 below · cited by 2 · depth 16 - Prime-to-p torsion of cut classes on the geometric p-fibre
ModularCurve.DRModelPackage.forall_fibre_exists_pow_eq_one_algEquivZeroGroupCut431 below · cited by 1 · depth 16 - Away from `compZero`, `compInf` restricts to an open immersion
ModularCurve.DRModelPackage.isOpenImmersion_restrict_compInf_compl_range_compZero2 below · cited by 4 · depth 16 - Smooth locus in the p-fibre: complement of the crossings
ModularCurve.DRModelPackage.mem_preimage_smoothLocus_iff_not_mem_range_compInf_inter_range_compZero55 below · cited by 17 · depth 16 - Each bad-fibre component meets every j-level set in one point
ModularCurve.DRModelPackage.compl_jNeLocus_inter_range_comp_eq_singleton233 below · cited by 9 · depth 17 - Ogg's unit detects the ∞-component mod p
ModularCurve.DRModelPackage.exists_coordinate_forall_mem_range_compInf_and_not_mem_range_compZero280 below · cited by 1 · depth 17 - Existence of a resolved Deligne–Rapoport model with étale crossing charts
ModularCurve.DRModelPackage.exists_dRResolvedModelPackageV4_and_dRResolvedModelCharts1,166 below · cited by 1 · depth 17 - Residue-field point above a crossing point of the mod-p fibre
ModularCurve.DRModelPackage.exists_residueField_point_baseChangeMap_eq_of_isAlgClosed_residueField270 below · cited by 3 · depth 17 - Zeros of g(v) in the finite chart lie in the smooth locus
ModularCurve.DRModelPackage.iotaFin_mem_smoothLocus_of_aeval_mem57 below · cited by 1 · depth 17 - Vanishing of g(v) forces membership in the ε_∞-component
ModularCurve.DRModelPackage.mem_connectedComponentIn_of_aeval_mem195 below · cited by 1 · depth 17 - Geometric-fibre transport of the Poincaré bundle with section twists
ModularCurve.DRModelPackage.nonempty_poincare_pullbackAlong_comp_iso_of_pullback_toDR_iso_of_sectionTwist17 below · cited by 1 · depth 17 - Recognising pts of a divisor class from its Poincaré fibre
ModularCurve.DRModelPackage.pts_pic0Mk_eq_comp_of_poincare_pullbackAlong_iso21 below · cited by 1 · depth 17 - Transporting a degree-zero divisor along a level equivalence
ModularCurve.DRModelPackage.sum_coef_eq_zero_and_exists_degZero_mapDomain_of_equiv_support145 below · cited by 1 · depth 17 - Components of the special fibre stay distinct over O
ModularCurve.DRModelPackage.baseChangeMap_compInf_genericPoint_ne_baseChangeMap_compZero_genericPoint206 below · cited by 11 · depth 18 - Generic points of the two special-fibre components are smooth
ModularCurve.DRModelPackage.baseChangeMap_genericPoint_mem_preimage_smoothLocus59 below · cited by 6 · depth 18 - Maximality of the two special-fibre component generic points
ModularCurve.DRModelPackage.eq_baseChangeMap_genericPoint_of_specializes44 below · cited by 10 · depth 18 - Minimal primes over p label the components of the bad fibre
ModularCurve.DRModelPackage.exists_index_forall_mem_range_compInf_of_not_le56 below · cited by 1 · depth 18 - Node coordinates at a supersingular crossing from a chart presentation
ModularCurve.DRModelPackage.exists_nodeCoordinates_and_forall_mem_support_iff_chainPos_of_chartPresentation_of_branch_of_jPin1,158 below · cited by 1 · depth 18 - Supersingular places enumerate the crossings, with width and j-pin
ModularCurve.DRModelPackage.exists_nodeEquiv_width_eq_and_jPin414 below · cited by 1 · depth 18 - Function field of the DR model over an unramified DVR
ModularCurve.DRModelPackage.exists_ringHom_functionField_pullback_eq_algebraMap_and_coe_eq_coeffEmb860 below · cited by 1 · depth 18 - An orientation bit matching strict places to branch components
ModularCurve.DRModelPackage.exists_swap_forall_isStrict_section_mem_range_comp_of_reading445 below · cited by 1 · depth 18 - Two-line degeneration with named components of the X₀(p) fibre
ModularCurve.DRModelPackage.exists_twoLineDegeneration_of_not_smooth_iso_comp_eq962 below · cited by 1 · depth 18 - Supersingular O-lift of j at each crossing
ModularCurve.DRModelPackage.forall_exists_lift_jFun_sub_mem_maximalIdeal_and_mem_ssJSet404 below · cited by 2 · depth 18 - Oriented crossing charts for the Deligne–Rapoport model at p
ModularCurve.DRModelPackage.forall_exists_orientedCrossingChart1,094 below · cited by 2 · depth 18 - Injectivity of the node map into the model over O
ModularCurve.DRModelPackage.injective_baseChangeMap_compInf_of_exists_section2 below · cited by 1 · depth 18 - Normality of stalks of the base-changed Deligne–Rapoport model
ModularCurve.DRModelPackage.isIntegrallyClosed_stalk_pullback_toBase1,102 below · cited by 1 · depth 18 - Regular stalks of X_O away from the crossing points
ModularCurve.DRModelPackage.isRegularLocalRing_stalk_of_forall_ne_baseChangeMap_crossing59 below · cited by 2 · depth 18 - Finite presentation of the Deligne–Rapoport model over ℤ
ModularCurve.DRModelPackage.locallyOfFinitePresentation_toBase1 below · cited by 7 · depth 18 - Trichotomy for points of the integral model over a DVR
ModularCurve.DRModelPackage.mem_preimage_basicOpen_or_mem_preimage_smoothLocus_or_exists_eq_of_pullback_toBase56 below · cited by 3 · depth 18 - Fibre points off the 0-component: smooth, in the cusp component
ModularCurve.DRModelPackage.mem_smoothLocus_and_mem_connectedComponentIn_of_mem_range_compInf56 below · cited by 2 · depth 18 - Image of the zero component equals its generic point's closure
ModularCurve.DRModelPackage.range_compZero_comp_baseChangeMap_eq_closure_and_isClosed252 below · cited by 8 · depth 18 - Stalks of X×_ℤSpec O have dimension at most two
ModularCurve.DRModelPackage.ringKrullDim_stalk_pullback_toBase_le_two2 below · cited by 3 · depth 18 - Sections through non-strict inertia-fixed places meet the matched crossing
ModularCurve.DRModelPackage.section_base_closedPoint_eq_crossing_of_reduceFst_mem337 below · cited by 1 · depth 18 - Crossing points specialise from both component generic points
ModularCurve.DRModelPackage.baseChangeMap_genericPoint_specializes_crossing0 below · cited by 5 · depth 19 - Special fibre is covered by the two component generic points
ModularCurve.DRModelPackage.baseChangeMap_genericPoint_specializes_or0 below · cited by 4 · depth 19 - Germ of p regular and stalk dimension ≥ 2 at a crossing
ModularCurve.DRModelPackage.baseGerm_mem_nonZeroDivisors_and_two_le_ringKrullDim_stalk945 below · cited by 1 · depth 19 - Branch ideals at a crossing meet in the ideal (p)
ModularCurve.DRModelPackage.branchIdeal_xiInf_inf_branchIdeal_xiZero_eq_span_baseGerm888 below · cited by 2 · depth 19 - Transversality at a crossing: branch ideals sum to the maximal ideal
ModularCurve.DRModelPackage.branchIdeal_xiInf_sup_branchIdeal_xiZero_eq_maximalIdeal270 below · cited by 2 · depth 19 - Uniqueness of the point over 𝒪 on a j-level set
ModularCurve.DRModelPackage.eq_of_forall_exists_comp_baseChangeMap_eq_of_not_mem_jNeLocus234 below · cited by 2 · depth 19 - Special-fibre points with stalk of dimension ≤ 1
ModularCurve.DRModelPackage.eq_or_eq_baseChangeMap_genericPoint_of_ringKrullDim_stalk_le_one904 below · cited by 1 · depth 19 - Germ readings are V-integral with value the section pull-back
ModularCurve.DRModelPackage.evalAt_eq_stalkClosedPointTo_of_schemeHomOver3 below · cited by 3 · depth 19 - Points in both component closures are crossing points
ModularCurve.DRModelPackage.exists_eq_baseChangeMap_crossing_of_mem_closure_of_mem_closure253 below · cited by 3 · depth 19 - Germ reading j(qᵖ)-j(q)ᵖ at a supersingular crossing
ModularCurve.DRModelPackage.exists_germ_jq_sub_pow_and_stalkSpecializes_mem_maximalIdeal_of_swap568 below · cited by 2 · depth 19 - Vanishing of j(qᵖ)-j(q)ᵖ on the p-fibre components
ModularCurve.DRModelPackage.exists_range_comp_subset_zeroLocus_jq_sub_pow198 below · cited by 2 · depth 19 - Each branch ideal at a crossing is generated by two elements
ModularCurve.DRModelPackage.exists_span_pair_baseGerm_eq_branchIdeal968 below · cited by 1 · depth 19 - Ogg's unit at a crossing: tt' = p¹²
ModularCurve.DRModelPackage.exists_stalk_mul_eq_baseGerm_pow_and_isUnit_stalkSpecializes_of_crossing402 below · cited by 1 · depth 19 - (p) is radical on affine opens of X_O
ModularCurve.DRModelPackage.isRadical_span_natCast_sections_pullback_toBase185 below · cited by 1 · depth 19 - Maximal ideals generate along base change to the geometric fibre
ModularCurve.DRModelPackage.map_maximalIdeal_stalkMap_baseChangeMap_eq_of_inertia_grain0 below · cited by 2 · depth 19 - Germs at a supersingular crossing are node-integral
ModularCurve.DRModelPackage.mem_nodeIntegers_of_stalk_of_specializes_of_exists_sub_mem998 below · cited by 2 · depth 19 - Evaluation at an A-integral place via an 𝒪-section
ModularCurve.DRModelPackage.mem_preimage_and_forall_evalAt_eq_stalkClosedPointTo_of_ord_sub_pos126 below · cited by 2 · depth 19 - Branch residues at a supersingular crossing: kernels and orders
ModularCurve.DRModelPackage.nodeResidue_eq_zero_iff_and_ord_eq_of_specializes_of_mem_maximalIdeal549 below · cited by 2 · depth 19 - Branch residues and orders at a supersingular crossing, swapped labelling
ModularCurve.DRModelPackage.nodeResidue_eq_zero_iff_and_ord_eq_of_specializes_of_mem_maximalIdeal_swap549 below · cited by 2 · depth 19 - Incomparable branch ideals at a crossing point
ModularCurve.DRModelPackage.not_branchIdeal_le_branchIdeal_crossingPt253 below · cited by 3 · depth 19 - Transversal parameters at a crossing are uniformisers on each branch
ModularCurve.DRModelPackage.ord_placeOfPoint_stalkMap_eq_one_of_span_eq_maximalIdeal0 below · cited by 2 · depth 19 - Saturation of the two geometric p-fibre components under X-morphisms
ModularCurve.DRModelPackage.preimage_closure_image_range_compInf_eq_of_comp_fst_eq189 below · cited by 5 · depth 19 - Image of the ∞-component is the closure of its generic point
ModularCurve.DRModelPackage.range_compInf_comp_baseChangeMap_eq_closure_and_isClosed252 below · cited by 7 · depth 19 - An O/𝔪-rational crossing point is closed with residue field O/𝔪
ModularCurve.DRModelPackage.residue_baseGerm_surjective_and_isClosed_crossingPt0 below · cited by 1 · depth 19 - Crossings lie over the j-finite chart
ModularCurve.DRModelPackage.crossingPt_mem_preimage_chartFin244 below · cited by 3 · depth 20 - At a crossing, primes over p are the two branch primes
ModularCurve.DRModelPackage.eq_comap_or_eq_comap_of_mem_minimalPrimes_natCast_of_specializes0 below · cited by 1 · depth 20 - Points in both component closures are crossing points
ModularCurve.DRModelPackage.exists_eq_baseChangeMap_crossing_of_ne_of_mem_closure_of_mem_closure45 below · cited by 1 · depth 20 - Crossings of the mod p fibre have supersingular j-invariants
ModularCurve.DRModelPackage.exists_equiv_pullback_compInf_compZero_ssJSet_germ_jCoordBC_sub_constSection_mem393 below · cited by 1 · depth 20 - Maximal ideal at a crossing: branch ideal plus one generator
ModularCurve.DRModelPackage.exists_maximalIdeal_eq_branchIdeal_sup_span_singleton275 below · cited by 1 · depth 20 - Reading the compInf branch of the p-fibre as k(X₀(1))
ModularCurve.DRModelPackage.exists_ringEquiv_ratFunc_forall_stalkMap_genericPoint_compInf_eq_residueSnd275 below · cited by 1 · depth 20 - The `compZero` branch of the p-fibre as a j-line
ModularCurve.DRModelPackage.exists_ringEquiv_ratFunc_forall_stalkMap_genericPoint_compZero_eq_residueFst275 below · cited by 1 · depth 20 - First mod p branch reads the level-one fibre field
ModularCurve.DRModelPackage.exists_ringEquiv_ratFunc_forall_stalkMap_genericPoint_eq_residueFst275 below · cited by 1 · depth 20 - Second branch reading of the p-fibre function field
ModularCurve.DRModelPackage.exists_ringEquiv_ratFunc_forall_stalkMap_genericPoint_eq_residueSnd275 below · cited by 1 · depth 20 - Germs at a point below both branches lie in both prolongations
ModularCurve.DRModelPackage.mem_integers_and_mem_integers_of_stalk_of_specializes919 below · cited by 1 · depth 20 - Branch local rings at a crossing map into the two Gauss rings
ModularCurve.DRModelPackage.phi_algebraMap_stalk_mem_integers_and_exists_eq_jFun_of_specializes_of_mem_maximalIdeal289 below · cited by 1 · depth 20 - Branch integrality and j-attainment at a supersingular crossing (exchanged labels)
ModularCurve.DRModelPackage.phi_algebraMap_stalk_mem_integers_and_exists_eq_jFun_of_specializes_of_mem_maximalIdeal_swap289 below · cited by 1 · depth 20 - A common domain for the branch function field and k(jmath̃)
ModularCurve.DRModelPackage.exists_isDomain_ringHom_functionField_and_ringHom_modularFunctionFieldC_of_residueField_compInf272 below · cited by 2 · depth 21 - Common domain generating both branch and level-one function fields
ModularCurve.DRModelPackage.exists_isDomain_ringHom_functionField_and_ringHom_modularFunctionFieldC_of_residueField_compZero272 below · cited by 2 · depth 21 - Points over the closed point are closed or branch generic points
ModularCurve.DRModelPackage.isClosed_or_eq_generic_of_snd_eq_closedPoint911 below · cited by 1 · depth 21 - Stalk dominated by R₁ forces a non-closed point
ModularCurve.DRModelPackage.not_isClosed_of_forall_stalk_mem_integersFst1 below · cited by 1 · depth 21 - Stalk dominated by R₂ gives a non-closed point
ModularCurve.DRModelPackage.not_isClosed_of_forall_stalk_mem_integersSnd77 below · cited by 1 · depth 21 - Units at the two fibre generic points: Q(j) for ̄ Q≠ 0
ModularCurve.DRModelPackage.polynomialEval_mem_range_algebraMap_stalk_and_inv_mem_of_map_ne_zero250 below · cited by 2 · depth 21 - Off the image of i_∞, i₀ is an open immersion
ModularCurve.DRModelPackage.isOpenImmersion_restrict_compZero_compl_range_compInf2 below · cited by 2 · depth 22 - Generic points of the special fibre give minimal primes over (p)
ModularCurve.DRModelPackage.mem_minimalPrimes_of_fst_baseChangeMap_genericPoint_eq_iotaFin47 below · cited by 1 · depth 22 - Stalks at the two special-fibre generic points have dimension one
ModularCurve.DRModelPackage.ringKrullDim_stalk_baseChangeMap_genericPoint_eq_one906 below · cited by 1 · depth 22 - Images of the two special-fibre generic points are p-fibre-maximal
ModularCurve.DRModelPackage.eq_fst_baseChangeMap_genericPoint_of_specializes45 below · cited by 1 · depth 23
ModularCurve.DRModelPackageLevel 187
- Package cusp ∞ followed by π is the Igusa cusp
ModularCurve.DRModelPackageLevel.epsInf_comp_pi_eq0 below · cited by 1 · depth 12 - Néron object of J₀(N₀p) at p from a level model
ModularCurve.DRModelPackageLevel.exists_jZeroNeronObjectAtP_and_bridge_representsRelSubPic_abqFibre_of_levelModel4,507 below · cited by 1 · depth 12 - Strict points reduce to reduceFst and reduceSnd
ModularCurve.DRModelPackageLevel.compat_reduceFst_reduceSnd_of_sp_eq_spPlace1,848 below · cited by 5 · depth 13 - Degeneracy morphisms on Pic⁰ representing schemes as norm maps
ModularCurve.DRModelPackageLevel.exists_degeneracyHom_classifies_normModule78 below · cited by 2 · depth 13 - Representability of relative Pic⁰ of the level-N₀q model
ModularCurve.DRModelPackageLevel.exists_representsRelSubPic1,562 below · cited by 1 · depth 13 - Points dictionary for the relative Pic⁰ of the level-N₀p model
ModularCurve.DRModelPackageLevel.exists_representsRelSubPic_abelJacobi_pts_of_representsRelSubPic388 below · cited by 1 · depth 13 - Special-fibre torus, abelian quotient and pins at level N₀p
ModularCurve.DRModelPackageLevel.exists_torusFibre_abqFibre_degeneracy_specialFibre_pins_of_levelModel1,857 below · cited by 1 · depth 13 - Extension to an A-point of relative Pic⁰ versus good classes
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_iff_isGoodClass3,053 below · cited by 1 · depth 13 - Inertia displacements σ x-x extend over the place
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_smul_sub2,607 below · cited by 1 · depth 13 - Hecke algebra acts by homomorphic endomorphisms of D
ModularCurve.DRModelPackageLevel.forall_heckeAlg_exists_hom_mul_and_pts_smul_eq_comp2,300 below · cited by 1 · depth 13 - Finiteness, flatness and rank q+1 of π
ModularCurve.DRModelPackageLevel.isFinite_flat_finrank_pi1,207 below · cited by 4 · depth 13 - Properness and geometric connectedness of the generic Picard fibre
ModularCurve.DRModelPackageLevel.isProper_and_geometricallyConnected_pullback_snd_rat_of_representsRelSubPic392 below · cited by 1 · depth 13 - Multiplication by n on relative Pic⁰: flat, surjective, quasi-finite
ModularCurve.DRModelPackageLevel.nsmul_flat_surjective_locallyQuasiFinite_of_representsRelSubPic2,149 below · cited by 1 · depth 13 - Norm morphisms realise the degeneracy pushforwards on ℚ̄-points
ModularCurve.DRModelPackageLevel.pts_degeneracyPushforwardPair_eq_comp_degeneracyHom298 below · cited by 2 · depth 13 - Universal c_*𝒪=𝒪 for the level-N₀q Igusa model
ModularCurve.DRModelPackageLevel.bijective_algebraMap_sections_baseChange203 below · cited by 1 · depth 14 - Unique A-section of the model through a given place
ModularCurve.DRModelPackageLevel.existsUnique_section_comp_eq_pointEquivPlace_symm0 below · cited by 6 · depth 14 - Existence of the norm endomorphism on the special-fibre Pic⁰
ModularCurve.DRModelPackageLevel.exists_frobHom_classifies_normModule_baseChange78 below · cited by 3 · depth 14 - Hecke operator T_ℓ, ℓ≠ p, on relative Pic⁰
ModularCurve.DRModelPackageLevel.exists_hom_mul_and_pts_heckeOperatorBar_eq_comp_of_ne1,593 below · cited by 1 · depth 14 - Uₚ on J₀(N₀p) induced by an endomorphism of D
ModularCurve.DRModelPackageLevel.exists_hom_mul_and_pts_heckeOperatorBar_self_eq_comp1,914 below · cited by 1 · depth 14 - Finite subsets of the smooth locus over an affine open lie in an affine open
ModularCurve.DRModelPackageLevel.exists_isAffineOpen_of_finset_smoothLocus6 below · cited by 1 · depth 14 - Special fibre of Pic⁰ of the Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.exists_representsRelSubPic_torus_abq_specialFibre284 below · cited by 3 · depth 14 - Two-sided étale pools in the smooth locus for q≥ 5
ModularCurve.DRModelPackageLevel.exists_twoSidedPool_smoothLocus_closedPrime_of_five_le965 below · cited by 2 · depth 14 - Two-sided étale pools in the smooth locus at q=3
ModularCurve.DRModelPackageLevel.exists_twoSidedPool_smoothLocus_closedPrime_three965 below · cited by 2 · depth 14 - Two-sided étale pools in the smooth locus at q=2
ModularCurve.DRModelPackageLevel.exists_twoSidedPool_smoothLocus_closedPrime_two965 below · cited by 2 · depth 14 - Two-sided étale multisection pools at the generic prime
ModularCurve.DRModelPackageLevel.exists_twoSidedPool_smoothLocus_genericPrime968 below · cited by 1 · depth 14 - Closure under addition of points extending to a place
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_add0 below · cited by 1 · depth 14 - Inertia displacement at a non-crossing point extends to A
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_mk_smul_single_sub_single_of_not_mem_range_comp_inter1,146 below · cited by 1 · depth 14 - Negation preserves extendability of Picard points to a place
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_neg0 below · cited by 1 · depth 14 - Good classes extend to A-points of relative Pic⁰
ModularCurve.DRModelPackageLevel.extendsToPlace_pts_of_isGoodClass1,152 below · cited by 2 · depth 14 - Constant rank q+1 of the degeneracy morphism π
ModularCurve.DRModelPackageLevel.finrank_pi_eq883 below · cited by 1 · depth 14 - Flatness of π for a Deligne–Rapoport level package
ModularCurve.DRModelPackageLevel.flat_pi1,175 below · cited by 1 · depth 14 - The package map π is finite and locally of finite presentation
ModularCurve.DRModelPackageLevel.isFinite_and_locallyOfFinitePresentation_pi123 below · cited by 2 · depth 14 - Second component leg is finite flat of rank q
ModularCurve.DRModelPackageLevel.isFinite_flat_finrank_comp_one_pi1,217 below · cited by 2 · depth 14 - Extension over A implies good class for P
ModularCurve.DRModelPackageLevel.isGoodClass_of_extendsToPlace_pts3,008 below · cited by 1 · depth 14 - Reducedness of the joint kernel of the two degeneracy maps mod p
ModularCurve.DRModelPackageLevel.isReduced_pullback_ker_fibreRestrictAlong_normHom_of_comp_eq1,399 below · cited by 1 · depth 14 - Geometric fibres of the Deligne–Rapoport level model are reduced
ModularCurve.DRModelPackageLevel.isReduced_pullback_toBase_of_isAlgClosed2 below · cited by 5 · depth 14 - Locally quasi-finite [n] on a fibre where n is non-invertible
ModularCurve.DRModelPackageLevel.locallyQuasiFinite_fibre_schemeNsmul_of_not_isUnit2,139 below · cited by 1 · depth 14 - Smoothness and ε_∞-component for points off the second component
ModularCurve.DRModelPackageLevel.mem_smoothLocus_and_mem_connectedComponentIn_of_mem_range_comp_zero878 below · cited by 3 · depth 14 - Ogg's unit and q¹²u⁻¹ in the finite-j chart algebra
ModularCurve.DRModelPackageLevel.modularUnitSeries_mem_chartAlgFin_mul101 below · cited by 8 · depth 14 - Degeneracy norm morphism versus Abel–Jacobi on ℚ̄-points
ModularCurve.DRModelPackageLevel.mul_degeneracyHom_ajbar_abelJacobi_eq85 below · cited by 1 · depth 14 - Second degeneracy norm map versus Abel–Jacobi on ℚ̄-points
ModularCurve.DRModelPackageLevel.mul_degeneracyHom_one_ajbar_abelJacobi_eq85 below · cited by 1 · depth 14 - Algebraically trivial bundles with a section on geometric fibres
ModularCurve.DRModelPackageLevel.nonempty_iso_unit_fibre_of_isAlgEquivZero_of_ne_zero363 below · cited by 1 · depth 14 - Crossing special point: non-strict place, supersingular first reduction
ModularCurve.DRModelPackageLevel.not_isStrict_and_reduceFst_mem_of_range_subset_range_comp_inter1,910 below · cited by 1 · depth 14 - Strict places reduce to reduceFst, reduceSnd on the Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.placeOfPoint_eq_reduce_of_isModel_of_orderLawFixed1,896 below · cited by 2 · depth 14 - Package fibre dictionary and centre-pinned model read equal places
ModularCurve.DRModelPackageLevel.pointEquivPlace_efib_inv_eq_congrRingEquiv_pointEquivPlace_of_finChart_centrePin126 below · cited by 3 · depth 14 - Degeneracy map on ℚ̄-points restricts places along ᾱ
ModularCurve.DRModelPackageLevel.pointEquivPlace_eq_restrictAlong_heckeAlphaBar_of_comp_pi96 below · cited by 5 · depth 14 - Second degeneracy morphism: places restrict along `heckeBetaBar`
ModularCurve.DRModelPackageLevel.pointEquivPlace_eq_restrictAlong_heckeBetaBar_of_comp_piw131 below · cited by 4 · depth 14 - Abelian coordinates of the reduction equal the glued specialisation pair
ModularCurve.DRModelPackageLevel.ptsSp_symm_abq_reduction_eq_toPic0Pair_of_isGluedSpecialization1,470 below · cited by 1 · depth 14 - Inertia fixes the reduction of the section attached to a place
ModularCurve.DRModelPackageLevel.residue_comp_section_smul_eq_of_mem_inertia1 below · cited by 2 · depth 14 - Ribet's matrix for the two degeneracy maps mod p
ModularCurve.DRModelPackageLevel.symm_schemeHomOverComp_degeneracyHom_eq_add_frobeniusPushforwardModL_of_dictionary928 below · cited by 1 · depth 14 - Non-smooth fibres of the Deligne–Rapoport model are two glued curves
ModularCurve.DRModelPackageLevel.twoGluedSmoothCurveDegenerations246 below · cited by 1 · depth 14 - Reduction killed by both restriction maps when glued Pic⁰-pair vanishes
ModularCurve.DRModelPackageLevel.abq_reduction_eq_one_of_toPic0Pair_glueData_eq_zero_residueField1,404 below · cited by 1 · depth 15 - Twists of the geometric point commute with the component maps
ModularCurve.DRModelPackageLevel.baseChangeSnd_comp_comp856 below · cited by 1 · depth 15 - Ribet's matrix on κ-points of Pic⁰
ModularCurve.DRModelPackageLevel.baseChange_normHom_eq_restrict_mul_frob_restrict_points922 below · cited by 3 · depth 15 - Integral D-points have vanishing component invariant
ModularCurve.DRModelPackageLevel.comp_eq_zero_of_exists_schemeHomOver_of_depthCompLaw_of_abelJacobiPin_of_surjective_red_of_sp_eq_spPlace2,485 below · cited by 1 · depth 15 - Atkin–Lehner endomorphism of the relative Pic⁰ representing scheme
ModularCurve.DRModelPackageLevel.exists_atkinLehnerHom_classifies_pullback4 below · cited by 1 · depth 15 - An Ogg unit separating the two components of the q-fibre
ModularCurve.DRModelPackageLevel.exists_chartAlgFin_forall_mem_range_comp_zero_and_not_mem_range_comp_one236 below · cited by 4 · depth 15 - Existence of the degeneracy pullback homomorphism β^*
ModularCurve.DRModelPackageLevel.exists_degeneracyPullbackHom_classifies_pullback4 below · cited by 1 · depth 15 - Existence of q-expansion-pinned degeneracy pairs at level N₀ℓ
ModularCurve.DRModelPackageLevel.exists_heckeDegeneracyPair937 below · cited by 1 · depth 15 - Norm–pullback Hecke endomorphism of the Pic⁰ representing scheme
ModularCurve.DRModelPackageLevel.exists_heckeHom_classifies_norm_pullback_poincare_of_flat536 below · cited by 1 · depth 15 - Level polynomials for Ogg's unit on the Igusa chart
ModularCurve.DRModelPackageLevel.exists_levelPolynomials_of_chartAlgFin320 below · cited by 3 · depth 15 - Two minimal primes of (q) in the finite-j chart ring
ModularCurve.DRModelPackageLevel.exists_minimalPrimes_chartAlgFin_span_eq_pair_of_valuationSubring_pair173 below · cited by 7 · depth 15 - One-sided pool over R[1/f] from level polynomials
ModularCurve.DRModelPackageLevel.exists_oneSidedPool_baseChange_of_levelPolynomials892 below · cited by 3 · depth 15 - Strict places of the first kind reduce into the first component
ModularCurve.DRModelPackageLevel.exists_placeOfPoint_eq_reduceFst_of_isStrictFst1 below · cited by 6 · depth 15 - Strict second-kind places reduce onto the second DR component
ModularCurve.DRModelPackageLevel.exists_placeOfPoint_eq_reduceSnd_of_isStrictSnd1 below · cited by 6 · depth 15 - Non-smooth geometric fibres of the level model lie over q
ModularCurve.DRModelPackageLevel.exists_ringHom_charP_of_not_smooth_fibre7 below · cited by 2 · depth 15 - Bidegree-zero section twists give A-points of relative Pic⁰
ModularCurve.DRModelPackageLevel.exists_schemeHomOver_poincare_pullbackAlong_iso_rigidify_sectionTwist_of_sum_eq_zero1,134 below · cited by 4 · depth 15 - Strict places reduce onto one Deligne–Rapoport component, off the other
ModularCurve.DRModelPackageLevel.exists_swap_forall_isStrict_range_subset_range_comp3 below · cited by 1 · depth 15 - Two-sided pools from one-sided pools via the involution w
ModularCurve.DRModelPackageLevel.exists_twoSidedPool_of_oneSided0 below · cited by 3 · depth 15 - Ribet's matrix as an identity of morphisms on special fibres
ModularCurve.DRModelPackageLevel.fibreRestrictAlong_normHom_eq_lift_abq_comp_ribetMatrix928 below · cited by 1 · depth 15 - Base-changed w moves the ∞-component; cusp 0 lies off it
ModularCurve.DRModelPackageLevel.fibre_wL_mem_diff_connectedComponentIn_and_cuspZero_mem_baseChange882 below · cited by 3 · depth 15 - Finiteness of the crossings in the special fibre
ModularCurve.DRModelPackageLevel.finite_crossings121 below · cited by 16 · depth 15 - Degeneracy morphism on generic points is Spec of α
ModularCurve.DRModelPackageLevel.fromSpecStalk_genericPoint_comp_eq_spec_map_heckeAlphaBar2 below · cited by 1 · depth 15 - Second degeneracy map on generic points is Specβ
ModularCurve.DRModelPackageLevel.fromSpecStalk_genericPoint_comp_eq_spec_map_heckeBetaBar77 below · cited by 1 · depth 15 - Chart inclusions of a Deligne–Rapoport level package are finite
ModularCurve.DRModelPackageLevel.isFinite_and_locallyOfFinitePresentation_specMap_iota121 below · cited by 2 · depth 15 - Generic fibre degeneracy maps are finite flat of constant rank
ModularCurve.DRModelPackageLevel.isFinite_flat_finrank_curveChange_heckeDegeneracy_rat890 below · cited by 1 · depth 15 - Fibrewise finiteness and rank q+1 of π
ModularCurve.DRModelPackageLevel.isFinite_flat_finrank_fibreMap0_pi1,208 below · cited by 1 · depth 15 - Smooth locus meets the geometric fibre off the crossings
ModularCurve.DRModelPackageLevel.mem_preimage_smoothLocus_iff_not_mem_range_comp_inter15 below · cited by 10 · depth 15 - Geometric generic points lie in the smooth locus
ModularCurve.DRModelPackageLevel.mem_smoothLocus_of_mem_range_fst_geomGeneric0 below · cited by 3 · depth 15 - Poincaré bundle at geometric Abel–Jacobi points equals 𝒪(̄ y-∞)
ModularCurve.DRModelPackageLevel.nonempty_poincare_pullbackAlong_iso_ofPoint_tensor_ofPoint_idealModule_of_eq_comp_ajbar15 below · cited by 5 · depth 15 - Two-affine open cover of the fibre `fibre0`
ModularCurve.DRModelPackageLevel.nonempty_twoAffineOpenCover_fibre00 below · cited by 3 · depth 15 - Abelian-quotient reduction of a good class matches the glued specialisation
ModularCurve.DRModelPackageLevel.ptsSp_symm_abq_reduction_pair_eq_toPic0Pair_of_isGluedSpecialization1,211 below · cited by 1 · depth 15 - Atkin–Lehner involution acts as w_* on ℚ̄-points
ModularCurve.DRModelPackageLevel.pts_atkinLehner_smul_eq_comp_atkinLehnerHom173 below · cited by 1 · depth 15 - Second degeneracy pullback agrees with β^* on ℚ̄-points
ModularCurve.DRModelPackageLevel.pts_degeneracyPullbackPair_one_eq_comp_degeneracyPullbackHom1,238 below · cited by 1 · depth 15 - Geometric generic restriction of a smooth-locus A-point of Pic⁰
ModularCurve.DRModelPackageLevel.pts_pic0Mk_eq_comp_of_poincare_pullbackAlong_iso_rigidify_sectionTwist_of_range_subset_smoothLocus35 below · cited by 4 · depth 15 - Points through a crossing have supersingular first reduction
ModularCurve.DRModelPackageLevel.reduceFst_mem_ssPlaces_of_specialPoint_eq_crossing126 below · cited by 1 · depth 15 - Norm endomorphism of Pic⁰ annihilates tangent vectors
ModularCurve.DRModelPackageLevel.schemeHomOverComp_frob_eq_of_dualNumber951 below · cited by 1 · depth 15 - Degeneracy maps commute with forgetting the Γ₀(q)-structure
ModularCurve.DRModelPackageLevel.comp_pi_eq_pi_comp_of_pinned122 below · cited by 1 · depth 16 - Injectivity on closed points of πcirccomp₁ in characteristic p
ModularCurve.DRModelPackageLevel.eq_of_isClosed_of_comp_one_fibreMap0_pi_apply_eq0 below · cited by 1 · depth 16 - Existence of a resolved Deligne–Rapoport model with place–component dictionary
ModularCurve.DRModelPackageLevel.exists_dRResolvedModelPackageLevel_nodeEquiv_swap_nodeCoordinates_of_surjective_of_sp_eq_spPlace2,427 below · cited by 1 · depth 16 - Local points over a crossing factor through the finite-j chart
ModularCurve.DRModelPackageLevel.exists_eq_spec_map_comp_iotaFin_of_comp_base_eq1 below · cited by 5 · depth 16 - Étale level sets of the modular unit Δ(τ)/Δ(qτ)
ModularCurve.DRModelPackageLevel.exists_finite_etale_quotient_span_aeval317 below · cited by 1 · depth 16 - Minimal primes over q in the Igusa chart select one component
ModularCurve.DRModelPackageLevel.exists_index_forall_mem_range_comp_zero_of_not_le19 below · cited by 1 · depth 16 - Finite flat locus of π₂ in codimension ≤ 1
ModularCurve.DRModelPackageLevel.exists_opens_flat_morphismRestrict_heckeDegeneracy_and_finrank_eq_and_mem_of_ringKrullDim_le_one2 below · cited by 2 · depth 16 - Flatness of a finite surjection over the regular locus
ModularCurve.DRModelPackageLevel.exists_opens_flat_morphismRestrict_of_isFinite882 below · cited by 1 · depth 16 - Sections of the resolved X₀(N₀p) model with depth-prescribed components
ModularCurve.DRModelPackageLevel.exists_sections_multidegree_eq_depth_of_exists_schemeHomOver_of_branch186 below · cited by 1 · depth 16 - Atkin–Lehner image and cusp 0 off the ∞-component
ModularCurve.DRModelPackageLevel.fibreMap_w_mem_diff_connectedComponentIn_and_sectionFibre_cuspZero_mem878 below · cited by 1 · depth 16 - Norm of the pulled-back Poincaré bundle is fibrewise Pic⁰
ModularCurve.DRModelPackageLevel.fibrewiseAlgEquivZero_ofInvertible_norm_pullback_poincare530 below · cited by 1 · depth 16 - Finite-j chart level sets lie in the smooth locus
ModularCurve.DRModelPackageLevel.iotaFin_mem_smoothLocus_of_le_of_sup_span_singleton_eq_top881 below · cited by 1 · depth 16 - Disjointness of a chart level set from a w-translate
ModularCurve.DRModelPackageLevel.iotaFin_ne_w_iotaFin_of_span_singleton_sup_span_singleton_theta_eq_top75 below · cited by 1 · depth 16 - Bidegree-zero section twists are algebraically trivial on geometric fibres
ModularCurve.DRModelPackageLevel.isAlgEquivZero_fibreAt_sectionTwist_of_closedPoint_mem_range1,129 below · cited by 1 · depth 16 - Degree-zero section twists vanish away from the closed point
ModularCurve.DRModelPackageLevel.isAlgEquivZero_fibreAt_sectionTwist_of_closedPoint_notMem_range1,043 below · cited by 1 · depth 16 - Invertibility of section twists at smooth A-points
ModularCurve.DRModelPackageLevel.isInvertible_sectionTwist16 below · cited by 3 · depth 16 - Chart points off v lie in the cusp component of geometric fibres
ModularCurve.DRModelPackageLevel.mem_connectedComponentIn_baseChange_of_fst_eq_iotaFin880 below · cited by 1 · depth 16 - Atkin–Lehner endomorphism on Abel–Jacobi points of Pic⁰
ModularCurve.DRModelPackageLevel.mul_atkinLehnerHom_ajbar_ajbar_eq_of_comp_w23 below · cited by 1 · depth 16 - The two abq coordinates of a reduced Abel–Jacobi point
ModularCurve.DRModelPackageLevel.nonempty_poincare_pullbackAlong_abq_reduction_iso_pointTwist1,142 below · cited by 1 · depth 16 - Restricting a rigidified section twist to the geometric generic fibre
ModularCurve.DRModelPackageLevel.nonempty_poincare_pullbackAlong_iso_pointTwist_of_iso_rigidify_sectionTwist29 below · cited by 1 · depth 16 - Poincaré bundle at a degree-zero class as a point twist
ModularCurve.DRModelPackageLevel.nonempty_poincare_pullbackAlong_pts_pic0Mk_iso_pointTwist23 below · cited by 1 · depth 16 - Vanishing divisor class trivialises a point twist on the special fibre
ModularCurve.DRModelPackageLevel.nonempty_pointTwist_comp0_iso_unit_of_pic0Mk_eq_zero629 below · cited by 1 · depth 16 - Primitivity of the normed Poincaré bundle on T-points
ModularCurve.DRModelPackageLevel.nonempty_pullbackAlong_mul_iso_tensor_ofInvertible_norm_pullback_poincare10 below · cited by 1 · depth 16 - Triviality along the zero section of the normed Poincaré bundle
ModularCurve.DRModelPackageLevel.nonempty_pullbackAlong_zeroSection_ofInvertible_norm_pullback_poincare_iso_unit73 below · cited by 2 · depth 16 - Rigidified section twist restricted to the zeroth special-fibre component
ModularCurve.DRModelPackageLevel.nonempty_pullbackCurve_comp0_sectionTwist_iso854 below · cited by 3 · depth 16 - Point twists of trivial divisor class are rigidly trivial
ModularCurve.DRModelPackageLevel.nonempty_rigidify_pointTwist_comp1_iso_unit_of_pic0Mk_eq_zero629 below · cited by 1 · depth 16 - Restriction of the rigidified section twist to the second component
ModularCurve.DRModelPackageLevel.nonempty_rigidify_pullbackCurve_comp1_sectionTwist_iso854 below · cited by 3 · depth 16 - Positivity of ord_W(j-φ(j)) for a chart A-point
ModularCurve.DRModelPackageLevel.ord_jFun_sub_pos_of_eq_spec_map_comp_iotaFin1 below · cited by 1 · depth 16 - Atkin–Lehner involution restricts places along the geometric automorphism
ModularCurve.DRModelPackageLevel.pointEquivPlace_eq_restrictAlong_geomAut_of_comp_w4 below · cited by 1 · depth 16 - Second degeneracy pullback of [x]-[s₀] equals β^∗circaj₀
ModularCurve.DRModelPackageLevel.pts_degeneracyPullbackPair_one_mk_eq_abelJacobi_comp_degeneracyPullbackHom1,237 below · cited by 1 · depth 16 - Cusp sections miss the finite-j chart
ModularCurve.DRModelPackageLevel.range_cuspInf_inter_range_iotaFin_eq_empty_and_range_cuspZero_inter_range_iotaFin_eq_empty1 below · cited by 2 · depth 16 - Reduction of the chart value of j at a crossing
ModularCurve.DRModelPackageLevel.red_jChartFin_eq_evalAt_jGeomGen_nodeEquiv1 below · cited by 1 · depth 16 - Chart-pinned degeneracy maps commute with w_q
ModularCurve.DRModelPackageLevel.w_hom_comp_eq_comp_w_hom_of_pinned76 below · cited by 1 · depth 16 - Unramified nonempty level sets of the modular unit mod q
ModularCurve.DRModelPackageLevel.exists_avoid_forall_formallyUnramified_quotient_quotient_span_aeval226 below · cited by 1 · depth 17 - Existence of a resolved model with étale crossing charts
ModularCurve.DRModelPackageLevel.exists_dRResolvedModelPackageLevel_and_dRResolvedModelChartsLevelRam414 below · cited by 1 · depth 17 - Finite level sets of Ogg's unit with linear rank bound
ModularCurve.DRModelPackageLevel.exists_forall_finite_quotient_span_aeval_and_finrank_le276 below · cited by 1 · depth 17 - Generic unramifiedness of the j-chart over the modular unit line
ModularCurve.DRModelPackageLevel.exists_forall_isUnramifiedAt_polynomial_of_aeval_notMem126 below · cited by 1 · depth 17 - Minimal primes over q in the pole chart are the centres of W₀,W₁
ModularCurve.DRModelPackageLevel.exists_minimalPrimes_chartAlgInf_span_eq_pair_of_valuationSubring_pair186 below · cited by 3 · depth 17 - Depth-to-component dictionary for the resolved Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.exists_nodeCoordinates_and_forall_mem_support_iff_chainPos_of_charts_of_sp_eq_spPlace2,052 below · cited by 1 · depth 17 - Crossing points rational over an algebraically closed residue field
ModularCurve.DRModelPackageLevel.exists_residueField_point_baseChangeMap_eq_of_isAlgClosed_residueField16 below · cited by 1 · depth 17 - Inertia-fixed places give sections over the inertia valuation ring
ModularCurve.DRModelPackageLevel.exists_section_generic_eq_pointEquivPlace_symm_of_forall_inertia_smul_eq0 below · cited by 2 · depth 17 - Strict places specialise into one labelled component
ModularCurve.DRModelPackageLevel.exists_swap_forall_isStrict_section_mem_range_comp_of_sp_eq_spPlace1,861 below · cited by 1 · depth 17 - Oriented crossing charts uv=q^e on the Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.forall_exists_orientedCrossingChart303 below · cited by 1 · depth 17 - Node widths of the resolved model equal place widths
ModularCurve.DRModelPackageLevel.forall_width_eq_of_charts_of_sp_eq_spPlace2,299 below · cited by 1 · depth 17 - Bidegree-zero section twist: algebraic triviality on the special fibre
ModularCurve.DRModelPackageLevel.isAlgEquivZero_baseChange_rigidify_sectionTwist_residueField1,126 below · cited by 1 · depth 17 - Degree-zero section twists on geometric fibres are algebraically equivalent to zero
ModularCurve.DRModelPackageLevel.isAlgEquivZero_fibreAt_sectionTwist_algebraicClosure1,040 below · cited by 1 · depth 17 - Characteristic-p fibres of the rigidified bundle lie in Pic⁰
ModularCurve.DRModelPackageLevel.isAlgEquivZero_fibre_ofInvertible_of_pullback_zeroSection_iso_unit_of_charP481 below · cited by 1 · depth 17 - Characteristic-zero fibres of the level-N₀q model are integral
ModularCurve.DRModelPackageLevel.isIntegral_fibre_of_charZero1 below · cited by 1 · depth 17 - Reducedness mod q of the finite-j chart ring
ModularCurve.DRModelPackageLevel.isReduced_chartAlgFin_quotient_span_natCast3 below · cited by 1 · depth 17 - Poincaré pullback of an O-point as a twist of sections
ModularCurve.DRModelPackageLevel.nonempty_pullback_toDR_poincare_pullbackAlong_iso_foldr_sectionTwist180 below · cited by 1 · depth 17 - No isolated points in the geometric fibre at q
ModularCurve.DRModelPackageLevel.not_isOpen_singleton_fibre3 below · cited by 1 · depth 17 - Conorm divisors map to aj₀ followed by β^∗
ModularCurve.DRModelPackageLevel.pts_mk_pullbackAlong_heckeBetaBar_single_sub_eq_abelJacobi_comp_degeneracyPullbackHom1,236 below · cited by 1 · depth 17 - Stalks of the generic fibre have Krull dimension at most one
ModularCurve.DRModelPackageLevel.ringKrullDim_stalk_pullback_specMap_rat_le_one1 below · cited by 1 · depth 17 - Non-strict inertia-fixed places specialise to crossings
ModularCurve.DRModelPackageLevel.section_base_closedPoint_eq_crossing_of_reduceFst_mem_of_sp_eq_spPlace1,891 below · cited by 2 · depth 17 - Germ of q regular and stalk dimension ≥ 2 at a crossing
ModularCurve.DRModelPackageLevel.baseGerm_mem_nonZeroDivisors_and_two_le_ringKrullDim_stalk191 below · cited by 1 · depth 18 - Crossing points specialise from ξ_∞ and ξ₀
ModularCurve.DRModelPackageLevel.bcMap_genericPoint_specializes_crossingPt0 below · cited by 6 · depth 18 - Points of the q-fibre specialise from ξ_∞ or ξ₀
ModularCurve.DRModelPackageLevel.bcMap_genericPoint_specializes_or0 below · cited by 3 · depth 18 - Branch ideals at a crossing intersect in (q)
ModularCurve.DRModelPackageLevel.branchIdeal_xiInf_inf_branchIdeal_xiZero_eq_span_baseGerm10 below · cited by 2 · depth 18 - Transversality of the two branches at a crossing point
ModularCurve.DRModelPackageLevel.branchIdeal_xiInf_sup_branchIdeal_xiZero_eq_maximalIdeal202 below · cited by 2 · depth 18 - No proper generalisations of ξ_∞, ξ₀ in the special fibre
ModularCurve.DRModelPackageLevel.eq_xi_of_specializes125 below · cited by 6 · depth 18 - Node coordinates and chain position at one supersingular crossing
ModularCurve.DRModelPackageLevel.exists_nodeCoordinates_and_forall_mem_support_iff_chainPos_of_chartPresentation2,007 below · cited by 2 · depth 18 - Function field of the O-model inside ℚ̄(X₀(N₀q))
ModularCurve.DRModelPackageLevel.exists_ringHom_functionField_pullback_forall_eq_algebraMap_and_coe_eq_coeffEmb0 below · cited by 2 · depth 18 - Each branch ideal at a crossing equals (a,q)
ModularCurve.DRModelPackageLevel.exists_span_pair_baseGerm_eq_branchIdeal279 below · cited by 1 · depth 18 - Ogg's element at a crossing point: tt'=q¹²
ModularCurve.DRModelPackageLevel.exists_stalk_mul_eq_baseGerm_pow_and_isUnit_stalkSpecializes_of_crossing243 below · cited by 1 · depth 18 - Strict places orient sections onto the two special-fibre components
ModularCurve.DRModelPackageLevel.forall_isStrict_section_mem_range_comp_zero_comp_one_of_sp_eq_spPlace1,861 below · cited by 1 · depth 18 - Injectivity of crossing points under residue-field rationality
ModularCurve.DRModelPackageLevel.injective_crossingPt_of_exists_section2 below · cited by 1 · depth 18 - Integrality of the q-adic base change of the Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.isIntegral_pullback_toBase_specMap3 below · cited by 10 · depth 18 - Regularity of stalks off D(q) away from crossing points
ModularCurve.DRModelPackageLevel.isRegularLocalRing_stalk_of_forall_ne_crossingPt204 below · cited by 3 · depth 18 - Poincaré pullback at an effective divisor class as ideal-power modules
ModularCurve.DRModelPackageLevel.nonempty_poincare_pullbackAlong_pts_mk_iso_invModule_prod_pow_tensor_module_pow24 below · cited by 1 · depth 18 - Non-emptiness of the finite Igusa chart over O
ModularCurve.DRModelPackageLevel.nonempty_preimage_iotaFin_pullback_toBase_specMap0 below · cited by 2 · depth 18 - Incomparable branch ideals at a crossing point
ModularCurve.DRModelPackageLevel.not_branchIdeal_le_branchIdeal_crossingPt186 below · cited by 3 · depth 18 - Image of a special-fibre component is closed
ModularCurve.DRModelPackageLevel.range_comp_bcMap_eq_closure_and_isClosed185 below · cited by 4 · depth 18 - Rational crossing points are closed with residue field O/𝔪
ModularCurve.DRModelPackageLevel.residue_baseGerm_surjective_and_isClosed_crossingPt0 below · cited by 1 · depth 18 - Stalks of the base-changed level model have dimension at most two
ModularCurve.DRModelPackageLevel.ringKrullDim_stalk_XO_le_two2 below · cited by 3 · depth 18 - The two branch points of the special fibre are distinct
ModularCurve.DRModelPackageLevel.xiInf_ne_xiZero133 below · cited by 8 · depth 18 - The points ξ_∞ and ξ₀ lie over the smooth locus
ModularCurve.DRModelPackageLevel.xi_mem_preimage_smoothLocus141 below · cited by 5 · depth 18 - Crossing points lie over the j-finite chart
ModularCurve.DRModelPackageLevel.crossingPt_mem_preimage_iotaFin3 below · cited by 1 · depth 19 - Minimal primes over q at a crossing point are the two branch primes
ModularCurve.DRModelPackageLevel.eq_comap_or_eq_comap_of_mem_minimalPrimes_natCast_of_specializes0 below · cited by 1 · depth 19 - Special-fibre points of codimension ≤ 1 are ξ_∞ or ξ₀
ModularCurve.DRModelPackageLevel.eq_xi_of_ringKrullDim_stalk_le_one52 below · cited by 1 · depth 19 - Points in both component closures are crossing points
ModularCurve.DRModelPackageLevel.exists_eq_crossingPt_of_mem_closure_of_mem_closure185 below · cited by 3 · depth 19 - Germ of j(q^q)-j^q vanishing along the first component
ModularCurve.DRModelPackageLevel.exists_germ_jq_sub_pow_and_stalkSpecializes_mem_maximalIdeal_comp_zero299 below · cited by 1 · depth 19 - Maximal ideal at a crossing is a branch ideal plus one element
ModularCurve.DRModelPackageLevel.exists_maximalIdeal_eq_branchIdeal_sup_span_singleton267 below · cited by 1 · depth 19 - Radicality of (q) on the base-changed level model
ModularCurve.DRModelPackageLevel.isRadical_span_natCast_sections_XO5 below · cited by 1 · depth 19 - Maximal ideals preserved at crossings under base change
ModularCurve.DRModelPackageLevel.map_maximalIdeal_stalkMap_bcMap_eq_of_inertia_grain0 below · cited by 1 · depth 19 - Minimality over (q) and distinctness of the two generic points
ModularCurve.DRModelPackageLevel.mem_minimalPrimes_of_fst_comp_genericPoint_eq_and_fst_comp_ne132 below · cited by 4 · depth 19 - Germs at a supersingular crossing lie in the node ring
ModularCurve.DRModelPackageLevel.mem_nodeIntegers_of_stalk_of_specializes_of_nodeEquiv_eq1,993 below · cited by 1 · depth 19 - Trichotomy for points of the base-changed Deligne–Rapoport model
ModularCurve.DRModelPackageLevel.mem_preimage_basicOpen_or_mem_preimage_smoothLocus_or_exists_eq_crossingPt201 below · cited by 1 · depth 19 - Branch residues and orders at a supersingular crossing
ModularCurve.DRModelPackageLevel.nodeResidue_eq_zero_iff_and_ord_eq_of_specializes_of_mem_maximalIdeal364 below · cited by 1 · depth 19 - Crossing coordinates are uniformisers on the two branches
ModularCurve.DRModelPackageLevel.ord_placeOfPoint_stalkMap_eq_one_of_span_eq_maximalIdeal0 below · cited by 1 · depth 19 - Evaluation at a place equals pull-back along an O-section
ModularCurve.DRModelPackageLevel.phi_mem_and_evalAt_eq_stalkClosedPointTo_of_section3 below · cited by 1 · depth 19 - Saturation of the geometric q-fibre components under morphisms over X
ModularCurve.DRModelPackageLevel.preimage_closure_image_range_comp_eq_of_comp_fst_eq235 below · cited by 2 · depth 19 - Crossing points are rational over the inertia ring O
ModularCurve.DRModelPackageLevel.surjective_residue_comp_germ_comp_appTop_of_inertia_grain10 below · cited by 1 · depth 19 - A-points above supersingular places specialise to the crossing
ModularCurve.DRModelPackageLevel.base_closedPoint_eq_crossing_of_reduceFst_eq_of_sp_eq_spPlace1,891 below · cited by 1 · depth 20 - The zeroth fibre component meets the finite chart in the Gauss prime
ModularCurve.DRModelPackageLevel.exists_fst_comp_zero_genericPoint_eq_iotaFin_and_mem_asIdeal_iff288 below · cited by 2 · depth 20 - Branch germs read as Gauss residues on X₀(N₀)_{κ_A}
ModularCurve.DRModelPackageLevel.ffEquiv_symm_stalkMap_genericPoint_eq_residue_phi363 below · cited by 1 · depth 20 - Germs at a point met by both branches lie in both prolongations
ModularCurve.DRModelPackageLevel.mem_integers_and_mem_integers_of_stalk_of_specializes361 below · cited by 1 · depth 20 - Branch generic stalks map into the two Gauss prolongations
ModularCurve.DRModelPackageLevel.phi_algebraMap_stalk_mem_integers_comp_genericPoint360 below · cited by 3 · depth 20 - Chart-pinned readings agree at the generic point
ModularCurve.DRModelPackageLevel.specMap_comp_fromSpecStalk_genericPoint_comp_fst_eq_of_coe_eq_coeffEmb0 below · cited by 1 · depth 20 - Retraction ρ_∞ kills W₀-nonunits, misses the second component
ModularCurve.DRModelPackageLevel.apply_rhoInf_eq_zero_of_mem_nonunits_and_not_ker_le_of_fst_comp_one_eq_iotaInf210 below · cited by 1 · depth 21 - Generic points of both fibre components lie in both Igusa charts
ModularCurve.DRModelPackageLevel.exists_fst_comp_genericPoint_eq_iotaFin_and_eq_iotaInf212 below · cited by 3 · depth 21
ModularCurve.DRResolvedModelPackage 22
- Base change along τ of a section and its geometric generic point
ModularCurve.DRResolvedModelPackage.eEta_comp_pullbackMap_eq_comp_toDR_of_comp_fst_eq0 below · cited by 1 · depth 17 - Divisorial descent along the resolved Deligne–Rapoport model
ModularCurve.DRResolvedModelPackage.exists_isInvertible_and_pullback_toDR_iso_of_forall_exceptional_degree_eq_zero1,192 below · cited by 1 · depth 17 - Node bijection and orientation bit for the component dictionary
ModularCurve.DRResolvedModelPackage.exists_nodeEquiv_swap_forall_comp_eq_dict_of_sections_of_charts1,640 below · cited by 1 · depth 17 - An O-point of D whose Poincaré class is M
ModularCurve.DRResolvedModelPackage.exists_schemeHomOver_poincare_pullbackAlong_iso_of_generic_sectionTwist_of_forall_isAlgEquivZero36 below · cited by 1 · depth 17 - Inertia-fixed places give sections of the resolved model
ModularCurve.DRResolvedModelPackage.exists_section_toDR_generic_eq_pointEquivPlace_symm_of_forall_inertia_smul_eq0 below · cited by 2 · depth 17 - Vertical twist to multidegree zero fixing the generic fibre
ModularCurve.DRResolvedModelPackage.exists_verticalTwist_multidegree_eq_zero_and_generic_iso_sectionTwist46 below · cited by 1 · depth 17 - Multidegree zero forces algebraic equivalence to zero on fibres
ModularCurve.DRResolvedModelPackage.isAlgEquivZero_fibre_of_pullback_toDR_iso_divisorial_of_multidegree_eq_zero1,241 below · cited by 1 · depth 17 - Vanishing component class puts the multidegree in α's image
ModularCurve.DRResolvedModelPackage.multidegree_mem_range_intersectionAlpha_of_comp_eq_zero482 below · cited by 1 · depth 17 - Sections of the resolved model avoid the edge points
ModularCurve.DRResolvedModelPackage.ne_edgePt_and_mem_smoothOffEdges_and_existsUnique_mem_comp_support_of_section1 below · cited by 1 · depth 17 - Generic fibre of a bundle descended through the resolution
ModularCurve.DRResolvedModelPackage.nonempty_pullback_comp_toDR_iso_sectionTwist_of_iso_divisorial0 below · cited by 1 · depth 17 - Local chart presentation at a node of the resolved model
ModularCurve.DRResolvedModelPackage.DRResolvedModelCharts.exists_chartPresentation_stalk255 below · cited by 2 · depth 18 - Local triviality at a crossing of a degree-zero divisorial sheaf
ModularCurve.DRResolvedModelPackage.DRResolvedModelCharts.exists_open_pullback_iso_unit_of_forall_exceptional_degree_eq_zero94 below · cited by 1 · depth 18 - Crossing points have stalk Krull dimension greater than one
ModularCurve.DRResolvedModelPackage.DRResolvedModelCharts.not_ringKrullDim_stalk_crossingPt_le_one4 below · cited by 1 · depth 18 - Strict transforms detected on the Deligne–Rapoport closed fibre
ModularCurve.DRResolvedModelPackage.eq_inl_iff_toDR_base_mem_range_compInf_of_mem_comp_support56 below · cited by 1 · depth 18 - Multidegree zero gives χ=χ(𝒪) on a strict transform
ModularCurve.DRResolvedModelPackage.eulerChar_sectionsOf_pullback_strictTransform_eq_of_multidegree_eq_zero_of_surjective237 below · cited by 1 · depth 18 - Node width equals the j-width of the reduced j-value
ModularCurve.DRResolvedModelPackage.width_eq_jWidth_of_exists_jFun_sub_mem_maximalIdeal_of_prolongationTuple1,630 below · cited by 1 · depth 18 - Euler characteristic of ι_w^*𝒪(Cᵥ) shifts by adjacency number
ModularCurve.DRResolvedModelPackage.eulerChar_sectionsOf_pullback_invModule_comp_eq_add_x0MqAdjV4107 below · cited by 1 · depth 19 - Nodes of the resolved model index supersingular j-invariants
ModularCurve.DRResolvedModelPackage.exists_node_equiv_ssJSet_germ_sub_mem_maximalIdeal_iff394 below · cited by 2 · depth 19 - Edges between distinct components equal the adjacency x0MqAdj
ModularCurve.DRResolvedModelPackage.natCard_edge_eq_x0MqAdjV40 below · cited by 3 · depth 19 - Inertia-fixed strict place with an 𝒪-section of the resolved model
ModularCurve.DRResolvedModelPackage.exists_isStrictFst_forall_inertia_smul_eq_and_section_toDR_generic_eq444 below · cited by 1 · depth 20 - Finiteness and degree of Cᵥ ∩ C_w over k
ModularCurve.DRResolvedModelPackage.isFinite_and_finrank_subscheme_comap_comp_eq_natCardV42 below · cited by 1 · depth 20 - Invertibility of the ideal of Cᵥ restricted to C_w
ModularCurve.DRResolvedModelPackage.isInvertible_comap_comp_subschemeIotaV41 below · cited by 1 · depth 20
ModularCurve.DRResolvedModelPackageLevel 13
- Multidegree lies in the image of the intersection pairing
ModularCurve.DRResolvedModelPackageLevel.multidegree_mem_range_intersectionAlpha_of_pullbackAlong_generic_iso_of_surjective272 below · cited by 1 · depth 16 - Divisorial presentation of a Pic⁰-point on the regular model
ModularCurve.DRResolvedModelPackageLevel.exists_pullback_toDR_iso_foldr_ker_tensor_invModule_prod_comp53 below · cited by 1 · depth 17 - Sections avoiding the zeroth branch meet only the component inl 1
ModularCurve.DRResolvedModelPackageLevel.mem_support_comp_inl_one_of_not_mem_range0 below · cited by 1 · depth 17 - Section off the branch meets only `comp (Sum.inl 0)`
ModularCurve.DRResolvedModelPackageLevel.mem_support_comp_inl_zero_of_not_mem_range0 below · cited by 1 · depth 17 - Vanishing multidegree of a divisorially presented bundle on the resolved model
ModularCurve.DRResolvedModelPackageLevel.sum_single_add_intersectionAlpha_eq_zero_of_pullback_toDR_iso251 below · cited by 1 · depth 17 - Euler characteristic of the Poincaré pullback on a component
ModularCurve.DRResolvedModelPackageLevel.eulerChar_sectionsOf_pullback_comp_toDR_poincare_tensor_unit_eq82 below · cited by 1 · depth 18 - Euler characteristic of a divisorial twist on a fibre component
ModularCurve.DRResolvedModelPackageLevel.eulerChar_sectionsOf_pullback_foldr_ker_tensor_prod_comp_eq_add_sum_single_add_intersectionAlpha135 below · cited by 1 · depth 18 - Special-fibre components as proper κ-curves with two affine charts
ModularCurve.DRResolvedModelPackageLevel.exists_toSpec_comp_eq_and_isProper_and_twoAffineOpenCover_and_sections_of_surjective95 below · cited by 1 · depth 18 - Sections of a resolved model avoid the edge points
ModularCurve.DRResolvedModelPackageLevel.ne_edgePt_and_mem_smoothOffEdges_and_existsUnique_mem_comp_support_of_section1 below · cited by 1 · depth 18 - Sections miss generic points of special-fibre components
ModularCurve.DRResolvedModelPackageLevel.eta_notMem_range_of_comp_toBase_eq_id3 below · cited by 1 · depth 19 - Euler characteristics of ± a C_c twists on the resolved model
ModularCurve.DRResolvedModelPackageLevel.eulerChar_sectionsOf_pullback_pow_comp_invModule_tensor_and_module_tensor_self126 below · cited by 1 · depth 19 - Component intersection on the resolved model: finite, degree the edge count
ModularCurve.DRResolvedModelPackageLevel.isFinite_and_finrank_subscheme_comap_comp_eq_natCard2 below · cited by 2 · depth 19 - Restriction of a component ideal to a distinct component is invertible
ModularCurve.DRResolvedModelPackageLevel.isInvertible_comap_comp_subschemeIota1 below · cited by 2 · depth 19
ModularCurve.DRResolvedModelPackageLevelRam 1
- Local equation uv=q^ew at a node of the resolved model
ModularCurve.DRResolvedModelPackageLevelRam.DRResolvedModelChartsLevelRam.exists_chartPresentation_stalk189 below · cited by 2 · depth 18
ModularCurve.FifteenA1 4
- Rational affine points of the elliptic curve 15a1
ModularCurve.FifteenA1.coords_of_equation14 below · cited by 1 · depth 9 - The 2-descent pair of 15a1 is multiplicative modulo squares
ModularCurve.FifteenA1.deltaPairHom0 below · cited by 2 · depth 10 - Square certificate for halving on 15a1: kerδ=2E(ℚ)
ModularCurve.FifteenA1.secondDescentInput0 below · cited by 1 · depth 10 - Selmer bound for the 2-descent image on 15a1
ModularCurve.FifteenA1.selmerBound1 below · cited by 1 · depth 10
ModularCurve.FrobeniusQuadratic 1
- Frobenius quadratic relation from specialisation data
ModularCurve.FrobeniusQuadratic.of_specializationExists0 below · cited by 5 · depth 9
ModularCurve.FullLevel 1518
- GL₂ generator laws on Jac(q,M') when q∤ M'
ModularCurve.FullLevel.gl2Laws_of_not_dvd34 below · cited by 4 · depth 16 - Hecke generators on the full-level Jacobian commute
ModularCurve.FullLevel.heckeGenCommute278 below · cited by 2 · depth 16 - Existence of level automorphisms when q ∤ M'
ModularCurve.FullLevel.levelAutInputs_of_not_dvd31 below · cited by 20 · depth 16 - Inertia at ℓ ∤ qM'λ acts trivially on T_λ(Jac)
ModularCurve.FullLevel.tateGal_eq_one_of_mem_inertiaSubgroupIn974 below · cited by 2 · depth 16 - Galois and GL₂(ℤ/q) actions commute on the Tate module
ModularCurve.FullLevel.tateGal_mul_tateGL2_comm32 below · cited by 6 · depth 16 - Hecke and GL₂(mathbb F_q) actions commute on the Tate module
ModularCurve.FullLevel.tateHecke_mul_tateGL2_comm86 below · cited by 2 · depth 16 - Hecke and Galois operators commute on the Tate module
ModularCurve.FullLevel.tateHecke_mul_tateGal_comm11 below · cited by 2 · depth 16 - Scalars act by inverse diamond and squared component shift
ModularCurve.FullLevel.eval_gl2Jac_scalarElem_eq_diamondHBar_inv_eval_pow67 below · cited by 2 · depth 17 - Level automorphisms lift to the Hecke top curve
ModularCurve.FullLevel.exists_algEquiv_intertwinesAlong_heckeAlphaHBar_heckeBetaHBar_levelAutBar33 below · cited by 1 · depth 17 - Finite freeness and base-changed actions on T_λ(Jac(q,M'))
ModularCurve.FullLevel.exists_galoisRep_isAdicContinuous_heckeRep_gl2Rep_baseChange_tateModule_jac303 below · cited by 3 · depth 17 - Cuspidal type of a newform in the full-level Tate module
ModularCurve.FullLevel.exists_ringHom_heckeGen_eq_and_exists_ne_zero_comm_baseChange_tateModule_jac790 below · cited by 2 · depth 17 - Integral form of Shimura reciprocity for γ^sharp-slashing
ModularCurve.FullLevel.exists_smul_slash_conjElem_eq_sum_exp_pow_smul_of_mem_Gamma030 below · cited by 2 · depth 17 - Level automorphisms compose contravariantly: τ_{αβ}=τ_β∘τ_α
ModularCurve.FullLevel.levelAutBar_mul32 below · cited by 40 · depth 17 - Level automorphisms under diagonal conjugation modulo q
ModularCurve.FullLevel.levelAutBar_pow_inv_eq_levelAutBar_of_diag_conj32 below · cited by 9 · depth 17 - Eichler–Shimura relation with scalar diamond at full level q
ModularCurve.FullLevel.tateGal_mul_tateGal_sub_tateHecke_mul_tateGal_add_smul_tateGL2_scalarElem_eq_zero1,028 below · cited by 2 · depth 17 - Newform K(q)-invariants embed into the dual Tate module
ModularCurve.FullLevel.exists_injective_dual_baseChange_tateModule_jac_of_isNewform_of_range_eq_span613 below · cited by 1 · depth 18 - Shimura reciprocity for f∣_kγ^sharp at full level
ModularCurve.FullLevel.exists_ratCast_slash_conjElem_eq_sum_exp_pow_smul_of_mem_Gamma019 below · cited by 1 · depth 18 - Cusp forms embed in the dual complexified Tate module of J_H
ModularCurve.FullLevel.exists_injective_cuspForm_dual_baseChange_tateModule_jacComp_comm529 below · cited by 1 · depth 19 - Equivariant injection of λ-adic Tate modules for M' ∣ M''
ModularCurve.FullLevel.exists_injective_linearMap_tateModule_jac_comp_tateGal_eq_and_comp_tateGL2_eq_of_dvd68 below · cited by 2 · depth 20 - Drinfeld specialisation of the full-level Tate module
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_inertiaIgusa1,288 below · cited by 1 · depth 20 - Semistable covering, model and descent at full level q
ModularCurve.FullLevel.exists_semistableCovering_semistableModel_descent_equiv_w2_guards_inertiaInfty4,821 below · cited by 1 · depth 20 - Level-M' q-expansion field inside the full-level field
ModularCurve.FullLevel.laurentBaseChange_gamma0_le_fieldBar0 below · cited by 8 · depth 20 - Transport of semistable model to the Fin-indexed telescope
ModularCurve.FullLevel.SemistableCovering.exists_semistableModel_telescope0 below · cited by 6 · depth 21 - Telescope laws of a semistable covering at full level
ModularCurve.FullLevel.SemistableCovering.telescope_laws0 below · cited by 6 · depth 21 - Compatibility of level automorphisms under divisibility of levels
ModularCurve.FullLevel.coe_levelAutBar_apply_eq_coe_levelAutBar_apply_of_dvd_of_coe_eq0 below · cited by 1 · depth 21 - Existence of the Igusa valuation rings at full level q
ModularCurve.FullLevel.exists_igusaValuationSubrings1,098 below · cited by 9 · depth 21 - Tate-module specialisation from a semistable covering, q=3
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_inertiaIgusa_of_eq_three1,288 below · cited by 1 · depth 21 - Drinfeld specialisation of the λ-adic Tate module, q=2
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_inertiaIgusa_of_eq_two1,288 below · cited by 1 · depth 21 - Drinfeld specialisation of the full-level Tate module over a model
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_reduction_of_inertiaIgusa1,243 below · cited by 1 · depth 21 - Assembly of the full-level semistable covering, model and descent
ModularCurve.FullLevel.exists_semistableCovering_equivClauses_of_valuationSubrings_semistableModel_inertiaInfty_charted4,820 below · cited by 1 · depth 21 - Semistable covering, model and descent at full level q=3
ModularCurve.FullLevel.exists_semistableCovering_semistableModel_descent_equiv_perPoint_w2_guards_inertiaInfty_of_eq_three_of_dvd4,597 below · cited by 1 · depth 21 - Semistable covering, model and descent at q=2, per-point clauses
ModularCurve.FullLevel.exists_semistableCovering_semistableModel_descent_equiv_perPoint_w2_guards_inertiaInfty_of_eq_two_of_dvd4,583 below · cited by 1 · depth 21 - Supersingular component chart over a supersingular place: existence and uniqueness
ModularCurve.FullLevel.exists_supersingularChart3,878 below · cited by 2 · depth 21 - Tame inertia frame for the full-level telescope
ModularCurve.FullLevel.telescope_frame_of_semistableCovering876 below · cited by 2 · depth 21 - Chart residue of a level function equals its value at s
ModularCurve.FullLevel.ComponentChart.exists_residue_inclusion_eq_algebraMap_evalAt_of_integers_eq0 below · cited by 3 · depth 22 - Inertia of tame value α sends ζ to ζ^{N(α)}
ModularCurve.FullLevel.Idx.smul_eq_pow_of_tameCharacter_eq_of_algebraMap_eq_pow_succ0 below · cited by 7 · depth 22 - Semistable model with descent from a disc-charted covering
ModularCurve.FullLevel.SemistableCovering.exists_semistableModel_descent_of_discCharts_of_noCuspFreePackageSS_of_jPins_of_nodeRings_nodeCharts4,476 below · cited by 1 · depth 22 - Igusa unipotent clause at ∞ from a Gauss presentation
ModularCurve.FullLevel.SemistableCovering.igusaUnipotentClause_of_gaussPresentation35 below · cited by 6 · depth 22 - Inertia clause from Gauss presentation and residue discs
ModularCurve.FullLevel.SemistableCovering.inertiaClause_of_gaussPresentation_of_integers_eq_comap_of_discs44 below · cited by 3 · depth 22 - Anchored Γ₀(M')-equivariance of the Igusa charts
ModularCurve.FullLevel.SemistableCovering.naturality_anchor_of_equivClauses_of_igusaLabelling1,094 below · cited by 1 · depth 22 - Inertia naturality on the supersingular charts
ModularCurve.FullLevel.SemistableCovering.naturality_inertia_supersingular_of_discTransport_of_inTube_of_perm_drinfeld3,874 below · cited by 1 · depth 22 - Naturality of level automorphisms on the supersingular charts
ModularCurve.FullLevel.SemistableCovering.naturality_levelAut_supersingular_of_discFamily1,094 below · cited by 1 · depth 22 - W2 clauses of a semistable covering from its chart rings
ModularCurve.FullLevel.SemistableCovering.w2Clauses_of_gaussPresentation_of_valuationSubring_over_fixed3,870 below · cited by 1 · depth 22 - Tame inertia with trivial character commutes with ℚ̄-level automorphisms
ModularCurve.FullLevel.arithmeticGalois_mul_ofAlgAut_levelAutBar_of_tameCharacter_eq_one40 below · cited by 19 · depth 22 - Borel-fixing level automorphisms preserve the Gauss valuation ring
ModularCurve.FullLevel.comap_levelAutBar_eq_of_redQ_smul_lineInfty_eq1,092 below · cited by 8 · depth 22 - Uniqueness of level-fixed component charts over a supersingular place
ModularCurve.FullLevel.componentChart_integers_eq_of_isCurveOver_of_over_of_comap_levelAutBar_eq3,864 below · cited by 6 · depth 22 - Equivariance of the cuspidal specialisation map from component laws
ModularCurve.FullLevel.cuspidalSpecialization_comp_eq_tateProdRep_comp_of_laws0 below · cited by 3 · depth 22 - Injectivity of the cuspidal specialisation at full level q
ModularCurve.FullLevel.eq_zero_of_cuspidalSpecialization_eq_zero_of_forall_sum_unipotent_eq_zero_of_semistableCovering_of_reduction312 below · cited by 1 · depth 22 - Inertia on supersingular charts: induced automorphism and naturality of reduction
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_arithmeticGalois_and_red_eq_of_semistableCovering_of_igusaDom42 below · cited by 1 · depth 22 - Naturality of λ-adic reduction under level automorphisms
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_levelAutBar_and_red_eq_of_semistableCovering42 below · cited by 1 · depth 22 - Separating supersingular places by reductions of cusp-regular integral functions
ModularCurve.FullLevel.exists_cuspRegular_separating755 below · cited by 4 · depth 22 - Inertia permutes the Igusa chart domains of a semistable covering
ModularCurve.FullLevel.exists_forall_mem_dom_teleChart_eIg_arithmeticGalois_smul_of_inertiaIgusaInftyClause38 below · cited by 1 · depth 22 - Full-level test function: Igusa unit, vanishing over s, annulus-unit
ModularCurve.FullLevel.exists_fullLevelFunction_residue_zero_unit_igusa_ord_zero_tubeAnnulus_jE46 below · cited by 1 · depth 22 - Igusa nodes and residue-disc family for the Gauss prolongation
ModularCurve.FullLevel.exists_igusaNodes_discFamily_of_igusaGaussRing_allInertia2,035 below · cited by 1 · depth 22 - Igusa valuation rings at q = 3 for full level
ModularCurve.FullLevel.exists_igusaValuationSubrings_of_eq_three381 below · cited by 9 · depth 22 - Igusa valuation subrings at full level for q = 2
ModularCurve.FullLevel.exists_igusaValuationSubrings_of_eq_two381 below · cited by 9 · depth 22 - Labelled level automorphisms suffice to reach an Igusa disc
ModularCurve.FullLevel.exists_levelAut_smul_mem_igusaDisc_of_forall_not_inTube1,104 below · cited by 1 · depth 22 - Tate module of the full-level Jacobian splits over its components
ModularCurve.FullLevel.exists_linearEquiv_tateModule_jac_pi_tateGal_slJac_diagJac0 below · cited by 3 · depth 22 - Drinfeld intertwining on the supersingular charts, rational Tate modules
ModularCurve.FullLevel.exists_linearMap_rationalTateModule_drinfeld_injective_comp_eq_of_inducesOnChart_of_semistableCovering64 below · cited by 1 · depth 22 - Drinfeld specialisation sp₀ over a semistable covering, q=3
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_reduction_of_inertiaIgusa_of_eq_three1,243 below · cited by 1 · depth 22 - Drinfeld specialisation sp₀ from a semistable covering, q=2
ModularCurve.FullLevel.exists_linearMap_tateProd_comp_baseChange_eq_and_eq_zero_of_semistableCovering_of_semistableModel_of_reduction_of_inertiaIgusa_of_eq_two1,243 below · cited by 1 · depth 22 - Igusa nodes over supersingular places along level transports
ModularCurve.FullLevel.exists_mem_igusaNodes_over_of_levelAut_transport_linear_nodes_iff292 below · cited by 1 · depth 22 - Regular prolongation whose integers are the Igusa ring at ∞
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing1,112 below · cited by 12 · depth 22 - Semistable covering of the full-level modular curve at q=3
ModularCurve.FullLevel.exists_semistableCovering_equivClauses_of_valuationSubrings_semistableModel_inertiaInfty_charted_of_eq_three_of_dvd4,596 below · cited by 1 · depth 22 - Semistable covering, model and descent at q=2
ModularCurve.FullLevel.exists_semistableCovering_equivClauses_of_valuationSubrings_semistableModel_inertiaInfty_charted_of_eq_two_of_dvd4,582 below · cited by 1 · depth 22 - Supersingular component chart with q+1 exceptional places
ModularCurve.FullLevel.exists_supersingularChart_local3,698 below · cited by 1 · depth 22 - Supersingular component chart at q=3, existence and uniqueness
ModularCurve.FullLevel.exists_supersingularChart_of_eq_three_of_dvd3,870 below · cited by 2 · depth 22 - Existence and uniqueness of the supersingular chart at q=2
ModularCurve.FullLevel.exists_supersingularChart_of_eq_two_of_dvd3,868 below · cited by 2 · depth 22 - Tube annuli at a supersingular place, with discs and crossing models
ModularCurve.FullLevel.exists_tubeAnnuli_width_inertia_discs_charted_inertNodes_nodeRings_nodeCharts_moduliHasse4,080 below · cited by 1 · depth 22 - Genus identity for the semistable covering of X_H(q²M')
ModularCurve.FullLevel.genusFF_fieldBar_add_eq_of_igusa_supersingular_charts4,108 below · cited by 1 · depth 22 - Tame inertia stabilises the transported Igusa discs
ModularCurve.FullLevel.igusaDiscs_inertia_stable_of_stalkInert42 below · cited by 1 · depth 22 - Cusp-free residue discs lie in the supersingular tube
ModularCurve.FullLevel.inTube_of_mem_drinfeldDisc_of_cuspFree1,114 below · cited by 1 · depth 22 - Level automorphisms preserve the j-shadow of a supersingular place
ModularCurve.FullLevel.jShadow_levelAutBar_smul_iff523 below · cited by 3 · depth 22 - Places in a supersingular tube lie in its j-shadow
ModularCurve.FullLevel.jShadow_of_inTube46 below · cited by 3 · depth 22 - Level-M' reduction integers are the Igusa Gauss ring at ∞
ModularCurve.FullLevel.mem_constantReduction_integers_iff_inclusion_mem_igusaGaussRing2 below · cited by 10 · depth 22 - Unipotent naturality at the Igusa chart of ∞
ModularCurve.FullLevel.naturality_unipotent_igusaInfty_of_discFamily1,094 below · cited by 1 · depth 22 - No smooth-point package at a reciprocal annulus pair
ModularCurve.FullLevel.not_smoothPointPackage_of_annulusPair_attached_igusaEnd_of_testFunction_fullLevel1 below · cited by 3 · depth 22 - The q-scaled j-expansion is regular where j is
ModularCurve.FullLevel.ord_nonneg_of_ord_jBar_nonneg_of_coe_eq_jqNModC40 below · cited by 12 · depth 22 - Covering and exclusivity for Igusa charts, Drinfeld charts and annuli
ModularCurve.FullLevel.partition_of_cover_of_disjoint0 below · cited by 3 · depth 22 - Vanishing cycles: (τ-1)V inside unipotent-fixed GL₂(𝔽_q)-translates
ModularCurve.FullLevel.range_tateGal_sub_one_le_span_unipotent_fixed_of_semistableCovering_of_semistableModel1,174 below · cited by 1 · depth 22 - Coordinate law for diag(1,e)σ when σζ=ζ^{1/e}
ModularCurve.FullLevel.ratCoord_comp_baseChange_tateGL2_diagOneElem_mul_tateGal_eq_rationalGaloisRep_arithmeticGalois_comp0 below · cited by 3 · depth 22 - Coordinatewise Γ₀(M')-equivariance of the full-level Tate module
ModularCurve.FullLevel.ratCoord_comp_baseChange_tateGL2_redQ_eq_rationalGaloisRep_levelAutBar_comp0 below · cited by 3 · depth 22 - Cuspidal vectors have tame-inertia-invariant components at full level
ModularCurve.FullLevel.ratCoord_mem_iInf_ker_sub_one_of_forall_sum_unipotent_eq_zero34 below · cited by 3 · depth 22 - Gauss-ring stability forces fixing of [1:0] modulo q
ModularCurve.FullLevel.redQ_smul_lineInfty_eq_of_comap_levelAutBar_eq1,096 below · cited by 2 · depth 22 - Telescope frame for the full-level semistable covering, q=3
ModularCurve.FullLevel.telescope_frame_of_semistableCovering_of_eq_three876 below · cited by 2 · depth 22 - Telescope frame for the full-level semistable covering, q=2
ModularCurve.FullLevel.telescope_frame_of_semistableCovering_of_eq_two876 below · cited by 2 · depth 22 - Drinfeld clause for supersingular charts from their valuation rings
ModularCurve.FullLevel.SemistableCovering.drinfeldClause_of_valuationSubring_over_fixed3,866 below · cited by 2 · depth 23 - Annuli correspond bijectively to nodes of the special fibre
ModularCurve.FullLevel.SemistableCovering.exists_node_forall_mem_annulus_dom_iff_of_charts_eq_smoothFibres_of_annulusFibre_of_nodeFibre47 below · cited by 3 · depth 23 - Semistable model with descent from a disc-charted covering, q=3
ModularCurve.FullLevel.SemistableCovering.exists_semistableModel_descent_of_discCharts_of_noCuspFreePackageSS_of_jPins_of_nodeRings_nodeCharts_of_eq_three_of_dvd4,246 below · cited by 1 · depth 23 - Semistable model with descent from a disc-charted covering, q=2
ModularCurve.FullLevel.SemistableCovering.exists_semistableModel_descent_of_discCharts_of_noCuspFreePackageSS_of_jPins_of_nodeRings_nodeCharts_of_eq_two_of_dvd4,246 below · cited by 1 · depth 23 - Tame inertia acts trivially on transported Igusa charts
ModularCurve.FullLevel.SemistableCovering.inducesOnChart_CIg_arithmeticGalois_of_integers_eq_comap43 below · cited by 1 · depth 23 - Tame-character-one inertia induces the identity on a Drinfeld chart
ModularCurve.FullLevel.SemistableCovering.inducesOnChart_refl_of_drinfeldClause_of_tameCharacter_eq_one0 below · cited by 3 · depth 23 - Anchoring label for the semistable covering at q=3
ModularCurve.FullLevel.SemistableCovering.naturality_anchor_of_equivClauses_of_igusaLabelling_of_eq_three_of_dvd377 below · cited by 1 · depth 23 - Naturality of the Igusa labelling at an anchoring index (q=2)
ModularCurve.FullLevel.SemistableCovering.naturality_anchor_of_equivClauses_of_igusaLabelling_of_eq_two_of_dvd377 below · cited by 1 · depth 23 - Inertia naturality on the supersingular charts, q=3
ModularCurve.FullLevel.SemistableCovering.naturality_inertia_supersingular_of_discTransport_of_inTube_of_perm_drinfeld_of_eq_three_of_dvd3,868 below · cited by 1 · depth 23 - Inertia naturality on the supersingular charts, q=2
ModularCurve.FullLevel.SemistableCovering.naturality_inertia_supersingular_of_discTransport_of_inTube_of_perm_drinfeld_of_eq_two_of_dvd3,866 below · cited by 1 · depth 23 - Naturality of level automorphisms on supersingular charts, q=3
ModularCurve.FullLevel.SemistableCovering.naturality_levelAut_supersingular_of_discFamily_of_eq_three_of_dvd377 below · cited by 1 · depth 23 - Level automorphisms act on the supersingular charts (q=2)
ModularCurve.FullLevel.SemistableCovering.naturality_levelAut_supersingular_of_discFamily_of_eq_two_of_dvd377 below · cited by 1 · depth 23 - Drinfeld and Igusa clauses at q=3 from chart rings
ModularCurve.FullLevel.SemistableCovering.perPoint_w2Clauses_of_gaussPresentation_of_valuationSubring_over_fixed_of_eq_three_of_dvd3,864 below · cited by 1 · depth 23 - Drinfeld and Igusa clauses of a semistable covering at q=2
ModularCurve.FullLevel.SemistableCovering.perPoint_w2Clauses_of_gaussPresentation_of_valuationSubring_over_fixed_of_eq_two_of_dvd3,862 below · cited by 1 · depth 23 - Galois covariance of the full-level automorphisms τ_{ζ,γ}
ModularCurve.FullLevel.arithmeticGalois_mul_ofAlgAut_levelAutBar_inv_smul32 below · cited by 13 · depth 23 - Inertia stabilises the supersingular valuation rings O_{SS}(s)
ModularCurve.FullLevel.arithmeticGalois_smul_mem_drinfeldRing_iff_of_componentChart3,870 below · cited by 1 · depth 23 - Level automorphism acts by Q ↦ Q^{q^2} when q ∣ a
ModularCurve.FullLevel.coe_levelAutBar_apply_eq_qExpand_sq_of_dvd_of_mem_laurentBaseChange_gamma035 below · cited by 19 · depth 23 - Unipotent γ acts on q-expansions as the ζᵇ-twist
ModularCurve.FullLevel.coe_levelAutBar_apply_eq_qTwist_of_redQ_eq_unipotent32 below · cited by 3 · depth 23 - Gauss ring at ∞ is stable under Borel level automorphisms, q=3
ModularCurve.FullLevel.comap_levelAutBar_eq_of_redQ_smul_lineInfty_eq_of_eq_three375 below · cited by 7 · depth 23 - Borel-fixed level automorphisms preserve the Gauss ring, q=2
ModularCurve.FullLevel.comap_levelAutBar_eq_of_redQ_smul_lineInfty_eq_of_eq_two375 below · cited by 7 · depth 23 - Level automorphism with q ∣ δ₀₀ moves the Gauss ring
ModularCurve.FullLevel.comap_levelAutBar_ne_of_dvd50 below · cited by 1 · depth 23 - Uniqueness of the level-fixed component chart over s (q=3)
ModularCurve.FullLevel.componentChart_integers_eq_of_isCurveOver_of_over_of_comap_levelAutBar_eq_of_eq_three_of_dvd3,858 below · cited by 6 · depth 23 - Level-fixed component charts over a supersingular place agree
ModularCurve.FullLevel.componentChart_integers_eq_of_isCurveOver_of_over_of_comap_levelAutBar_eq_of_eq_two_of_dvd3,856 below · cited by 6 · depth 23 - Reduction fixing [1:0] forces q ∣ c
ModularCurve.FullLevel.dvd_of_redQ_smul_lineInfty_eq0 below · cited by 5 · depth 23 - Uniqueness of the place over a supersingular place
ModularCurve.FullLevel.eq_of_forall_mem_iff_map_mem_of_integers_eq_igusaRing1,144 below · cited by 1 · depth 23 - Cuspidal specialisation is injective at full level, q=3
ModularCurve.FullLevel.eq_zero_of_cuspidalSpecialization_eq_zero_of_forall_sum_unipotent_eq_zero_of_semistableCovering_of_reduction_of_eq_three312 below · cited by 1 · depth 23 - Injectivity of cuspidal specialisation along a semistable covering, q=2
ModularCurve.FullLevel.eq_zero_of_cuspidalSpecialization_eq_zero_of_forall_sum_unipotent_eq_zero_of_semistableCovering_of_reduction_of_eq_two312 below · cited by 1 · depth 23 - Unique place above each supersingular place of X₀(M')_κ
ModularCurve.FullLevel.existsUnique_place_restrictAlong_eq_of_mem_ssPlaces1,119 below · cited by 5 · depth 23 - Inertia on supersingular charts; naturality of reduction, q=3
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_arithmeticGalois_and_red_eq_of_semistableCovering_of_igusaDom_of_eq_three42 below · cited by 1 · depth 23 - Inertia on supersingular charts: induced automorphism and equivariance of red, q=2
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_arithmeticGalois_and_red_eq_of_semistableCovering_of_igusaDom_of_eq_two42 below · cited by 1 · depth 23 - Level automorphisms on supersingular charts commute with λ-adic reduction, q=3
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_levelAutBar_and_red_eq_of_semistableCovering_of_eq_three42 below · cited by 1 · depth 23 - Level automorphisms commute with λ-adic reduction on supersingular charts (q=2)
ModularCurve.FullLevel.exists_algEquiv_inducesOnChart_levelAutBar_and_red_eq_of_semistableCovering_of_eq_two42 below · cited by 1 · depth 23 - Level automorphism acts by root-of-unity twist when q∤ a
ModularCurve.FullLevel.exists_coe_levelAutBar_apply_eq_qTwist_of_not_dvd_of_mem_laurentBaseChange_gamma038 below · cited by 18 · depth 23 - Valuation rings over A dominating R₀ are Igusa rings
ModularCurve.FullLevel.exists_eq_igusaRing_of_forall_mem_integers_mem1,116 below · cited by 2 · depth 23 - Integral functions independent over the level-M' field
ModularCurve.FullLevel.exists_family_liftIndep_gamma01,039 below · cited by 1 · depth 23 - Inertia permutes the Igusa charts' domains (q=3)
ModularCurve.FullLevel.exists_forall_mem_dom_teleChart_eIg_arithmeticGalois_smul_of_inertiaIgusaInftyClause_of_eq_three38 below · cited by 1 · depth 23 - Inertia transports Igusa chart domains, case q=2
ModularCurve.FullLevel.exists_forall_mem_dom_teleChart_eIg_arithmeticGalois_smul_of_inertiaIgusaInftyClause_of_eq_two38 below · cited by 1 · depth 23 - A test function vanishing on the supersingular component, q=3
ModularCurve.FullLevel.exists_fullLevelFunction_residue_zero_unit_igusa_ord_zero_tubeAnnulus_jE_of_eq_three_of_dvd46 below · cited by 1 · depth 23 - Full-level test function at q=2: Igusa unit, zero residue
ModularCurve.FullLevel.exists_fullLevelFunction_residue_zero_unit_igusa_ord_zero_tubeAnnulus_jE_of_eq_two_of_dvd46 below · cited by 1 · depth 23 - Igusa nodes and residue discs at q=3
ModularCurve.FullLevel.exists_igusaNodes_discFamily_of_igusaGaussRing_allInertia_of_eq_three_of_dvd1,770 below · cited by 1 · depth 23 - Igusa nodes and residue discs for q=2
ModularCurve.FullLevel.exists_igusaNodes_discFamily_of_igusaGaussRing_allInertia_of_eq_two_of_dvd1,770 below · cited by 1 · depth 23 - Smooth-point charts for the Igusa Gauss ring at ∞
ModularCurve.FullLevel.exists_igusaSmoothPointCharts_of_igusaGaussRing_allInertia2,033 below · cited by 1 · depth 23 - The j-invariant is R-integral with residue of order -1
ModularCurve.FullLevel.exists_jInvariant_mem_integers3 below · cited by 5 · depth 23 - Labelled level automorphisms suffice to reach Igusa discs, q=3
ModularCurve.FullLevel.exists_levelAut_smul_mem_igusaDisc_of_forall_not_inTube_of_eq_three_of_dvd387 below · cited by 1 · depth 23 - Labelled level automorphisms suffice for Igusa discs, q=2
ModularCurve.FullLevel.exists_levelAut_smul_mem_igusaDisc_of_forall_not_inTube_of_eq_two_of_dvd387 below · cited by 1 · depth 23 - Drinfeld identification on supersingular charts, case q=3
ModularCurve.FullLevel.exists_linearMap_rationalTateModule_drinfeld_injective_comp_eq_of_inducesOnChart_of_semistableCovering_of_eq_three64 below · cited by 1 · depth 23 - Drinfeld identification on supersingular charts at q=2
ModularCurve.FullLevel.exists_linearMap_rationalTateModule_drinfeld_injective_comp_eq_of_inducesOnChart_of_semistableCovering_of_eq_two64 below · cited by 1 · depth 23 - Transitivity of Γ₀(M') on P¹(𝔽_q) via reduction mod q
ModularCurve.FullLevel.exists_mem_gamma0_redQ_inv_smul_eq1 below · cited by 6 · depth 23 - Igusa nodes over supersingular places under level transport, q=3
ModularCurve.FullLevel.exists_mem_igusaNodes_over_of_levelAut_transport_linear_nodes_iff_of_eq_three_of_dvd292 below · cited by 1 · depth 23 - Transported Igusa nodes over supersingular places, q = 2
ModularCurve.FullLevel.exists_mem_igusaNodes_over_of_levelAut_transport_linear_nodes_iff_of_eq_two_of_dvd292 below · cited by 1 · depth 23 - Completeness of reduction on Riemann–Roch spaces, full level
ModularCurve.FullLevel.exists_mem_integers_riemannRochSpace_residue_eq_of_mem_riemannRochSpace_placeMap750 below · cited by 2 · depth 23 - Residue of a q^q-expanded Γ₀(qM') function is q^q-expanded
ModularCurve.FullLevel.exists_mem_qExpFunctionFieldC_gamma0_and_residue_eq_qExpand_of_coe_eq_qExpand134 below · cited by 5 · depth 23 - Presented Gauss ring as a regular prolongation at full level
ModularCurve.FullLevel.exists_regularProlongation_fieldBar_integers_eq2 below · cited by 3 · depth 23 - Regular prolongation on the Igusa Gauss ring at q=3
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_of_eq_three_of_dvd492 below · cited by 1 · depth 23 - Regular prolongation with Igusa Gauss ring at ∞, q=2
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_of_eq_two_of_dvd492 below · cited by 1 · depth 23 - Semistable model with Igusa–Drinfeld components and henselian descent
ModularCurve.FullLevel.exists_semistableScheme_descent_of_valuationSubrings_and_smoothLocus_iff_of_isUnit_width_jDich4,403 below · cited by 1 · depth 23 - Supersingular component chart with node annuli, charted exhaustion
ModularCurve.FullLevel.exists_supersingularChart_local_affinoid_nodeAnnuli_charted3,860 below · cited by 1 · depth 23 - Local supersingular chart at q=3 with q+1 exceptional places
ModularCurve.FullLevel.exists_supersingularChart_local_of_eq_three_of_dvd3,690 below · cited by 1 · depth 23 - Supersingular component chart at q=2 with q+1 nodes
ModularCurve.FullLevel.exists_supersingularChart_local_of_eq_two_of_dvd3,688 below · cited by 1 · depth 23 - Supersingular regular prolongation with smooth-point charts at full level q
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts3,695 below · cited by 1 · depth 23 - Supersingular prolongation with smooth charts, node presentations, Hasse J
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_igusaOverS_inertia_nodeCharts_hasseJ3,844 below · cited by 2 · depth 23 - Supersingular prolongation: charts, node annuli, cross units, inertia
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_inertia3,845 below · cited by 4 · depth 23 - Tube annuli and residue discs at a supersingular place (q=3)
ModularCurve.FullLevel.exists_tubeAnnuli_width_inertia_discs_charted_inertNodes_nodeRings_nodeCharts_moduliHasse_of_eq_three_of_dvd3,933 below · cited by 1 · depth 23 - Tube annuli, discs and node rings at one supersingular place (q=2)
ModularCurve.FullLevel.exists_tubeAnnuli_width_inertia_discs_charted_inertNodes_nodeRings_nodeCharts_moduliHasse_of_eq_two_of_dvd3,933 below · cited by 1 · depth 23 - Level-fixed charts over a supersingular place are affinoid-bounded
ModularCurve.FullLevel.forall_tubeBounded_mem_of_fixed_of_typeII_exhaustion_charted1 below · cited by 1 · depth 23 - Genus identity for the semistable covering at q=3
ModularCurve.FullLevel.genusFF_fieldBar_add_eq_of_igusa_supersingular_charts_of_eq_three_of_dvd3,952 below · cited by 1 · depth 23 - Genus identity for the semistable covering at q=2
ModularCurve.FullLevel.genusFF_fieldBar_add_eq_of_igusa_supersingular_charts_of_eq_two_of_dvd3,932 below · cited by 1 · depth 23 - Genus of the full level-q curve over X₀(M')
ModularCurve.FullLevel.genusFF_fieldBar_eq414 below · cited by 1 · depth 23 - Genus of the level Γ_H(q²M') function field in characteristic q
ModularCurve.FullLevel.genusFF_xHFunctionFieldC_levelH_eq1,313 below · cited by 1 · depth 23 - Tame-1 inertia stabilises the transported Igusa discs, q=3
ModularCurve.FullLevel.igusaDiscs_inertia_stable_of_stalkInert_of_eq_three_of_dvd42 below · cited by 1 · depth 23 - Tame-1 inertia stabilises transported Igusa discs, q=2
ModularCurve.FullLevel.igusaDiscs_inertia_stable_of_stalkInert_of_eq_two_of_dvd42 below · cited by 1 · depth 23 - Cusp-free Drinfeld discs lie in the supersingular tube (q=3)
ModularCurve.FullLevel.inTube_of_mem_drinfeldDisc_of_cuspFree_of_eq_three_of_dvd495 below · cited by 1 · depth 23 - Drinfeld residue discs lie in the supersingular tube, q=2
ModularCurve.FullLevel.inTube_of_mem_drinfeldDisc_of_cuspFree_of_eq_two_of_dvd495 below · cited by 1 · depth 23 - Level automorphisms fix q-expansion images from Γ₀(qM')
ModularCurve.FullLevel.levelAutBar_apply_eq_self_of_coe_eq_qExpand34 below · cited by 3 · depth 23 - Constant reduction integers equal the Igusa Gauss ring at ∞ (q=3)
ModularCurve.FullLevel.mem_constantReduction_integers_iff_inclusion_mem_igusaGaussRing_of_eq_three_of_dvd2 below · cited by 1 · depth 23 - Igusa Gauss ring at ∞ cuts out the level-M' reduction (q=2)
ModularCurve.FullLevel.mem_constantReduction_integers_iff_inclusion_mem_igusaGaussRing_of_eq_two_of_dvd2 below · cited by 1 · depth 23 - Unipotent naturality at the Igusa chart of ∞, q=3
ModularCurve.FullLevel.naturality_unipotent_igusaInfty_of_discFamily_of_eq_three_of_dvd377 below · cited by 1 · depth 23 - Unipotent level automorphisms at the Igusa chart of ∞, q=2
ModularCurve.FullLevel.naturality_unipotent_igusaInfty_of_discFamily_of_eq_two_of_dvd377 below · cited by 1 · depth 23 - No smooth-point package at an Igusa end, q=3
ModularCurve.FullLevel.not_smoothPointPackage_of_annulusPair_attached_igusaEnd_of_testFunction_fullLevel_of_eq_three_of_dvd1 below · cited by 3 · depth 23 - No smooth-point package at an Igusa-end node, q=2
ModularCurve.FullLevel.not_smoothPointPackage_of_annulusPair_attached_igusaEnd_of_testFunction_fullLevel_of_eq_two_of_dvd1 below · cited by 3 · depth 23 - Substitution t ↦ t^q carries the Γ₀(qM') field into `fieldBar`
ModularCurve.FullLevel.qExpand_coe_mem_fieldBar_of_mem34 below · cited by 23 · depth 23 - Inertia image spanned by unipotent-fixed translates, q=3
ModularCurve.FullLevel.range_tateGal_sub_one_le_span_unipotent_fixed_of_semistableCovering_of_semistableModel_of_eq_three1,174 below · cited by 1 · depth 23 - Unipotent-fixed GL₂(𝔽_q)-translates span (τ-1)V at q=2
ModularCurve.FullLevel.range_tateGal_sub_one_le_span_unipotent_fixed_of_semistableCovering_of_semistableModel_of_eq_two1,174 below · cited by 1 · depth 23 - Invariance of the Gauss ring at ∞ forces [1:0] fixed (q=3)
ModularCurve.FullLevel.redQ_smul_lineInfty_eq_of_comap_levelAutBar_eq_of_eq_three379 below · cited by 2 · depth 23 - Gauss-ring stability at ∞ forces the Borel condition (q=2)
ModularCurve.FullLevel.redQ_smul_lineInfty_eq_of_comap_levelAutBar_eq_of_eq_two379 below · cited by 2 · depth 23 - Relative degree at most q(q-1)/2 over the q-expanded Γ₀(qM) field
ModularCurve.FullLevel.relfinrank_adjoin_qExpand_image_laurentBaseChange_gamma0_fieldBar_le261 below · cited by 3 · depth 23 - Finiteness of the full-level field over the q-scaled Γ₀(qM) field
ModularCurve.FullLevel.relfinrank_adjoin_qExpand_image_laurentBaseChange_gamma0_fieldBar_ne_zero261 below · cited by 8 · depth 23 - Residues of Gauss-integral level-M' functions give F_κ(Γ₀(M'))
ModularCurve.FullLevel.residue_mem_qExpFunctionFieldC_gamma0_and_surj6 below · cited by 4 · depth 23 - Genus equation for a supersingular component chart
ModularCurve.FullLevel.two_mul_placeWidthChar_mul_genusFF_add_of_chart_over3,873 below · cited by 1 · depth 23 - Drinfeld clause from a regular prolongation on a supersingular chart
ModularCurve.FullLevel.SemistableCovering.drinfeldClause_of_regularProlongation_of_exists_algEquiv_quotField0 below · cited by 1 · depth 24 - Drinfeld clause at q=3 from supersingular chart valuation rings
ModularCurve.FullLevel.SemistableCovering.drinfeldClause_of_valuationSubring_over_fixed_of_eq_three_of_dvd3,860 below · cited by 2 · depth 24 - Drinfeld clause for the supersingular charts, q=2
ModularCurve.FullLevel.SemistableCovering.drinfeldClause_of_valuationSubring_over_fixed_of_eq_two_of_dvd3,858 below · cited by 2 · depth 24 - Annulus nodes read by their attachment places
ModularCurve.FullLevel.SemistableCovering.exists_node_forall_mem_annulus_dom_sp_eq_of_smoothFibres_subset_charts_of_annulusFibre46 below · cited by 1 · depth 24 - Annulus separation from cross-unit test functions
ModularCurve.FullLevel.annulus_separation_of_crossUnits5 below · cited by 3 · depth 24 - Inertia fixes the supersingular Drinfeld rings, q=3
ModularCurve.FullLevel.arithmeticGalois_smul_mem_drinfeldRing_iff_of_componentChart_of_eq_three_of_dvd3,864 below · cited by 1 · depth 24 - Inertia stabilises the supersingular valuation rings (q=2)
ModularCurve.FullLevel.arithmeticGalois_smul_mem_drinfeldRing_iff_of_componentChart_of_eq_two_of_dvd3,862 below · cited by 1 · depth 24 - Level automorphism with q ∣ a sends j to j(q^{q^2})
ModularCurve.FullLevel.coe_levelAutBar_apply_eq_qExpand_sq_jqModC_of_dvd43 below · cited by 5 · depth 24 - Level-equivariance of the supersingular chart and its node annuli
ModularCurve.FullLevel.comap_dom_eq_and_exists_comap_annulus_dom_eq_of_levelOrbits0 below · cited by 3 · depth 24 - At full level q=3, an antipodal level automorphism moves the Gauss ring
ModularCurve.FullLevel.comap_levelAutBar_ne_of_dvd_of_eq_three50 below · cited by 1 · depth 24 - A level automorphism at q=2 moves the Gauss ring
ModularCurve.FullLevel.comap_levelAutBar_ne_of_dvd_of_eq_two50 below · cited by 1 · depth 24 - Uniqueness of a place over a supersingular place, q=3
ModularCurve.FullLevel.eq_of_forall_mem_iff_map_mem_of_integers_eq_igusaRing_of_eq_three449 below · cited by 1 · depth 24 - Unique place of an Igusa component over a supersingular place, q=2
ModularCurve.FullLevel.eq_of_forall_mem_iff_map_mem_of_integers_eq_igusaRing_of_eq_two_of_dvd399 below · cited by 1 · depth 24 - Unique place over a supersingular place via q↦ q^{q^2}
ModularCurve.FullLevel.existsUnique_place_forall_mem_iff_mem_of_coe_eq_qExpand_sq_of_mem_ssPlaces1,123 below · cited by 2 · depth 24 - Unique place over a supersingular place when q=3
ModularCurve.FullLevel.existsUnique_place_restrictAlong_eq_of_mem_ssPlaces_of_eq_three423 below · cited by 5 · depth 24 - Unique place over a supersingular place at q=2
ModularCurve.FullLevel.existsUnique_place_restrictAlong_eq_of_mem_ssPlaces_of_eq_two300 below · cited by 3 · depth 24 - Supersingular charts as μ_{q+1}-quotients of the Drinfeld curve
ModularCurve.FullLevel.exists_algEquiv_quotField_of_chart_over3,867 below · cited by 1 · depth 24 - Level-automorphism invariants are Γ₀(M')-expansions in q^q
ModularCurve.FullLevel.exists_coe_eq_qExpand_of_forall_levelAutBar_apply_eq1,104 below · cited by 1 · depth 24 - Valuation rings over A dominating R₀ are Igusa rings, q=3
ModularCurve.FullLevel.exists_eq_igusaRing_of_forall_mem_integers_mem_of_eq_three655 below · cited by 2 · depth 24 - Valuation rings over A containing R₀ are Igusa rings (q=2)
ModularCurve.FullLevel.exists_eq_igusaRing_of_forall_mem_integers_mem_of_eq_two655 below · cited by 2 · depth 24 - A unit of the full-level field reducing to E₄E₆/Δ
ModularCurve.FullLevel.exists_fieldBar_mul_intSeriesC_eq_and_reduction_eisensteinRatio8 below · cited by 1 · depth 24 - Level automorphisms over Γ₀(M') form a finite group
ModularCurve.FullLevel.exists_finite_subgroup_forall_levelAutBar_mem34 below · cited by 21 · depth 24 - Smooth-point charts on the Igusa ∞-component, q=3
ModularCurve.FullLevel.exists_igusaSmoothPointCharts_of_igusaGaussRing_allInertia_of_eq_three1,768 below · cited by 1 · depth 24 - Smooth Igusa charts off the supersingular locus, q=2
ModularCurve.FullLevel.exists_igusaSmoothPointCharts_of_igusaGaussRing_allInertia_of_eq_two1,768 below · cited by 1 · depth 24 - Igusa smooth-point data at each layer of a constants tower
ModularCurve.FullLevel.exists_igusaTower_smoothPointData_of_stable2,008 below · cited by 1 · depth 24 - Supersingular valuation ring over k₀ with smooth-point packages
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages3,681 below · cited by 1 · depth 24 - Separating a place from its level translate by a chart function
ModularCurve.FullLevel.exists_levelAut_ord_residue_pos_and_not_of_levelOrbits0 below · cited by 3 · depth 24 - Semistable model and henselian descent at q=3
ModularCurve.FullLevel.exists_semistableScheme_descent_of_valuationSubrings_and_smoothLocus_iff_of_isUnit_width_jDich_of_eq_three_of_dvd4,173 below · cited by 1 · depth 24 - Semistable A-model of the full level-2 modular curve
ModularCurve.FullLevel.exists_semistableScheme_descent_of_valuationSubrings_and_smoothLocus_iff_of_isUnit_width_jDich_of_eq_two_of_dvd4,173 below · cited by 1 · depth 24 - Semistable normal model over the henselian descent base A₀
ModularCurve.FullLevel.exists_semistableScheme_over_descentBase_of_valuationSubrings_of_eq_pi_relDimOne_jDich4,361 below · cited by 1 · depth 24 - Supersingular chart with node annuli at q=3, charted
ModularCurve.FullLevel.exists_supersingularChart_local_affinoid_nodeAnnuli_charted_of_eq_three_of_dvd3,854 below · cited by 1 · depth 24 - Supersingular chart and node annuli at q=2, charted
ModularCurve.FullLevel.exists_supersingularChart_local_affinoid_nodeAnnuli_charted_of_eq_two_of_dvd3,852 below · cited by 1 · depth 24 - Supersingular regular prolongation: charts, node models, affine chart
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_cover_nodeCharts_hasseJ_drinfeldInertia_affineChart3,767 below · cited by 1 · depth 24 - Supersingular prolongation at q=3: charts, annuli, node models, Drinfeld identification
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_igusaOverS_inertia_nodeCharts_hasseJ_of_eq_three_of_dvd3,838 below · cited by 2 · depth 24 - Supersingular prolongation, node package and Drinfeld inertia at q=2
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_igusaOverS_inertia_nodeCharts_hasseJ_of_eq_two_of_dvd3,836 below · cited by 2 · depth 24 - Supersingular prolongation at q=3: charts, node annuli, Drinfeld action
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_inertia_of_eq_three_of_dvd3,839 below · cited by 4 · depth 24 - Supersingular Gauss prolongation at q=2: charts, node annuli, Drinfeld
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_crossUnits_inertia_of_eq_two_of_dvd3,837 below · cited by 4 · depth 24 - Supersingular regular prolongation with smooth-point charts, q=3
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_of_eq_three_of_dvd3,687 below · cited by 1 · depth 24 - Supersingular prolongation with smooth-point charts, level q=2
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_of_eq_two_of_dvd3,685 below · cited by 1 · depth 24 - Fixed charted type II points contain the supersingular affinoid, q=3
ModularCurve.FullLevel.forall_tubeBounded_mem_of_fixed_of_typeII_exhaustion_charted_of_eq_three_of_dvd1 below · cited by 1 · depth 24 - Level-fixed charted rings over a supersingular place contain the affinoid
ModularCurve.FullLevel.forall_tubeBounded_mem_of_fixed_of_typeII_exhaustion_charted_of_eq_two_of_dvd1 below · cited by 1 · depth 24 - Genus of the full level-3 modular function field
ModularCurve.FullLevel.genusFF_fieldBar_eq_of_eq_three413 below · cited by 1 · depth 24 - Genus of the full level-2 curve over X₀(M')
ModularCurve.FullLevel.genusFF_fieldBar_eq_of_eq_two555 below · cited by 1 · depth 24 - Genus of the Γ_H(q²M') function field at q=3
ModularCurve.FullLevel.genusFF_xHFunctionFieldC_levelH_eq_of_eq_three_of_dvd928 below · cited by 1 · depth 24 - Characteristic-2 genus of the Γ_H(4M') q-expansion field
ModularCurve.FullLevel.genusFF_xHFunctionFieldC_levelH_eq_of_eq_two872 below · cited by 1 · depth 24 - Genus of the Igusa-level field at a place of ℚ̄ over q
ModularCurve.FullLevel.genusFF_xHFunctionFieldC_levelH_eq_of_liesOverPrime1,298 below · cited by 1 · depth 24 - Igusa level field as a degree (q-1)/2 Kummer extension
ModularCurve.FullLevel.isKummerGenerator_eisensteinRatio_and_igusaFunctionField_eq_xHFunctionFieldC_levelH1,085 below · cited by 7 · depth 24 - Level automorphism fixes the Γ₀(M') subfield when q∣ b
ModularCurve.FullLevel.levelAutBar_apply_eq_self_of_dvd_of_coe_mem_laurentBaseChange_gamma035 below · cited by 1 · depth 24 - Valuation ring agrees with the Igusa Gauss ring on q-scaled functions
ModularCurve.FullLevel.mem_iff_mem_igusaGaussRing_of_coe_eq_qExpand_of_forall_mem_integers_mem569 below · cited by 1 · depth 24 - Supersingular chart genus identity at q=2
ModularCurve.FullLevel.placeWidthChar_mul_genusFF_add_of_chart_over_of_eq_two_of_dvd3,865 below · cited by 1 · depth 24 - Ramification of the Igusa level field over X₀(M') in characteristic q
ModularCurve.FullLevel.ramificationIndex_xHFunctionFieldC_levelH_modularFunctionFieldC_eq_of_liesOverPrime1,115 below · cited by 2 · depth 24 - Relative degree bound ≤ q(q-1)/2 at q=3
ModularCurve.FullLevel.relfinrank_adjoin_qExpand_image_laurentBaseChange_gamma0_fieldBar_le_of_eq_three261 below · cited by 3 · depth 24 - Degree at most 2 at q=2 over the q-expanded Γ₀-field
ModularCurve.FullLevel.relfinrank_adjoin_qExpand_image_laurentBaseChange_gamma0_fieldBar_le_of_eq_two261 below · cited by 2 · depth 24 - Level orbits and generators for a supersingular prolongation
ModularCurve.FullLevel.supersingularProlongation_drinfeldQuotient_levelOrbits_generators_inertia_of_drinfeldIdentification_of_affineChart144 below · cited by 1 · depth 24 - Node annuli and crossing data over a supersingular place
ModularCurve.FullLevel.supersingularProlongation_exists_nodeAnnuli_nodePresentations_cover_crossUnits_zeroFree_of_nodePresentations_nodeCharts_hasseJ215 below · cited by 1 · depth 24 - No cusp-free smooth-point chart at an end of the supersingular component
ModularCurve.FullLevel.supersingularProlongation_not_smoothPointPackage_of_mem_ends_nodeCharts_hasseJ_cuspFree197 below · cited by 1 · depth 24 - Semilinear transport of smooth-point packages off the ends
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_semilinearTransport_of_cuspFree0 below · cited by 1 · depth 24 - Uniqueness of the smooth-point package on the supersingular component
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_unique64 below · cited by 1 · depth 24 - Index of ±Γ_H(q²M') for H the kernel mod q
ModularCurve.FullLevel.two_mul_index_gammaH_levelH_sup_zpowers_neg_one_eq5 below · cited by 2 · depth 24 - Cusp count for Γ_H(q²M') with H=ker to (ℤ/q)^×
ModularCurve.FullLevel.two_mul_natCard_doubleCoset_gammaH_levelH_zpowers_T_eq8 below · cited by 2 · depth 24 - Genus of a supersingular component chart at q=3
ModularCurve.FullLevel.two_mul_placeWidthChar_mul_genusFF_add_of_chart_over_of_eq_three_of_dvd3,867 below · cited by 1 · depth 24 - Type-II exhaustion of the supersingular tube, charted form
ModularCurve.FullLevel.typeII_exhaustion_of_placeCover_of_componentChart771 below · cited by 1 · depth 24 - Drinfeld clause from a regular prolongation, hedged exponent
ModularCurve.FullLevel.SemistableCovering.exists_drinfeldClause_of_regularProlongation_of_exists_algEquiv_quotField_hedged0 below · cited by 2 · depth 25 - Degree of the full-level field over scaled Γ₀(M') functions
ModularCurve.FullLevel.adjoin_qExpand_image_le_fieldBar_and_relfinrank_pos_and_le263 below · cited by 1 · depth 25 - Unique place over a supersingular place via q↦ q^{q^2}, q=3
ModularCurve.FullLevel.existsUnique_place_forall_mem_iff_mem_of_coe_eq_qExpand_sq_of_mem_ssPlaces_of_eq_three429 below · cited by 2 · depth 25 - Unique place reading a supersingular place at q=2
ModularCurve.FullLevel.existsUnique_place_forall_mem_iff_mem_of_coe_eq_qExpand_sq_of_mem_ssPlaces_of_eq_two302 below · cited by 2 · depth 25 - Unique place over supersingular places in the q=2 Igusa cover
ModularCurve.FullLevel.existsUnique_place_restrictAlong_eq_of_mem_ssPlaces_of_eq_two_of_dvd300 below · cited by 1 · depth 25 - Supersingular chart is a Drinfeld quotient field, q=3
ModularCurve.FullLevel.exists_algEquiv_quotField_of_chart_over_of_eq_three_of_dvd3,861 below · cited by 1 · depth 25 - Supersingular component charts as Drinfeld-curve quotients, q=2
ModularCurve.FullLevel.exists_algEquiv_quotField_of_chart_over_of_eq_two_of_dvd3,859 below · cited by 1 · depth 25 - Fixed field of the level automorphisms for q = 3
ModularCurve.FullLevel.exists_coe_eq_qExpand_of_forall_levelAutBar_apply_eq_of_eq_three387 below · cited by 1 · depth 25 - Fixed elements are q-rescaled Γ₀(M') functions (q=2)
ModularCurve.FullLevel.exists_coe_eq_qExpand_of_forall_levelAutBar_apply_eq_of_eq_two387 below · cited by 1 · depth 25 - Smoothness and Drinfeld affine charts of the descended model
ModularCurve.FullLevel.exists_drinfeldChart_localRing_eq_localization_formallySmooth_of_normalModel_gen_j3,951 below · cited by 1 · depth 25 - Smooth-point stalks of the Igusa base model at level q²M'
ModularCurve.FullLevel.exists_igusaBaseModel_smoothPointStalks1,994 below · cited by 1 · depth 25 - Smooth Igusa charts on the descended full-level model
ModularCurve.FullLevel.exists_igusaChart_localRing_eq_localization_formallySmooth_and_crossing_unique_of_normalModel_gen_j4,287 below · cited by 1 · depth 25 - Igusa smooth-point data over all constant layers, q=3
ModularCurve.FullLevel.exists_igusaTower_smoothPointData_of_stable_of_eq_three1,743 below · cited by 1 · depth 25 - Igusa tower smooth-point data for q=2
ModularCurve.FullLevel.exists_igusaTower_smoothPointData_of_stable_of_eq_two1,743 below · cited by 1 · depth 25 - k₀-level supersingular package: smooth charts, nodes, Drinfeld action
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages_nodePresentations_nodeCharts_hasseJ_drinfeldInertia_affineChart3,749 below · cited by 1 · depth 25 - Supersingular DVR with smooth-point packages at q=3
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages_of_eq_three_of_dvd3,673 below · cited by 1 · depth 25 - Supersingular valuation ring over small constants with smooth-point packages (q=2)
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages_of_eq_two_of_dvd3,671 below · cited by 1 · depth 25 - Supersingular valuation ring with smooth-point stalks at every layer
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointStalks3,673 below · cited by 1 · depth 25 - Affine node chart at an Igusa–Drinfeld crossing
ModularCurve.FullLevel.exists_nodeChart_point_specializes_iff_adicCompletion_stalk_of_normalModel_gen_j_local3,917 below · cited by 3 · depth 25 - Semistable normal model over a henselian descent base, q=3
ModularCurve.FullLevel.exists_semistableScheme_over_descentBase_of_valuationSubrings_of_eq_pi_relDimOne_jDich_of_eq_three_of_dvd4,130 below · cited by 1 · depth 25 - Semistable descent model at q=2 with Igusa–Drinfeld components
ModularCurve.FullLevel.exists_semistableScheme_over_descentBase_of_valuationSubrings_of_eq_pi_relDimOne_jDich_of_eq_two_of_dvd4,130 below · cited by 1 · depth 25 - Supersingular prolongation: charts, node annuli, level orbits, Drinfeld quotient
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodeFrames_levelOrbits_generators3,846 below · cited by 1 · depth 25 - Supersingular prolongation at q=3: charts, nodes, Drinfeld inertia
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_cover_nodeCharts_hasseJ_drinfeldInertia_of_eq_three_of_dvd_affineChart3,761 below · cited by 1 · depth 25 - Supersingular prolongation with charts, nodes and Drinfeld inertia, q=2
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodePresentations_cover_nodeCharts_hasseJ_drinfeldInertia_of_eq_two_of_dvd_affineChart3,759 below · cited by 1 · depth 25 - Test functions cutting out the residue tube of a supersingular place
ModularCurve.FullLevel.exists_testFamily_forall_isRational_tube_of_forall_evalAt_mem_maximalIdeal768 below · cited by 3 · depth 25 - Order of the Igusa Kummer radicand away from supersingular places
ModularCurve.FullLevel.gcd_natAbs_ord_eisensteinRatio_pow_eq_div_placeWidth_of_not_mem_ssPlaces480 below · cited by 1 · depth 25 - Order of the Eisenstein radicand at supersingular places
ModularCurve.FullLevel.gcd_natAbs_ord_eisensteinRatio_pow_eq_one_of_mem_ssPlaces479 below · cited by 1 · depth 25 - Integral level-M' functions with non-zero reduction are Igusa units
ModularCurve.FullLevel.inclusion_mem_igusaRing_and_inv_mem_of_residue_ne_zero227 below · cited by 6 · depth 25 - Level automorphisms fix level-Γ₀(M') functions read in q^q
ModularCurve.FullLevel.levelAutBar_apply_eq_self_of_coe_eq_qExpand_of_mem_laurentBaseChange_gamma034 below · cited by 13 · depth 25 - Valuation ring meets Igusa Gauss ring on q-power expansions (q=3)
ModularCurve.FullLevel.mem_iff_mem_igusaGaussRing_of_coe_eq_qExpand_of_forall_mem_integers_mem_of_eq_three569 below · cited by 1 · depth 25 - Case q=2: 𝒪 agrees with the Igusa ring on q^q-expansions
ModularCurve.FullLevel.mem_iff_mem_igusaGaussRing_of_coe_eq_qExpand_of_forall_mem_integers_mem_of_eq_two568 below · cited by 1 · depth 25 - Lower bound q(q²-1)≤ 2|G| for level automorphism groups
ModularCurve.FullLevel.mul_card_ge_of_forall_levelAutBar_mem1,100 below · cited by 1 · depth 25 - Level orbits and affine generators for the supersingular prolongation at q=3
ModularCurve.FullLevel.supersingularProlongation_drinfeldQuotient_levelOrbits_generators_inertia_of_drinfeldIdentification_of_affineChart_of_eq_three_of_dvd144 below · cited by 1 · depth 25 - Level orbits and generators for the q=2 supersingular prolongation
ModularCurve.FullLevel.supersingularProlongation_drinfeldQuotient_levelOrbits_generators_inertia_of_drinfeldIdentification_of_affineChart_of_eq_two_of_dvd144 below · cited by 1 · depth 25 - Annulus pairs at the nodes of a supersingular prolongation
ModularCurve.FullLevel.supersingularProlongation_exists_annulusPair_of_nodePresentation153 below · cited by 3 · depth 25 - Cross-units separating two nodes on the supersingular fibre
ModularCurve.FullLevel.supersingularProlongation_exists_crossUnit_nodePlaces_of_sep154 below · cited by 1 · depth 25 - R-integral generators regular off the ends, from an affine chart
ModularCurve.FullLevel.supersingularProlongation_exists_generators_regular_off_ends_of_affineChart107 below · cited by 1 · depth 25 - Node annuli at supersingular reduction for q=3
ModularCurve.FullLevel.supersingularProlongation_exists_nodeAnnuli_nodePresentations_cover_crossUnits_zeroFree_of_nodePresentations_nodeCharts_hasseJ_of_eq_three_of_dvd215 below · cited by 1 · depth 25 - Node annuli at the q+1 ends, q=2
ModularCurve.FullLevel.supersingularProlongation_exists_nodeAnnuli_nodePresentations_cover_crossUnits_zeroFree_of_nodePresentations_nodeCharts_hasseJ_of_eq_two_of_dvd215 below · cited by 1 · depth 25 - Level automorphisms: transitive on ends, no fixed smooth place
ModularCurve.FullLevel.supersingularProlongation_levelAut_transitive_ends_moves_smoothPlaces124 below · cited by 1 · depth 25 - Node place-sets avoid the smooth residue discs
ModularCurve.FullLevel.supersingularProlongation_nodePlaces_disjoint_smoothDiscs_of_sep0 below · cited by 1 · depth 25 - No cusp-free smooth chart at an end of the supersingular fibre, q=3
ModularCurve.FullLevel.supersingularProlongation_not_smoothPointPackage_of_mem_ends_nodeCharts_hasseJ_cuspFree_of_eq_three_of_dvd197 below · cited by 1 · depth 25 - No cusp-free smooth-point package at an end (q=2)
ModularCurve.FullLevel.supersingularProlongation_not_smoothPointPackage_of_mem_ends_nodeCharts_hasseJ_cuspFree_of_eq_two_of_dvd197 below · cited by 1 · depth 25 - Containment of smooth-point package discs at a supersingular place
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_disc_subset63 below · cited by 1 · depth 25 - Semilinear transport of smooth-point packages, q=3
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_semilinearTransport_of_cuspFree_of_eq_three_of_dvd0 below · cited by 1 · depth 25 - Semilinear transport of supersingular smooth-point packages (q=2)
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_semilinearTransport_of_cuspFree_of_eq_two_of_dvd0 below · cited by 1 · depth 25 - Uniqueness of the smooth-point package away from N, q=3
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_unique_of_eq_three_of_dvd64 below · cited by 1 · depth 25 - Uniqueness of smooth-point packages on the supersingular component, q=2
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_unique_of_eq_two_of_dvd64 below · cited by 1 · depth 25 - Type II exhaustion of the supersingular tube, q = 3
ModularCurve.FullLevel.typeII_exhaustion_of_placeCover_of_componentChart_of_eq_three_of_dvd771 below · cited by 1 · depth 25 - Charted type II exhaustion of the supersingular tube, q=2
ModularCurve.FullLevel.typeII_exhaustion_of_placeCover_of_componentChart_of_eq_two_of_dvd771 below · cited by 1 · depth 25 - In characteristic 3, the Igusa-level q-expansion field is κ(X₀(M'))
ModularCurve.FullLevel.xHFunctionFieldC_levelH_eq_modularFunctionFieldC_of_eq_three369 below · cited by 5 · depth 25 - At q=2 the level-4M' q-expansion field descends to level M'
ModularCurve.FullLevel.xHFunctionFieldC_levelH_eq_modularFunctionFieldC_of_eq_two314 below · cited by 1 · depth 25 - Full level 2: Γ_H(4M') field equals level-M' modular field
ModularCurve.FullLevel.xHFunctionFieldC_levelH_eq_modularFunctionFieldC_of_liesOverPrime_of_eq_two299 below · cited by 6 · depth 25 - Full-level field over the q-scaled Γ₀(M') field: inclusion and degree bound, q=3
ModularCurve.FullLevel.adjoin_qExpand_image_le_fieldBar_and_relfinrank_pos_and_le_of_eq_three263 below · cited by 1 · depth 26 - Full-level field over q-scaled Γ₀(M') field: q=2
ModularCurve.FullLevel.adjoin_qExpand_image_le_fieldBar_and_relfinrank_pos_and_le_of_eq_two263 below · cited by 1 · depth 26 - A place is centred at at most one good point
ModularCurve.FullLevel.eq_of_centred_of_centred_twoChartIntegralModel0 below · cited by 1 · depth 26 - Uniqueness of the Igusa-ring refinement above a supersingular place
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent1,478 below · cited by 2 · depth 26 - Bottom-layer admissibility and identification of the level field
ModularCurve.FullLevel.exists_admissible_smallConstants_botLayer_levelField_ringEquiv_of_descentBase38 below · cited by 15 · depth 26 - The q-adic place is centred at a good point
ModularCurve.FullLevel.exists_centred_of_toValuationSubring_eq_qIntegersBar_twoChartIntegralModel15 below · cited by 1 · depth 26 - Smooth Drinfeld charts at a supersingular point, q=3
ModularCurve.FullLevel.exists_drinfeldChart_localRing_eq_localization_formallySmooth_of_normalModel_gen_j_of_eq_three_of_dvd3,799 below · cited by 1 · depth 26 - Smooth Drinfeld charts on the descended full-level model, q=2
ModularCurve.FullLevel.exists_drinfeldChart_localRing_eq_localization_formallySmooth_of_normalModel_gen_j_of_eq_two_of_dvd3,798 below · cited by 1 · depth 26 - Smooth Igusa base model off the supersingular locus, q=3
ModularCurve.FullLevel.exists_igusaBaseModel_smoothPointStalks_of_eq_three1,726 below · cited by 1 · depth 26 - Smooth Igusa base model off the supersingular locus, q=2
ModularCurve.FullLevel.exists_igusaBaseModel_smoothPointStalks_of_eq_two1,726 below · cited by 1 · depth 26 - Igusa charts and crossing uniqueness on the descended curve, q=3
ModularCurve.FullLevel.exists_igusaChart_localRing_eq_localization_formallySmooth_and_crossing_unique_of_normalModel_gen_j_of_eq_three_of_dvd4,054 below · cited by 1 · depth 26 - Smooth Igusa charts and unique crossings, q=2
ModularCurve.FullLevel.exists_igusaChart_localRing_eq_localization_formallySmooth_and_crossing_unique_of_normalModel_gen_j_of_eq_two_of_dvd4,054 below · cited by 1 · depth 26 - Nodes of the ∞-Igusa component over the supersingular places
ModularCurve.FullLevel.exists_igusaNodes_card_eq_of_igusaGaussRing1,139 below · cited by 2 · depth 26 - Supersingular DVR over small constants with base-layer stalks
ModularCurve.FullLevel.exists_klevel_supersingularDVR_baseSmoothPointStalks3,655 below · cited by 1 · depth 26 - Supersingular k₀-level model at q=3: smooth and nodal data
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages_nodePresentations_nodeCharts_hasseJ_drinfeldInertia_of_eq_three_of_dvd_affineChart3,743 below · cited by 1 · depth 26 - k-level supersingular charts, nodes and Drinfeld inertia at q=2
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointPackages_nodePresentations_nodeCharts_hasseJ_drinfeldInertia_of_eq_two_of_dvd_affineChart3,741 below · cited by 1 · depth 26 - Supersingular k₀-level valuation ring and smooth-point stalks, q=3
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointStalks_of_eq_three_of_dvd3,665 below · cited by 1 · depth 26 - Smooth-point stalks at supersingular places of the k₀-level model, q=2
ModularCurve.FullLevel.exists_klevel_supersingularDVR_smoothPointStalks_of_eq_two_of_dvd3,663 below · cited by 1 · depth 26 - Centring non-supersingular places at good points after a level automorphism
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_twoChartIntegralModel_of_forall_not_ssTube1,268 below · cited by 1 · depth 26 - A k₀-rational level field inside the full-level function field
ModularCurve.FullLevel.exists_levelField_coeff_mem_sup_eq_top_levelAutBar_stable_linearDisjoint33 below · cited by 5 · depth 26 - A k₀-form F₀ of the full-level function field
ModularCurve.FullLevel.exists_levelField_sup_eq_top_levelAutBar_stable_regular_rat33 below · cited by 1 · depth 26 - Node chart at (ℓ,s) of the descended full-level model, q=3
ModularCurve.FullLevel.exists_nodeChart_point_specializes_iff_adicCompletion_stalk_of_normalModel_gen_j_local_of_eq_three_of_dvd3,765 below · cited by 3 · depth 26 - Node chart at (ℓ,s) of the descended model, q=2
ModularCurve.FullLevel.exists_nodeChart_point_specializes_iff_adicCompletion_stalk_of_normalModel_gen_j_local_of_eq_two_of_dvd3,764 below · cited by 3 · depth 26 - Igusa components of the descended two-chart model
ModularCurve.FullLevel.exists_primes_chartAlg_localization_eq_igusaRing_minimal_injective_descent1,260 below · cited by 8 · depth 26 - Separating two supersingular places by a rational level-M' function
ModularCurve.FullLevel.exists_rational_integral_evalAt_ne_of_ne_ssPlaces795 below · cited by 3 · depth 26 - Level automorphisms permute good points of the two-chart model
ModularCurve.FullLevel.exists_reads_resAut_smul_and_centred_iff_of_mem_closure_levelAutBar_twoChartIntegralModel250 below · cited by 1 · depth 26 - Étale coordinate and residue character at a good off-branch point
ModularCurve.FullLevel.exists_stalk_etaleCoordinate_residueChar_of_offBranch_of_reads_twoChartIntegralModel1,914 below · cited by 1 · depth 26 - Smooth chart or supersingular alternative for Igusa refinements
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_igusaRing_descent_local2,013 below · cited by 2 · depth 26 - Affine Drinfeld chart at a supersingular place of X_H(q²M')
ModularCurve.FullLevel.exists_subalgebra_drinfeldRing_iff_localization_formallySmooth_card_le_descent3,675 below · cited by 1 · depth 26 - Supersingular chart over the level-q field: nodes, Hasse, inertia
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_inertia_of_levelField3,606 below · cited by 2 · depth 26 - Supersingular chart of the level-q model with its q+1 nodes
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_nodes_of_levelField3,609 below · cited by 1 · depth 26 - Supersingular model at q=3: charts, node annuli, level orbits, generators
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodeFrames_levelOrbits_generators_of_eq_three_of_dvd3,840 below · cited by 1 · depth 26 - Supersingular chart data and Drinfeld identification at q=2
ModularCurve.FullLevel.exists_supersingularRegularProlongation_smoothPointCharts_nodeFrames_levelOrbits_generators_of_eq_two_of_dvd3,838 below · cited by 1 · depth 26 - Formal smoothness of the descended Igusa ring over A₀
ModularCurve.FullLevel.formallySmooth_subalgebra_of_mem_iff_mem_igusaRing_descent2,014 below · cited by 1 · depth 26 - Tame inertia in the Drinfeld identification at level field
ModularCurve.FullLevel.klevel_drinfeldInertia_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_inertia11 below · cited by 1 · depth 26 - Node places, crossing models and residue-disc cover at full level
ModularCurve.FullLevel.klevel_nodePresentations_nodeCharts_hasseJ_of_affineChart_poles_hasse_commonChart_nodes_igusaSep1,031 below · cited by 1 · depth 26 - Smooth-point stalks yield smooth-point packages at every layer
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_chart59 below · cited by 1 · depth 26 - Smooth-point stalks at every layer from the base layer
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_chart72 below · cited by 1 · depth 26 - Affine chart of the descended model, read on given data
ModularCurve.FullLevel.klevel_supersingularDVR_affineChart_of_levelField_affineChart48 below · cited by 1 · depth 26 - Base smooth-point stalks and Drinfeld identification from an affine chart
ModularCurve.FullLevel.klevel_supersingularDVR_baseSmoothPointStalks_of_affineChart_chart178 below · cited by 1 · depth 26 - Affine Drinfeld chart with q+1 nodes forces W₀=𝒪'∩ F₀
ModularCurve.FullLevel.mem_iff_coe_mem_drinfeldRing_of_affineChart_nodes_cover126 below · cited by 2 · depth 26 - Lower bound q(q²-1)≤ 2|G| for level automorphisms, q=3
ModularCurve.FullLevel.mul_card_ge_of_forall_levelAutBar_mem_of_eq_three383 below · cited by 1 · depth 26 - Order bound q(q²-1)≤|G| for level automorphisms at q=2
ModularCurve.FullLevel.mul_card_ge_of_forall_levelAutBar_mem_of_eq_two383 below · cited by 1 · depth 26 - Level automorphisms moving the Igusa Gauss ring displace centres
ModularCurve.FullLevel.not_centred_smul_of_comap_igusaGaussRing_ne_of_offBranch_twoChartIntegralModel240 below · cited by 1 · depth 26 - Places centred at good points avoid supersingular tubes
ModularCurve.FullLevel.not_mem_ssTube_of_centred_twoChartIntegralModel0 below · cited by 1 · depth 26 - Good points of the two-chart model read unique non-node places
ModularCurve.FullLevel.reads_unique_and_exists_offBranch_of_not_mem_igusaNodes_twoChartIntegralModel1,881 below · cited by 1 · depth 26 - Inertia fixes good points of the two-chart model to first order
ModularCurve.FullLevel.sub_inMax_of_mem_inertiaSubgroupIn_of_inStalk_twoChartIntegralModel2 below · cited by 1 · depth 26 - Reciprocal annulus pair at each node, case q=3
ModularCurve.FullLevel.supersingularProlongation_exists_annulusPair_of_nodePresentation_of_eq_three_of_dvd153 below · cited by 3 · depth 26 - Reciprocal annulus pair at each node place, q = 2
ModularCurve.FullLevel.supersingularProlongation_exists_annulusPair_of_nodePresentation_of_eq_two_of_dvd153 below · cited by 3 · depth 26 - Cross-units separating two nodes, q=3 rigid level
ModularCurve.FullLevel.supersingularProlongation_exists_crossUnit_nodePlaces_of_sep_of_eq_three_of_dvd154 below · cited by 1 · depth 26 - Cross-units separating two nodes, case q=2
ModularCurve.FullLevel.supersingularProlongation_exists_crossUnit_nodePlaces_of_sep_of_eq_two_of_dvd154 below · cited by 1 · depth 26 - Finite generation of functions regular off the q+1 ends
ModularCurve.FullLevel.supersingularProlongation_exists_finite_generators_regular_off_ends_residueField105 below · cited by 1 · depth 26 - R-integral generators regular off the ends, q=3
ModularCurve.FullLevel.supersingularProlongation_exists_generators_regular_off_ends_of_affineChart_of_eq_three_of_dvd107 below · cited by 1 · depth 26 - Generators regular off the ends from an affine chart, q=2
ModularCurve.FullLevel.supersingularProlongation_exists_generators_regular_off_ends_of_affineChart_of_eq_two_of_dvd107 below · cited by 1 · depth 26 - Lifting functions regular off the ends, given an affine chart
ModularCurve.FullLevel.supersingularProlongation_exists_lift_regular_on_smoothDiscs_of_regular_off_ends_of_affineChart0 below · cited by 1 · depth 26 - Level automorphisms: transitivity on ends, no fixed smooth place (q=3)
ModularCurve.FullLevel.supersingularProlongation_levelAut_transitive_ends_moves_smoothPlaces_of_eq_three_of_dvd124 below · cited by 1 · depth 26 - Level automorphisms: transitive on the q+1 ends, no fixed smooth place (q=2)
ModularCurve.FullLevel.supersingularProlongation_levelAut_transitive_ends_moves_smoothPlaces_of_eq_two_of_dvd124 below · cited by 1 · depth 26 - Node places of the supersingular component avoid smooth residue discs (q=3)
ModularCurve.FullLevel.supersingularProlongation_nodePlaces_disjoint_smoothDiscs_of_sep_of_eq_three_of_dvd0 below · cited by 1 · depth 26 - Node place-sets avoid smooth residue discs, case q=2
ModularCurve.FullLevel.supersingularProlongation_nodePlaces_disjoint_smoothDiscs_of_sep_of_eq_two_of_dvd0 below · cited by 1 · depth 26 - Ends off N are the affine places, moved by the level
ModularCurve.FullLevel.supersingularProlongation_not_mem_ends_iff_affine_and_exists_levelAut_smul_ne121 below · cited by 1 · depth 26 - Disc inclusion for two smooth-point packages, q=3
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_disc_subset_of_eq_three_of_dvd63 below · cited by 1 · depth 26 - Inclusion of smooth-point package discs, q=2 case
ModularCurve.FullLevel.supersingularProlongation_smoothPointPackage_disc_subset_of_eq_two_of_dvd63 below · cited by 1 · depth 26 - Level field of the full-level model is a function field in j
ModularCurve.FullLevel.transcendental_and_finiteDimensional_adjoin_levelField126 below · cited by 42 · depth 26 - Supersingular chart with q+1 ends and linked inertia
ModularCurve.FullLevel.AuxLevel.exists_supersingularAffineChart_ends_moduliHasse_commonChart_orbitPoles_chartAlgFin_igusaSep_deckSep_linkedScalars_linkedInertia_of_tame3,449 below · cited by 1 · depth 27 - At most one good point centres a place (q=3)
ModularCurve.FullLevel.eq_of_centred_of_centred_twoChartIntegralModel_of_eq_three0 below · cited by 1 · depth 27 - Separation of good points centring a place (q=2)
ModularCurve.FullLevel.eq_of_centred_of_centred_twoChartIntegralModel_of_eq_two0 below · cited by 1 · depth 27 - Uniqueness of a good point reading a given place
ModularCurve.FullLevel.eq_of_goodPt_of_reads_twoChartIntegralModel0 below · cited by 1 · depth 27 - Refinements of the traced Gauss ring over a supersingular place agree
ModularCurve.FullLevel.eq_of_le_gaussRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent1,470 below · cited by 1 · depth 27 - Transport of the Igusa branch uniqueness to every line
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_gaussBranch_descent248 below · cited by 1 · depth 27 - Uniqueness of the refinement of a traced Igusa ring (q=3)
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent_of_eq_three1,152 below · cited by 2 · depth 27 - Uniqueness of the Drinfeld-pinned refinement of an Igusa ring, q=2
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent_of_eq_two1,153 below · cited by 2 · depth 27 - Uniqueness of the place read by a good point
ModularCurve.FullLevel.eq_of_reads_of_reads_of_goodPt_twoChartIntegralModel1,852 below · cited by 1 · depth 27 - Bottom-layer admissible constants and level field, q=3
ModularCurve.FullLevel.exists_admissible_smallConstants_botLayer_levelField_ringEquiv_of_descentBase_of_eq_three38 below · cited by 15 · depth 27 - Admissible bottom layer and level field at q=2
ModularCurve.FullLevel.exists_admissible_smallConstants_botLayer_levelField_ringEquiv_of_descentBase_of_eq_two38 below · cited by 15 · depth 27 - Admissible small-constant field k₀ for the descended Igusa base
ModularCurve.FullLevel.exists_admissible_smallConstants_of_descentBase3 below · cited by 3 · depth 27 - Rational places are centred at closed special points
ModularCurve.FullLevel.exists_centred_of_isRational_of_isProper_twoChartIntegralModel0 below · cited by 2 · depth 27 - q-adic cusp place centred at a good point, q=3
ModularCurve.FullLevel.exists_centred_of_toValuationSubring_eq_qIntegersBar_twoChartIntegralModel_of_eq_three15 below · cited by 1 · depth 27 - Centring the q-adic cusp at a good Igusa point, q=2
ModularCurve.FullLevel.exists_centred_of_toValuationSubring_eq_qIntegersBar_twoChartIntegralModel_of_eq_two15 below · cited by 1 · depth 27 - Constant-term retraction on the pole chart at ∞
ModularCurve.FullLevel.exists_constantTerm_chartAlgInf_twoChartIntegralModel_levelField7 below · cited by 1 · depth 27 - Identification of branches with Igusa valuation rings
ModularCurve.FullLevel.exists_forall_coe_mem_igusa_iff_of_valuationSubring_levelField1,221 below · cited by 8 · depth 27 - Centred closed special points lie on an Igusa branch
ModularCurve.FullLevel.exists_forall_mem_nonunits_igusa_mem_asIdeal_of_centred_twoChartIntegralModel1,225 below · cited by 2 · depth 27 - A good off-branch point reading each non-node Igusa place
ModularCurve.FullLevel.exists_goodPt_and_offBranch_and_reads_of_not_mem_igusaNodes_twoChartIntegralModel1,870 below · cited by 1 · depth 27 - Nodes over the supersingular places on the ∞-Igusa component, q=3
ModularCurve.FullLevel.exists_igusaNodes_card_eq_of_igusaGaussRing_of_eq_three442 below · cited by 2 · depth 27 - Igusa nodes over the supersingular places at q=2
ModularCurve.FullLevel.exists_igusaNodes_card_eq_of_igusaGaussRing_of_eq_two396 below · cited by 2 · depth 27 - Level automorphisms act transitively on the supersingular chart fibre
ModularCurve.FullLevel.exists_isLevelAutAt_map_chartAlgFin_eq_of_over_of_over_of_not_dvd2,870 below · cited by 3 · depth 27 - Supersingular closed point of the j-chart above a given place
ModularCurve.FullLevel.exists_isMaximal_chartAlgFin_mem_ssJSet_over_of_ssPlaces354 below · cited by 2 · depth 27 - Drinfeld rings read jmatĥ as an A₀-constant
ModularCurve.FullLevel.exists_jInvariant_sub_mem_maximalIdeal_drinfeldRing_descent0 below · cited by 5 · depth 27 - Affine chart at a supersingular place over small constants
ModularCurve.FullLevel.exists_klevel_supersingularDVR_affineChart3,615 below · cited by 1 · depth 27 - Supersingular valuation ring with base-layer stalks, q=3
ModularCurve.FullLevel.exists_klevel_supersingularDVR_baseSmoothPointStalks_of_eq_three_of_dvd3,647 below · cited by 1 · depth 27 - Supersingular base-layer stalks over small constants at q=2
ModularCurve.FullLevel.exists_klevel_supersingularDVR_baseSmoothPointStalks_of_eq_two_of_dvd3,645 below · cited by 1 · depth 27 - Moving places centred off the j-chart to good points
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_goodPt_of_centred_of_not_mem_range_iotaFin_twoChartIntegralModel1,234 below · cited by 1 · depth 27 - Centring rational places at good points, q=3
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_twoChartIntegralModel_of_forall_not_ssTube_of_eq_three1,083 below · cited by 1 · depth 27 - Centring non-supersingular rational places at good points, q=2
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_twoChartIntegralModel_of_forall_not_ssTube_of_eq_two1,082 below · cited by 1 · depth 27 - Level field of k₀-rational q-expansions, q=3
ModularCurve.FullLevel.exists_levelField_coeff_mem_sup_eq_top_levelAutBar_stable_linearDisjoint_of_eq_three33 below · cited by 5 · depth 27 - Coefficient level field over k₀, q = 2 case
ModularCurve.FullLevel.exists_levelField_coeff_mem_sup_eq_top_levelAutBar_stable_linearDisjoint_of_eq_two33 below · cited by 5 · depth 27 - A k₀-form F₀ of the full-level field, case q=3
ModularCurve.FullLevel.exists_levelField_sup_eq_top_levelAutBar_stable_regular_rat_of_eq_three33 below · cited by 1 · depth 27 - A k₀-form F₀ of the full-level field, q=2
ModularCurve.FullLevel.exists_levelField_sup_eq_top_levelAutBar_stable_regular_rat_of_eq_two33 below · cited by 1 · depth 27 - Good points are not generic on the ∞-Igusa component
ModularCurve.FullLevel.exists_mem_asIdeal_and_residue_ne_zero_of_goodPt_twoChartIntegralModel2 below · cited by 2 · depth 27 - Smooth neighbourhood of a good point of the integral model
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel1,857 below · cited by 2 · depth 27 - Igusa components of the descended two-chart model, q=3
ModularCurve.FullLevel.exists_primes_chartAlg_localization_eq_igusaRing_minimal_injective_descent_of_eq_three1,111 below · cited by 8 · depth 27 - Igusa components of the descended two-chart model, q=2
ModularCurve.FullLevel.exists_primes_chartAlg_localization_eq_igusaRing_minimal_injective_descent_of_eq_two1,111 below · cited by 8 · depth 27 - Pole charts of j and j(q^q) over a level field
ModularCurve.FullLevel.exists_qExpand_forall_mem_chartAlgInf_exists_mul_mem_levelField241 below · cited by 3 · depth 27 - Level automorphisms of K₁· F₀ fixing j(qτ)
ModularCurve.FullLevel.exists_qExpand_mem_chartAlgFin_and_forall_mem_closure_levelAutBar_exists_algEquiv_levelField240 below · cited by 8 · depth 27 - Rational cusp-regular function separating two reduced rational places
ModularCurve.FullLevel.exists_rational_integral_cuspRegular_evalAt_ne_of_isRational_of_ne794 below · cited by 14 · depth 27 - Rational integral function separating two supersingular places, q=3
ModularCurve.FullLevel.exists_rational_integral_evalAt_ne_of_ne_ssPlaces_of_eq_three795 below · cited by 3 · depth 27 - Rational integral function separating two supersingular places, q=2
ModularCurve.FullLevel.exists_rational_integral_evalAt_ne_of_ne_ssPlaces_of_eq_two795 below · cited by 3 · depth 27 - A good point of the integral model reads a place
ModularCurve.FullLevel.exists_reads_of_goodPt_twoChartIntegralModel17 below · cited by 1 · depth 27 - Equivariance of level automorphisms on good points (q=3)
ModularCurve.FullLevel.exists_reads_resAut_smul_and_centred_iff_of_mem_closure_levelAutBar_twoChartIntegralModel_of_eq_three250 below · cited by 1 · depth 27 - Level automorphisms act equivariantly on good points, q=2
ModularCurve.FullLevel.exists_reads_resAut_smul_and_centred_iff_of_mem_closure_levelAutBar_twoChartIntegralModel_of_eq_two250 below · cited by 1 · depth 27 - Igusa Gauss ring prolongation compatible with constant reduction
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_coe_residue_eq_coe_residue1,099 below · cited by 1 · depth 27 - Regular prolongation of A extending a given discrete valuation
ModularCurve.FullLevel.exists_regularProlongation_integers_inter_levelField_eq_of_isDiscreteValuationRing5 below · cited by 5 · depth 27 - Coefficientwise transport of the level field K₁· F₀
ModularCurve.FullLevel.exists_ringEquiv_laurentBaseChange_levelField_coe_eq_coeffMap0 below · cited by 35 · depth 27 - Étale coordinate on the stalk at a good off-branch point, q=3
ModularCurve.FullLevel.exists_stalk_etaleCoordinate_residueChar_of_offBranch_of_reads_twoChartIntegralModel_of_eq_three1,639 below · cited by 1 · depth 27 - Étale coordinate and residue character at a good point (q=2)
ModularCurve.FullLevel.exists_stalk_etaleCoordinate_residueChar_of_offBranch_of_reads_twoChartIntegralModel_of_eq_two1,638 below · cited by 1 · depth 27 - Transporting the Igusa-chart dichotomy to every line
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_forall_gaussBranch_descent_local250 below · cited by 1 · depth 27 - Dichotomy for the Gauss component: smooth chart or supersingular place
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_gaussRing_descent2,009 below · cited by 1 · depth 27 - Igusa-chart dichotomy for the descended model at q=3
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_igusaRing_descent_local_of_eq_three1,761 below · cited by 2 · depth 27 - Igusa-chart dichotomy for the descended model, q=2
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_igusaRing_descent_local_of_eq_two1,760 below · cited by 2 · depth 27 - Affine Drinfeld chart at a supersingular place, q=3
ModularCurve.FullLevel.exists_subalgebra_drinfeldRing_iff_localization_formallySmooth_card_le_descent_of_eq_three_of_dvd3,669 below · cited by 1 · depth 27 - Affine Drinfeld chart at a supersingular place, q=2
ModularCurve.FullLevel.exists_subalgebra_drinfeldRing_iff_localization_formallySmooth_card_le_descent_of_eq_two_of_dvd3,667 below · cited by 1 · depth 27 - Supersingular chart, valuation ring and q+1 nodes over a level field
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_nodes_of_levelField3,609 below · cited by 2 · depth 27 - Level descent of the rigid supersingular chart, linked inertia
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_linkedInertia_of_rigidChart3,152 below · cited by 1 · depth 27 - Supersingular chart at q=3: DVR, Drinfeld chart, nodes, inertia
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_inertia_of_levelField_of_eq_three_of_dvd3,600 below · cited by 3 · depth 27 - Supersingular DVR, Drinfeld chart, nodes and inertia at q=2
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_inertia_of_levelField_of_eq_two_of_dvd3,598 below · cited by 3 · depth 27 - Supersingular DVR, affine chart and q+1 nodes at level q
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_of_levelField3,608 below · cited by 2 · depth 27 - Supersingular DVR, Drinfeld affine chart and q+1 nodes, q=3
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_nodes_of_levelField_of_eq_three_of_dvd3,601 below · cited by 1 · depth 27 - Supersingular chart and its q+1 nodes at q=2
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_nodes_of_levelField_of_eq_two_of_dvd3,599 below · cited by 1 · depth 27 - Formal smoothness of the descended Igusa ring, q=3
ModularCurve.FullLevel.formallySmooth_subalgebra_of_mem_iff_mem_igusaRing_descent_of_eq_three1,762 below · cited by 1 · depth 27 - Formal smoothness of the descended Igusa ring at q=2
ModularCurve.FullLevel.formallySmooth_subalgebra_of_mem_iff_mem_igusaRing_descent_of_eq_two1,761 below · cited by 1 · depth 27 - Stalk dictionary for the chart algebras of `TwoChartIntegralModel`
ModularCurve.FullLevel.inStalk_and_inMax_iff_mem_asIdeal_chartAlg_twoChartIntegralModel0 below · cited by 4 · depth 27 - Algebraicity over A₀[j] inside full-level modular function fields
ModularCurve.FullLevel.isAlgebraic_adjoin_j_of_mem_subfield_fieldBar5 below · cited by 3 · depth 27 - Properness of the two-chart integral model over A₁
ModularCurve.FullLevel.isProper_toBase_twoChartIntegralModel_levelField129 below · cited by 3 · depth 27 - Level-automorphism translates of j are Gauss units
ModularCurve.FullLevel.isUnit_levelAutBar_jBar_sub_algebraMap_of_gauss46 below · cited by 12 · depth 27 - Drinfeld inertia law at the level field, case q=3
ModularCurve.FullLevel.klevel_drinfeldInertia_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_inertia_of_eq_three_of_dvd11 below · cited by 1 · depth 27 - Tame inertia on the Drinfeld identification at level field, q=2
ModularCurve.FullLevel.klevel_drinfeldInertia_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_inertia_of_eq_two_of_dvd11 below · cited by 1 · depth 27 - Layered node charts and Hasse germs at a supersingular place
ModularCurve.FullLevel.klevel_nodeCore_nodeCharts_hasseGerm_nodeCentre_of_affineChart_poles_hasse_commonChart_nodes_igusaSep954 below · cited by 1 · depth 27 - Node charts and j-datum at a supersingular place (q=3)
ModularCurve.FullLevel.klevel_nodePresentations_nodeCharts_hasseJ_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_of_eq_three_of_dvd1,031 below · cited by 1 · depth 27 - Node places of the semistable fibre with charts and Hasse j-datum, q=2
ModularCurve.FullLevel.klevel_nodePresentations_nodeCharts_hasseJ_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_of_eq_two_of_dvd1,031 below · cited by 1 · depth 27 - From smooth-point stalks to smooth-point packages at q=3
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_of_eq_three_of_dvd_chart59 below · cited by 1 · depth 27 - From smooth-point stalks to smooth-point packages at q=2
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_of_eq_two_of_dvd_chart59 below · cited by 1 · depth 27 - Smooth-point stalks at all layers from the base layer (q=3)
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_of_eq_three_of_dvd_chart72 below · cited by 1 · depth 27 - Layerwise smooth-point stalks from the base layer, q=2
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_of_eq_two_of_dvd_chart72 below · cited by 1 · depth 27 - Affine chart with residual transcendence at a supersingular place, q=3
ModularCurve.FullLevel.klevel_supersingularDVR_affineChart_of_levelField_affineChart_of_eq_three_of_dvd48 below · cited by 1 · depth 27 - Assembling the q=2 supersingular affine chart with residual transcendence
ModularCurve.FullLevel.klevel_supersingularDVR_affineChart_of_levelField_affineChart_of_eq_two_of_dvd48 below · cited by 1 · depth 27 - Smooth-point stalks from the affine chart at q=3
ModularCurve.FullLevel.klevel_supersingularDVR_baseSmoothPointStalks_of_affineChart_of_eq_three_of_dvd_chart178 below · cited by 1 · depth 27 - Smooth-point stalks from the affine chart at q=2
ModularCurve.FullLevel.klevel_supersingularDVR_baseSmoothPointStalks_of_affineChart_of_eq_two_of_dvd_chart178 below · cited by 1 · depth 27 - Transport of non-supersingularity along σ fixing j(qτ)
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_algEquiv_apply_qExpand_eq_twoChartIntegralModel56 below · cited by 1 · depth 27 - Centres of places avoiding all supersingular tubes have non-supersingular j
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_centred_of_forall_not_ssTube_twoChartIntegralModel284 below · cited by 1 · depth 27 - Node cover identifies W₀ with the trace of 𝒪' (q=3)
ModularCurve.FullLevel.mem_iff_coe_mem_drinfeldRing_of_affineChart_nodes_cover_of_eq_three_of_dvd126 below · cited by 2 · depth 27 - Exceptional valuation ring recovered from the affine chart, q=2
ModularCurve.FullLevel.mem_iff_coe_mem_drinfeldRing_of_affineChart_nodes_cover_of_eq_two_of_dvd126 below · cited by 2 · depth 27 - Igusa rings trace to branch rings over (A₁,j₁), injectively
ModularCurve.FullLevel.mem_igusaRing_coe_levelField_and_injective_twoChartIntegralModel236 below · cited by 8 · depth 27 - Moving the ∞-Igusa ring decentres off-branch good points (q=3)
ModularCurve.FullLevel.not_centred_smul_of_comap_igusaGaussRing_ne_of_offBranch_twoChartIntegralModel_of_eq_three240 below · cited by 1 · depth 27 - Level automorphisms moving the ∞-Igusa ring decentre good points (q=2)
ModularCurve.FullLevel.not_centred_smul_of_comap_igusaGaussRing_ne_of_offBranch_twoChartIntegralModel_of_eq_two240 below · cited by 1 · depth 27 - Places read at good points avoid the Igusa node set
ModularCurve.FullLevel.not_mem_igusaNodes_of_reads_of_goodPt_twoChartIntegralModel52 below · cited by 1 · depth 27 - Places centred at good points avoid supersingular tubes, q=3
ModularCurve.FullLevel.not_mem_ssTube_of_centred_twoChartIntegralModel_of_eq_three0 below · cited by 1 · depth 27 - Places centred at good Igusa points avoid supersingular tubes, q=2
ModularCurve.FullLevel.not_mem_ssTube_of_centred_twoChartIntegralModel_of_eq_two0 below · cited by 1 · depth 27 - j(mathsf q^ℓ) lies in the level-(qℓ)²M' field and is integral over A[j(mathsf q)]
ModularCurve.FullLevel.qExpand_coeffEmb_jq_mem_and_mem_chartAlgFin_laurentBaseChange_xHFunctionField188 below · cited by 22 · depth 27 - Good points of the Igusa ∞-branch read non-node places (q=3)
ModularCurve.FullLevel.reads_unique_and_exists_offBranch_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_three1,607 below · cited by 1 · depth 27 - Reading bijection: good points versus non-node Igusa places, q=2
ModularCurve.FullLevel.reads_unique_and_exists_offBranch_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_two1,607 below · cited by 1 · depth 27 - Rigid-chart decomposition order equals 2 placeWidthChar
ModularCurve.FullLevel.rigidChart_decompositionOrder_eq_two_mul_placeWidthChar_of_decompositionUnique_linkedScalars2,909 below · cited by 1 · depth 27 - Inertia acts trivially to first order at good points, q=3
ModularCurve.FullLevel.sub_inMax_of_mem_inertiaSubgroupIn_of_inStalk_twoChartIntegralModel_of_eq_three2 below · cited by 1 · depth 27 - Inertia acts trivially to first order at good points (q=2)
ModularCurve.FullLevel.sub_inMax_of_mem_inertiaSubgroupIn_of_inStalk_twoChartIntegralModel_of_eq_two2 below · cited by 1 · depth 27 - Residual transcendence of the supersingular valuation ring
ModularCurve.FullLevel.supersingularDVR_residuallyTranscendental_of_affineChart47 below · cited by 2 · depth 27 - Residue discs from an affine chart: disjointness, cusps, equivariance
ModularCurve.FullLevel.supersingularProlongation_discRiders_of_affineChart36 below · cited by 3 · depth 27 - Ends and smooth-point stalks from an affine chart
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_localization_of_affineChart171 below · cited by 2 · depth 27 - Ends and smooth-point stalks from an affine chart
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_of_affineChart171 below · cited by 1 · depth 27 - Supersingular component from an affine chart has q+1 ends
ModularCurve.FullLevel.supersingularProlongation_ends_of_affineChart148 below · cited by 3 · depth 27 - Ends, residue discs and nodal cover over a supersingular place
ModularCurve.FullLevel.supersingularProlongation_ends_residueDiscs_cover_of_affineChart_poles_hasse_commonChart_nodes221 below · cited by 1 · depth 27 - Supersingular reduced field as a Drinfeld quotient field
ModularCurve.FullLevel.supersingularProlongation_existDL_of_affineChart9 below · cited by 2 · depth 27 - Functions regular off the q+1 ends are finitely generated (q=3)
ModularCurve.FullLevel.supersingularProlongation_exists_finite_generators_regular_off_ends_residueField_of_eq_three_of_dvd105 below · cited by 1 · depth 27 - Functions regular off the ends are finitely generated (q=2)
ModularCurve.FullLevel.supersingularProlongation_exists_finite_generators_regular_off_ends_residueField_of_eq_two_of_dvd105 below · cited by 1 · depth 27 - Functions regular off the ends lift to disc-regular integral functions, q=3
ModularCurve.FullLevel.supersingularProlongation_exists_lift_regular_on_smoothDiscs_of_regular_off_ends_of_affineChart_of_eq_three_of_dvd0 below · cited by 1 · depth 27 - Lifting functions regular off the ends from an affine chart, q=2
ModularCurve.FullLevel.supersingularProlongation_exists_lift_regular_on_smoothDiscs_of_regular_off_ends_of_affineChart_of_eq_two_of_dvd0 below · cited by 1 · depth 27 - Ends are non-affine Drinfeld places and level moves them, q=3
ModularCurve.FullLevel.supersingularProlongation_not_mem_ends_iff_affine_and_exists_levelAut_smul_ne_of_eq_three_of_dvd121 below · cited by 1 · depth 27 - Ends of the supersingular component for q = 2
ModularCurve.FullLevel.supersingularProlongation_not_mem_ends_iff_affine_and_exists_levelAut_smul_ne_of_eq_two_of_dvd122 below · cited by 1 · depth 27 - Residue field of the supersingular prolongation from the affine chart
ModularCurve.FullLevel.supersingularProlongation_residue_surjective_ker_of_affineChart7 below · cited by 5 · depth 27 - Étale coordinate and residue character on a smooth-point stalk
ModularCurve.FullLevel.supersingularProlongation_smoothPointStalk_of_affineChart8 below · cited by 3 · depth 27 - Level automorphisms for Γ₀(M') form a finite subgroup
ModularCurve.FullLevel.AuxLevel.exists_finite_subgroup_forall_mem_iff_exists_isLevelAutAt_of_exists_ringHom63 below · cited by 18 · depth 28 - Level automorphisms over L for γ∈Γ₀(M')
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_of_mem_gamma0_of_exists_ringHom30 below · cited by 52 · depth 28 - A G-invariant chart element avoiding all Igusa valuation rings
ModularCurve.FullLevel.AuxLevel.exists_mem_invariants_forall_igusaValuation_not_mem_of_rigidChart_framed258 below · cited by 1 · depth 28 - Invariant element avoiding every valuation ring over another supersingular place
ModularCurve.FullLevel.AuxLevel.exists_mem_invariants_forall_valuationSubring_over_not_mem_of_rigidChart_framed2,878 below · cited by 1 · depth 28 - Level and tame-inertia laws on the invariant chart
ModularCurve.FullLevel.AuxLevel.exists_quotField_ringHom_invariants_levelLaw_inertiaLaw_linkedInertia_of_rigidChart_framed262 below · cited by 1 · depth 28 - Completed stalk at a supersingular point as Drinfeld chart
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_mem_ssJSet_of_pow_eq_mul_moduliHasse_of_isAlgClosed3,198 below · cited by 3 · depth 28 - Supersingular affine chart with ends and linked tame inertia
ModularCurve.FullLevel.AuxLevel.exists_supersingularAffineChart_ends_moduliHasse_commonChart_orbitPoles_inertia_chartAlgFin_of_stalk_drinfeldChart_moduliHasse_igusaSep_deckSep_linkedScalars_linkedInertia794 below · cited by 1 · depth 28 - Tame inertia on the completed Drinfeld chart, general constants
ModularCurve.FullLevel.AuxLevel.inertia_drinfeldChart_semilinear_linearPart_tameCharacter_diagOneElem_of_levelAut_linearPart_of_pow_eq_mul_of_isAlgClosed3,229 below · cited by 1 · depth 28 - Level automorphisms over Γ₀(M'): uniqueness, composition, triviality
ModularCurve.FullLevel.AuxLevel.isLevelAutAt_unique_mul_one_of_exists_ringHom63 below · cited by 43 · depth 28 - Level automorphisms of Γ₀(M') normalise G and preserve K₀
ModularCurve.FullLevel.AuxLevel.mul_mul_inv_mem_and_map_fixedField_of_isLevelAutAt_gamma064 below · cited by 3 · depth 28 - Descended chart and supersingular valuation ring as G-invariants
ModularCurve.FullLevel.AuxLevel.supersingularDVR_affineChart_invariants_of_rigidChart_framed922 below · cited by 1 · depth 28 - Supersingular chart with q+1 ends, Drinfeld quotient, tame inertia
ModularCurve.FullLevel.AuxLevelOne.exists_supersingularAffineChart_ends_moduliHasse_commonChart_orbitPoles_chartAlgFin_igusaSep_deckSep_linkedScalars_linkedInertia_of_tame_of_dvd3,434 below · cited by 2 · depth 28 - Level automorphisms matching two ideals over a supersingular place (q=3)
ModularCurve.FullLevel.Diamond.exists_isLevelAutAt_map_chartAlgFin_eq_of_over_of_over_of_not_dvd_of_eq_three_of_dvd2,861 below · cited by 2 · depth 28 - Level automorphism carrying one chart point to another, q=2
ModularCurve.FullLevel.Diamond.exists_isLevelAutAt_map_chartAlgFin_eq_of_over_of_over_of_not_dvd_of_eq_two_of_dvd2,861 below · cited by 2 · depth 28 - Supersingular closed point on the j-chart, diamond frame, q=3
ModularCurve.FullLevel.Diamond.exists_isMaximal_chartAlgFin_mem_ssJSet_over_of_ssPlaces_of_eq_three_of_dvd29 below · cited by 2 · depth 28 - Supersingular maximal ideal of the j-chart at q=2
ModularCurve.FullLevel.Diamond.exists_isMaximal_chartAlgFin_mem_ssJSet_over_of_ssPlaces_of_eq_two_of_dvd29 below · cited by 2 · depth 28 - Level descent of the rigid Γ_{H_1} chart at q=3
ModularCurve.FullLevel.Diamond.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_linkedInertia_of_rigidChart_of_eq_three_of_dvd3,146 below · cited by 1 · depth 28 - Level descent of the rigid chart at q = 2
ModularCurve.FullLevel.Diamond.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_linkedInertia_of_rigidChart_of_eq_two_of_dvd3,148 below · cited by 1 · depth 28 - Rigid-chart rotation order equals place width at q=2
ModularCurve.FullLevel.Diamond.rigidChart_decompositionOrder_eq_placeWidthChar_of_decompositionUnique_linkedScalars_of_eq_two_of_dvd2,899 below · cited by 1 · depth 28 - Rigid-chart decomposition order is twice the place width (q=3)
ModularCurve.FullLevel.Diamond.rigidChart_decompositionOrder_eq_two_mul_placeWidthChar_of_decompositionUnique_linkedScalars_of_eq_three_of_dvd2,899 below · cited by 1 · depth 28 - Igusa-dominated valuation subrings of F₀ contain the constants
ModularCurve.FullLevel.algebraMap_mem_of_le_igusaRing_descent0 below · cited by 3 · depth 28 - Chart algebras inside the Igusa prolongation; constant residues attained
ModularCurve.FullLevel.coe_chartAlg_mem_integers_and_exists_residue_algebraMap_eq_twoChartIntegralModel0 below · cited by 2 · depth 28 - Igusa branch rings stay distinct on the level field
ModularCurve.FullLevel.eq_of_forall_coe_levelField_mem_igusaRing_iff_twoChartIntegralModel199 below · cited by 1 · depth 28 - Uniqueness of good points reading the same place, q=3
ModularCurve.FullLevel.eq_of_goodPt_of_reads_twoChartIntegralModel_of_eq_three0 below · cited by 1 · depth 28 - Uniqueness of a good point reading a given Igusa place, q=2
ModularCurve.FullLevel.eq_of_goodPt_of_reads_twoChartIntegralModel_of_eq_two0 below · cited by 1 · depth 28 - Descended Gauss ring: uniqueness of supersingular refinement, q=3
ModularCurve.FullLevel.eq_of_le_gaussRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent_of_eq_three1,143 below · cited by 1 · depth 28 - Refinements of the Gauss ring over a supersingular place, q=2
ModularCurve.FullLevel.eq_of_le_gaussRing_of_forall_isIntegral_mem_maximalIdeal_drinfeldRing_mem_nonunits_descent_of_eq_two1,144 below · cited by 1 · depth 28 - Level transport of Igusa-branch uniqueness at q = 3
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_gaussBranch_descent_of_eq_three248 below · cited by 1 · depth 28 - Transport of Igusa-branch uniqueness from ℓ=∞, q=2
ModularCurve.FullLevel.eq_of_le_igusaRing_of_forall_gaussBranch_descent_of_eq_two248 below · cited by 1 · depth 28 - Uniqueness of the place read by a good point, q=3
ModularCurve.FullLevel.eq_of_reads_of_reads_of_goodPt_twoChartIntegralModel_of_eq_three1,570 below · cited by 1 · depth 28 - Good points of the integral model read a unique place (q=2)
ModularCurve.FullLevel.eq_of_reads_of_reads_of_goodPt_twoChartIntegralModel_of_eq_two1,570 below · cited by 1 · depth 28 - Uniqueness of an Igusa-component valuation subring over a supersingular place
ModularCurve.FullLevel.eq_of_valuationSubring_residueField_igusaRing_of_floorTrace_descent1,468 below · cited by 2 · depth 28 - Maximum principle on a node tube: integrality at the node places
ModularCurve.FullLevel.evalAt_mem_of_mem_integers_igusaEnd_of_forall_mem_nodePlaces104 below · cited by 2 · depth 28 - Descent field K₀ as admissible small constants, q=3
ModularCurve.FullLevel.exists_admissible_smallConstants_of_descentBase_of_eq_three3 below · cited by 3 · depth 28 - Admissible field of small constants, case q=2
ModularCurve.FullLevel.exists_admissible_smallConstants_of_descentBase_of_eq_two3 below · cited by 3 · depth 28 - Descended fixed field of the rigid level model equals F₀
ModularCurve.FullLevel.exists_algEquiv_fixedField_levelField_coeffMap_eq_qExpand_framed77 below · cited by 1 · depth 28 - Igusa-side end of a node: branch prolongation and its centre
ModularCurve.FullLevel.exists_branchPlace_igusaEnd_integral_overS_of_node_crossingPresentation_igusaBranch855 below · cited by 2 · depth 28 - Unique centre place of a crossing-presented node
ModularCurve.FullLevel.exists_centrePlace_ord_residue_eq_one_of_node_crossingPresentation48 below · cited by 2 · depth 28 - Rational places centre on the proper two-chart model, q=3
ModularCurve.FullLevel.exists_centred_of_isRational_of_isProper_twoChartIntegralModel_of_eq_three0 below · cited by 2 · depth 28 - Centring rational places on the proper two-chart model, q = 2
ModularCurve.FullLevel.exists_centred_of_isRational_of_isProper_twoChartIntegralModel_of_eq_two0 below · cited by 2 · depth 28 - Chart-pole and other-pole clauses transported to the level field
ModularCurve.FullLevel.exists_chartPole_otherPole_levelField_of_fixedField_of_algEquiv_framed276 below · cited by 1 · depth 28 - Constant-term retraction on the ∞-chart algebra, q=3
ModularCurve.FullLevel.exists_constantTerm_chartAlgInf_twoChartIntegralModel_levelField_of_eq_three7 below · cited by 1 · depth 28 - Constant-term section on the ∞-chart algebra at q=2
ModularCurve.FullLevel.exists_constantTerm_chartAlgInf_twoChartIntegralModel_levelField_of_eq_two7 below · cited by 1 · depth 28 - The level Γ_H(q²M') q-expansion field is finite over a rational subfield
ModularCurve.FullLevel.exists_finiteDimensional_adjoin_xHFunctionFieldC_levelH9 below · cited by 4 · depth 28 - Branches of the level-field model are Igusa rings (q=3)
ModularCurve.FullLevel.exists_forall_coe_mem_igusa_iff_of_valuationSubring_levelField_of_eq_three1,036 below · cited by 8 · depth 28 - Branch identification in the Igusa tower at q=2
ModularCurve.FullLevel.exists_forall_coe_mem_igusa_iff_of_valuationSubring_levelField_of_eq_two1,035 below · cited by 8 · depth 28 - Supersingular special points of the descended chart are Drinfeld centres
ModularCurve.FullLevel.exists_forall_mem_maximalIdeal_drinfeldRing_mem_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent1,591 below · cited by 1 · depth 28 - Centred closed points lie on an Igusa branch, q=3
ModularCurve.FullLevel.exists_forall_mem_nonunits_igusa_mem_asIdeal_of_centred_twoChartIntegralModel_of_eq_three1,040 below · cited by 2 · depth 28 - Centred closed special points lie on an Igusa branch (q=2)
ModularCurve.FullLevel.exists_forall_mem_nonunits_igusa_mem_asIdeal_of_centred_twoChartIntegralModel_of_eq_two1,039 below · cited by 2 · depth 28 - Good off-branch point reading a non-node place, q=3
ModularCurve.FullLevel.exists_goodPt_and_offBranch_and_reads_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_three1,596 below · cited by 1 · depth 28 - Good off-branch points reading non-nodal Igusa places, q=2
ModularCurve.FullLevel.exists_goodPt_and_offBranch_and_reads_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_two1,596 below · cited by 1 · depth 28 - Changing the primitive root twists a level automorphism by diag(1,d)
ModularCurve.FullLevel.exists_isLevelAutAt_conj_of_isLevelAutAt_of_isPrimitiveRoot21 below · cited by 7 · depth 28 - A maximal ideal of the j-chart lying over a supersingular place
ModularCurve.FullLevel.exists_isMaximal_chartAlgFin_over_of_ssPlaces87 below · cited by 2 · depth 28 - Drinfeld rings see jmatĥ as an A₀-constant: q=3
ModularCurve.FullLevel.exists_jInvariant_sub_mem_maximalIdeal_drinfeldRing_descent_of_eq_three0 below · cited by 5 · depth 28 - Drinfeld rings read j as an A₀-constant (q=2)
ModularCurve.FullLevel.exists_jInvariant_sub_mem_maximalIdeal_drinfeldRing_descent_of_eq_two0 below · cited by 5 · depth 28 - Affine chart at a supersingular place over k₀, q=3
ModularCurve.FullLevel.exists_klevel_supersingularDVR_affineChart_of_eq_three_of_dvd3,607 below · cited by 1 · depth 28 - Affine chart at a supersingular place, full level, q=2
ModularCurve.FullLevel.exists_klevel_supersingularDVR_affineChart_of_eq_two_of_dvd3,605 below · cited by 1 · depth 28 - Layers of constants: discrete valuation rings, uniformisers, completions
ModularCurve.FullLevel.exists_layerConstants_uniformizers_completions_of_constantsTower4 below · cited by 3 · depth 28 - Layered rational node rings with crossing presentations
ModularCurve.FullLevel.exists_layeredRationalNodeRings_of_node_ends_layers129 below · cited by 2 · depth 28 - Moving a cusp-centred place to a good point (q=3)
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_goodPt_of_centred_of_not_mem_range_iotaFin_twoChartIntegralModel_of_eq_three1,049 below · cited by 1 · depth 28 - Moving a centred place onto a good point (q=2)
ModularCurve.FullLevel.exists_levelAutBar_smul_centred_goodPt_of_centred_of_not_mem_range_iotaFin_twoChartIntegralModel_of_eq_two1,048 below · cited by 1 · depth 28 - Good points of the two-chart model are not Igusa-generic (q=3)
ModularCurve.FullLevel.exists_mem_asIdeal_and_residue_ne_zero_of_goodPt_twoChartIntegralModel_of_eq_three2 below · cited by 2 · depth 28 - Chart primes carry functions with non-zero Igusa reduction, q=2
ModularCurve.FullLevel.exists_mem_asIdeal_and_residue_ne_zero_of_goodPt_twoChartIntegralModel_of_eq_two2 below · cited by 2 · depth 28 - Nodes of the rigid model descended to K_ℓ^G
ModularCurve.FullLevel.exists_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_fixedField_of_rigidChart_ends1,290 below · cited by 1 · depth 28 - Transport of the node block along the level-field identification
ModularCurve.FullLevel.exists_moduliHasse_commonChart_nodes_igusaSep_levelField_of_nodes_fixedField_of_algEquiv_framed276 below · cited by 1 · depth 28 - At a centred Igusa datum, jmatĥ or jmatĥ⁻¹ lies in B_𝔪
ModularCurve.FullLevel.exists_mul_eq_or_inv_mul_eq_of_centred_igusaChart_descent249 below · cited by 1 · depth 28 - Node centre, Igusa end and layered node rings
ModularCurve.FullLevel.exists_nodeCentre_igusaEnd_layeredNodeRings_of_mem_nodes_igusaSep_layerExponent917 below · cited by 1 · depth 28 - Smoothness of the full-level integral model on the Igusa branch
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_forall_mem_nonunits_gauss_of_forall_not_mem_ssJSet_twoChartIntegralModel_xH_of_isAlgebraic1,851 below · cited by 1 · depth 28 - Smooth neighbourhood at a good point of the descended model
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel_descent1,876 below · cited by 1 · depth 28 - Good points of the q=3 two-chart integral model are smooth
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel_of_eq_three1,575 below · cited by 2 · depth 28 - Smooth neighbourhood at a good point, q=2
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel_of_eq_two1,575 below · cited by 2 · depth 28 - Powers on the Igusa branch at a crossing node
ModularCurve.FullLevel.exists_pow_eq_mul_nonunit_of_mem_maximalIdeal_igusaBranch_of_node_crossingPresentation2 below · cited by 2 · depth 28 - Pole charts of j and j(q^q) compare, q=3
ModularCurve.FullLevel.exists_qExpand_forall_mem_chartAlgInf_exists_mul_mem_levelField_of_eq_three241 below · cited by 3 · depth 28 - Pole charts of j and j(q^q) compare, case q=2
ModularCurve.FullLevel.exists_qExpand_forall_mem_chartAlgInf_exists_mul_mem_levelField_of_eq_two241 below · cited by 3 · depth 28 - Descent of level automorphisms to the level field, q=3
ModularCurve.FullLevel.exists_qExpand_mem_chartAlgFin_and_forall_mem_closure_levelAutBar_exists_algEquiv_levelField_of_eq_three240 below · cited by 8 · depth 28 - Level automorphisms on the level field and its j-charts, q=2
ModularCurve.FullLevel.exists_qExpand_mem_chartAlgFin_and_forall_mem_closure_levelAutBar_exists_algEquiv_levelField_of_eq_two240 below · cited by 8 · depth 28 - Good points read a place of the Igusa reduction (q=3)
ModularCurve.FullLevel.exists_reads_of_goodPt_twoChartIntegralModel_of_eq_three17 below · cited by 1 · depth 28 - Good points of the ∞-Igusa branch read a place (q=2)
ModularCurve.FullLevel.exists_reads_of_goodPt_twoChartIntegralModel_of_eq_two17 below · cited by 1 · depth 28 - Regular prolongation onto the Igusa Gauss ring, q=3
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_coe_residue_eq_coe_residue_of_eq_three382 below · cited by 1 · depth 28 - Regular prolongation on the Igusa Gauss ring at q=2
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_coe_residue_eq_coe_residue_of_eq_two382 below · cited by 1 · depth 28 - Igusa Gauss ring at ∞ as a regular prolongation (q=3)
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_of_eq_three492 below · cited by 12 · depth 28 - Regular prolongation with Igusa Gauss ring at ∞, q=2
ModularCurve.FullLevel.exists_regularProlongation_integers_eq_igusaGaussRing_of_eq_two492 below · cited by 12 · depth 28 - Gauss valuation on a finite level field is unramified over A'
ModularCurve.FullLevel.exists_smul_mem_and_inv_mem_gauss_levelField192 below · cited by 6 · depth 28 - Every rational place is read at a closed special point
ModularCurve.FullLevel.exists_special_closed_branch_reads_of_isRational_twoChartIntegralModel130 below · cited by 1 · depth 28 - Special-fibre points of the rigid chart as level structures
ModularCurve.FullLevel.exists_ssFibreDictionary_chartAlgFin_rigidDataPow2,868 below · cited by 1 · depth 28 - Rational places with supersingular j lie in a supersingular tube
ModularCurve.FullLevel.exists_ssTube_of_residue_evalAt_mem_ssJSet282 below · cited by 1 · depth 28 - Formally smooth chart package at a smooth non-supersingular centre
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgFin_descent1,266 below · cited by 1 · depth 28 - Formally smooth centred subalgebra at a smooth pole-chart centre
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgInf_descent1,266 below · cited by 1 · depth 28 - Igusa-chart dichotomy transported to every line, q=3
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_forall_gaussBranch_descent_local_of_eq_three250 below · cited by 1 · depth 28 - Igusa-chart dichotomy transported from the Gauss line, q=2
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_forall_gaussBranch_descent_local_of_eq_two250 below · cited by 1 · depth 28 - Gauss-component chart dichotomy for the descended model, q=3
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_gaussRing_descent_of_eq_three1,757 below · cited by 1 · depth 28 - Gauss-component chart dichotomy for the descended model, q=2
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_or_exists_forall_mem_nonunits_of_le_gaussRing_descent_of_eq_two1,756 below · cited by 1 · depth 28 - Supersingular descended model for q=3: affine chart and nodes
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_nodes_of_levelField_of_eq_three_of_dvd3,601 below · cited by 2 · depth 28 - Descended chart and q+1 nodes at a supersingular place, q=2
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_nodes_of_levelField_of_eq_two_of_dvd3,599 below · cited by 2 · depth 28 - Affine chart at a supersingular point over the level field
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_of_levelField3,610 below · cited by 1 · depth 28 - Supersingular chart and q+1 nodes over the level-q field
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_poles_moduliHasse_commonChart_nodes_igusaSep_of_levelField3,607 below · cited by 1 · depth 28 - Localisations of the descended chart algebra are valuation rings
ModularCurve.FullLevel.exists_valuationSubring_localization_chartAlg_of_not_mem_descent165 below · cited by 1 · depth 28 - Good-reduction integers of the level-M' floor lie in O
ModularCurve.FullLevel.forall_mem_integers_inclusion_mem_of_algebraMap_mem_iff_of_forall_aeval_mem945 below · cited by 1 · depth 28 - Chart functions: stalks and maximal ideals, q=3
ModularCurve.FullLevel.inStalk_and_inMax_iff_mem_asIdeal_chartAlg_twoChartIntegralModel_of_eq_three0 below · cited by 4 · depth 28 - Chart functions of the two-chart model: stalks and primes (q=2)
ModularCurve.FullLevel.inStalk_and_inMax_iff_mem_asIdeal_chartAlg_twoChartIntegralModel_of_eq_two0 below · cited by 4 · depth 28 - Integrality of the full-level field over the level-M' floor
ModularCurve.FullLevel.isIntegral_inclusion_modularFunctionFieldBar_fieldBar122 below · cited by 4 · depth 28 - The centre of V on the chart algebra is maximal
ModularCurve.FullLevel.isMaximal_of_forall_mem_iff_mem_nonunits_of_lt_gaussRing_descent1,261 below · cited by 3 · depth 28 - Properness of the two-chart integral model at q=3
ModularCurve.FullLevel.isProper_toBase_twoChartIntegralModel_levelField_of_eq_three129 below · cited by 3 · depth 28 - Properness of the level-field two-chart integral model, q=2
ModularCurve.FullLevel.isProper_toBase_twoChartIntegralModel_levelField_of_eq_two129 below · cited by 3 · depth 28 - Regularity of the special fibre at a good point
ModularCurve.FullLevel.isRegularLocalRing_stalk_quotient_span_of_goodPt_twoChartIntegralModel1,846 below · cited by 2 · depth 28 - Node charts, Hasse germ and node centre for q=3
ModularCurve.FullLevel.klevel_nodeCore_nodeCharts_hasseGerm_nodeCentre_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_of_eq_three_of_dvd954 below · cited by 1 · depth 28 - Per-node core of the layered node presentations at q=2
ModularCurve.FullLevel.klevel_nodeCore_nodeCharts_hasseGerm_nodeCentre_of_affineChart_poles_hasse_commonChart_nodes_igusaSep_of_eq_two_of_dvd954 below · cited by 1 · depth 28 - Smooth-point stalks yield smooth-point packages at full level
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_linked59 below · cited by 1 · depth 28 - All-layer smooth-point stalks from base-layer stalks, shared ends
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_linked72 below · cited by 1 · depth 28 - Level automorphisms transport node place-sets and Igusa ends
ModularCurve.FullLevel.levelAut_transport_nodePlaces_igusaEnds_of_nodeCentre_igusaBranch79 below · cited by 1 · depth 28 - Non-supersingularity transports along σ on the j-finite chart (q=3)
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_algEquiv_apply_qExpand_eq_twoChartIntegralModel_of_eq_three56 below · cited by 1 · depth 28 - Non-supersingularity transports along an automorphism fixing j(q^q) (q=2)
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_algEquiv_apply_qExpand_eq_twoChartIntegralModel_of_eq_two56 below · cited by 1 · depth 28 - Centred rational places outside supersingular tubes avoid supersingular j (q=3)
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_centred_of_forall_not_ssTube_twoChartIntegralModel_of_eq_three284 below · cited by 1 · depth 28 - Tube-free rational places centre at non-supersingular j-values (q=2)
ModularCurve.FullLevel.map_jChartFin_not_mem_ssJSet_of_centred_of_forall_not_ssTube_twoChartIntegralModel_of_eq_two284 below · cited by 1 · depth 28 - Level automorphisms preserve the j-finite chart algebra
ModularCurve.FullLevel.map_mem_chartAlgFin_of_isLevelAutAt_of_mem_Gamma085 below · cited by 14 · depth 28 - Igusa Gauss ring at ∞ cuts out R₀-integers (q=3)
ModularCurve.FullLevel.mem_constantReduction_integers_iff_inclusion_mem_igusaGaussRing_of_eq_three2 below · cited by 9 · depth 28 - Igusa Gauss ring at ∞ cuts out the constant reduction (q=2)
ModularCurve.FullLevel.mem_constantReduction_integers_iff_inclusion_mem_igusaGaussRing_of_eq_two2 below · cited by 9 · depth 28 - Igusa rings on the level field: branch conditions and separation, q=3
ModularCurve.FullLevel.mem_igusaRing_coe_levelField_and_injective_twoChartIntegralModel_of_eq_three236 below · cited by 8 · depth 28 - Igusa rings traced on the level field, q=2
ModularCurve.FullLevel.mem_igusaRing_coe_levelField_and_injective_twoChartIntegralModel_of_eq_two236 below · cited by 8 · depth 28 - Drinfeld centre at a supersingular place has supersingular j
ModularCurve.FullLevel.mem_ssJSet_of_forall_mem_maximalIdeal_drinfeldRing_mem_chartAlgFin_descent21 below · cited by 1 · depth 28 - No Igusa node is read at a good point (q=3)
ModularCurve.FullLevel.not_mem_igusaNodes_of_reads_of_goodPt_twoChartIntegralModel_of_eq_three52 below · cited by 1 · depth 28 - Good points read places off the Igusa nodes (q=2)
ModularCurve.FullLevel.not_mem_igusaNodes_of_reads_of_goodPt_twoChartIntegralModel_of_eq_two52 below · cited by 1 · depth 28 - Points reading a non-node place have non-supersingular j
ModularCurve.FullLevel.not_mem_ssJSet_of_reads_of_not_mem_igusaNodes_twoChartIntegralModel161 below · cited by 1 · depth 28 - Good points lie off the other Igusa branches
ModularCurve.FullLevel.offBranch_of_goodPt_twoChartIntegralModel1,859 below · cited by 1 · depth 28 - Decomposition order at a supersingular point equals #Aut(E,C)
ModularCurve.FullLevel.rigidChart_decompositionOrder_eq_natCard_rationalAut_of_moduliPlace_of_decompositionUnique_linkedScalars2,907 below · cited by 1 · depth 28 - Residual transcendence of the supersingular chart valuation ring (q=3)
ModularCurve.FullLevel.supersingularDVR_residuallyTranscendental_of_affineChart_of_eq_three_of_dvd47 below · cited by 2 · depth 28 - Residual transcendence at a supersingular place, Drinfeld chart, q=2
ModularCurve.FullLevel.supersingularDVR_residuallyTranscendental_of_affineChart_of_eq_two_of_dvd47 below · cited by 2 · depth 28 - Rational places over s meet a node ring or a disc
ModularCurve.FullLevel.supersingularProlongation_cover_of_affineChart_nodes_of_baseSmoothPointStalks_of_coeffEmb_mem52 below · cited by 1 · depth 28 - Supersingular residue discs from an affine chart, q=3
ModularCurve.FullLevel.supersingularProlongation_discRiders_of_affineChart_of_eq_three36 below · cited by 3 · depth 28 - Residue discs of an affine chart: disjoint, cusp-free, equivariant (q=2)
ModularCurve.FullLevel.supersingularProlongation_discRiders_of_affineChart_of_eq_two36 below · cited by 3 · depth 28 - Ends and smooth-point stalks of a supersingular chart, q=3
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_localization_of_affineChart_of_eq_three_of_dvd171 below · cited by 2 · depth 28 - Ends and smooth-point stalks of a supersingular chart, q=2
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_localization_of_affineChart_of_eq_two_of_dvd171 below · cited by 2 · depth 28 - Ends and smooth-point stalks on the supersingular chart, q=3
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_of_affineChart_of_eq_three_of_dvd171 below · cited by 1 · depth 28 - Ends and smooth-point stalks from an affine chart, q=2
ModularCurve.FullLevel.supersingularProlongation_ends_baseSmoothPointStalks_of_affineChart_of_eq_two_of_dvd171 below · cited by 1 · depth 28 - Supersingular chart has q+1 ends permuted by level automorphisms (q=3)
ModularCurve.FullLevel.supersingularProlongation_ends_of_affineChart_of_eq_three148 below · cited by 3 · depth 28 - Supersingular chart has q+1 ends, permuted by level automorphisms (q=2)
ModularCurve.FullLevel.supersingularProlongation_ends_of_affineChart_of_eq_two148 below · cited by 3 · depth 28 - Ends, residue discs and valuative cover at q=3
ModularCurve.FullLevel.supersingularProlongation_ends_residueDiscs_cover_of_affineChart_poles_hasse_commonChart_nodes_of_eq_three_of_dvd221 below · cited by 1 · depth 28 - Ends, residue discs and valuative cover at a supersingular place, q=2
ModularCurve.FullLevel.supersingularProlongation_ends_residueDiscs_cover_of_affineChart_poles_hasse_commonChart_nodes_of_eq_two_of_dvd221 below · cited by 1 · depth 28 - Supersingular reduced field as a Drinfeld quotient field, q=3
ModularCurve.FullLevel.supersingularProlongation_existDL_of_affineChart_of_eq_three_of_dvd9 below · cited by 2 · depth 28 - Supersingular reduced field as a Drinfeld quotient field, q=2
ModularCurve.FullLevel.supersingularProlongation_existDL_of_affineChart_of_eq_two_of_dvd9 below · cited by 2 · depth 28 - Residues of the affine chart generate Fₛₛ, q=3
ModularCurve.FullLevel.supersingularProlongation_residue_surjective_ker_of_affineChart_of_eq_three7 below · cited by 5 · depth 28 - Affine-chart residues generate the supersingular residue field, q=2
ModularCurve.FullLevel.supersingularProlongation_residue_surjective_ker_of_affineChart_of_eq_two7 below · cited by 5 · depth 28 - Smooth-point stalk from an affine chart, supersingular component, q=3
ModularCurve.FullLevel.supersingularProlongation_smoothPointStalk_of_affineChart_of_eq_three8 below · cited by 3 · depth 28 - Smooth supersingular point stalk from an affine chart, q=2
ModularCurve.FullLevel.supersingularProlongation_smoothPointStalk_of_affineChart_of_eq_two8 below · cited by 3 · depth 28 - Cusp centre on the ∞-chart is a closed special point
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_chartAlgInf_descent1,271 below · cited by 1 · depth 28 - Centre of a refined Gauss ring is a closed special point
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_descent1,271 below · cited by 1 · depth 28 - Drinfeld special fibre and level action on the blow-up chart
ModularCurve.FullLevel.AuxLevel.blowupChart_drinfeldFibre_levelAut_decomposition_linkedScalars_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_moduliHasse181 below · cited by 1 · depth 29 - Tame inertia on the exceptional Drinfeld fibre
ModularCurve.FullLevel.AuxLevel.blowupChart_drinfeldFibre_levelAut_linkedScalars_inertia_of_decomposition_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_inertia183 below · cited by 1 · depth 29 - Auxiliary-level fixed field is the k₀-rational level-q field
ModularCurve.FullLevel.AuxLevel.exists_algEquiv_fixedField_coeffMap_eq_qExpand_of_forall_mem_iff_coeff71 below · cited by 1 · depth 29 - Weighted blow-up chart C[J/varpiₜ] and its exceptional valuation ring
ModularCurve.FullLevel.AuxLevel.exists_blowupChart_eq_adjoin_exceptionalValuation_of_drinfeldChartWitness_of_stalk_drinfeldChart_moduliHasse189 below · cited by 1 · depth 29 - Base change of the j-chart along the cyclotomic constants
ModularCurve.FullLevel.AuxLevel.exists_chartAlgFin_tensorProduct_ringEquiv_of_cyclotomicConstants_of_isAlgClosed2,268 below · cited by 2 · depth 29 - Ends of the blown-up supersingular chart: cyclic decomposition and crossings
ModularCurve.FullLevel.AuxLevel.exists_cyclicDecomposition_ends_moduliHasse_igusaSepTranslate_commonChart_cover_blowupChart_linked_of_eq_adjoin_of_drinfeldChartWitness758 below · cited by 1 · depth 29 - Cyclotomic frame inside a general q-adic discrete valuation ring
ModularCurve.FullLevel.AuxLevel.exists_cyclotomicConstants_of_isPrimitiveRoot_of_pow_eq_mul1 below · cited by 2 · depth 29 - Level dictionary for the descent isomorphism φ
ModularCurve.FullLevel.AuxLevel.exists_map_fixedField_and_apply_eq_levelAutBar_of_isLevelAutAt_of_coeffMap_eq_qExpand71 below · cited by 1 · depth 29 - Poles of the blow-up chart along Igusa and off-orbit valuations
ModularCurve.FullLevel.AuxLevel.exists_mem_blowupChart_not_mem_igusaValuation_orbitPole_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart269 below · cited by 1 · depth 29 - Descended supersingular chart reduces to a quotient Drinfeld curve
ModularCurve.FullLevel.AuxLevel.exists_quotField_ringHom_invariants_of_rigidChart_framed259 below · cited by 2 · depth 29 - Base change of a cyclotomic Drinfeld chart witness
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_baseChange_of_cyclotomicWitness_of_isAlgClosed248 below · cited by 2 · depth 29 - Drinfeld chart for the completed stalk, with level and inertia riders
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_levelAut_linearPart_inertia_of_mem_ssJSet_of_pow_eq_mul_of_isAlgClosed3,194 below · cited by 1 · depth 29 - Drinfeld local chart at a supersingular point, full level
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_mem_ssJSet_twoChartIntegralModel3,137 below · cited by 1 · depth 29 - Initial form of the moduli j-invariant on Drinfeld charts
ModularCurve.FullLevel.AuxLevel.exists_sub_const_eq_mk_of_mem_pow_isUnit_homogeneous_drinfeldChart_of_ringEquiv_adicCompletion_stalk2,298 below · cited by 1 · depth 29 - Finiteness of the Γ(q)∩Γ₀(M') level-automorphism group
ModularCurve.FullLevel.AuxLevel.finite_and_natCard_dvd_of_eq_closure_isLevelAutAt_gamma65 below · cited by 11 · depth 29 - Formal smoothness of the descended chart B₀ over A∩ k₀
ModularCurve.FullLevel.AuxLevel.formallySmooth_invariants_of_rigidChart_framed321 below · cited by 1 · depth 29 - Transport of the semilinear tame-inertia law between Drinfeld charts
ModularCurve.FullLevel.AuxLevel.inertia_drinfeldChart_semilinear_linearPart_transport_of_levelAut_linearPart_of_pow_eq_mul_of_isAlgClosed3,209 below · cited by 1 · depth 29 - Supersingular fibre points of the two-chart model are maximal
ModularCurve.FullLevel.AuxLevel.isMaximal_asIdeal_and_algebraMap_mem_of_mem_ssJSet_of_exists_ringHom8 below · cited by 3 · depth 29 - Drinfeld chart for the completed supersingular stalk
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_mem_ssJSet_of_pow_eq_mul_moduliHasse_of_isPrimitiveRoot_mul_of_dvd3,185 below · cited by 3 · depth 29 - Supersingular chart with Drinfeld ends, decomposition and inertia
ModularCurve.FullLevel.AuxLevelOne.exists_supersingularAffineChart_ends_moduliHasse_commonChart_orbitPoles_inertia_chartAlgFin_of_stalk_drinfeldChart_moduliHasse_igusaSep_deckSep_linkedScalars_linkedInertia_of_dvd622 below · cited by 1 · depth 29 - Tame inertia on the Drinfeld chart: semilinear, linear part ctcdotdiag(1,d^{-q})
ModularCurve.FullLevel.AuxLevelOne.inertia_drinfeldChart_semilinear_linearPart_tameCharacter_diagOneElem_of_levelAut_linearPart_of_pow_eq_mul_of_isPrimitiveRoot_mul_of_dvd3,216 below · cited by 1 · depth 29 - Invariant chart element outside every Igusa valuation ring, q=3
ModularCurve.FullLevel.Diamond.AuxLevel.exists_mem_invariants_forall_igusaValuation_not_mem_of_rigidChart_linkedScalars_of_eq_three_of_dvd34 below · cited by 1 · depth 29 - Invariant chart element in no Igusa valuation ring of K₀
ModularCurve.FullLevel.Diamond.AuxLevel.exists_mem_invariants_forall_igusaValuation_not_mem_of_rigidChart_linkedScalars_of_eq_two_of_dvd34 below · cited by 1 · depth 29 - Invariant element avoiding valuation rings over the other supersingular places
ModularCurve.FullLevel.Diamond.AuxLevel.exists_mem_invariants_forall_valuationSubring_over_not_mem_of_rigidChart_linkedScalars_of_eq_three_of_dvd2,869 below · cited by 1 · depth 29 - Invariant function with poles over the other supersingular places
ModularCurve.FullLevel.Diamond.AuxLevel.exists_mem_invariants_forall_valuationSubring_over_not_mem_of_rigidChart_linkedScalars_of_eq_two_of_dvd2,869 below · cited by 1 · depth 29 - Drinfeld reduction of the invariant chart at q=3, Γ₁(ℓ_g) frame
ModularCurve.FullLevel.Diamond.AuxLevel.exists_quotField_ringHom_invariants_levelLaw_inertiaLaw_linkedInertia_of_rigidChart_of_eq_three_of_dvd46 below · cited by 1 · depth 29 - Level and tame-inertia laws on the invariant chart at q=2
ModularCurve.FullLevel.Diamond.AuxLevel.exists_quotField_ringHom_invariants_levelLaw_inertiaLaw_linkedInertia_of_rigidChart_of_eq_two_of_dvd47 below · cited by 1 · depth 29 - Descent of the rigid chart to the fixed field K₀ at q=3
ModularCurve.FullLevel.Diamond.AuxLevel.supersingularDVR_affineChart_invariants_of_rigidChart_linkedScalars_of_eq_three_of_dvd895 below · cited by 1 · depth 29 - Descent of the supersingular chart to G-invariants at q=2
ModularCurve.FullLevel.Diamond.AuxLevel.supersingularDVR_affineChart_invariants_of_rigidChart_linkedScalars_of_eq_two_of_dvd896 below · cited by 1 · depth 29 - The fixed field K₀ is the level field F₀ (q=3)
ModularCurve.FullLevel.Diamond.exists_algEquiv_fixedField_levelField_coeffMap_eq_linkedScalars_of_eq_three_of_dvd72 below · cited by 1 · depth 29 - Level-field descent of the rigid level-q model, q=2
ModularCurve.FullLevel.Diamond.exists_algEquiv_fixedField_levelField_coeffMap_eq_linkedScalars_of_eq_two_of_dvd72 below · cited by 1 · depth 29 - Transport of chart poles to the level field, q=3
ModularCurve.FullLevel.Diamond.exists_chartPole_otherPole_levelField_of_fixedField_of_algEquiv_linkedScalars_of_eq_three_of_dvd232 below · cited by 1 · depth 29 - Transport of chart-pole and other-pole clauses along the level-field identification
ModularCurve.FullLevel.Diamond.exists_chartPole_otherPole_levelField_of_fixedField_of_algEquiv_linkedScalars_of_eq_two_of_dvd232 below · cited by 1 · depth 29 - Maximal ideal of the j-finite chart over a supersingular place, q=3
ModularCurve.FullLevel.Diamond.exists_isMaximal_chartAlgFin_over_of_ssPlaces_of_eq_three_of_dvd1 below · cited by 1 · depth 29 - Special-fibre point of the j-finite chart over s, q=2
ModularCurve.FullLevel.Diamond.exists_isMaximal_chartAlgFin_over_of_ssPlaces_of_eq_two_of_dvd1 below · cited by 1 · depth 29 - The q+1 nodes of the descended supersingular model, q=3
ModularCurve.FullLevel.Diamond.exists_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_fixedField_of_rigidChart_ends_of_eq_three_of_dvd1,278 below · cited by 1 · depth 29 - Descent of the q+1 nodes to the fixed field, q=2
ModularCurve.FullLevel.Diamond.exists_moduliHasse_commonChart_nodes_igusaSep_deckSep_linkedScalars_fixedField_of_rigidChart_ends_of_eq_two_of_dvd1,278 below · cited by 1 · depth 29 - Transport of the node package along the level-field identification, q=3
ModularCurve.FullLevel.Diamond.exists_moduliHasse_commonChart_nodes_igusaSep_levelField_of_nodes_fixedField_of_algEquiv_of_eq_three_of_dvd232 below · cited by 1 · depth 29 - Transport of the node block to the level field, q=2
ModularCurve.FullLevel.Diamond.exists_moduliHasse_commonChart_nodes_igusaSep_levelField_of_nodes_fixedField_of_algEquiv_of_eq_two_of_dvd232 below · cited by 1 · depth 29 - Special-fibre dictionary for the rigid chart at level Γ(q)∩Γ₁(ℓ_g)∩Γ₀(M')
ModularCurve.FullLevel.Diamond.exists_ssFibreDictionary_chartAlgFin_rigidDataGamma1Pow2,857 below · cited by 2 · depth 29 - Decomposition order equals the number of rational automorphisms of (E,C)
ModularCurve.FullLevel.Diamond.rigidChart_decompositionOrder_eq_natCard_rationalAut_of_moduliPlace_of_decompositionUnique_linkedScalars_of_eq_three_of_dvd2,897 below · cited by 1 · depth 29 - Twice the decomposition order counts automorphisms of (E,Cyc), q=2
ModularCurve.FullLevel.Diamond.two_mul_rigidChart_decompositionOrder_eq_natCard_rationalAut_of_moduliPlace_of_decompositionUnique_linkedScalars_of_eq_two_of_dvd2,897 below · cited by 1 · depth 29 - Transcendence of jmatĥ and finiteness of the descended field
ModularCurve.FullLevel.aeval_eq_zero_imp_and_finiteDimensional_closure_descent164 below · cited by 1 · depth 29 - Polynomials in j with non-zero reduction are R₀-units
ModularCurve.FullLevel.aeval_jq_mem_integers_and_inv_mem_of_map_residue_ne_zero0 below · cited by 3 · depth 29 - Constants lie in valuation subrings of the Igusa ring, q=3
ModularCurve.FullLevel.algebraMap_mem_of_le_igusaRing_descent_of_eq_three493 below · cited by 3 · depth 29 - Constants lie in any valuation ring below an Igusa ring (q=2)
ModularCurve.FullLevel.algebraMap_mem_of_le_igusaRing_descent_of_eq_two493 below · cited by 3 · depth 29 - Chart algebras inside the ∞-Igusa prolongation, q=3
ModularCurve.FullLevel.coe_chartAlg_mem_integers_and_exists_residue_algebraMap_eq_twoChartIntegralModel_of_eq_three0 below · cited by 2 · depth 29 - Chart algebras lie in the ∞-Igusa ring, q=2
ModularCurve.FullLevel.coe_chartAlg_mem_integers_and_exists_residue_algebraMap_eq_twoChartIntegralModel_of_eq_two0 below · cited by 2 · depth 29 - Distinct Igusa lines have distinct rings on the level field (q=3)
ModularCurve.FullLevel.eq_of_forall_coe_levelField_mem_igusaRing_iff_twoChartIntegralModel_of_eq_three199 below · cited by 1 · depth 29 - Distinct Igusa lines differ on the level field (q=2)
ModularCurve.FullLevel.eq_of_forall_coe_levelField_mem_igusaRing_iff_twoChartIntegralModel_of_eq_two199 below · cited by 1 · depth 29 - Uniqueness of the Igusa-component valuation ring reading s (q=3)
ModularCurve.FullLevel.eq_of_valuationSubring_residueField_igusaRing_of_floorTrace_descent_of_eq_three1,141 below · cited by 2 · depth 29 - Uniqueness of Igusa-component valuation ring reading a supersingular place (q=2)
ModularCurve.FullLevel.eq_of_valuationSubring_residueField_igusaRing_of_floorTrace_descent_of_eq_two1,142 below · cited by 2 · depth 29 - Level-field node ring as base rational node-ring input
ModularCurve.FullLevel.exists_baseRationalNodeRing_input_of_node_ends_nodePlaces80 below · cited by 2 · depth 29 - Igusa-branch place at a node of the semistable model
ModularCurve.FullLevel.exists_branchPlace_of_igusaEnd_of_node_crossingPresentation_igusaBranch48 below · cited by 1 · depth 29 - Finite generation of the level-Γ_H(q²M') q-expansion field
ModularCurve.FullLevel.exists_finiteDimensional_adjoin_xHFunctionFieldC_levelH_of_eq_three9 below · cited by 4 · depth 29 - Finiteness over a simple subfield of the X_H q-expansion field, q=2
ModularCurve.FullLevel.exists_finiteDimensional_adjoin_xHFunctionFieldC_levelH_of_eq_two9 below · cited by 4 · depth 29 - Finiteness of the Γ₀(M') level automorphisms at guarded level
ModularCurve.FullLevel.exists_finite_subgroup_forall_mem_iff_exists_isLevelAutAt_of_exists_ringHom_of_eq_levelH_inf_ker30 below · cited by 26 · depth 29 - Drinfeld centre of a supersingular special point, q=3
ModularCurve.FullLevel.exists_forall_mem_maximalIdeal_drinfeldRing_mem_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent_of_eq_three1,435 below · cited by 1 · depth 29 - Supersingular special points centre a Drinfeld ring (q=2)
ModularCurve.FullLevel.exists_forall_mem_maximalIdeal_drinfeldRing_mem_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent_of_eq_two1,433 below · cited by 1 · depth 29 - Constancy of the level-ℓ' Weil pairing on the special fibre
ModularCurve.FullLevel.exists_forall_weilPairing0_eq_of_eq_map_classify_rigidDataPow42 below · cited by 3 · depth 29 - Coefficientwise conjugation preserves level automorphisms
ModularCurve.FullLevel.exists_isLevelAutAt_apply_conj_of_coeffMap_ringEquiv0 below · cited by 5 · depth 29 - Changing the q-th root of unity conjugates level automorphisms diagonally
ModularCurve.FullLevel.exists_isLevelAutAt_conj_of_isLevelAutAt_of_isPrimitiveRoot_of_eq_levelH_inf_ker21 below · cited by 12 · depth 29 - Existence of level automorphisms at the guarded level H₁
ModularCurve.FullLevel.exists_isLevelAutAt_of_mem_gamma0_of_eq_levelH_inf_ker30 below · cited by 59 · depth 29 - Unique maximal ideal over a supersingular place, descended chart
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_unique_chartAlgFin_descent1,579 below · cited by 1 · depth 29 - Layer step for rational node rings with crossing presentation
ModularCurve.FullLevel.exists_layerRationalNodeRing_of_baseRationalNodeRing_layer_chart127 below · cited by 1 · depth 29 - Full-level Weierstrass moduli package, integral over A[j₀]
ModularCurve.FullLevel.exists_levelModuliPackageAbs_isIntegral_adjoin_of_isSectionTransport_of_isNoetherianRing_of_isUnit_two_three_gamma0Pow139 below · cited by 14 · depth 29 - Half-unit g=cₓ v at a crossing node, Igusa-separated
ModularCurve.FullLevel.exists_mem_eq_cx_mul_unit_isUnit_of_commonChart_of_igusaSep3 below · cited by 1 · depth 29 - Centred Igusa datum: jmatĥ or jmatĥ⁻¹ regular (q=3)
ModularCurve.FullLevel.exists_mul_eq_or_inv_mul_eq_of_centred_igusaChart_descent_of_eq_three249 below · cited by 1 · depth 29 - Regularity of jmatĥ or its inverse at centred Igusa data, q=2
ModularCurve.FullLevel.exists_mul_eq_or_inv_mul_eq_of_centred_igusaChart_descent_of_eq_two249 below · cited by 1 · depth 29 - Node ring generates the full-level field over the constants
ModularCurve.FullLevel.exists_mul_mem_eq_sum_smul_of_adjoin_sup_eq_top_of_fractionRing0 below · cited by 2 · depth 29 - Node data at a supersingular place: centre, Igusa end, layers
ModularCurve.FullLevel.exists_nodeCentre_igusaEnd_layeredNodeRings_of_mem_nodes_igusaSep_layerExponent_of_prime917 below · cited by 1 · depth 29 - Node centre, Igusa-side end and layered node rings
ModularCurve.FullLevel.exists_nodeCentre_igusaEnd_layeredNodeRings_of_mem_nodes_igusaSep_layerExponent_of_prime_of_width917 below · cited by 1 · depth 29 - Smoothness of the two-chart model along the Gauss branch, q=3
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_forall_mem_nonunits_gauss_of_forall_not_mem_ssJSet_twoChartIntegralModel_xH_of_isAlgebraic_of_eq_three1,569 below · cited by 1 · depth 29 - Smoothness of the integral model along the Gauss branch, q=2
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_forall_mem_nonunits_gauss_of_forall_not_mem_ssJSet_twoChartIntegralModel_xH_of_isAlgebraic_of_eq_two1,569 below · cited by 1 · depth 29 - Smooth neighbourhood of an ordinary Igusa-branch point, q=3
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel_descent_of_eq_three1,620 below · cited by 1 · depth 29 - Smooth neighbourhood of a good ordinary point, descended model, q=2
ModularCurve.FullLevel.exists_opens_smooth_comp_toBase_of_goodPt_twoChartIntegralModel_descent_of_eq_two1,620 below · cited by 1 · depth 29 - Tate point of the Γ₀(M')-rigidified problem and level automorphisms
ModularCurve.FullLevel.exists_pt_forall_isLevelAutAt_map_eq_act_of_exists_ringHom_gamma0Pow_of_tate330 below · cited by 5 · depth 29 - Shimura reciprocity for slashing by diag(m,1)⁻¹gammadiag(m,1)
ModularCurve.FullLevel.exists_ratCast_slash_conjElemN_eq_sum_exp_pow_smul_of_mem_Gamma019 below · cited by 36 · depth 29 - Igusa component residue field reads in the reduced level-H field
ModularCurve.FullLevel.exists_ringHom_residueField_igusaRing_xHFunctionFieldC_reading1,110 below · cited by 2 · depth 29 - Level-M' reduction through the Igusa branch lies over s
ModularCurve.FullLevel.exists_ringHom_residue_eq_and_mem_iff_of_branchPlace_igusaEnd_igusaBranch802 below · cited by 1 · depth 29 - Every rational place is read at a closed special point (q=3)
ModularCurve.FullLevel.exists_special_closed_branch_reads_of_isRational_twoChartIntegralModel_of_eq_three130 below · cited by 1 · depth 29 - Reading rational Igusa places at closed special points, q=2
ModularCurve.FullLevel.exists_special_closed_branch_reads_of_isRational_twoChartIntegralModel_of_eq_two130 below · cited by 1 · depth 29 - Descended special point with supersingular j lies over a supersingular place
ModularCurve.FullLevel.exists_ssPlace_floorTrace_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent1,285 below · cited by 1 · depth 29 - Supersingular reduction forces a place into a supersingular tube, q=3
ModularCurve.FullLevel.exists_ssTube_of_residue_evalAt_mem_ssJSet_of_eq_three282 below · cited by 1 · depth 29 - Supersingular reduction puts a rational place in a supersingular tube (q=2)
ModularCurve.FullLevel.exists_ssTube_of_residue_evalAt_mem_ssJSet_of_eq_two282 below · cited by 1 · depth 29 - Formally smooth centred subalgebra at a non-supersingular smooth centre, q=3
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgFin_descent_of_eq_three1,117 below · cited by 1 · depth 29 - Formally smooth centred chart subalgebra at a non-supersingular centre (q=2)
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgFin_descent_of_eq_two1,117 below · cited by 1 · depth 29 - Centred formally smooth subalgebra on the pole chart, q=3
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgInf_descent_of_eq_three1,117 below · cited by 1 · depth 29 - Formally smooth centred subalgebra at a smooth pole-chart centre, q=2
ModularCurve.FullLevel.exists_subalgebra_centred_formallySmooth_of_smooth_opens_chartAlgInf_descent_of_eq_two1,117 below · cited by 1 · depth 29 - Supersingular affine chart over the level field, case q=3
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_of_levelField_of_eq_three_of_dvd3,602 below · cited by 1 · depth 29 - Supersingular affine chart over the level field at q=2
ModularCurve.FullLevel.exists_supersingularDVR_affineChart_of_levelField_of_eq_two_of_dvd3,600 below · cited by 1 · depth 29 - Localisations of the descended chart algebra are valuation rings (q=3)
ModularCurve.FullLevel.exists_valuationSubring_localization_chartAlg_of_not_mem_descent_of_eq_three165 below · cited by 1 · depth 29 - Chart-algebra localisations away from varpi₀ are valuation rings (q=2)
ModularCurve.FullLevel.exists_valuationSubring_localization_chartAlg_of_not_mem_descent_of_eq_two165 below · cited by 1 · depth 29 - Place over an Igusa component reads the supersingular place s
ModularCurve.FullLevel.forall_mem_iff_mem_of_place_reads_ssPlace_igusaRing_descent937 below · cited by 1 · depth 29 - Level-M' good-reduction ring lies in O (q=3)
ModularCurve.FullLevel.forall_mem_integers_inclusion_mem_of_algebraMap_mem_iff_of_forall_aeval_mem_of_eq_three945 below · cited by 1 · depth 29 - Level-M' reduction integers lie in every j-adapted ring (q=2)
ModularCurve.FullLevel.forall_mem_integers_inclusion_mem_of_algebraMap_mem_iff_of_forall_aeval_mem_of_eq_two945 below · cited by 1 · depth 29 - No q-torsion and alignment above a supersingular place
ModularCurve.FullLevel.forall_nsmul_eq_zero_and_exists_variableChange_of_over_of_eq_map_classify_rigidDataPow_of_tatePoint2,862 below · cited by 3 · depth 29 - Minimality of the Igusa branch at a crossing node
ModularCurve.FullLevel.igusaBranch_le_of_le_of_mem_maximalIdeal_of_not_mem_of_node_crossingPresentation38 below · cited by 1 · depth 29 - Igusa-side prolongation traces to the Gauss reduction at level M'
ModularCurve.FullLevel.inclusion_mem_integers_iff_mem_constantReduction_integers_of_igusaEnd_igusaBranch3 below · cited by 1 · depth 29 - Uniqueness, composition and triviality of guarded level automorphisms
ModularCurve.FullLevel.isLevelAutAt_unique_mul_one_of_exists_ringHom_of_eq_levelH_inf_ker30 below · cited by 53 · depth 29 - Maximality of the centre of V on an Igusa chart algebra (q=3)
ModularCurve.FullLevel.isMaximal_of_forall_mem_iff_mem_nonunits_of_lt_gaussRing_descent_of_eq_three1,112 below · cited by 3 · depth 29 - Maximality of the centre of V on a chart algebra, q=2
ModularCurve.FullLevel.isMaximal_of_forall_mem_iff_mem_nonunits_of_lt_gaussRing_descent_of_eq_two1,112 below · cited by 3 · depth 29 - Regularity of the special fibre at ordinary points of the Gauss branch
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isMaximal_of_not_mem_ssJSet_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_perfectField1,718 below · cited by 2 · depth 29 - Regularity of the fibre at the Gauss-component generic point
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_isMaximal_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH12 below · cited by 1 · depth 29 - Regularity of the special fibre at ∞-Igusa cusps
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_mem_range_iotaFin_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_embedding_of_isAlgebraic1,655 below · cited by 2 · depth 29 - Regularity of the special fibre at good points, q=3
ModularCurve.FullLevel.isRegularLocalRing_stalk_quotient_span_of_goodPt_twoChartIntegralModel_of_eq_three1,564 below · cited by 2 · depth 29 - Regular special fibre at a good point of the two-chart model (q=2)
ModularCurve.FullLevel.isRegularLocalRing_stalk_quotient_span_of_goodPt_twoChartIntegralModel_of_eq_two1,564 below · cited by 2 · depth 29 - j(q^q) lies in the level q²M' function field
ModularCurve.FullLevel.jqNModC_mem_fieldBar220 below · cited by 6 · depth 29 - Stalks to packages for the q=3 supersingular level model
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_linked_of_eq_three_of_dvd59 below · cited by 1 · depth 29 - From smooth-point stalks to smooth-point packages (q=2)
ModularCurve.FullLevel.klevel_smoothPointPackages_of_smoothPointStalks_linked_of_eq_two_of_dvd59 below · cited by 1 · depth 29 - Layer propagation of smooth-point stalks, q=3
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_linked_of_eq_three_of_dvd72 below · cited by 1 · depth 29 - All-layer smooth-point stalks from base-layer stalks (q=2)
ModularCurve.FullLevel.klevel_smoothPointStalks_of_baseSmoothPointStalks_linked_of_eq_two_of_dvd72 below · cited by 1 · depth 29 - Level automorphisms transport node place-sets and Igusa ends
ModularCurve.FullLevel.levelAut_transport_nodePlaces_igusaEnds_of_nodeCentre_igusaBranch_of_prime79 below · cited by 1 · depth 29 - Level automorphisms permute node places and Igusa ends
ModularCurve.FullLevel.levelAut_transport_nodePlaces_igusaEnds_of_nodeCentre_igusaBranch_of_prime_of_width79 below · cited by 1 · depth 29 - Level automorphisms preserve the j-finite integral chart
ModularCurve.FullLevel.map_mem_chartAlgFin_of_isLevelAutAt85 below · cited by 20 · depth 29 - Invariance of dominating place sets under the arithmetic Galois action
ModularCurve.FullLevel.mem_nodePlaces_iff_smul_mem_of_arithmeticGalois_smul_eq_of_mem_decompositionSubgroup0 below · cited by 3 · depth 29 - Supersingular j-value at a Drinfeld point, q = 3
ModularCurve.FullLevel.mem_ssJSet_of_forall_mem_maximalIdeal_drinfeldRing_mem_chartAlgFin_descent_of_eq_three21 below · cited by 1 · depth 29 - Drinfeld centres have supersingular j-value (q=2)
ModularCurve.FullLevel.mem_ssJSet_of_forall_mem_maximalIdeal_drinfeldRing_mem_chartAlgFin_descent_of_eq_two21 below · cited by 1 · depth 29 - Level-automorphism stabiliser of y counts Aut(E,Cyc)
ModularCurve.FullLevel.natCard_levelAut_stabilizer_eq_natCard_rationalAut_of_moduliPlace2,904 below · cited by 1 · depth 29 - Reading a non-node place forbids supersingular j (q=3)
ModularCurve.FullLevel.not_mem_ssJSet_of_reads_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_three161 below · cited by 1 · depth 29 - At q=2, points reading non-node places avoid supersingular j
ModularCurve.FullLevel.not_mem_ssJSet_of_reads_of_not_mem_igusaNodes_twoChartIntegralModel_of_eq_two161 below · cited by 1 · depth 29 - Good points lie off the other Igusa branches (q=3)
ModularCurve.FullLevel.offBranch_of_goodPt_twoChartIntegralModel_of_eq_three1,585 below · cited by 1 · depth 29 - A good point lies on no other Igusa branch (q=2)
ModularCurve.FullLevel.offBranch_of_goodPt_twoChartIntegralModel_of_eq_two1,585 below · cited by 1 · depth 29 - Per-node conclusion at the K₀-traces of all ends
ModularCurve.FullLevel.pernodeConclusion_traces_of_rigidDescentHyps1,284 below · cited by 1 · depth 29 - Integral cusp-regular level-M' functions lie in the j_ℓ-chart algebra
ModularCurve.FullLevel.qExpand_coeffEmb_mem_chartAlgFin_of_mem_integers_of_cuspRegular_of_not_dvd827 below · cited by 8 · depth 29 - Level-q functions embed in level-qℓ and are level-fixed
ModularCurve.FullLevel.qExpand_mem_and_apply_eq_of_isLevelAutAt_of_mem_Gamma_of_exists_ringHom3 below · cited by 10 · depth 29 - Classifying map's image is the integral closure of A[j]
ModularCurve.FullLevel.range_classify_eq_chartAlgFin_of_jOf_eq_jqNModC_of_exists_ringHom_gamma0Pow2,210 below · cited by 8 · depth 29 - Frobenius pin: ̄ a₀^{ q} equals the moduli value at a supersingular place
ModularCurve.FullLevel.residue_pow_eq_evalAt_of_jqNModC_sub_mem89 below · cited by 1 · depth 29 - Level automorphisms: stabilising Wₜ iff fixing the chart point
ModularCurve.FullLevel.rigidChart_decompositionAut_iff_fixesPoint_linkedScalars0 below · cited by 1 · depth 29 - Exactly n level automorphisms stabilise the exceptional valuation ring
ModularCurve.FullLevel.rigidChart_natCard_decompositionAut_eq_linkedScalars64 below · cited by 1 · depth 29 - Places over a supersingular point: node or residue disc (q=3)
ModularCurve.FullLevel.supersingularProlongation_cover_of_affineChart_nodes_of_baseSmoothPointStalks_of_coeffEmb_mem_of_eq_three_of_dvd52 below · cited by 1 · depth 29 - Rational places over a supersingular point: node or disc (q=2)
ModularCurve.FullLevel.supersingularProlongation_cover_of_affineChart_nodes_of_baseSmoothPointStalks_of_coeffEmb_mem_of_eq_two_of_dvd52 below · cited by 1 · depth 29 - Cusp centre on the pole chart of the descended model
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_chartAlgInf_descent_of_eq_three1,122 below · cited by 1 · depth 29 - Cusp centre on the pole chart: closed ∞-branch point, q=2
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_chartAlgInf_descent_of_eq_two1,122 below · cited by 1 · depth 29 - Centre of a refinement of the Igusa Gauss ring, q=3
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_descent_of_eq_three1,122 below · cited by 1 · depth 29 - Descended Igusa centre is a closed special point (q=2)
ModularCurve.FullLevel.toBase_eq_closedPoint_and_specializes_and_mem_asIdeal_of_centre_descent_of_eq_two1,122 below · cited by 1 · depth 29 - Rational test functions detect the tube of a supersingular place
ModularCurve.FullLevel.tube_of_isRational_of_forall_rational_cuspRegular_evalAt_sub_mem_maximalIdeal799 below · cited by 3 · depth 29 - Level automorphisms stabilise the blow-up centre and chart algebra
ModularCurve.FullLevel.AuxLevel.blowupChart_centre_levelAut_stable_of_eq_adjoin_of_drinfeldChartWitness66 below · cited by 20 · depth 30 - Coefficientwise automorphisms stabilise the blow-up centre J and chart B
ModularCurve.FullLevel.AuxLevel.blowupChart_centre_stable_of_coeffMap_ringEquiv_of_localCentre_stable_of_drinfeldChartWitness23 below · cited by 1 · depth 30 - Level automorphisms act on the Drinfeld fibre through H
ModularCurve.FullLevel.AuxLevel.blowupChart_drinfeldFibre_hAction_of_isLevelAutAt_of_fibrePackage67 below · cited by 2 · depth 30 - Semilinear chart automorphisms act on the Drinfeld fibre through H
ModularCurve.FullLevel.AuxLevel.blowupChart_drinfeldFibre_hAction_of_semilinear_chartAut_of_fibrePackage0 below · cited by 1 · depth 30 - Transitivity and reducedness for the blow-up chart above varpi
ModularCurve.FullLevel.AuxLevel.blowupChart_primes_transitive_reduced_of_levelAut_stable_of_exceptional_eq_span152 below · cited by 1 · depth 30 - Drinfeld chart as a flat, dense extension of the j-chart algebra
ModularCurve.FullLevel.AuxLevel.comap_eq_and_dense_and_flat_drinfeldChartWitness_chartAlgFin9 below · cited by 29 · depth 30 - Branch primes with distinct tangent directions contract to distinct stalk primes
ModularCurve.FullLevel.AuxLevel.comap_ne_comap_of_branchPrime_of_drinfeldChartWitness_of_mem_ssJSet_twoChartIntegralModel3,130 below · cited by 3 · depth 30 - Hasse germ at an end: j - a₀ is a unit times V^e
ModularCurve.FullLevel.AuxLevel.exists_apply_jqNModC_sub_eq_unit_mul_V_pow_of_end_blowupChart_of_moduliHasse_linked309 below · cited by 2 · depth 30 - Blow-up chart surjects onto the Drinfeld coordinate ring
ModularCurve.FullLevel.AuxLevel.exists_blowupChart_ringHom_localBlowupChart_surjective_ker_eq_span_of_dense_of_flat9 below · cited by 5 · depth 30 - Common chart, exceptional stability and valuative cover at the ends
ModularCurve.FullLevel.AuxLevel.exists_commonChart_and_stabilizer_and_valuativeCover_ends_blowupChart_of_drinfeldChartWitness_linked388 below · cited by 1 · depth 30 - Finite-type end chart with pole along the other Igusa components
ModularCurve.FullLevel.AuxLevel.exists_endChart_finiteType_isLocalization_pole_of_end_blowupChart_of_drinfeldChartWitness_linked679 below · cited by 1 · depth 30 - The q+1 ends of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevel.exists_finset_ends_iff_isLocalization_blowupChart_card_eq_of_eq_adjoin_of_drinfeldChartWitness_linked680 below · cited by 1 · depth 30 - Igusa branch through an end separating ends and level-q translates
ModularCurve.FullLevel.AuxLevel.exists_igusaValuationSubring_translateSep_of_mem_ends_blowupChart_of_drinfeldChartWitness_linked694 below · cited by 1 · depth 30 - Restricting level automorphisms along a cyclotomic coefficient map
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_restrict_coeffMap_of_isLevelAutAt31 below · cited by 2 · depth 30 - Cyclic Γ(q)-action of order dividing q+1 on the blow-up chart
ModularCurve.FullLevel.AuxLevel.exists_levelAut_pow_eq_and_forall_eq_pow_blowupChart_of_eq_adjoin_of_drinfeldChartWitness_linked64 below · cited by 1 · depth 30 - An element of C[J/varpiₜ] outside every Igusa valuation ring
ModularCurve.FullLevel.AuxLevel.exists_mem_blowupChart_not_mem_igusaValuation_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart262 below · cited by 1 · depth 30 - Other-orbit pole in the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevel.exists_mem_blowupChart_orbitPole_of_pow_mem_centre_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart66 below · cited by 1 · depth 30 - Powers of vanishing germs lie in (σ₁varpiₜ,X₀,X₁)
ModularCurve.FullLevel.AuxLevel.exists_pow_map_germ_mem_span_of_mem_asIdeal_of_ringEquiv_adicCompletion_stalk1 below · cited by 4 · depth 30 - Drinfeld chart with Hasse datum at a supersingular stalk
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_const_residueField_pow_hasse_of_mem_ssJSet2,292 below · cited by 1 · depth 30 - Drinfeld chart with level, branch and inertia riders
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_levelAut_riders_inertia_of_mem_ssJSet_twoChartIntegralModel3,154 below · cited by 1 · depth 30 - Drinfeld chart at a supersingular point: constants, equivariance, branch
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_levelAut_riders_of_mem_ssJSet_twoChartIntegralModel3,136 below · cited by 2 · depth 30 - Crossing presentation ̂ A[[U,V]]/(UV-varpi^m) at the ends, with diagonal action
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_uvCrossingModel_tangent_of_end_blowupChart_of_drinfeldChartWitness_linked318 below · cited by 1 · depth 30 - Weighted blow-up chart C[J/varpiₜ]: presentation, fibre dimension, exceptional valuation
ModularCurve.FullLevel.AuxLevel.finitePresentation_krullDimLE_exists_exceptionalValuation_blowupChart_of_drinfeldChartWitness179 below · cited by 1 · depth 30 - Fixed field of level automorphisms equals the level-q field
ModularCurve.FullLevel.AuxLevel.forall_isLevelAutAt_apply_eq_iff_exists_eq_qExpand_of_exists_ringHom63 below · cited by 8 · depth 30 - Base change of the Gauss-branch criterion on a Drinfeld chart
ModularCurve.FullLevel.AuxLevel.forall_mem_comap_drinfeldChart_iff_forall_coeff_mem_maximalIdeal_baseChange_of_cyclotomic4 below · cited by 1 · depth 30 - Formal smoothness of the blow-up chart C[J/varpiₜ]
ModularCurve.FullLevel.AuxLevel.formallySmooth_blowupChart_of_drinfeldChartWitness186 below · cited by 1 · depth 30 - Base change of the Drinfeld-chart tame inertia law
ModularCurve.FullLevel.AuxLevel.inertia_drinfeldChart_baseChange_semilinear_linearPart_of_cyclotomicWitness_inertia_of_isAlgClosed123 below · cited by 1 · depth 30 - Normality of the descended special fibre B₀/π₀B₀
ModularCurve.FullLevel.AuxLevel.isIntegrallyClosed_invariants_quotient_of_rigidChart_framed263 below · cited by 1 · depth 30 - Supersingular chart points are closed and carry the uniformiser
ModularCurve.FullLevel.AuxLevel.isMaximal_asIdeal_and_algebraMap_mem_of_mem_ssJSet8 below · cited by 3 · depth 30 - Local structure of the stalk at a supersingular point
ModularCurve.FullLevel.AuxLevel.isNoetherianRing_stalk_and_residue_and_dense_and_mem_iff_of_mem_ssJSet_of_exists_ringHom126 below · cited by 2 · depth 30 - Reducedness of the special fibre of the j-finite chart
ModularCurve.FullLevel.AuxLevel.isReduced_residueField_tensorProduct_chartAlgFin_of_exists_ringHom2,212 below · cited by 1 · depth 30 - Level automorphisms preserve the j-finite chart and fix y
ModularCurve.FullLevel.AuxLevel.levelAut_mem_chartAlgFin_and_sub_mem_of_isLevelAutAt_of_mem_ssJSet_twoChartIntegralModel3,028 below · cited by 6 · depth 30 - Orbit centre generates (σ₁varpiₜ,X₀,X₁) in the Drinfeld chart
ModularCurve.FullLevel.AuxLevel.map_orbitCentre_eq_span_drinfeldChartWitness_of_stabilizes_of_dense149 below · cited by 16 · depth 30 - Valuation subring invariance under level automorphisms fixing its centre
ModularCurve.FullLevel.AuxLevel.mem_iff_apply_mem_valuationSubring_of_isLevelAutAt_of_stabilizes_centre_of_least_prime64 below · cited by 1 · depth 30 - Centre of the exceptional valuation on the blow-up chart
ModularCurve.FullLevel.AuxLevel.mem_maximalIdeal_iff_mem_span_image_of_blowupChart_exceptionalValuation_of_isPrime0 below · cited by 3 · depth 30 - Transcendence of j and finiteness of K over L(j)
ModularCurve.FullLevel.AuxLevel.transcendental_and_finiteDimensional_and_isSeparable_adjoin_of_coe_eq_coeffEmb_jq114 below · cited by 8 · depth 30 - Drinfeld fibre and linked scalars on the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.blowupChart_drinfeldFibre_levelAut_decomposition_linkedScalars_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_moduliHasse_of_dvd150 below · cited by 1 · depth 30 - Linked scalars and tame inertia on the exceptional Drinfeld fibre
ModularCurve.FullLevel.AuxLevelOne.blowupChart_drinfeldFibre_levelAut_linkedScalars_inertia_of_decomposition_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_inertia_of_dvd152 below · cited by 1 · depth 30 - Weighted blow-up chart C[J/varpiₜ] and its exceptional valuation
ModularCurve.FullLevel.AuxLevelOne.exists_blowupChart_eq_adjoin_exceptionalValuation_of_drinfeldChartWitness_of_stalk_drinfeldChart_moduliHasse_of_dvd158 below · cited by 1 · depth 30 - Base change of the j-chart from the cyclotomic constants
ModularCurve.FullLevel.AuxLevelOne.exists_chartAlgFin_tensorProduct_ringEquiv_of_cyclotomicConstants_of_isAlgClosed_of_isPrimitiveRoot_mul_of_dvd2,237 below · cited by 2 · depth 30 - Ends of the blown-up supersingular chart: cyclic decomposition and crossings
ModularCurve.FullLevel.AuxLevelOne.exists_cyclicDecomposition_ends_moduliHasse_igusaSepTranslate_commonChart_cover_blowupChart_linked_of_eq_adjoin_of_drinfeldChartWitness_of_dvd586 below · cited by 1 · depth 30 - A cyclotomic frame inside a q-adic discrete valuation ring
ModularCurve.FullLevel.AuxLevelOne.exists_cyclotomicConstants_of_isPrimitiveRoot_of_pow_eq_mul_of_ne1 below · cited by 2 · depth 30 - Pole clauses for the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.exists_mem_blowupChart_not_mem_igusaValuation_orbitPole_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_of_dvd267 below · cited by 1 · depth 30 - Drinfeld formal chart at a supersingular point over general constants
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_baseChange_of_cyclotomicWitness_of_isAlgClosed_of_isPrimitiveRoot_mul_of_dvd248 below · cited by 2 · depth 30 - Drinfeld chart at a supersingular point with level and inertia riders
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_levelAut_linearPart_inertia_of_mem_ssJSet_of_pow_eq_mul_of_isPrimitiveRoot_mul_of_dvd3,185 below · cited by 1 · depth 30 - Drinfeld chart of the completed stalk at a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,128 below · cited by 1 · depth 30 - Purity of the moduli j-germ on a Drinfeld chart
ModularCurve.FullLevel.AuxLevelOne.exists_sub_const_eq_mk_of_mem_pow_isUnit_homogeneous_drinfeldChart_of_ringEquiv_adicCompletion_stalk_of_isPrimitiveRoot_mul_of_dvd2,209 below · cited by 1 · depth 30 - Deck group of order dividing (ℓ-1)/2 for q=2
ModularCurve.FullLevel.AuxLevelOne.finite_and_natCard_dvd_div_two_of_eq_closure_isLevelAutAt_gamma_of_eq_two_of_dvd33 below · cited by 5 · depth 30 - Level automorphisms generate a group of order dividing ℓ-1
ModularCurve.FullLevel.AuxLevelOne.finite_and_natCard_dvd_of_eq_closure_isLevelAutAt_gamma_of_dvd32 below · cited by 12 · depth 30 - Transport of the semilinear inertia law between two Drinfeld charts
ModularCurve.FullLevel.AuxLevelOne.inertia_drinfeldChart_semilinear_linearPart_transport_of_levelAut_linearPart_of_pow_eq_mul_of_isPrimitiveRoot_mul_of_dvd3,195 below · cited by 1 · depth 30 - Supersingular chart points are closed and contain varpi
ModularCurve.FullLevel.AuxLevelOne.isMaximal_asIdeal_and_algebraMap_mem_of_mem_ssJSet_of_exists_ringHom_of_dvd8 below · cited by 3 · depth 30 - Level automorphisms normalise the Γ(q) subgroup and its fixed field
ModularCurve.FullLevel.AuxLevelOne.mul_mul_inv_mem_and_map_fixedField_of_isLevelAutAt_gamma0_of_dvd32 below · cited by 6 · depth 30 - Drinfeld quotient model of the descended supersingular chart, q=3
ModularCurve.FullLevel.Diamond.AuxLevel.exists_quotField_ringHom_invariants_of_rigidChart_linkedScalars_of_eq_three_of_dvd878 below · cited by 2 · depth 30 - Drinfeld quotient field model of the invariant supersingular chart (q=2)
ModularCurve.FullLevel.Diamond.AuxLevel.exists_quotField_ringHom_invariants_of_rigidChart_linkedScalars_of_eq_two_of_dvd879 below · cited by 2 · depth 30 - Formal smoothness of the descended supersingular chart at q=3
ModularCurve.FullLevel.Diamond.AuxLevel.formallySmooth_invariants_of_rigidChart_linkedScalars_of_eq_three_of_dvd894 below · cited by 1 · depth 30 - Formal smoothness of the descended supersingular chart B₀
ModularCurve.FullLevel.Diamond.AuxLevel.formallySmooth_invariants_of_rigidChart_linkedScalars_of_eq_two_of_dvd895 below · cited by 1 · depth 30 - Mutual integrality of j and j(mathsf q^q) in the level-H₁ field
ModularCurve.FullLevel.Diamond.coeffEmb_jq_mem_and_qExpand_mem_and_mem_chartAlgFin_xHFunctionField190 below · cited by 10 · depth 30 - Cusp-regular integral level-M' functions lie in the j-chart
ModularCurve.FullLevel.Diamond.coeffEmb_mem_chartAlgFin_of_cuspRegular_of_mem_integers827 below · cited by 10 · depth 30 - Fixed field of level automorphisms at q=3
ModularCurve.FullLevel.Diamond.exists_algEquiv_fixedField_coeffMap_eq_of_forall_mem_iff_coeff_of_eq_three_of_dvd66 below · cited by 1 · depth 30 - Fixed field of the level automorphisms for q=2
ModularCurve.FullLevel.Diamond.exists_algEquiv_fixedField_coeffMap_eq_of_forall_mem_iff_coeff_of_eq_two_of_dvd66 below · cited by 1 · depth 30 - Level automorphisms at q=3 descend to the fixed field
ModularCurve.FullLevel.Diamond.exists_map_fixedField_and_apply_eq_levelAutBar_of_isLevelAutAt_of_coeffMap_eq_of_eq_three_of_dvd67 below · cited by 1 · depth 30 - Descent of level automorphisms to F̄ at q=2
ModularCurve.FullLevel.Diamond.exists_map_fixedField_and_apply_eq_levelAutBar_of_isLevelAutAt_of_coeffMap_eq_of_eq_two_of_dvd67 below · cited by 1 · depth 30 - Directed supersingular-fibre dictionary for the Γ₁(ℓ_g)-rigid moduli data
ModularCurve.FullLevel.Diamond.exists_ssFibreDictionary_chartAlgFin_rigidDataGamma1Pow_directedAt2,856 below · cited by 3 · depth 30 - Level-q functions fixed by Γ(q)∩Γ₀(M')-level automorphisms
ModularCurve.FullLevel.Diamond.mem_and_apply_eq_of_isLevelAutAt_of_mem_Gamma_of_eq_levelH_inf_ker0 below · cited by 9 · depth 30 - Level-automorphism stabiliser equals rational automorphism count, q=3
ModularCurve.FullLevel.Diamond.natCard_levelAut_stabilizer_eq_natCard_rationalAut_of_moduliPlace_of_eq_three_of_dvd2,894 below · cited by 1 · depth 30 - Per-node block for traces of ends on K₀ at q=3
ModularCurve.FullLevel.Diamond.pernodeConclusion_traces_of_rigidChart_linkedScalars_of_eq_three_of_dvd1,272 below · cited by 1 · depth 30 - Per-node block for traces of ends, q = 2
ModularCurve.FullLevel.Diamond.pernodeConclusion_traces_of_rigidChart_linkedScalars_of_eq_two_of_dvd1,272 below · cited by 1 · depth 30 - Stretched Γ₀(M') expansions lie in K and are level-fixed
ModularCurve.FullLevel.Diamond.qExpand_mem_and_apply_eq_of_isLevelAutAt_of_mem_gamma0_of_eq_levelH_inf_ker3 below · cited by 8 · depth 30 - Hasse relation: ̄ a₀^{ q}=jmatĥ(s) at a supersingular place
ModularCurve.FullLevel.Diamond.residue_pow_eq_evalAt_of_jqNModC_sub_mem_of_over89 below · cited by 3 · depth 30 - Stabilising the exceptional valuation iff fixing the supersingular point (q=3)
ModularCurve.FullLevel.Diamond.rigidChart_decompositionAut_iff_fixesPoint_linkedScalars_of_eq_three_of_dvd0 below · cited by 1 · depth 30 - Stabilising the exceptional valuation iff fixing the supersingular point (q=2)
ModularCurve.FullLevel.Diamond.rigidChart_decompositionAut_iff_fixesPoint_linkedScalars_of_eq_two_of_dvd0 below · cited by 1 · depth 30 - Exactly n level automorphisms stabilise the exceptional valuation, q=3
ModularCurve.FullLevel.Diamond.rigidChart_natCard_decompositionAut_eq_linkedScalars_of_eq_three_of_dvd31 below · cited by 1 · depth 30 - Exactly n chart-stabilising level automorphisms at q=2
ModularCurve.FullLevel.Diamond.rigidChart_natCard_decompositionAut_eq_linkedScalars_of_eq_two_of_dvd31 below · cited by 1 · depth 30 - Stabiliser of a supersingular point versus Aut(E,C) at q=2
ModularCurve.FullLevel.Diamond.two_mul_natCard_levelAut_stabilizer_eq_natCard_rationalAut_of_moduliPlace_of_eq_two_of_dvd2,894 below · cited by 1 · depth 30 - Transcendence of jmatĥ and finiteness of F₀ over K₀(jmatĥ), q=3
ModularCurve.FullLevel.aeval_eq_zero_imp_and_finiteDimensional_closure_descent_of_eq_three164 below · cited by 1 · depth 30 - Descended level field: jmatĥ transcendental and F₀ finite, q=2
ModularCurve.FullLevel.aeval_eq_zero_imp_and_finiteDimensional_closure_descent_of_eq_two164 below · cited by 1 · depth 30 - Units of the constant reduction: A-polynomials in j at q=3
ModularCurve.FullLevel.aeval_jq_mem_integers_and_inv_mem_of_map_residue_ne_zero_of_eq_three0 below · cited by 2 · depth 30 - Polynomials in j with non-zero reduction are R₀-units (q=2)
ModularCurve.FullLevel.aeval_jq_mem_integers_and_inv_mem_of_map_residue_ne_zero_of_eq_two0 below · cited by 2 · depth 30 - Density of the full-level classifying image at the Tate point
ModularCurve.FullLevel.dense_range_classify_of_jOf_eq_jqNModC_of_exists_ringHom_gamma0Pow_of_finiteType486 below · cited by 1 · depth 30 - Uniqueness of the closed point on an Igusa component over s
ModularCurve.FullLevel.eq_of_isMaximal_of_mem_nonunits_igusaRing_of_floorTrace_chartAlgFin_descent1,570 below · cited by 1 · depth 30 - Maximum principle on a node annulus: values lie in A
ModularCurve.FullLevel.evalAt_mem_of_mem_integers_igusaEnd_of_forall_mem_nodePlaces_of_prime104 below · cited by 2 · depth 30 - Igusa-branch prolongation at a node, with rational branch place
ModularCurve.FullLevel.exists_branchPlace_igusaEnd_integral_overS_of_node_crossingPresentation_igusaBranch_of_prime855 below · cited by 2 · depth 30 - Unique centre place of a crossing-presented node
ModularCurve.FullLevel.exists_centrePlace_ord_residue_eq_one_of_node_crossingPresentation_of_prime48 below · cited by 2 · depth 30 - Descent of a chart element avoiding Igusa-type valuation rings
ModularCurve.FullLevel.exists_invariant_notMem_of_endChartPole_of_rigidChart257 below · cited by 1 · depth 30 - Descended chart ring inside the Drinfeld ring, centre maximal
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_chartAlgFin_descent1,263 below · cited by 2 · depth 30 - Unique maximal ideal over a supersingular place (q=3)
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_unique_chartAlgFin_descent_of_eq_three1,422 below · cited by 1 · depth 30 - Unique chart maximal ideal over a supersingular place (q=2)
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_unique_chartAlgFin_descent_of_eq_two1,419 below · cited by 1 · depth 30 - Layered rational node rings from a node and its two ends
ModularCurve.FullLevel.exists_layeredRationalNodeRings_of_node_ends_layers_of_prime129 below · cited by 2 · depth 30 - Half-unit at a crossing node: cₓ-associate invertible in 𝒪'
ModularCurve.FullLevel.exists_mem_eq_cx_mul_unit_isUnit_of_commonChart_of_igusaSep_of_prime3 below · cited by 2 · depth 30 - One moduli place for all rigid-chart points over s
ModularCurve.FullLevel.exists_place_forall_isModuliPlaceOf_of_over_of_eq_map_classify_rigidDataPow_of_tatePoint2,858 below · cited by 1 · depth 30 - Rational affine place below a maximal ideal of the chart
ModularCurve.FullLevel.exists_place_isRational_floorTrace_of_isMaximal_chartAlgFin_descent1,278 below · cited by 1 · depth 30 - A rational place centred on a node of the full-level curve
ModularCurve.FullLevel.exists_place_mem_toValuationSubring_and_evalAt_mem_maximalIdeal_of_ringEquiv_uvCrossingModel79 below · cited by 2 · depth 30 - Transversal floor prime at elliptic j-values; horizontal unramifiedness elsewhere
ModularCurve.FullLevel.exists_prime_map_sup_span_eq_maximalIdeal_and_isUnramifiedAt_of_map_jChartFin_mem_xH_of_isAlgebraic1,259 below · cited by 1 · depth 30 - Rational decomposition of f∣γ'^{sharp} on Γ_{H_1}(m²M')
ModularCurve.FullLevel.exists_ratCast_slash_conjElemN_eq_sum_exp_pow_smul_of_mem_Gamma0_of_eq_levelH_inf_ker19 below · cited by 5 · depth 30 - Finite-type representability of raw full-level rigid Weierstrass data
ModularCurve.FullLevel.exists_represents_raw_rigidData_gamma0Pow76 below · cited by 1 · depth 30 - Residually transcendental trace along an Igusa branch
ModularCurve.FullLevel.exists_residuallyTranscendental_trace_of_igusaBranch_of_rigidChart307 below · cited by 1 · depth 30 - Coefficientwise identification of the level field over K₀
ModularCurve.FullLevel.exists_ringEquiv_laurentBaseChange_coe_eq_coeffMap_of_forall_mem_iff_forall_coeff_mem35 below · cited by 1 · depth 30 - Igusa component residue field embeds into reduced level-H field, q=3
ModularCurve.FullLevel.exists_ringHom_residueField_igusaRing_xHFunctionFieldC_reading_of_eq_three394 below · cited by 2 · depth 30 - Residue field of an Igusa component embeds, q=2
ModularCurve.FullLevel.exists_ringHom_residueField_igusaRing_xHFunctionFieldC_reading_of_eq_two394 below · cited by 2 · depth 30 - Supersingular special point lies over a supersingular place, q=3
ModularCurve.FullLevel.exists_ssPlace_floorTrace_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent_of_eq_three1,136 below · cited by 1 · depth 30 - Supersingular special points lie over supersingular places, q=2
ModularCurve.FullLevel.exists_ssPlace_floorTrace_of_isMaximal_chartAlgFin_of_mem_ssJSet_descent_of_eq_two1,136 below · cited by 1 · depth 30 - Level automorphisms act on the Tate datum by γ-relabelling
ModularCurve.FullLevel.exists_variableChange_act_mapRing_eq_relabel_of_isLevelAutAt_of_level_fst_gamma0Pow246 below · cited by 5 · depth 30 - Tate raw datum: weight-one twist, cusp levels, j=j(mathsf q^{qℓ})
ModularCurve.FullLevel.exists_variableChange_raw_rigidData_tate_weightOne_level_fst_gamma0Pow286 below · cited by 5 · depth 30 - Igusa-chart reading of a supersingular place, q=3
ModularCurve.FullLevel.forall_mem_iff_mem_of_place_reads_ssPlace_igusaRing_descent_of_eq_three937 below · cited by 1 · depth 30 - Igusa-chart place lies over a supersingular place (q=2)
ModularCurve.FullLevel.forall_mem_iff_mem_of_place_reads_ssPlace_igusaRing_descent_of_eq_two937 below · cited by 1 · depth 30 - Minimality of the Igusa branch at a crossing node
ModularCurve.FullLevel.igusaBranch_le_of_le_of_mem_maximalIdeal_of_not_mem_of_node_crossingPresentation_of_prime38 below · cited by 2 · depth 30 - Chart membership for an R₀-integral, cusp-regular level-M' function
ModularCurve.FullLevel.inclusion_mem_chartAlgFin_of_mem_integers_of_cuspRegular_descent873 below · cited by 1 · depth 30 - Integrality of the full-level moduli ring over A[j₀]
ModularCurve.FullLevel.isIntegral_adjoin_j0_levelModuliPackageAbs_of_isUnit_two_three_gamma0Pow103 below · cited by 12 · depth 30 - Regular special fibre at an ordinary point of the ∞-branch
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_off_section_xH_of_isSeparable1,675 below · cited by 1 · depth 30 - Regular special fibre from unramifiedness in codimension one
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_xH46 below · cited by 1 · depth 30 - Regularity at cusp centres on the Gauss branch of X_H
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_forall_mem_nonunits_gauss_xH_of_isAlgebraic1,653 below · cited by 1 · depth 30 - Regular special fibre at ordinary ∞-branch points, q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isMaximal_of_not_mem_ssJSet_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_perfectField_of_eq_three1,416 below · cited by 2 · depth 30 - Regular special fibre at ordinary ∞-branch points, q=2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isMaximal_of_not_mem_ssJSet_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_perfectField_of_eq_two1,416 below · cited by 2 · depth 30 - Regularity of the varpi-fibre along the Gauss branch, q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_isMaximal_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_eq_three12 below · cited by 1 · depth 30 - Regularity of the Gauss-centred special fibre at q=2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_isMaximal_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_eq_two12 below · cited by 1 · depth 30 - Regularity of the special fibre along the ∞-chart, q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_mem_range_iotaFin_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_embedding_of_isAlgebraic_of_eq_three1,465 below · cited by 2 · depth 30 - Regularity of the special fibre at ∞-branch cusps, q=2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_not_mem_range_iotaFin_of_forall_mem_nonunits_gauss_twoChartIntegralModel_xH_of_embedding_of_isAlgebraic_of_eq_two1,465 below · cited by 2 · depth 30 - Unramifiedness at vertical height-one primes over the X₀(M') floor
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_xH_of_isAlgebraic1,533 below · cited by 1 · depth 30 - Horizontal unramifiedness of the full-level cover away from j=0,1728
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_map_jChartFin_ne_zero_of_ne_1728_xH_of_isAlgebraic119 below · cited by 1 · depth 30 - Level-M' laws along an Igusa branch at a supersingular point
ModularCurve.FullLevel.levelLaws_trace_of_igusaBranch_of_rigidChart1,215 below · cited by 1 · depth 30 - Igusa non-units of the descended chart are Drinfeld-central
ModularCurve.FullLevel.mem_maximalIdeal_drinfeldRing_of_mem_nonunits_igusaRing_chartAlgFin_descent1,269 below · cited by 1 · depth 30 - Integral closedness of the q-expansion image of the moduli ring
ModularCurve.FullLevel.mem_range_of_isIntegral_range_levelModuliPackageAbs_qExpansion_of_isIntegral_of_dense_of_exists_ringHom_gamma0Pow2,176 below · cited by 1 · depth 30 - Supersingular fibre count times #Aut(E,Cyc) equals #SL₂(ℤ/ℓ')
ModularCurve.FullLevel.natCard_isMaximal_over_mul_natCard_rationalAut_eq_natCard_specialLinearGroup_of_moduliPlace2,896 below · cited by 1 · depth 30 - Orbit–stabiliser count for level automorphisms at a supersingular point
ModularCurve.FullLevel.natCard_levelAut_attached_eq_natCard_isMaximal_over_mul_natCard_stabilizer2,871 below · cited by 1 · depth 30 - Level automorphisms attached to Γ(q)∩Γ₀(M') number #SL₂(ℤ/ℓ')
ModularCurve.FullLevel.natCard_levelAut_attached_eq_natCard_specialLinearGroup_zmod414 below · cited by 1 · depth 30 - Per-node conclusion at one end of the descended model
ModularCurve.FullLevel.pernodeConclusion_of_pernodeHyps_of_rigidDescentHyps57 below · cited by 1 · depth 30 - Rigid-level valuation ring reads level-M' constant reductions
ModularCurve.FullLevel.qExpand_coeffEmb_mem_maximalIdeal_iff_residue_eq_zero_of_forall_aeval_jqNModC_mem955 below · cited by 2 · depth 30 - Frobenius pinning at a supersingular place
ModularCurve.FullLevel.residue_pow_eq_evalAt_of_jqNModC_sub_mem_of_over89 below · cited by 1 · depth 30 - Two-chart model of X₀(M') is smooth of relative dimension one
ModularCurve.FullLevel.smoothOfRelativeDimension_one_toBase_twoChartIntegralModel_laurentBaseChange_gamma0_of_not_dvd941 below · cited by 7 · depth 30 - Primes over a supersingular point: trichotomy, and one end per component
ModularCurve.FullLevel.AuxLevel.blowupChart_primes_over_supersingular_exceptional_generic_or_end_on_unique_component_of_drinfeldChartWitness_linked381 below · cited by 3 · depth 31 - Level automorphisms carrying an end into W stabilise W
ModularCurve.FullLevel.AuxLevel.exceptionalValuation_stable_of_end_le_translate_blowupChart_of_drinfeldChartWitness_linked64 below · cited by 1 · depth 31 - Chart map extends to the blow-up algebra C[J/varpiₜ]
ModularCurve.FullLevel.AuxLevel.exists_blowupChart_ringHom_away_extends_chartMap_of_eq_adjoin0 below · cited by 1 · depth 31 - Surjection from the blow-up chart onto the Drinfeld coordinate ring
ModularCurve.FullLevel.AuxLevel.exists_blowupChart_ringHom_coordRing_surjective_ker_eq_span_of_chartMap_of_localFibreMap0 below · cited by 1 · depth 31 - Common level-stable chart at the ends of the blow-up
ModularCurve.FullLevel.AuxLevel.exists_commonChart_ends_blowupChart_of_drinfeldChartWitness_linked167 below · cited by 1 · depth 31 - Crossing presentation at an end: Hasse germ along V
ModularCurve.FullLevel.AuxLevel.exists_crossingPresentation_apply_jqNModC_sub_eq_unit_mul_V_pow_of_end_blowupChart_of_moduliHasse_linked301 below · cited by 1 · depth 31 - Exactly q+1 components through a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_finset_isPrime_lt_supersingular_card_eq_succ_of_drinfeldChartWitness_linked20 below · cited by 2 · depth 31 - Weighted centre: tame exponent, reduction, orbit support and transport
ModularCurve.FullLevel.AuxLevel.exists_finset_prod_pow_le_weightedCentre_and_levelAut_transport_of_drinfeldChartWitness163 below · cited by 7 · depth 31 - Igusa branch through an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevel.exists_igusaValuationSubring_of_mem_ends_blowupChart_of_drinfeldChartWitness_linked413 below · cited by 1 · depth 31 - Level automorphisms translate the ends of the blown-up chart
ModularCurve.FullLevel.AuxLevel.exists_isEnd_blowupChart_map_of_isLevelAutAt_of_mem_valuationSubring_iff_of_drinfeldChartWitness_linked67 below · cited by 1 · depth 31 - Level automorphisms act transitively on branches below a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_mem_valuationSubring_iff_map_isPrime_lt_supersingular_of_drinfeldChartWitness_linked87 below · cited by 1 · depth 31 - Local blow-up chart of a Drinfeld chart presentation
ModularCurve.FullLevel.AuxLevel.exists_localBlowupChart_ringHom_coordRing_of_chartPresentation_of_mem_nonZeroDivisors6 below · cited by 2 · depth 31 - A uniform pole function in Bₓ along all Igusa valuations
ModularCurve.FullLevel.AuxLevel.exists_mem_blowupChart_mem_end_not_mem_igusaValuation_of_end_blowupChart10 below · cited by 1 · depth 31 - Distinct ends of the blown-up supersingular chart are separated
ModularCurve.FullLevel.AuxLevel.exists_not_isUnit_isUnit_of_isEnd_blowupChart_ne_of_drinfeldChartWitness_linked51 below · cited by 1 · depth 31 - Drinfeld chart at a supersingular point: constants and linear parts
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_const_linearPart_of_mem_ssJSet_twoChartIntegralModel2,299 below · cited by 2 · depth 31 - Completed stalk at a supersingular point: regular, with Drinfeld basis
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_regularLocalRing_isDrinfeldBasisAdic_const_hasseParam_of_mem_ssJSet2,281 below · cited by 1 · depth 31 - Crossing presentation at a τ₀-stable end with tangent character
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_uvCrossingModel_tangent_of_isEnd_of_levelAut_mem_iff_blowupChart_of_drinfeldChartWitness_linked316 below · cited by 1 · depth 31 - A finite-type equivariant affine neighbourhood of an end
ModularCurve.FullLevel.AuxLevel.exists_subalgebra_le_blowupChart_inf_end_finiteType_isLocalization_of_end_blowupChart671 below · cited by 1 · depth 31 - Branch prime with tangent X₀ detects mathfrak m_A-integral q-expansions
ModularCurve.FullLevel.AuxLevel.forall_mem_comap_drinfeldChart_iff_forall_coeff_mem_maximalIdeal_of_linearPart_riders_twoChartIntegralModel3,041 below · cited by 1 · depth 31 - Level-q automorphisms fix every Igusa-type valuation ring of K
ModularCurve.FullLevel.AuxLevel.forall_mem_iff_map_mem_igusaValuation_of_isLevelAutAt_gamma_of_drinfeldChartWitness550 below · cited by 3 · depth 31 - Formal smoothness of the generic fibre of the blow-up chart
ModularCurve.FullLevel.AuxLevel.formallySmooth_fiber_bot_blowupChart_of_drinfeldChartWitness14 below · cited by 1 · depth 31 - Formal smoothness of the special fibre of the blow-up chart
ModularCurve.FullLevel.AuxLevel.formallySmooth_fiber_maximalIdeal_blowupChart_of_drinfeldChartWitness181 below · cited by 1 · depth 31 - Tame inertia on the Drinfeld chart: linear part d diag(1,d^{-q})
ModularCurve.FullLevel.AuxLevel.inertia_drinfeldChart_semilinear_linearPart_diagOneElem_of_linearPart_riders_twoChartIntegralModel3,056 below · cited by 1 · depth 31 - Special-fibre components are discrete valuation rings at supersingular points
ModularCurve.FullLevel.AuxLevel.isDiscreteValuationRing_stalk_quotient_of_mem_of_not_isMaximal_of_mem_ssJSet_twoChartIntegralModel3,128 below · cited by 1 · depth 31 - Level automorphism attached to γ ≡ ± 1 (mod qℓ) is trivial
ModularCurve.FullLevel.AuxLevel.levelAut_eq_one_of_map_eq_one_or_eq_neg_one_of_exists_ringHom59 below · cited by 3 · depth 31 - Level automorphisms fix supersingular points of the finite j-chart
ModularCurve.FullLevel.AuxLevel.levelAut_sub_mem_of_isLevelAutAt_of_mem_Gamma_of_mem_ssJSet_twoChartIntegralModel3,027 below · cited by 1 · depth 31 - Faithfulness of the attached level automorphisms mod qℓ
ModularCurve.FullLevel.AuxLevel.map_eq_one_or_eq_neg_one_of_isLevelAutAt_one_of_exists_ringHom408 below · cited by 1 · depth 31 - Level-q automorphisms stabilise primes below a supersingular point
ModularCurve.FullLevel.AuxLevel.mem_iff_map_mem_isPrime_lt_supersingular_of_isLevelAutAt_gamma_of_drinfeldChartWitness_linked551 below · cited by 1 · depth 31 - Regularity of C[J/varpiₜ] at places where j is regular
ModularCurve.FullLevel.AuxLevel.ord_nonneg_of_mem_blowupChart_of_ord_j_nonneg_of_eq_adjoin_of_drinfeldChartWitness_linked1 below · cited by 1 · depth 31 - Transcendence of j and finiteness of K over L(j)
ModularCurve.FullLevel.AuxLevel.transcendental_and_finiteDimensional_and_isSeparable_adjoin_of_coe_eq_coeffEmb_jq_of_charZero114 below · cited by 6 · depth 31 - Valuative cover of the blown-up supersingular chart by its ends
ModularCurve.FullLevel.AuxLevel.valuativeCover_ends_blowupChart_of_drinfeldChartWitness_linked382 below · cited by 1 · depth 31 - Level automorphisms stabilise the blow-up centre and chart
ModularCurve.FullLevel.AuxLevelOne.blowupChart_centre_levelAut_stable_of_eq_adjoin_of_drinfeldChartWitness_of_dvd34 below · cited by 20 · depth 31 - Coefficientwise automorphisms preserve the blow-up centre and chart
ModularCurve.FullLevel.AuxLevelOne.blowupChart_centre_stable_of_coeffMap_ringEquiv_of_localCentre_stable_of_drinfeldChartWitness_of_dvd23 below · cited by 1 · depth 31 - Level automorphisms act on the Drinfeld fibre through H
ModularCurve.FullLevel.AuxLevelOne.blowupChart_drinfeldFibre_hAction_of_isLevelAutAt_of_fibrePackage_of_dvd35 below · cited by 2 · depth 31 - Semilinear chart automorphisms act through `hAction` on the Drinfeld fibre
ModularCurve.FullLevel.AuxLevelOne.blowupChart_drinfeldFibre_hAction_of_semilinear_chartAut_of_fibrePackage_of_dvd0 below · cited by 1 · depth 31 - Transitivity above varpi and reducedness for the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.blowupChart_primes_transitive_reduced_of_levelAut_stable_of_exceptional_eq_span_of_dvd121 below · cited by 1 · depth 31 - Drinfeld-chart reading of the j-finite chart algebra
ModularCurve.FullLevel.AuxLevelOne.comap_eq_and_dense_and_flat_drinfeldChartWitness_chartAlgFin_of_dvd9 below · cited by 30 · depth 31 - Branch primes with distinct tangent lines contract to distinct stalk primes
ModularCurve.FullLevel.AuxLevelOne.comap_ne_comap_of_branchPrime_of_drinfeldChartWitness_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,120 below · cited by 3 · depth 31 - Hasse germ at an end: j-translate equals unit times V^e
ModularCurve.FullLevel.AuxLevelOne.exists_apply_jqNModC_sub_eq_unit_mul_V_pow_of_end_blowupChart_of_moduliHasse_linked_of_dvd278 below · cited by 2 · depth 31 - Special fibre of the blow-up chart is the Drinfeld curve
ModularCurve.FullLevel.AuxLevelOne.exists_blowupChart_ringHom_localBlowupChart_surjective_ker_eq_span_of_dense_of_flat_of_dvd9 below · cited by 5 · depth 31 - Common chart, stabiliser and valuative cover at the ends
ModularCurve.FullLevel.AuxLevelOne.exists_commonChart_and_stabilizer_and_valuativeCover_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd357 below · cited by 1 · depth 31 - End-adapted finite-type chart with poles along the other ends
ModularCurve.FullLevel.AuxLevelOne.exists_endChart_finiteType_isLocalization_pole_of_end_blowupChart_of_drinfeldChartWitness_linked_of_dvd485 below · cited by 1 · depth 31 - The q+1 ends of the blow-up at a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_finset_ends_iff_isLocalization_blowupChart_card_eq_of_eq_adjoin_of_drinfeldChartWitness_linked_of_dvd474 below · cited by 1 · depth 31 - Separating Igusa branch through an end of the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_igusaValuationSubring_translateSep_of_mem_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd488 below · cited by 1 · depth 31 - Descending level automorphisms along a coefficient embedding
ModularCurve.FullLevel.AuxLevelOne.exists_isLevelAutAt_restrict_coeffMap_of_isLevelAutAt_of_isPrimitiveRoot_mul_of_dvd31 below · cited by 2 · depth 31 - Cyclic Γ(q)-action of order dividing q+1 on the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_levelAut_pow_eq_and_forall_eq_pow_blowupChart_of_eq_adjoin_of_drinfeldChartWitness_linked_of_dvd31 below · cited by 1 · depth 31 - An element of C[J/varpiₜ] outside every Igusa-type valuation ring
ModularCurve.FullLevel.AuxLevelOne.exists_mem_blowupChart_not_mem_igusaValuation_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_of_dvd261 below · cited by 1 · depth 31 - Orbit pole in the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.exists_mem_blowupChart_orbitPole_of_pow_mem_centre_of_eq_adjoin_of_drinfeldChartWitness_of_stalk_drinfeldChart_of_dvd33 below · cited by 1 · depth 31 - Drinfeld chart of a supersingular stalk carrying a Hasse datum
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_const_residueField_pow_hasse_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd2,203 below · cited by 1 · depth 31 - Drinfeld local chart with level, branch and inertia equivariance
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_levelAut_riders_inertia_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,146 below · cited by 1 · depth 31 - Drinfeld chart at a supersingular point, with equivariance riders
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_of_levelAut_riders_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,127 below · cited by 2 · depth 31 - Crossing model ̂ A[[U,V]]/(UV-varpi^m) at ends of the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_uvCrossingModel_tangent_of_end_blowupChart_of_drinfeldChartWitness_linked_of_dvd287 below · cited by 1 · depth 31 - Blow-up chart C[J/varpiₜ]: presentation, fibre dimension, exceptional valuation
ModularCurve.FullLevel.AuxLevelOne.finitePresentation_krullDimLE_exists_exceptionalValuation_blowupChart_of_drinfeldChartWitness_of_dvd148 below · cited by 1 · depth 31 - Fixed field of the Γ(q)∩Γ₀(M') level automorphisms on K
ModularCurve.FullLevel.AuxLevelOne.forall_isLevelAutAt_apply_eq_iff_exists_of_exists_ringHom_of_dvd59 below · cited by 8 · depth 31 - Gauss-valuation anchor transported from the cyclotomic witness
ModularCurve.FullLevel.AuxLevelOne.forall_mem_comap_drinfeldChart_iff_forall_coeff_mem_maximalIdeal_baseChange_of_cyclotomic_of_isPrimitiveRoot_mul_of_dvd4 below · cited by 1 · depth 31 - Formal smoothness of the weighted blow-up chart B=C[J/varpiₜ] over A
ModularCurve.FullLevel.AuxLevelOne.formallySmooth_blowupChart_of_drinfeldChartWitness_of_dvd155 below · cited by 1 · depth 31 - Base change of a cyclotomic Drinfeld-chart inertia witness
ModularCurve.FullLevel.AuxLevelOne.inertia_drinfeldChart_baseChange_semilinear_linearPart_of_cyclotomicWitness_inertia_of_isAlgClosed_of_isPrimitiveRoot_mul_of_dvd123 below · cited by 1 · depth 31 - Supersingular points of the j-finite chart are closed
ModularCurve.FullLevel.AuxLevelOne.isMaximal_asIdeal_and_algebraMap_mem_of_mem_ssJSet_of_ringHom_of_dvd8 below · cited by 3 · depth 31 - Noetherian stalk, A-residues and chart density at supersingular points
ModularCurve.FullLevel.AuxLevelOne.isNoetherianRing_stalk_and_residue_and_dense_and_mem_iff_of_mem_ssJSet_of_exists_ringHom_of_dvd126 below · cited by 2 · depth 31 - Level automorphisms preserve the j-chart algebra and fix y
ModularCurve.FullLevel.AuxLevelOne.levelAut_mem_chartAlgFin_and_sub_mem_of_isLevelAutAt_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,018 below · cited by 6 · depth 31 - Orbit centre generates the Drinfeld chart witness ideal
ModularCurve.FullLevel.AuxLevelOne.map_orbitCentre_eq_span_drinfeldChartWitness_of_stabilizes_of_dense_of_dvd118 below · cited by 16 · depth 31 - Level automorphisms preserve the localised valuation subring
ModularCurve.FullLevel.AuxLevelOne.mem_iff_apply_mem_valuationSubring_of_isLevelAutAt_of_stabilizes_centre_of_least_prime_of_dvd32 below · cited by 1 · depth 31 - Centre of the exceptional valuation is yB on the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.mem_maximalIdeal_iff_mem_span_image_of_blowupChart_exceptionalValuation_of_isPrime_of_dvd0 below · cited by 3 · depth 31 - Normality of the descended supersingular special fibre, q=3
ModularCurve.FullLevel.Diamond.AuxLevel.isIntegrallyClosed_invariants_quotient_of_rigidChart_linkedScalars_of_eq_three_of_dvd882 below · cited by 1 · depth 31 - Special fibre of the descended chart is integrally closed (q=2)
ModularCurve.FullLevel.Diamond.AuxLevel.isIntegrallyClosed_invariants_quotient_of_rigidChart_linkedScalars_of_eq_two_of_dvd883 below · cited by 1 · depth 31 - Descent of a chart element avoiding Igusa-type valuation rings
ModularCurve.FullLevel.Diamond.exists_invariant_notMem_of_endChartPole_of_rigidChart_of_eq_levelH_inf_ker89 below · cited by 2 · depth 31 - Representability of the H₁=Γ₀(M')∩Γ₁(ℓ) Weierstrass problem
ModularCurve.FullLevel.Diamond.exists_levelModuliPackageAbs_isIntegral_adjoin_of_isSectionTransport_of_isNoetherianRing_rigidDataH1Pow129 below · cited by 13 · depth 31 - Level automorphisms act on the Tate point by diamond relabelling
ModularCurve.FullLevel.Diamond.exists_pt_forall_isLevelAutAt_map_eq_act_of_exists_ringHom_rigidDataH1Pow_of_tate_pinGamma1350 below · cited by 6 · depth 31 - Residually transcendental trace on the fixed field K₀
ModularCurve.FullLevel.Diamond.exists_residuallyTranscendental_trace_of_igusaBranch_of_rigidChart_of_eq_levelH_inf_ker188 below · cited by 2 · depth 31 - Vanishing q-torsion and line alignment at supersingular places
ModularCurve.FullLevel.Diamond.forall_nsmul_eq_zero_and_exists_variableChange_and_inLine_of_over_of_eq_map_classify_rigidDataH1Pow_of_tatePoint_pinGamma12,852 below · cited by 2 · depth 31 - Reduced special fibre of the j-finite chart algebra of X_{H_1}(q²M')
ModularCurve.FullLevel.Diamond.isReduced_residueField_tensorProduct_chartAlgFin_of_exists_ringHom_of_eq_levelH_inf_ker2,181 below · cited by 1 · depth 31 - Level-M' laws for traces along an Igusa branch
ModularCurve.FullLevel.Diamond.levelLaws_trace_of_igusaBranch_of_rigidChart_of_eq_levelH_inf_ker1,192 below · cited by 2 · depth 31 - Descended per-node crossing data at a supersingular end, q=3
ModularCurve.FullLevel.Diamond.pernodeConclusion_of_pernodeHyps_of_rigidChart_linkedScalars_of_eq_three_of_dvd57 below · cited by 1 · depth 31 - Descended per-node block at one end, q=2
ModularCurve.FullLevel.Diamond.pernodeConclusion_of_pernodeHyps_of_rigidChart_linkedScalars_of_eq_two_of_dvd57 below · cited by 1 · depth 31 - Range of the H₁-classifying map is the finite chart algebra
ModularCurve.FullLevel.Diamond.range_classify_eq_chartAlgFin_of_jOf_eq_jqNModC_of_exists_ringHom_rigidDataH1Pow2,131 below · cited by 7 · depth 31 - Transcendence of j and finiteness of K over L(j)
ModularCurve.FullLevel.Diamond.transcendental_and_finiteDimensional_and_isSeparable_adjoin_of_coe_eq_coeffEmb_jq_of_eq_levelH_inf_ker114 below · cited by 2 · depth 31 - Transported μ_{p^k} kernel has coefficients in the level field
ModularCurve.FullLevel.coeff_kernelVariableChangeDeg_mem_range_of_variableChange_cuspData_xP_mem_range_gamma0Pow58 below · cited by 2 · depth 31 - Uniqueness of the closed point over a floor place, q=3
ModularCurve.FullLevel.eq_of_isMaximal_of_mem_nonunits_igusaRing_of_floorTrace_chartAlgFin_descent_of_eq_three1,413 below · cited by 1 · depth 31 - Uniqueness on an Igusa component over a supersingular place, q=2
ModularCurve.FullLevel.eq_of_isMaximal_of_mem_nonunits_igusaRing_of_floorTrace_chartAlgFin_descent_of_eq_two1,410 below · cited by 1 · depth 31 - Unique transversal prime through a j=c point on the Γ₀(M') chart
ModularCurve.FullLevel.existsUnique_prime_le_jChartFin_sub_mem_map_sup_span_eq_maximalIdeal_chartAlgFin_gamma0_of_not_dvd1,256 below · cited by 1 · depth 31 - Level-field node ring as a base rational node ring input
ModularCurve.FullLevel.exists_baseRationalNodeRing_input_of_node_ends_nodePlaces_of_prime80 below · cited by 2 · depth 31 - Branch place on the Igusa side of a node
ModularCurve.FullLevel.exists_branchPlace_of_igusaEnd_of_node_crossingPresentation_igusaBranch_of_prime48 below · cited by 1 · depth 31 - Branches over O₀' are Gauss conjugates, unramified over O₀
ModularCurve.FullLevel.exists_comap_eq_and_ramificationIdx_eq_one_and_isSeparable_of_over_gauss_gamma0_mul_xH1,111 below · cited by 3 · depth 31 - Descended chart ring inside the Drinfeld ring, q=3
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_chartAlgFin_descent_of_eq_three1,114 below · cited by 2 · depth 31 - Descended chart algebra inside the Drinfeld ring, q=2
ModularCurve.FullLevel.exists_isMaximal_mem_iff_mem_maximalIdeal_drinfeldRing_chartAlgFin_descent_of_eq_two1,114 below · cited by 2 · depth 31 - One layer of the layered node rings at a node of X(q²M')
ModularCurve.FullLevel.exists_layerRationalNodeRing_of_baseRationalNodeRing_layer_chart_of_prime127 below · cited by 1 · depth 31 - Galois translate of the Tate point relabels its Drinfeld pair
ModularCurve.FullLevel.exists_level_snd_snd_act_mapRing_eq_relabel_gamma0Pow230 below · cited by 1 · depth 31 - Diamond operators act transitively on primes of the pole chart
ModularCurve.FullLevel.exists_mulSemiringAction_chartAlgInf_isPrime_conj_of_forall_mem_floor_iff_xH242 below · cited by 1 · depth 31 - Cyclic tame inertia for the Borel action on the j-chart
ModularCurve.FullLevel.exists_mulSemiringAction_isInvariant_chartAlgFin_isCyclic_inertia_of_not_mem_ssJSet_xH1,578 below · cited by 1 · depth 31 - Rational affine floor place below a maximal ideal of the jmatĥ-chart (q=3)
ModularCurve.FullLevel.exists_place_isRational_floorTrace_of_isMaximal_chartAlgFin_descent_of_eq_three1,129 below · cited by 1 · depth 31 - Floor place under a maximal ideal of the jmatĥ-chart, q=2
ModularCurve.FullLevel.exists_place_isRational_floorTrace_of_isMaximal_chartAlgFin_descent_of_eq_two1,129 below · cited by 1 · depth 31 - Divisibility of Igusa-branch functions by cₓ at a node
ModularCurve.FullLevel.exists_pow_eq_mul_nonunit_of_mem_maximalIdeal_igusaBranch_of_node_crossingPresentation_of_prime2 below · cited by 1 · depth 31 - Level automorphisms act on the Tate point by relabelling
ModularCurve.FullLevel.exists_pt_forall_isLevelAutAt_map_eq_act_of_exists_ringHom_gamma0Pow330 below · cited by 1 · depth 31 - Igusa branch reduction factors through level M' above s
ModularCurve.FullLevel.exists_ringHom_residue_eq_and_mem_iff_of_branchPlace_igusaEnd_igusaBranch_of_prime802 below · cited by 1 · depth 31 - Supersingular fibre dictionary with automorphism count at s
ModularCurve.FullLevel.exists_ssFibreDictionary_autCount_chartAlgFin_rigidDataPow2,889 below · cited by 1 · depth 31 - A unit μ with ⟨μ,0,0,0⟩·τ_*x having the curve of x
ModularCurve.FullLevel.exists_units_curve_act_mapRing_eq_of_isLevelAutAt_gamma0Pow147 below · cited by 1 · depth 31 - Weight-one change of variables: Tate curve and cusps over K
ModularCurve.FullLevel.exists_variableChange_weightOne_tateBase_mem_laurentBaseChange_and_cuspData_mem_of_exists_ringHom114 below · cited by 2 · depth 31 - Degree bound over L(j(q^N)) at level N²M'
ModularCurve.FullLevel.finiteDimensional_and_finrank_adjoin_le_index_of_coe_eq_jqNModC_of_eq_laurentBaseChange_levelH175 below · cited by 1 · depth 31 - Ordinary special points: valuation rings over O₀ lie over O₀'
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_not_mem_ssJSet_xH_of_isAlgebraic1,032 below · cited by 2 · depth 31 - Closed points above a supersingular place carry no q-torsion
ModularCurve.FullLevel.forall_nsmul_eq_zero_of_over_of_eq_map_classify_rigidDataPow104 below · cited by 3 · depth 31 - Cusp-regular R₀-integral functions lie in the jmatĥ-chart, q=3
ModularCurve.FullLevel.inclusion_mem_chartAlgFin_of_mem_integers_of_cuspRegular_descent_of_eq_three873 below · cited by 1 · depth 31 - Chart membership for cusp-regular descended functions, q=2
ModularCurve.FullLevel.inclusion_mem_chartAlgFin_of_mem_integers_of_cuspRegular_descent_of_eq_two873 below · cited by 1 · depth 31 - Igusa-branch prolongation traces to the Gauss reduction at level M'
ModularCurve.FullLevel.inclusion_mem_integers_iff_mem_constantReduction_integers_of_igusaEnd_igusaBranch_of_prime3 below · cited by 1 · depth 31 - Igusa bound: [±Γ_H: SL₂(ℤ)] index bounds [T:L(j)]
ModularCurve.FullLevel.index_le_finrank_adjoin_jOf_of_transcendental_jOf_rigidDataPow411 below · cited by 1 · depth 31 - Identity level automorphism forces d_γ ≡ 1 (mod ℓ_g)
ModularCurve.FullLevel.intCast_apply_eq_one_of_isLevelAutAt_one_of_eq_levelH_inf_ker4 below · cited by 1 · depth 31 - Components of the full-level moduli ring are normal
ModularCurve.FullLevel.isIntegrallyClosed_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_gamma0Pow2,174 below · cited by 1 · depth 31 - Sign invariance of `IsLevelAutAt` in γ
ModularCurve.FullLevel.isLevelAutAt_neg_iff0 below · cited by 4 · depth 31 - Components of the full-level moduli ring have reduced special fibre
ModularCurve.FullLevel.isReduced_residueField_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_gamma0Pow2,167 below · cited by 2 · depth 31 - Regular special fibre at an ordinary point, q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_xH_of_eq_three46 below · cited by 1 · depth 31 - Regular special fibre at ordinary points, q=2 case
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_xH_of_eq_two46 below · cited by 1 · depth 31 - Regularity on the ∞-Igusa branch of X_H(q²M') for q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_forall_mem_nonunits_gauss_xH_of_isAlgebraic_of_eq_three1,463 below · cited by 1 · depth 31 - Regularity at ∞-branch cusp centres of X_H(q²M'), q=2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_forall_mem_nonunits_gauss_xH_of_isAlgebraic_of_eq_two1,463 below · cited by 1 · depth 31 - Regularity of the 1/j-chart at cusps off the floor cusp ∞
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_not_forall_mem_floor_iff_xH_of_isAlgebraic1,628 below · cited by 1 · depth 31 - Regularity of the pole chart of X_H(q²M') at ∞
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_cuspInfty_xH346 below · cited by 1 · depth 31 - Transport of pole-chart regularity along automorphisms fixing j
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_of_algEquiv_apply_eq_xH0 below · cited by 1 · depth 31 - Change of floor X₀(qM')→ X₀(M') unramified at an ordinary point
ModularCurve.FullLevel.isUnramifiedAt_chartAlgFin_gamma0_mul_comap_of_not_mem_ssJSet_xH1,170 below · cited by 1 · depth 31 - Unramifiedness at vertical height-one primes over the floor, q=3
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_xH_of_isAlgebraic_of_eq_three1,383 below · cited by 1 · depth 31 - Unramified over X₀(M') at ordinary Igusa-branch height-one primes, q=2
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_xH_of_isAlgebraic_of_eq_two1,383 below · cited by 1 · depth 31 - Horizontal unramifiedness over the Γ₀(M') floor, q=3
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_map_jChartFin_ne_zero_of_ne_1728_xH_of_isAlgebraic_of_eq_three119 below · cited by 1 · depth 31 - Horizontal unramifiedness over the Γ₀(M') floor, case q=2
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_map_jChartFin_ne_zero_of_ne_1728_xH_of_isAlgebraic_of_eq_two119 below · cited by 1 · depth 31 - Minimal primes of the full-level moduli ring are conjugate
ModularCurve.FullLevel.ker_classify_mem_minimalPrimes_and_forall_exists_algEquiv_comap_eq_gamma0Pow2,100 below · cited by 2 · depth 31 - Kernel of the q-expansion map is a minimal prime
ModularCurve.FullLevel.ker_mem_minimalPrimes_of_levelModuliPackageAbs_qExpansion_of_dense_of_exists_ringHom_gamma0Pow0 below · cited by 1 · depth 31 - Valuative criterion over a DVR for the full-level moduli ring
ModularCurve.FullLevel.levelModuliPackageAbs_apply_mem_valuationSubring_of_isDiscreteValuationRing_of_j0_mem_of_isUnit_two_three_gamma0Pow94 below · cited by 1 · depth 31 - Level automorphisms fix the Γ₀-slot at the Tate point
ModularCurve.FullLevel.level_fst_act_mapRing_eq_of_curve_eq_units_of_level_fst_gamma0Pow128 below · cited by 1 · depth 31 - Level-ℓ slot of the twisted τ-transport is the γ-relabelling
ModularCurve.FullLevel.level_snd_fst_act_mapRing_eq_relabel_gamma0Pow141 below · cited by 1 · depth 31 - Igusa non-units lie in each Drinfeld maximal ideal (q=3)
ModularCurve.FullLevel.mem_maximalIdeal_drinfeldRing_of_mem_nonunits_igusaRing_chartAlgFin_descent_of_eq_three1,120 below · cited by 1 · depth 31 - Igusa components pass through every Drinfeld point, q=2
ModularCurve.FullLevel.mem_maximalIdeal_drinfeldRing_of_mem_nonunits_igusaRing_chartAlgFin_descent_of_eq_two1,120 below · cited by 1 · depth 31 - Equal floor readings and supersingular fibre force equal Γ₀(M')-class
ModularCurve.FullLevel.moduliPoint_mk_eq_of_forall_apply_qExpand_eq_of_eq_map_classify_rigidDataPow_of_tatePoint2,857 below · cited by 1 · depth 31 - Unramifiedness of X_H(q²M')→ X₀(M') off j=0,1728
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_eq_zero_of_ord_sub_eq_zero_xH_levelH111 below · cited by 3 · depth 31 - Level automorphism determined on ratios of modular forms
ModularCurve.FullLevel.AuxLevel.apply_eq_of_isLevelAutAt_of_coeffMap_mul_qExpansion_slash_eq63 below · cited by 3 · depth 32 - Inertial coefficientwise automorphisms preserve the j-chart and fix y
ModularCurve.FullLevel.AuxLevel.coeffMap_mem_chartAlgFin_and_sub_mem_asIdeal_of_drinfeldChartWitness_anchor_of_mem_ssJSet_twoChartIntegralModel10 below · cited by 1 · depth 32 - Independent tangent directions give distinct contracted branch primes
ModularCurve.FullLevel.AuxLevel.comap_ne_comap_of_branchPrime_of_exists_ne_of_drinfeldChartWitness_riders_twoChartIntegralModel193 below · cited by 1 · depth 32 - Two ends of the blow-up chart on one component coincide
ModularCurve.FullLevel.AuxLevel.end_blowupChart_eq_of_on_same_component_of_drinfeldChartWitness_linked299 below · cited by 1 · depth 32 - Ends of the blow-up chart are incomparable
ModularCurve.FullLevel.AuxLevel.eq_of_forall_mem_of_isEnd_blowupChart_of_drinfeldChartWitness_linked50 below · cited by 2 · depth 32 - End stabiliser acting on the UV-crossing model
ModularCurve.FullLevel.AuxLevel.exists_algEquiv_uvCrossingModel_compat_levelAut_pow_eq_of_isEnd_of_levelAut_mem_iff_blowupChart_of_drinfeldChartWitness_linked39 below · cited by 1 · depth 32 - Integrality of q-expansion coefficients on the finite chart
ModularCurve.FullLevel.AuxLevel.exists_coeff_eq_algebraMap_of_mem_chartAlgFin1 below · cited by 1 · depth 32 - Common level-stable finite-type chart dominated by all ends
ModularCurve.FullLevel.AuxLevel.exists_commonChart_ends_of_invariant_section_blowupChart_of_drinfeldChartWitness_linked74 below · cited by 1 · depth 32 - Blown-up chart has an end on each component through y
ModularCurve.FullLevel.AuxLevel.exists_end_blowupChart_on_component_of_isPrime_lt_supersingular_of_drinfeldChartWitness_linked229 below · cited by 1 · depth 32 - Igusa branch valuation ring at an end of the blow-up
ModularCurve.FullLevel.AuxLevel.exists_igusaBranch_valuationSubring_of_mem_ends_crossing_linked354 below · cited by 1 · depth 32 - Igusa valuation centre lies strictly inside a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_isMaximal_mem_ssJSet_centre_le_of_igusaValuation45 below · cited by 2 · depth 32 - Primes of the j-finite chart lift to the Drinfeld chart
ModularCurve.FullLevel.AuxLevel.exists_isPrime_comap_drinfeldChart_eq_of_isPrime_le_ne_twoChartIntegralModel122 below · cited by 1 · depth 32 - Primes below a supersingular point come from the Drinfeld chart
ModularCurve.FullLevel.AuxLevel.exists_isPrime_forall_mem_iff_germ_mem_comap_of_le_of_ne_of_drinfeldChartWitness10 below · cited by 5 · depth 32 - Residually non-trivial constants for θ₀ on the crossing model
ModularCurve.FullLevel.AuxLevel.exists_isUnit_iterate_apply_V_mul_sub_const_mem_sq_of_ringEquiv_compat_levelAut_of_isEnd_blowupChart_of_drinfeldChartWitness_linked82 below · cited by 1 · depth 32 - Moduli reading of a supersingular point of the integral model
ModularCurve.FullLevel.AuxLevel.exists_levelModuliPackageAbs_gamma0Pow_ringEquiv_adicCompletion_stalk_const_of_mem_ssJSet2,261 below · cited by 1 · depth 32 - Primes under a supersingular point as Γ(ℓ)-translates of the Gauss prime
ModularCurve.FullLevel.AuxLevel.exists_mem_Gamma_isLevelAutAt_forall_mem_iff_coeff_mem_maximalIdeal_of_le_of_ne_of_drinfeldChartWitness86 below · cited by 5 · depth 32 - Modular forms realising Tate cusp coordinates and c₄,c₆
ModularCurve.FullLevel.AuxLevel.exists_modularForm_mul_qExpansion_eq_cuspPoint_and_slash_conjElemN_eq122 below · cited by 4 · depth 32 - Toric generator-kernel coefficients as q-expansions of modular forms
ModularCurve.FullLevel.AuxLevel.exists_modularForm_qExpansion_eq_coeff_toricGenKernel_and_slash_conjElemN_eq4 below · cited by 1 · depth 32 - Invariant element of Jⁿ generating Jⁿ at every end
ModularCurve.FullLevel.AuxLevel.exists_pow_mem_invariant_generates_ends_blowupChart_of_drinfeldChartWitness_linked165 below · cited by 1 · depth 32 - Primes over varpi at an end of the blow-up chart
ModularCurve.FullLevel.AuxLevel.exists_primes_exceptional_igusa_trichotomy_of_end_blowupChart_linked306 below · cited by 4 · depth 32 - Supersingular moduli chart with Drinfeld basis and level action
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_adicCompletion_stalk_regularLocalRing_isDrinfeldBasisAdic_const_hasseParam_levelAut_linearPart_of_mem_ssJSet2,295 below · cited by 1 · depth 32 - Semilinear chart automorphism at a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_ringEquiv_drinfeldChart_semilinear_of_coeffMap_of_sub_mem_asIdeal_of_drinfeldChartWitness_const_twoChartIntegralModel132 below · cited by 1 · depth 32 - Completed end ring of blown-up supersingular chart as crossing model
ModularCurve.FullLevel.AuxLevel.exists_ringHom_ringEquiv_adicCompletion_uvCrossingModel_tangent_coords_of_end_blowupChart_of_drinfeldChartWitness_linked298 below · cited by 3 · depth 32 - A cyclotomic sub-DVR with residue degree data and ξ integral
ModularCurve.FullLevel.AuxLevel.exists_subDVR_isFractionRing_adjoin_of_isPrimitiveRoot1 below · cited by 2 · depth 32 - Two special-fibre components through a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_two_primes_chartAlgFin_le_asIdeal_of_mem_ssJSet_twoChartIntegralModel3,046 below · cited by 1 · depth 32 - Vertical unit on the blown-up supersingular chart adapted to an end
ModularCurve.FullLevel.AuxLevel.exists_verticalUnit_pow_centre_of_end_blowupChart669 below · cited by 1 · depth 32 - Finiteness of residually transcendental valuation subrings over A
ModularCurve.FullLevel.AuxLevel.finite_setOf_igusaValuation_of_drinfeldChartWitness9 below · cited by 1 · depth 32 - Residual transcendence of j(mathsf q^{qℓ}) along the Igusa branch
ModularCurve.FullLevel.AuxLevel.forall_aeval_jqNModC_mem_maximalIdeal_of_igusaBranch_linked88 below · cited by 1 · depth 32 - Unipotent level automorphisms preserve coefficients in mathfrak m_A
ModularCurve.FullLevel.AuxLevel.forall_coeff_mem_maximalIdeal_iff_of_isLevelAutAt_T_zpow_inv_of_exists_ringHom0 below · cited by 1 · depth 32 - Level automorphisms at Γ(q)∩Γ₀(M') preserve the Gauss prime
ModularCurve.FullLevel.AuxLevel.forall_coeff_mem_maximalIdeal_iff_of_isLevelAutAt_gamma_of_drinfeldChartWitness495 below · cited by 1 · depth 32 - Distinct ends are not contained in the Igusa branch Wₓ
ModularCurve.FullLevel.AuxLevel.forall_ends_exists_not_mem_igusaBranch_of_mem_ends_linked383 below · cited by 1 · depth 32 - Tame inertia acts on the Drinfeld chart by diag(d,1)
ModularCurve.FullLevel.AuxLevel.inertia_drinfeldChart_linearPart_diagOneElem_of_semilinear_of_drinfeldChartWitness_linearPart_riders_twoChartIntegralModel3,049 below · cited by 1 · depth 32 - Discreteness and localisation of valuation subrings containing the j-chart algebra
ModularCurve.FullLevel.AuxLevel.isDiscreteValuationRing_and_mem_iff_of_igusaValuation_of_drinfeldChartWitness125 below · cited by 2 · depth 32 - Blow-up charts: primes over the supersingular point are maximal
ModularCurve.FullLevel.AuxLevel.isMaximal_of_isPrime_of_supersingular_le_of_div_mem_blowupChart_of_drinfeldChartWitness_linked165 below · cited by 1 · depth 32 - Chart functions with mathfrak m_A-integral q-expansion vanish at a supersingular point
ModularCurve.FullLevel.AuxLevel.mem_of_forall_coeff_mem_maximalIdeal_of_mem_ssJSet_twoChartIntegralModel3,028 below · cited by 1 · depth 32 - Trichotomy and end classification on the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.blowupChart_primes_over_supersingular_exceptional_generic_or_end_on_unique_component_of_drinfeldChartWitness_linked_of_dvd350 below · cited by 3 · depth 32 - Ends carried into W force τ-stability of W
ModularCurve.FullLevel.AuxLevelOne.exceptionalValuation_stable_of_end_le_translate_blowupChart_of_drinfeldChartWitness_linked_of_dvd31 below · cited by 1 · depth 32 - Extension of the chart map to the affine blow-up algebra
ModularCurve.FullLevel.AuxLevelOne.exists_blowupChart_ringHom_away_extends_chartMap_of_eq_adjoin_of_dvd0 below · cited by 1 · depth 32 - Blow-up chart surjects onto Drinfeld coordinate ring, kernel yB
ModularCurve.FullLevel.AuxLevelOne.exists_blowupChart_ringHom_coordRing_surjective_ker_eq_span_of_chartMap_of_localFibreMap_of_dvd0 below · cited by 1 · depth 32 - A common level-stable chart dominating the ends of the blow-up
ModularCurve.FullLevel.AuxLevelOne.exists_commonChart_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd136 below · cited by 1 · depth 32 - Crossing presentation at an end off the exceptional chart
ModularCurve.FullLevel.AuxLevelOne.exists_crossingPresentation_apply_jqNModC_sub_eq_unit_mul_V_pow_of_end_blowupChart_of_moduliHasse_linked_of_dvd270 below · cited by 1 · depth 32 - Exactly q+1 special-fibre branches through a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_finset_isPrime_lt_supersingular_card_eq_succ_of_drinfeldChartWitness_linked_of_dvd20 below · cited by 2 · depth 32 - Tame exponent and finite orbit support of the weighted blow-up centre
ModularCurve.FullLevel.AuxLevelOne.exists_finset_prod_pow_le_weightedCentre_and_levelAut_transport_of_drinfeldChartWitness_of_dvd132 below · cited by 7 · depth 32 - Igusa branch through an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.exists_igusaValuationSubring_of_mem_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd382 below · cited by 1 · depth 32 - Level automorphisms transport ends of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.exists_isEnd_blowupChart_map_of_isLevelAutAt_of_mem_valuationSubring_iff_of_drinfeldChartWitness_linked_of_dvd35 below · cited by 1 · depth 32 - Level automorphisms act transitively on branches at a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_isLevelAutAt_mem_valuationSubring_iff_map_isPrime_lt_supersingular_of_drinfeldChartWitness_linked_of_dvd55 below · cited by 1 · depth 32 - Uniform element of Bₓ avoiding all admissible valuation subrings
ModularCurve.FullLevel.AuxLevelOne.exists_mem_blowupChart_mem_end_not_mem_igusaValuation_of_end_blowupChart_of_dvd10 below · cited by 1 · depth 32 - Distinct blow-up ends at a supersingular point are separated
ModularCurve.FullLevel.AuxLevelOne.exists_not_isUnit_isUnit_of_isEnd_blowupChart_ne_of_drinfeldChartWitness_linked_of_dvd51 below · cited by 1 · depth 32 - Drinfeld chart at a supersingular point with equivariant linear parts
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_drinfeldChart_const_linearPart_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd2,258 below · cited by 2 · depth 32 - Regular stalk with Drinfeld basis and Hasse parameter, all primes
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_regularLocalRing_isDrinfeldBasisAdic_const_hasseParam_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd2,189 below · cited by 1 · depth 32 - Crossing presentation at a τ₀-stable end of the blown-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_uvCrossingModel_tangent_of_isEnd_of_levelAut_mem_iff_blowupChart_of_drinfeldChartWitness_linked_of_dvd285 below · cited by 1 · depth 32 - Finite-type stable subalgebra localising to an end of the blow-up
ModularCurve.FullLevel.AuxLevelOne.exists_subalgebra_le_blowupChart_inf_end_finiteType_isLocalization_of_end_blowupChart_of_dvd479 below · cited by 1 · depth 32 - The (1,0)-branch prime is cut out by vanishing q-expansions
ModularCurve.FullLevel.AuxLevelOne.forall_mem_comap_drinfeldChart_iff_forall_coeff_mem_maximalIdeal_of_linearPart_riders_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd2,893 below · cited by 1 · depth 32 - Level-q automorphisms stabilise residually transcendental valuation subrings
ModularCurve.FullLevel.AuxLevelOne.forall_mem_iff_map_mem_igusaValuation_of_isLevelAutAt_gamma_of_drinfeldChartWitness_of_dvd324 below · cited by 3 · depth 32 - Generic fibre of blow-up chart C[J/varpiₜ] is formally smooth
ModularCurve.FullLevel.AuxLevelOne.formallySmooth_fiber_bot_blowupChart_of_drinfeldChartWitness_of_dvd14 below · cited by 1 · depth 32 - Formal smoothness of the special fibre of C[J/varpiₜ]
ModularCurve.FullLevel.AuxLevelOne.formallySmooth_fiber_maximalIdeal_blowupChart_of_drinfeldChartWitness_of_dvd150 below · cited by 1 · depth 32 - Inertial semilinear action on the Drinfeld chart with diagonal linear part
ModularCurve.FullLevel.AuxLevelOne.inertia_drinfeldChart_semilinear_linearPart_diagOneElem_of_linearPart_riders_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,047 below · cited by 1 · depth 32 - Supersingular stalk quotients are discrete valuation rings
ModularCurve.FullLevel.AuxLevelOne.isDiscreteValuationRing_stalk_quotient_of_mem_of_not_isMaximal_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,118 below · cited by 1 · depth 32 - Level automorphism acts trivially at a supersingular chart point
ModularCurve.FullLevel.AuxLevelOne.levelAut_sub_mem_of_isLevelAutAt_of_mem_Gamma_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,017 below · cited by 1 · depth 32 - Level-q automorphisms fix every prime below a supersingular point
ModularCurve.FullLevel.AuxLevelOne.mem_iff_map_mem_isPrime_lt_supersingular_of_isLevelAutAt_gamma_of_drinfeldChartWitness_linked_of_dvd325 below · cited by 1 · depth 32 - Regularity of the blow-up chart at places integral for j
ModularCurve.FullLevel.AuxLevelOne.ord_nonneg_of_mem_blowupChart_of_ord_j_nonneg_of_eq_adjoin_of_drinfeldChartWitness_linked_of_dvd1 below · cited by 1 · depth 32 - Valuative cover of the blown-up supersingular chart by B and its ends
ModularCurve.FullLevel.AuxLevelOne.valuativeCover_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd351 below · cited by 1 · depth 32 - Dense image of the H₁ classifying map at j(q^q)
ModularCurve.FullLevel.Diamond.dense_range_classify_of_jOf_eq_jqNModC_of_exists_ringHom_rigidDataH1Pow_of_finiteType492 below · cited by 3 · depth 32 - A single moduli place above a supersingular place, H₁ level
ModularCurve.FullLevel.Diamond.exists_place_forall_isModuliPlaceOf_of_over_of_eq_map_classify_rigidDataH1Pow_of_tatePoint_pinGamma12,847 below · cited by 1 · depth 32 - Representability of raw H₁-level data over A
ModularCurve.FullLevel.Diamond.exists_represents_raw_rigidDataH1Pow71 below · cited by 1 · depth 32 - Level automorphism at γ⁻¹ realises the diamond relabelling
ModularCurve.FullLevel.Diamond.exists_variableChange_act_mapRing_eq_relabel_of_isLevelAutAt_of_level_fst_rigidDataH1Pow252 below · cited by 3 · depth 32 - Tate point of the H₁ moduli problem over K
ModularCurve.FullLevel.Diamond.exists_variableChange_raw_rigidData_tate_weightOne_level_fst_rigidDataH1Pow300 below · cited by 2 · depth 32 - Integrality of the H₁-moduli ring over A[j₀]
ModularCurve.FullLevel.Diamond.isIntegral_adjoin_j0_levelModuliPackageAbs_rigidDataH1Pow99 below · cited by 12 · depth 32 - Reduced special fibre on each component of the H₁ moduli ring
ModularCurve.FullLevel.Diamond.isReduced_residueField_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_rigidDataH1Pow2,116 below · cited by 2 · depth 32 - Every minimal prime is a j-fixing translate of the Tate kernel
ModularCurve.FullLevel.Diamond.ker_classify_mem_minimalPrimes_and_forall_exists_algEquiv_comap_eq_rigidDataH1Pow1,974 below · cited by 2 · depth 32 - Integral closedness of the q-expansion image, Γ₁(ℓ_g) edition
ModularCurve.FullLevel.Diamond.mem_range_of_isIntegral_range_levelModuliPackageAbs_qExpansion_of_isIntegral_of_dense_of_exists_ringHom_rigidDataH1Pow2,124 below · cited by 1 · depth 32 - Level automorphisms move the Gauss ring by P¹(𝔽_q)-translation
ModularCurve.FullLevel.comap_gauss_eq_comap_gauss_iff_redQ_inv_smul_lineInfty_eq_of_isLevelAutAt_of_isAlgebraic1,119 below · cited by 3 · depth 32 - Density of the classifying map at the pinned Tate point
ModularCurve.FullLevel.dense_range_classify_of_muTuple_pin_gamma0Pow722 below · cited by 2 · depth 32 - Branches over the Γ₀(3M') Gauss ring: conjugacy and tameness
ModularCurve.FullLevel.exists_comap_eq_and_ramificationIdx_eq_one_and_isSeparable_of_over_gauss_gamma0_mul_xH_of_eq_three344 below · cited by 2 · depth 32 - Branches over the Γ₀(qM') Gauss place are conjugates of W₀, q=2
ModularCurve.FullLevel.exists_comap_eq_and_ramificationIdx_eq_one_and_isSeparable_of_over_gauss_gamma0_mul_xH_of_eq_two344 below · cited by 2 · depth 32 - Supersingular places read off injectively from Γ₀(M')-classes
ModularCurve.FullLevel.exists_injective_forall_place_eq_of_forall_evalAt_eq_of_eq_map_classify_rigidDataPow_of_tatePoint2,847 below · cited by 1 · depth 32 - Existence of level-q automorphisms for γ∈Γ₀(M')
ModularCurve.FullLevel.exists_isLevelAutAt_of_mem_gamma030 below · cited by 4 · depth 32 - Local inclusion and separable residue extension over the Gauss ring
ModularCurve.FullLevel.exists_isLocalHom_and_isSeparable_residueField_of_eq_comap_gauss_of_levelH1,089 below · cited by 1 · depth 32 - Base change of the abstract full-level moduli package
ModularCurve.FullLevel.exists_levelModuliPackageAbs_restrictScalars_gamma0Pow0 below · cited by 6 · depth 32 - Reading rigid full-level structures as point-level data, Galois-equivariantly
ModularCurve.FullLevel.exists_levelReading_baseChange_of_isAlgClosed40 below · cited by 1 · depth 32 - Descent of full-level K-points to a discrete valuation ring
ModularCurve.FullLevel.exists_map_eq_of_isDiscreteValuationRing_of_jOf_mem_range_gamma0Pow92 below · cited by 1 · depth 32 - Other cusps avoid the ∞-cusp prime on the pole chart
ModularCurve.FullLevel.exists_mem_forall_coeff_zero_ne_of_mem_minimalPrimes_span_jInvChartInf_xH344 below · cited by 1 · depth 32 - Toric products as modular forms on Γ_H(N²M)
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_qExpansion_eq_smul_prod_toricPoint_sub_gamma0Pow4 below · cited by 1 · depth 32 - Finite action on the ∞-chart, transitive over the floor (q=3)
ModularCurve.FullLevel.exists_mulSemiringAction_chartAlgInf_isPrime_conj_of_forall_mem_floor_iff_xH_of_eq_three242 below · cited by 1 · depth 32 - Transitivity of diamond operators on pole-chart primes, q=2
ModularCurve.FullLevel.exists_mulSemiringAction_chartAlgInf_isPrime_conj_of_forall_mem_floor_iff_xH_of_eq_two242 below · cited by 1 · depth 32 - Finite group action on X_H(q²M') with invariants the Γ₀-field
ModularCurve.FullLevel.exists_mulSemiringAction_isInvariant_laurentBaseChange_gamma0_smul_j_eq_xH241 below · cited by 2 · depth 32 - Closed points of the rigid j-chart read rational floor places
ModularCurve.FullLevel.exists_place_forall_evalAt_eq_apply_qExpand_of_eq_map_classify_rigidDataPow865 below · cited by 2 · depth 32 - Level automorphisms act on the Tate point as relabelling
ModularCurve.FullLevel.exists_pt_forall_isLevelAutAt_map_eq_act_of_exists_ringHom_gamma0Pow_of_tate_of_algebra_of_isScalarTower332 below · cited by 3 · depth 32 - Transport of level automorphism and supersingular point to j-chart
ModularCurve.FullLevel.exists_ringHom_chartAlgFin_levelAut_comap_eq_of_isLevelAutAt_of_ringHom_cyclotomic_rigidDataPow968 below · cited by 3 · depth 32 - Residue degree one for the ∞ Gauss branch of X₀(qM')
ModularCurve.FullLevel.exists_sub_inclusion_mem_nonunits_gauss_gamma0_of_mem_gauss_gamma0_mul135 below · cited by 1 · depth 32 - Finiteness of the group of level automorphisms attached to Γ₀(M')
ModularCurve.FullLevel.exists_subgroup_finite_mem_iff_exists_eq_of_isLevelAutAt63 below · cited by 3 · depth 32 - Admissible constants over a cyclotomic discrete valuation ring
ModularCurve.FullLevel.exists_valuationSubring_admissibleConstants_over_cyclotomic11 below · cited by 6 · depth 32 - Flatness of the full-level moduli ring over a discrete valuation ring
ModularCurve.FullLevel.flat_levelModuliPackageAbs_gamma0Pow_of_isDiscreteValuationRing_of_five_le1,383 below · cited by 9 · depth 32 - Minimal primes of the full-level moduli ring as relabelling translates
ModularCurve.FullLevel.forall_exists_algEquiv_comap_ker_classify_eq_of_dense_gamma0Pow1,747 below · cited by 1 · depth 32 - Borel-invariant elements form the Γ₀(qM') function field
ModularCurve.FullLevel.forall_isLevelAutAt_apply_eq_of_dvd_iff_mem_laurentBaseChange_gamma0_mul63 below · cited by 4 · depth 32 - Ordinary branches lie over the Γ₀(qM') Gauss ring, q=3
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_not_mem_ssJSet_xH_of_isAlgebraic_of_eq_three1,032 below · cited by 1 · depth 32 - Descent of the Gauss valuation to Γ₀(qM') at ordinary points, q=2
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_not_mem_ssJSet_xH_of_isAlgebraic_of_eq_two1,032 below · cited by 1 · depth 32 - Cyclicity of the torus of level automorphisms at q
ModularCurve.FullLevel.isCyclic_and_natCard_dvd_sub_one_of_forall_comap_gauss_eq_of_forall_apply_eq_of_isLevelAutAt1,121 below · cited by 1 · depth 32 - Dedekind special fibre of the j-finite chart of X₀(M')
ModularCurve.FullLevel.isDedekindDomain_chartAlgFin_quotient_span_algebraMap_gamma0_of_not_dvd1,209 below · cited by 1 · depth 32 - Normality of generic fibres of the full-level moduli ring
ModularCurve.FullLevel.isDomain_and_isIntegrallyClosed_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_gamma0Pow1,387 below · cited by 1 · depth 32 - Geometric integrality of components of the full-level moduli ring
ModularCurve.FullLevel.isDomain_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_gamma0Pow_of_isPrimitiveRoot2,104 below · cited by 1 · depth 32 - Reduced special fibre at a supersingular point of a full-level component
ModularCurve.FullLevel.isReduced_residueField_tensorProduct_adicCompletion_quotient_of_nthSeries_eq_mul_X_pow_mul_of_pow_sub_one_eq_mul_levelModuliPackageAbs_gamma0Pow1,474 below · cited by 1 · depth 32 - Reduced special fibre at an ordinary point
ModularCurve.FullLevel.isReduced_residueField_tensorProduct_adicCompletion_quotient_of_nthSeries_eq_mul_X_pow_of_isPrimitiveRoot_levelModuliPackageAbs_gamma0Pow1,459 below · cited by 1 · depth 32 - Regularity of the special fibre on the pole chart via purity
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_chartAlgInf_xH47 below · cited by 1 · depth 32 - Regular fibre of the pole chart at ∞-branch cusps, q=3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_not_forall_mem_floor_iff_xH_of_isAlgebraic_of_eq_three1,429 below · cited by 1 · depth 32 - Regularity at non-∞ floor cusps of the pole chart, q=2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_isLocalization_atPrime_chartAlgInf_of_not_forall_mem_floor_iff_xH_of_isAlgebraic_of_eq_two1,429 below · cited by 1 · depth 32 - Pole chart of X_H(q²M') regular at cusp ∞, q=3
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_cuspInfty_xH_of_eq_three346 below · cited by 1 · depth 32 - Regularity at the cusp ∞ of the pole chart, q=2
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_cuspInfty_xH_of_eq_two346 below · cited by 1 · depth 32 - Transport of pole-chart regularity along an automorphism fixing j
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_of_algEquiv_apply_eq_xH_of_eq_three0 below · cited by 1 · depth 32 - Transport of pole-chart regularity along an automorphism fixing j, q=2
ModularCurve.FullLevel.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_of_algEquiv_apply_eq_xH_of_eq_two0 below · cited by 1 · depth 32 - Unramifiedness at vertical height-one primes over cusps of the pole chart
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_chartAlgInf_of_jInvChartInf_mem_xH1,551 below · cited by 1 · depth 32 - Unramifiedness at horizontal height-one primes below an ∞-cusp
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_not_mem_chartAlgInf_of_jInvChartInf_mem_xH677 below · cited by 1 · depth 32 - Kernel of the classifying map at j(q^{qℓ}) is minimal
ModularCurve.FullLevel.ker_classify_mem_minimalPrimes_of_jOf_eq_jqNModC_gamma0Pow104 below · cited by 3 · depth 32 - Level automorphisms in Γ(ℓ')∩Γ₀(M') fix supersingular closed points
ModularCurve.FullLevel.levelAut_sub_self_mem_of_isLevelAutAt_of_mem_Gamma_of_over_ssPlace_rigidDataPow2,871 below · cited by 3 · depth 32 - A Γ₀(M') element fixing the Tate point is ± 1 mod qℓ
ModularCurve.FullLevel.map_eq_one_or_eq_neg_one_of_act_eq_self_gamma0Pow106 below · cited by 2 · depth 32 - Crossing Igusa branches force a supersingular j-value
ModularCurve.FullLevel.map_jChartFin_mem_ssJSet_of_comap_gauss_ne_gauss_of_forall_mem_nonunits_xH1,575 below · cited by 1 · depth 32 - Crossing points on the special fibre of X₀(M'q) are supersingular
ModularCurve.FullLevel.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_le_chartAlgFin_laurentBaseChange_gamma0_mul1,026 below · cited by 4 · depth 32 - Automorphisms of the Γ₀(M')-fibre datum over a supersingular place
ModularCurve.FullLevel.natCard_variableChange_act_curve_eq_and_level_fst_eq_eq_two_mul_placeWidthChar_of_over_of_eq_map_classify_rigidDataPow_of_tatePoint2,872 below · cited by 1 · depth 32 - Degeneracy image of a cusp-regular integral function is chart-integral
ModularCurve.FullLevel.qExpand_mem_chartAlgFin_of_cuspRegular_of_mem_integers828 below · cited by 4 · depth 32 - Conjugated Gauss rings are unramified over the base Gauss ring
ModularCurve.FullLevel.ramificationIdx_comap_gauss_eq_one_of_levelH0 below · cited by 1 · depth 32 - Ramification index one away from j=0,1728 (q=3)
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_eq_zero_of_ord_sub_eq_zero_xH_levelH_of_eq_three111 below · cited by 2 · depth 32 - Unramifiedness away from j=0,1728 and the cusps, q=2
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_eq_zero_of_ord_sub_eq_zero_xH_levelH_of_eq_two111 below · cited by 2 · depth 32 - Residue of a cusp-regular integral function is w-integral
ModularCurve.FullLevel.residue_mem_toValuationSubring_of_cuspRegular_of_isRational268 below · cited by 9 · depth 32 - Lower bound for the number of level structures in a G-orbit
ModularCurve.FullLevel.two_mul_index_gammaH_levelH_sup_zpowers_neg_one_le_ncard_orbit10 below · cited by 1 · depth 32 - Level automorphism acts on transported cusp data through γ
ModularCurve.FullLevel.zsmul_toPoint_add_zsmul_toPoint_eq_toPoint_levelAut_of_map_eq_cuspData_of_exists_ringHom140 below · cited by 2 · depth 32 - Level automorphisms attached to Γ₀(M') form a finite group
ModularCurve.FullLevel.AuxLevel.exists_finite_subgroup_forall_mem_iff_exists_isLevelAutAt63 below · cited by 1 · depth 33 - A level automorphism moving the Gauss prime of the finite chart
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_mem_chartAlgFin_mem_nonunits_gaussValuationSubring_and_apply_not_mem2,226 below · cited by 1 · depth 33 - Level automorphisms over ℚ(ζ_{qℓ}) for γ∈Γ₀(M')
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_of_mem_gamma030 below · cited by 20 · depth 33 - Gauss prime strictly inside a supersingular chart-algebra point
ModularCurve.FullLevel.AuxLevel.exists_isPrime_mem_iff_forall_coeff_mem_maximalIdeal_le_ne_of_drinfeldChartWitness19 below · cited by 2 · depth 33 - Moduli reading of a supersingular point of the cyclotomic two-chart model
ModularCurve.FullLevel.AuxLevel.exists_levelModuliPackageAbs_gamma0Pow_ringEquiv_adicCompletion_stalk_const_classify_levelAut_of_mem_ssJSet2,259 below · cited by 1 · depth 33 - Element of y outside (U,varpi) at a crossing end
ModularCurve.FullLevel.AuxLevel.exists_mem_not_mem_span_U_const_of_mem_ends_crossing_linked310 below · cited by 1 · depth 33 - Weight-four forms with Tate cusp and c₄ q-expansions
ModularCurve.FullLevel.AuxLevel.exists_modularForm_weight_four_qExpansion_eq_cuspPoint_sq_and_cFour110 below · cited by 1 · depth 33 - A weight-six form with q-expansion c₆ of the Tate curve
ModularCurve.FullLevel.AuxLevel.exists_modularForm_weight_six_qExpansion_eq_cSix_tateBase64 below · cited by 1 · depth 33 - Weight-three forms with Tate cusp-point q-expansions
ModularCurve.FullLevel.AuxLevel.exists_modularForm_weight_three_qExpansion_eq_cuspPoint72 below · cited by 1 · depth 33 - Generators of the blow-up chart C[J/a] are integral modulo y+(varpiₜ/a)
ModularCurve.FullLevel.AuxLevel.exists_monic_aeval_div_eq_of_mem_centre_blowupChart_supersingular_fibre_of_drinfeldChartWitness_linked164 below · cited by 1 · depth 33 - Distinct ends have distinct centres on the j-chart algebra
ModularCurve.FullLevel.AuxLevel.exists_not_mem_iff_mem_of_ne_of_mem_ends_igusaBranch_linked382 below · cited by 1 · depth 33 - Local structure of an end: widehat𝒪≅ W₁[[U,V]]/(UV-(σ₁varpi)^m)
ModularCurve.FullLevel.AuxLevel.exists_ringHom_ringEquiv_adicCompletion_uvCrossingModel_coeff_tangent_centre_of_end_blowupChart_of_drinfeldChartWitness_linked281 below · cited by 2 · depth 33 - Divisibility dichotomy modulo a component through a supersingular point
ModularCurve.FullLevel.AuxLevel.exists_sub_mul_mem_or_of_isPrime_lt_supersingular_of_drinfeldChartWitness_linked146 below · cited by 1 · depth 33 - An unramified DVR subring A₀ with A=A₀[ζ_q]
ModularCurve.FullLevel.AuxLevel.exists_unramified_subDVR_adjoin_eq_top2 below · cited by 2 · depth 33 - Vertical unit in Jⁿ adapted to an end of the blown-up chart
ModularCurve.FullLevel.AuxLevel.exists_verticalUnit_chartAlgFin_mem_maximalIdeal_iff_of_end_blowupChart667 below · cited by 1 · depth 33 - Translation level automorphism preserves maximal-ideal q-expansion coefficients
ModularCurve.FullLevel.AuxLevel.forall_coeff_mem_maximalIdeal_iff_of_isLevelAutAt_T_zpow_inv0 below · cited by 1 · depth 33 - Fixed field of the Γ₀(M') level automorphisms at level qℓ
ModularCurve.FullLevel.AuxLevel.forall_isLevelAutAt_apply_eq_iff_exists_eq_qExpand_gamma0_of_exists_ringHom69 below · cited by 3 · depth 33 - Level automorphisms preserve supersingular maximal ideals of the j-chart
ModularCurve.FullLevel.AuxLevel.isMaximal_comap_restrict_and_mem_ssJSet_of_isLevelAutAt472 below · cited by 3 · depth 33 - Unit criterion at an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevel.isUnit_of_notMem_maximalIdeal_exceptional_igusa_of_end_blowupChart308 below · cited by 1 · depth 33 - Transport of twisted Tate torsion coordinates by a level automorphism
ModularCurve.FullLevel.AuxLevel.levelAut_apply_eq_unit_pow_mul_of_coe_eq_cuspPoint_variableChange124 below · cited by 1 · depth 33 - Off-diagonal linear part of an inertial automorphism lies in mathfrak m_W
ModularCurve.FullLevel.AuxLevel.linearPart_offDiag_mem_maximalIdeal_of_coeffMap_of_drinfeldChartWitness_linearPart_twoChartIntegralModel3,036 below · cited by 1 · depth 33 - Inertia is trivial on the Gauss branch: second row
ModularCurve.FullLevel.AuxLevel.linearPart_row_mem_maximalIdeal_of_coeffMap_of_drinfeldChartWitness_anchor_twoChartIntegralModel130 below · cited by 1 · depth 33 - Gauss-nonunits of the finite chart lie in supersingular primes
ModularCurve.FullLevel.AuxLevel.mem_asIdeal_of_coe_mem_nonunits_gaussValuationSubring_of_mem_ssJSet_twoChartIntegralModel3,028 below · cited by 1 · depth 33 - End local ring 𝒪 lies in the exceptional valuation ring
ModularCurve.FullLevel.AuxLevel.mem_exceptionalValuation_of_mem_end_of_ringEquiv_adicCompletion_uvCrossingModel_centre_of_end_blowupChart_linked193 below · cited by 1 · depth 33 - Inertia acts on a uniformiser by the mod q cyclotomic character
ModularCurve.FullLevel.AuxLevel.ringEquiv_uniformizer_sub_mul_mem_maximalIdeal_sq_of_isCyclotomicExtension_of_apply_eq_pow0 below · cited by 2 · depth 33 - Inertial coefficientwise automorphism fixes the supersingular chart point
ModularCurve.FullLevel.AuxLevelOne.coeffMap_mem_chartAlgFin_and_sub_mem_asIdeal_of_drinfeldChartWitness_anchor_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd10 below · cited by 1 · depth 33 - Branches with independent tangents contract to distinct stalk primes
ModularCurve.FullLevel.AuxLevelOne.comap_ne_comap_of_branchPrime_of_exists_ne_of_drinfeldChartWitness_riders_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd163 below · cited by 1 · depth 33 - Ends of the blown-up supersingular chart on one component coincide
ModularCurve.FullLevel.AuxLevelOne.end_blowupChart_eq_of_on_same_component_of_drinfeldChartWitness_linked_of_dvd268 below · cited by 1 · depth 33 - Incomparability of ends over the supersingular point
ModularCurve.FullLevel.AuxLevelOne.eq_of_forall_mem_of_isEnd_blowupChart_of_drinfeldChartWitness_linked_of_dvd50 below · cited by 2 · depth 33 - Action of an end stabiliser on the UV-crossing model
ModularCurve.FullLevel.AuxLevelOne.exists_algEquiv_uvCrossingModel_compat_levelAut_pow_eq_of_isEnd_of_levelAut_mem_iff_blowupChart_of_drinfeldChartWitness_linked_of_dvd39 below · cited by 1 · depth 33 - Integrality of Laurent coefficients on the j-finite chart
ModularCurve.FullLevel.AuxLevelOne.exists_coeff_eq_algebraMap_of_mem_chartAlgFin_of_isPrimitiveRoot_mul_of_dvd1 below · cited by 1 · depth 33 - Common level-stable affine chart for the ends of the blow-up
ModularCurve.FullLevel.AuxLevelOne.exists_commonChart_ends_of_invariant_section_blowupChart_of_drinfeldChartWitness_linked_of_dvd42 below · cited by 1 · depth 33 - Ends of the blow-up on components through the supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_end_blowupChart_on_component_of_isPrime_lt_supersingular_of_drinfeldChartWitness_linked_of_dvd198 below · cited by 1 · depth 33 - Igusa branch valuation subring at a crossing end
ModularCurve.FullLevel.AuxLevelOne.exists_igusaBranch_valuationSubring_of_mem_ends_crossing_linked_of_dvd323 below · cited by 1 · depth 33 - Supersingular point strictly above the centre of an Igusa valuation
ModularCurve.FullLevel.AuxLevelOne.exists_isMaximal_mem_ssJSet_centre_le_of_igusaValuation_of_dvd45 below · cited by 2 · depth 33 - Primes under the supersingular point lift to the Drinfeld chart
ModularCurve.FullLevel.AuxLevelOne.exists_isPrime_comap_drinfeldChart_eq_of_isPrime_le_ne_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd122 below · cited by 1 · depth 33 - Primes below a supersingular point come from Drinfeld-chart primes
ModularCurve.FullLevel.AuxLevelOne.exists_isPrime_forall_mem_iff_germ_mem_comap_of_le_of_ne_of_drinfeldChartWitness_of_dvd10 below · cited by 6 · depth 33 - First-order reading of the end action on the crossing model
ModularCurve.FullLevel.AuxLevelOne.exists_isUnit_iterate_apply_V_mul_sub_const_mem_sq_of_ringEquiv_compat_levelAut_of_isEnd_blowupChart_of_drinfeldChartWitness_linked_of_dvd49 below · cited by 1 · depth 33 - Moduli package at a supersingular point of the two-chart model
ModularCurve.FullLevel.AuxLevelOne.exists_levelModuliPackageAbs_rigidDataH1Pow_ringEquiv_adicCompletion_stalk_const_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd2,163 below · cited by 1 · depth 33 - Components through a supersingular point as Γ(ℓ)-translates
ModularCurve.FullLevel.AuxLevelOne.exists_mem_Gamma_isLevelAutAt_forall_mem_iff_coeff_mem_maximalIdeal_of_le_of_ne_of_drinfeldChartWitness_of_dvd54 below · cited by 4 · depth 33 - Invariant generator of a power of the centre at every end
ModularCurve.FullLevel.AuxLevelOne.exists_pow_mem_invariant_generates_ends_blowupChart_of_drinfeldChartWitness_linked_of_dvd134 below · cited by 1 · depth 33 - Primes over varpi at an end of the blown-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_primes_exceptional_igusa_trichotomy_of_end_blowupChart_linked_of_dvd275 below · cited by 4 · depth 33 - Drinfeld chart at a supersingular point with level action
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_adicCompletion_stalk_regularLocalRing_isDrinfeldBasisAdic_const_hasseParam_monic_levelAut_linearPart_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd2,252 below · cited by 1 · depth 33 - Semilinearity of Drinfeld charts at supersingular points
ModularCurve.FullLevel.AuxLevelOne.exists_ringEquiv_drinfeldChart_semilinear_of_coeffMap_of_sub_mem_asIdeal_of_drinfeldChartWitness_const_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd132 below · cited by 1 · depth 33 - Completed end ring as the UV=varpi^m crossing model
ModularCurve.FullLevel.AuxLevelOne.exists_ringHom_ringEquiv_adicCompletion_uvCrossingModel_tangent_coords_of_end_blowupChart_of_drinfeldChartWitness_linked_of_dvd267 below · cited by 3 · depth 33 - Two special-fibre components through a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_two_primes_chartAlgFin_le_asIdeal_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,035 below · cited by 1 · depth 33 - Invariant vertical unit in Jⁿ adapted to an end
ModularCurve.FullLevel.AuxLevelOne.exists_verticalUnit_pow_centre_of_end_blowupChart_of_dvd477 below · cited by 1 · depth 33 - Finiteness of Igusa-type valuation subrings of a level field
ModularCurve.FullLevel.AuxLevelOne.finite_setOf_igusaValuation_of_drinfeldChartWitness_of_dvd9 below · cited by 1 · depth 33 - Residual transcendence of j(mathsf q^q) along an Igusa branch
ModularCurve.FullLevel.AuxLevelOne.forall_aeval_jqNModC_mem_maximalIdeal_of_igusaBranch_linked_of_dvd88 below · cited by 1 · depth 33 - Unipotent level automorphisms preserve mathfrak m_A-integral q-expansion coefficients
ModularCurve.FullLevel.AuxLevelOne.forall_coeff_mem_maximalIdeal_iff_of_isLevelAutAt_T_zpow_inv_of_exists_ringHom_of_isPrimitiveRoot_mul_of_dvd0 below · cited by 1 · depth 33 - Level automorphisms in Γ(q)∩Γ₀(M') preserve coefficientwise mathfrak m_A-integrality
ModularCurve.FullLevel.AuxLevelOne.forall_coeff_mem_maximalIdeal_iff_of_isLevelAutAt_gamma_of_drinfeldChartWitness_of_dvd267 below · cited by 1 · depth 33 - Ends other than O escape the branch Wₓ
ModularCurve.FullLevel.AuxLevelOne.forall_ends_exists_not_mem_igusaBranch_of_mem_ends_linked_of_dvd352 below · cited by 1 · depth 33 - Linear part of semilinear inertia on an anchored Drinfeld chart
ModularCurve.FullLevel.AuxLevelOne.inertia_drinfeldChart_linearPart_diagOneElem_of_semilinear_of_drinfeldChartWitness_linearPart_riders_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,042 below · cited by 1 · depth 33 - Igusa-type valuation subrings: discreteness and chart-algebra localisation
ModularCurve.FullLevel.AuxLevelOne.isDiscreteValuationRing_and_mem_iff_of_igusaValuation_of_drinfeldChartWitness_of_dvd125 below · cited by 2 · depth 33 - Maximality of primes over the supersingular point on blow-up charts
ModularCurve.FullLevel.AuxLevelOne.isMaximal_of_isPrime_of_supersingular_le_of_div_mem_blowupChart_of_drinfeldChartWitness_linked_of_dvd134 below · cited by 1 · depth 33 - Chart functions vanishing mod mathfrak m_A lie in a supersingular point
ModularCurve.FullLevel.AuxLevelOne.mem_of_forall_coeff_mem_maximalIdeal_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd2,879 below · cited by 1 · depth 33 - Transported μ_{p^k} kernel has coefficients in the level-H₁ field
ModularCurve.FullLevel.Diamond.coeff_kernelVariableChangeDeg_mem_range_of_variableChange_tateToricPoint_fst_mem_range_rigidDataH1Pow58 below · cited by 2 · depth 33 - Base change of the abstract H₁ level-moduli package
ModularCurve.FullLevel.Diamond.exists_levelModuliPackageAbs_restrictScalars_rigidDataH1Pow0 below · cited by 6 · depth 33 - Transport of the Drinfeld Γ(q)-pair is relabelling by γ
ModularCurve.FullLevel.Diamond.exists_level_snd_snd_act_mapRing_eq_relabel_rigidDataH1Pow236 below · cited by 1 · depth 33 - Raw Γ₀(M')–Γ₁(ℓ)–Γ(q) data are representable by a finite-type algebra
ModularCurve.FullLevel.Diamond.exists_represents_raw_rigidDataGamma1Pow70 below · cited by 1 · depth 33 - Transport of a cyclotomic level automorphism to the k₀-chart
ModularCurve.FullLevel.Diamond.exists_ringHom_chartAlgFin_levelAut_comap_eq_of_isLevelAutAt_of_ringHom_cyclotomic_rigidDataGamma1Pow939 below · cited by 3 · depth 33 - Level automorphism rescales the Tate datum by a weight-one unit
ModularCurve.FullLevel.Diamond.exists_units_curve_act_mapRing_eq_of_isLevelAutAt_rigidDataH1Pow151 below · cited by 1 · depth 33 - K-rationality of the weight-one twist of Tate(mathsf q^q)
ModularCurve.FullLevel.Diamond.exists_variableChange_weightOne_tateBase_mem_laurentBaseChange_and_cuspData_mem_of_exists_ringHom_pinGamma1114 below · cited by 3 · depth 33 - Flatness over a DVR of the H₁-level fine moduli ring
ModularCurve.FullLevel.Diamond.flat_levelModuliPackageAbs_rigidDataH1Pow_of_isDiscreteValuationRing1,398 below · cited by 9 · depth 33 - Minimal primes as j₀-fixing translates of the Tate kernel
ModularCurve.FullLevel.Diamond.forall_exists_algEquiv_comap_ker_classify_eq_of_dense_rigidDataH1Pow1,745 below · cited by 1 · depth 33 - Supersingular specialisations of the H₁ datum have no q-torsion
ModularCurve.FullLevel.Diamond.forall_nsmul_eq_zero_of_over_of_eq_map_classify_rigidDataH1Pow104 below · cited by 2 · depth 33 - Igusa bound: [SL₂(ℤ):±Γ_{H_1}]≤[T:L(j(x))]
ModularCurve.FullLevel.Diamond.index_le_finrank_adjoin_jOf_of_transcendental_jOf_rigidDataH1Pow417 below · cited by 1 · depth 33 - Components of the H₁ fine moduli ring stay integral over L
ModularCurve.FullLevel.Diamond.isDomain_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_rigidDataH1Pow_of_isPrimitiveRoot2,056 below · cited by 1 · depth 33 - Minimal-prime quotients of the H₁ moduli ring are normal
ModularCurve.FullLevel.Diamond.isIntegrallyClosed_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_rigidDataH1Pow2,123 below · cited by 1 · depth 33 - Reduced special fibre at a supersingular point, H₁ level
ModularCurve.FullLevel.Diamond.isReduced_residueField_tensorProduct_adicCompletion_quotient_of_nthSeries_eq_mul_X_pow_mul_of_pow_sub_one_eq_mul_levelModuliPackageAbs_rigidDataH1Pow1,485 below · cited by 1 · depth 33 - Reduced special fibre at an ordinary point, H₁ level
ModularCurve.FullLevel.Diamond.isReduced_residueField_tensorProduct_adicCompletion_quotient_of_nthSeries_eq_mul_X_pow_of_isPrimitiveRoot_levelModuliPackageAbs_rigidDataH1Pow1,471 below · cited by 1 · depth 33 - Dense q-expansion kernel is a minimal prime (H₁ level)
ModularCurve.FullLevel.Diamond.ker_mem_minimalPrimes_of_levelModuliPackageAbs_qExpansion_of_dense_of_exists_ringHom_rigidDataH1Pow0 below · cited by 4 · depth 33 - Trivial-diamond level automorphisms fix supersingular chart points
ModularCurve.FullLevel.Diamond.levelAut_sub_self_mem_of_isLevelAutAt_of_mem_gamma0_of_apply_eq_one_of_over_ssPlace_rigidDataGamma1Pow2,859 below · cited by 1 · depth 33 - Valuative criterion over the j-line for the H₁ moduli ring
ModularCurve.FullLevel.Diamond.levelModuliPackageAbs_apply_mem_valuationSubring_of_isDiscreteValuationRing_of_j0_mem_rigidDataH1Pow93 below · cited by 1 · depth 33 - Γ₀(M')-component fixed by the rescaled level automorphism
ModularCurve.FullLevel.Diamond.level_fst_act_mapRing_eq_of_curve_eq_units_of_level_fst_rigidDataH1Pow14 below · cited by 1 · depth 33 - Equal floor readings force equal Γ₀(M')-moduli class
ModularCurve.FullLevel.Diamond.moduliPoint_mk_eq_of_forall_apply_eq_of_eq_map_classify_rigidDataH1Pow_of_tatePoint_pinGamma12,846 below · cited by 1 · depth 33 - Diamond action: Γ₁(ℓ_g)-point of the τ-transport is γ₀₀-fold
ModularCurve.FullLevel.Diamond.toPoint_level_snd_fst_act_mapRing_eq_zsmul_toPoint_of_curve_eq_units_rigidDataH1Pow142 below · cited by 1 · depth 33 - Conjugated Gauss ring restricts to the floor valuation ring
ModularCurve.FullLevel.algebraMap_mem_comap_gauss_iff_of_isLevelAutAt970 below · cited by 1 · depth 33 - Gauss ring of K contracts to that of K₀
ModularCurve.FullLevel.algebraMap_mem_gauss_iff_of_levelH0 below · cited by 1 · depth 33 - Minimal primes of the moduli ring dominate the j-line
ModularCurve.FullLevel.comap_adjoin_jZero_eq_bot_of_mem_minimalPrimes_gamma0Pow1,397 below · cited by 2 · depth 33 - Stability of the Gauss ring under a level automorphism
ModularCurve.FullLevel.comap_gauss_eq_iff_redQ_smul_lineInfty_eq_of_levelAutBar_apply1,096 below · cited by 1 · depth 33 - Weil pairings separate relabelled full-level components
ModularCurve.FullLevel.det_eq_of_ker_classify_act_eq_of_relabel_gamma0Pow281 below · cited by 2 · depth 33 - Automorphisms fixing the floor and the cusp prime are trivial
ModularCurve.FullLevel.eq_refl_of_forall_apply_eq_of_forall_coeff_zero_mem_iff_chartAlgInf_xH262 below · cited by 1 · depth 33 - Closed points of the rigid chart read through R₀
ModularCurve.FullLevel.exists_algHom_forall_apply_residue_eq_apply_qExpand_of_eq_map_classify_rigidDataPow860 below · cited by 1 · depth 33 - Minimal primes of the full-level moduli ring are q-expansion kernels
ModularCurve.FullLevel.exists_algHom_laurentSeries_ker_eq_of_mem_minimalPrimes_levelModuliPackageAbs_gamma0Pow_of_isPrimitiveRoot2,101 below · cited by 1 · depth 33 - Constants of A=A₀[ζ_q] lie in the classifying map's image
ModularCurve.FullLevel.exists_classify_eq_algebraMap_of_adjoin_eq_top_gamma0Pow_of_finite_residueField1,417 below · cited by 2 · depth 33 - Γ₀(M')-layer lies in fractions of classify-values at the Tate point
ModularCurve.FullLevel.exists_classify_eq_of_coe_eq_qExpand_of_mem_laurentBaseChange_gamma0Pow_tatePoint456 below · cited by 1 · depth 33 - Level automorphisms at q=3 for Γ₀(M')
ModularCurve.FullLevel.exists_isLevelAutAt_of_mem_gamma0_of_eq_three30 below · cited by 1 · depth 33 - Existence of level automorphisms at q=2
ModularCurve.FullLevel.exists_isLevelAutAt_of_mem_gamma0_of_eq_two30 below · cited by 1 · depth 33 - Gauss branches over the Γ₀(M') floor, case q=3
ModularCurve.FullLevel.exists_isLocalHom_and_isSeparable_residueField_of_eq_comap_gauss_of_levelH_of_eq_three320 below · cited by 1 · depth 33 - Gauss branch over the Γ₀(M') floor is separable, q=2
ModularCurve.FullLevel.exists_isLocalHom_and_isSeparable_residueField_of_eq_comap_gauss_of_levelH_of_eq_two320 below · cited by 1 · depth 33 - Supersingular closed point lifts to the chart over admissible constants
ModularCurve.FullLevel.exists_isMaximal_chartAlgFin_comap_eq_of_coeffMap_cyclotomic_rigidDataPow197 below · cited by 1 · depth 33 - Read place is a moduli place of the Frobenius-twisted fibre
ModularCurve.FullLevel.exists_isModuliPlaceOf_map_frobenius_of_forall_evalAt_eq_of_eq_map_classify_rigidDataPow_of_tatePoint2,828 below · cited by 2 · depth 33 - Cusps other than ∞ carry a unit constant term (q=3)
ModularCurve.FullLevel.exists_mem_forall_coeff_zero_ne_of_mem_minimalPrimes_span_jInvChartInf_xH_of_eq_three344 below · cited by 1 · depth 33 - Non-∞ cusps carry unit constant terms, q=2
ModularCurve.FullLevel.exists_mem_forall_coeff_zero_ne_of_mem_minimalPrimes_span_jInvChartInf_xH_of_eq_two344 below · cited by 1 · depth 33 - Diamond group action on the X_H(q²M') function field, q=3
ModularCurve.FullLevel.exists_mulSemiringAction_isInvariant_laurentBaseChange_gamma0_smul_j_eq_xH_of_eq_three241 below · cited by 2 · depth 33 - Diamond action on X_H(q²M') over X₀(q²M') for q=2
ModularCurve.FullLevel.exists_mulSemiringAction_isInvariant_laurentBaseChange_gamma0_smul_j_eq_xH_of_eq_two241 below · cited by 2 · depth 33 - Tate point of the full-level moduli datum over K
ModularCurve.FullLevel.exists_pt_laurentBaseChange_jOf_eq_jqNModC_gamma0Pow_of_algebra228 below · cited by 1 · depth 33 - A Γ₀(qM')-pullback separating the Gauss ring from its level conjugate
ModularCurve.FullLevel.exists_qExpand_mem_gauss_xor_mem_comap_gauss_of_dvd_of_not_dvd_of_isLevelAutAt1,125 below · cited by 1 · depth 33 - Transport of level automorphisms along a cyclotomic coefficient map
ModularCurve.FullLevel.exists_ringHom_chartAlgFin_isLevelAutAt_restrict_comp_eq_of_isLevelAutAt_cyclotomic_rigidDataPow242 below · cited by 1 · depth 33 - Transport of the Gauss valuation ring along an embedding of constants
ModularCurve.FullLevel.exists_ringHom_fieldBar_comap_gauss_iff_of_isAlgebraic1,099 below · cited by 3 · depth 33 - Supersingular fibre dictionary with Γ₀(M') relabelling
ModularCurve.FullLevel.exists_ssFibreDictionary_relabel_of_isLevelAutAt_chartAlgFin_rigidDataPow2,868 below · cited by 1 · depth 33 - Supersingular points of the j-chart lie over supersingular places
ModularCurve.FullLevel.exists_ssPlace_under_of_isMaximal_chartAlgFin_of_mem_ssJSet_rigidDataPow891 below · cited by 1 · depth 33 - Finiteness of the Γ₀(M') level automorphisms, case q=3
ModularCurve.FullLevel.exists_subgroup_finite_mem_iff_exists_eq_of_isLevelAutAt_of_eq_three63 below · cited by 1 · depth 33 - Finiteness of the Γ₀(M') level automorphisms, case q=2
ModularCurve.FullLevel.exists_subgroup_finite_mem_iff_exists_eq_of_isLevelAutAt_of_eq_two63 below · cited by 1 · depth 33 - Degree bound over L(j(q^N)) at level Γ_H(N²M')
ModularCurve.FullLevel.finiteDimensional_and_finrank_adjoin_le_index_of_coe_eq_jqNModC_of_eq_laurentBaseChange_gammaH175 below · cited by 1 · depth 33 - Generic rank of the moduli component through a dense point
ModularCurve.FullLevel.finrank_fractionRing_tensorProduct_quotient_ker_classify_eq_of_dense_gamma0Pow346 below · cited by 2 · depth 33 - Flatness and normal components of the full-level moduli ring
ModularCurve.FullLevel.flat_and_isIntegrallyClosed_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_of_maximalIdeal_eq_span_natCast_gamma0Pow1,368 below · cited by 2 · depth 33 - Classifying map has image the j-finite chart algebra
ModularCurve.FullLevel.forall_classify_mem_chartAlgFin_and_forall_exists_classify_eq_of_jOf_eq_jqNModC_gamma0Pow_of_isScalarTower2,213 below · cited by 2 · depth 33 - Fixed field of the Γ₀(M') level automorphisms is qτ-expansions
ModularCurve.FullLevel.forall_isLevelAutAt_apply_eq_iff_exists_eq_qExpand63 below · cited by 3 · depth 33 - Fixed field of the Borel level automorphisms, q=3
ModularCurve.FullLevel.forall_isLevelAutAt_apply_eq_of_dvd_iff_mem_laurentBaseChange_gamma0_mul_of_eq_three63 below · cited by 1 · depth 33 - Borel-fixed field of the level automorphisms: the case q=2
ModularCurve.FullLevel.forall_isLevelAutAt_apply_eq_of_dvd_iff_mem_laurentBaseChange_gamma0_mul_of_eq_two63 below · cited by 1 · depth 33 - Branches through an ∞-chart cusp lie over the Γ₀(qM') Gauss ring
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_jInvChartInf_mem_xH_of_isAlgebraic959 below · cited by 1 · depth 33 - Supersingular completion of full-level moduli ring modulo 1-ζ is reduced
ModularCurve.FullLevel.isReduced_adicCompletion_quotient_span_one_sub_of_isPrimitiveRoot_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_gamma0Pow1,035 below · cited by 1 · depth 33 - Reducedness of (1-ζ)-quotient at an ordinary point of the full-level package
ModularCurve.FullLevel.isReduced_adicCompletion_quotient_span_one_sub_of_pow_eq_one_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_gamma0Pow1,164 below · cited by 1 · depth 33 - Generic fibre of the full-level moduli ring: reduced, of rank ψ(M')|GL₂(𝔽_ℓ)||GL₂(𝔽_q)|/2
ModularCurve.FullLevel.isReduced_and_finrank_fractionRing_tensorProduct_levelModuliPackageAbs_eq_gamma0Pow301 below · cited by 2 · depth 33 - Regular special fibre on the pole chart, q = 3
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_chartAlgInf_xH_of_eq_three47 below · cited by 1 · depth 33 - Regular special fibre on the pole chart, q = 2
ModularCurve.FullLevel.isRegularLocalRing_fibre_of_forall_height_one_isUnramifiedAt_chartAlgInf_xH_of_eq_two47 below · cited by 1 · depth 33 - Separability of the level-H q-expansion field over κ(̄ j,̄ j_{M'})
ModularCurve.FullLevel.isSeparable_modularFunctionFieldC_of_mem_xHFunctionFieldC_levelH_of_charP1,087 below · cited by 1 · depth 33 - Unramifiedness at vertical height-one primes over cusps, q=3
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_chartAlgInf_of_jInvChartInf_mem_xH_of_eq_three1,348 below · cited by 1 · depth 33 - Unramifiedness at vertical height-one primes at ∞-chart cusps, q=2
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_mem_chartAlgInf_of_jInvChartInf_mem_xH_of_eq_two1,348 below · cited by 1 · depth 33 - Horizontal unramifiedness over the X₀(M') floor at ∞-branch cusps, q=3
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_not_mem_chartAlgInf_of_jInvChartInf_mem_xH_of_eq_three677 below · cited by 1 · depth 33 - Horizontal unramifiedness at a cusp over the X₀(M') floor, q=2
ModularCurve.FullLevel.isUnramifiedAt_of_height_one_of_algebraMap_not_mem_chartAlgInf_of_jInvChartInf_mem_xH_of_eq_two677 below · cited by 1 · depth 33 - Kernel of the Tate-point classifying map lies in every prime power
ModularCurve.FullLevel.ker_classify_le_pow_of_isPrime_of_jOf_eq_jqNModC_gamma0Pow_of_adjoin_eq_top2,123 below · cited by 2 · depth 33 - Level automorphism over L agrees coefficientwise with `levelAutBar`
ModularCurve.FullLevel.levelAutBar_apply_eq_of_isLevelAutAt_of_coe_eq_coeffMap66 below · cited by 4 · depth 33 - Level automorphism fixing the Tate point's classifying image is trivial
ModularCurve.FullLevel.levelAut_eq_one_of_forall_apply_classify_eq_gamma0Pow_tatePoint312 below · cited by 1 · depth 33 - Crossing points of the special fibre are supersingular (transported)
ModularCurve.FullLevel.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_le_chartAlgFin_of_algEquiv_laurentBaseChange_gamma0_mul1,027 below · cited by 1 · depth 33 - Vanishing q-expansion forces vanishing at a supersingular point
ModularCurve.FullLevel.mem_of_forall_coeff_mem_maximalIdeal_of_isMaximal_of_mem_ssJSet_chartAlgFin2,339 below · cited by 2 · depth 33 - Unramified descent of completed local rings at an ordinary point
ModularCurve.FullLevel.nonempty_ringEquiv_adicCompletion_quotient_adicCompletion_unramified_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_gamma0Pow1,455 below · cited by 1 · depth 33 - Completed local ring at a supersingular point descends to an unramified base
ModularCurve.FullLevel.nonempty_ringEquiv_adicCompletion_quotient_adicCompletion_unramified_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_gamma0Pow1,455 below · cited by 1 · depth 33 - Level-q field maps into level-qℓ field, fixed by Γ(q)
ModularCurve.FullLevel.qExpand_mem_and_apply_eq_of_isLevelAutAt_of_mem_Gamma3 below · cited by 23 · depth 33 - Relabelling by γ∈Γ(ℓ')∩Γ₀(M') fixes a supersingular class
ModularCurve.FullLevel.quotMk_eq_of_relabel_of_mem_Gamma_of_forall_smul_eq_zero_rigidDataPow10 below · cited by 1 · depth 33 - Conjugate Gauss rings unramified over Γ₀(M'): case q=3
ModularCurve.FullLevel.ramificationIdx_comap_gauss_eq_one_of_levelH_of_eq_three0 below · cited by 1 · depth 33 - Conjugate Gauss rings are unramified over Γ₀(M'): case q=2
ModularCurve.FullLevel.ramificationIdx_comap_gauss_eq_one_of_levelH_of_eq_two0 below · cited by 1 · depth 33 - Cusps on the Gauss branch are unramified over X₀(M')
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_jInvChartInf_pos_of_forall_ord_pos_mem_of_forall_mem_nonunits_gauss_xH662 below · cited by 1 · depth 33 - Smoothness of the generic fibre of the full-level moduli ring
ModularCurve.FullLevel.smooth_tensorProduct_levelModuliPackageAbs_gamma0Pow_of_isFractionRing213 below · cited by 3 · depth 33 - Level automorphisms are compatible with q ↦ q^ℓ
ModularCurve.FullLevel.AuxLevel.coe_apply_eq_qExpand_coe_apply_of_isLevelAutAt_of_exists_ringHom59 below · cited by 2 · depth 34 - Transfer of the blow-up chart C[J/a] to the Drinfeld chart
ModularCurve.FullLevel.AuxLevel.exists_blowupChart_ringHom_away_eq_div_add_sum_of_eq_add_sum_of_drinfeldChartWitness_linked163 below · cited by 1 · depth 34 - A level automorphism and a chart element with Tate-slot expansions
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_mem_chartAlgFin_coe_eq_slotSubst_sub_and_apply_eq_laurent2,224 below · cited by 1 · depth 34 - Level automorphisms attached to Γ₀(M'), case q=3
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_of_mem_gamma0_of_eq_three30 below · cited by 2 · depth 34 - Level automorphisms for Γ₀(M') at q=2
ModularCurve.FullLevel.AuxLevel.exists_isLevelAutAt_of_mem_gamma0_of_eq_two30 below · cited by 2 · depth 34 - Label unit at an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevel.exists_labelUnit_chartAlgFin_of_end_blowupChart653 below · cited by 1 · depth 34 - Some element of J is a unit at Igusa-type valuations
ModularCurve.FullLevel.AuxLevel.exists_mem_centre_not_mem_maximalIdeal_of_igusaType_blowupChart164 below · cited by 1 · depth 34 - Stretching Γ(N)-forms by diag(N,1) to Γ_H(N²M')
ModularCurve.FullLevel.AuxLevel.exists_modularForm_gammaH_coe_eq_slash_heckeDiagMatrix63 below · cited by 6 · depth 34 - Weight-four form with q-expansion c₄ of the Tate curve
ModularCurve.FullLevel.AuxLevel.exists_modularForm_weight_four_qExpansion_eq_c463 below · cited by 1 · depth 34 - A power of an invariant unit lies in Jⁿ
ModularCurve.FullLevel.AuxLevel.exists_pow_mem_centre_pow_of_labelUnit_of_end_blowupChart327 below · cited by 1 · depth 34 - Completed blow-up chart C[J/a] matches S[M/Ψ a]
ModularCurve.FullLevel.AuxLevel.exists_ringHom_ringEquiv_adicCompletion_blowupChartAt_localChart_of_drinfeldChartWitness_linked167 below · cited by 1 · depth 34 - Basic properties of the blow-up chart C[J/a] at an end
ModularCurve.FullLevel.AuxLevel.isNoetherianRing_and_mem_iff_and_exists_sub_of_end_blowupChart_of_drinfeldChartWitness_linked230 below · cited by 1 · depth 34 - A level automorphism moving the Gauss prime
ModularCurve.FullLevel.AuxLevelOne.exists_isLevelAutAt_mem_chartAlgFin_mem_nonunits_gaussValuationSubring_and_apply_not_mem_of_isPrimitiveRoot_mul_of_dvd2,199 below · cited by 1 · depth 34 - Gauss ideal is prime and strictly below a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_isPrime_mem_iff_forall_coeff_mem_maximalIdeal_le_ne_of_drinfeldChartWitness_of_dvd19 below · cited by 2 · depth 34 - Moduli reading of a supersingular stalk with level-automorphism dictionary
ModularCurve.FullLevel.AuxLevelOne.exists_levelModuliPackageAbs_rigidDataH1Pow_ringEquiv_adicCompletion_stalk_const_classify_levelAut_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd2,214 below · cited by 1 · depth 34 - An element of y outside (U,varpi) at a crossing end
ModularCurve.FullLevel.AuxLevelOne.exists_mem_not_mem_span_U_const_of_mem_ends_crossing_linked_of_dvd279 below · cited by 1 · depth 34 - Monic relation for generators i/a of the blow-up chart
ModularCurve.FullLevel.AuxLevelOne.exists_monic_aeval_div_eq_of_mem_centre_blowupChart_supersingular_fibre_of_drinfeldChartWitness_linked_of_dvd133 below · cited by 1 · depth 34 - Distinct ends give distinct centres on the finite chart
ModularCurve.FullLevel.AuxLevelOne.exists_not_mem_iff_mem_of_ne_of_mem_ends_igusaBranch_linked_of_dvd351 below · cited by 1 · depth 34 - Node structure of the end local ring: crossing model, tangent, centre
ModularCurve.FullLevel.AuxLevelOne.exists_ringHom_ringEquiv_adicCompletion_uvCrossingModel_coeff_tangent_centre_of_end_blowupChart_of_drinfeldChartWitness_linked_of_dvd250 below · cited by 2 · depth 34 - Components through a supersingular point have valuation local rings
ModularCurve.FullLevel.AuxLevelOne.exists_sub_mul_mem_or_of_isPrime_lt_supersingular_of_drinfeldChartWitness_linked_of_dvd146 below · cited by 1 · depth 34 - Unramified sub-DVR A₀ of A with A=A₀[ζ_q]
ModularCurve.FullLevel.AuxLevelOne.exists_unramified_subDVR_adjoin_eq_top_of_ne2 below · cited by 2 · depth 34 - Vertical Γ(q)-invariant unit in the weighted centre at an end
ModularCurve.FullLevel.AuxLevelOne.exists_verticalUnit_chartAlgFin_mem_maximalIdeal_iff_of_end_blowupChart_of_dvd475 below · cited by 1 · depth 34 - Unit criterion at an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.isUnit_of_notMem_maximalIdeal_exceptional_igusa_of_end_blowupChart_of_dvd277 below · cited by 1 · depth 34 - Off-diagonal vanishing in the linear part of an inertial automorphism
ModularCurve.FullLevel.AuxLevelOne.linearPart_offDiag_mem_maximalIdeal_of_coeffMap_of_drinfeldChartWitness_linearPart_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd3,025 below · cited by 1 · depth 34 - Second row of the Drinfeld linear part is (0,1)
ModularCurve.FullLevel.AuxLevelOne.linearPart_row_mem_maximalIdeal_of_coeffMap_of_drinfeldChartWitness_anchor_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd142 below · cited by 1 · depth 34 - Gauss non-units lie in supersingular primes of the finite chart
ModularCurve.FullLevel.AuxLevelOne.mem_asIdeal_of_coe_mem_nonunits_gaussValuationSubring_of_mem_ssJSet_twoChartIntegralModel_of_isPrimitiveRoot_mul_of_dvd2,878 below · cited by 1 · depth 34 - Containment O ⊆ W for an end of the blow-up
ModularCurve.FullLevel.AuxLevelOne.mem_exceptionalValuation_of_mem_end_of_ringEquiv_adicCompletion_uvCrossingModel_centre_of_end_blowupChart_linked_of_dvd162 below · cited by 1 · depth 34 - Level automorphism determined by q-expansion form ratios
ModularCurve.FullLevel.Diamond.apply_eq_of_isLevelAutAt_of_coeffMap_mul_qExpansion_slash_eq0 below · cited by 4 · depth 34 - Minimal primes of the H₁ moduli ring contract to zero
ModularCurve.FullLevel.Diamond.comap_adjoin_jZero_eq_bot_of_mem_minimalPrimes_rigidDataH1Pow1,409 below · cited by 2 · depth 34 - Drinfeld relabellings with equal classifying kernels have congruent determinants
ModularCurve.FullLevel.Diamond.det_eq_of_ker_classify_act_eq_of_relabel_drinfeld_rigidDataH1Pow270 below · cited by 2 · depth 34 - Minimal primes of the H₁ moduli ring as q-expansion kernels
ModularCurve.FullLevel.Diamond.exists_algHom_laurentSeries_ker_eq_of_mem_minimalPrimes_levelModuliPackageAbs_rigidDataH1Pow_of_isPrimitiveRoot2,053 below · cited by 1 · depth 34 - Supersingular places injectively indexed by Γ₀(M')-moduli points
ModularCurve.FullLevel.Diamond.exists_injective_forall_place_eq_of_forall_evalAt_eq_of_eq_map_classify_rigidDataH1Pow_of_tatePoint_pinGamma12,836 below · cited by 1 · depth 34 - A maximal ideal of the k₀-chart contracting to y₁
ModularCurve.FullLevel.Diamond.exists_isMaximal_chartAlgFin_comap_eq_of_coeffMap_cyclotomic_rigidDataGamma1Pow198 below · cited by 1 · depth 34 - Reading H₁-level structures over an algebraically closed field
ModularCurve.FullLevel.Diamond.exists_levelReading_baseChange_of_isAlgClosed_rigidDataH1Pow43 below · cited by 1 · depth 34 - Points with integral j over a DVR lift, H₁-level
ModularCurve.FullLevel.Diamond.exists_map_eq_of_isDiscreteValuationRing_of_jOf_mem_range_rigidDataH1Pow91 below · cited by 1 · depth 34 - Modular forms on Γ_{H_1}(q²M') with prescribed toric q-expansions
ModularCurve.FullLevel.Diamond.exists_modularForm_gammaH_qExpansion_eq_smul_prod_toricPoint_sub_rigidDataH1Pow4 below · cited by 1 · depth 34 - Tate q-torsion coordinates and c₄,c₆ via forms on Γ_{H_1}
ModularCurve.FullLevel.Diamond.exists_modularForm_mul_qExpansion_eq_cuspPoint_and_slash_conjElemN_eq122 below · cited by 3 · depth 34 - Forms on Γ_{H_1}(q²M') realising toric Tate-point coordinates
ModularCurve.FullLevel.Diamond.exists_modularForm_mul_qExpansion_eq_tateToricPoint_and_slash_conjElemN_eq10 below · cited by 4 · depth 34 - Toric generator-kernel coefficients as modular forms on Γ_{H_1}
ModularCurve.FullLevel.Diamond.exists_modularForm_qExpansion_eq_coeff_toricGenKernel_and_slash_conjElemN_eq4 below · cited by 1 · depth 34 - One rational place reads all admissible functions at H₁ level
ModularCurve.FullLevel.Diamond.exists_place_forall_evalAt_eq_apply_of_eq_map_classify_rigidDataH1Pow864 below · cited by 1 · depth 34 - Transfer of level automorphisms to the j-finite chart
ModularCurve.FullLevel.Diamond.exists_ringHom_chartAlgFin_isLevelAutAt_restrict_comp_eq_of_isLevelAutAt_cyclotomic_rigidDataGamma1Pow212 below · cited by 1 · depth 34 - Directed supersingular-fibre dictionary under Γ₀(M')-level automorphisms
ModularCurve.FullLevel.Diamond.exists_ssFibreDictionary_chartAlgFin_rigidDataGamma1Pow_directedAt_of_mem_gamma02,856 below · cited by 1 · depth 34 - Supersingular chart points lie over supersingular places (Γ₁(ℓ_g) frame)
ModularCurve.FullLevel.Diamond.exists_ssPlace_under_of_isMaximal_chartAlgFin_of_mem_ssJSet_rigidDataGamma1Pow890 below · cited by 1 · depth 34 - Rank of the H₁ classifying quotient at a dense j(mathsf q^q) point
ModularCurve.FullLevel.Diamond.finrank_fractionRing_tensorProduct_quotient_ker_classify_eq_of_dense_rigidDataH1Pow341 below · cited by 2 · depth 34 - Flatness and normal components of the H₁ moduli ring
ModularCurve.FullLevel.Diamond.flat_and_isIntegrallyClosed_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_of_maximalIdeal_eq_span_natCast_rigidDataH1Pow1,381 below · cited by 2 · depth 34 - Normality of generic fibres of the H₁ moduli components
ModularCurve.FullLevel.Diamond.isDomain_and_isIntegrallyClosed_tensorProduct_quotient_of_mem_minimalPrimes_levelModuliPackageAbs_rigidDataH1Pow1,402 below · cited by 1 · depth 34 - Level automorphisms preserve supersingular maximal ideals of the j-chart
ModularCurve.FullLevel.Diamond.isMaximal_comap_restrict_and_mem_ssJSet_of_isLevelAutAt_of_eq_levelH_inf_ker247 below · cited by 3 · depth 34 - Reducedness modulo 1-ζ_q at a supersingular point (H₁ level)
ModularCurve.FullLevel.Diamond.isReduced_adicCompletion_quotient_span_one_sub_of_isPrimitiveRoot_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_rigidDataH1Pow1,045 below · cited by 1 · depth 34 - Reducedness of widehatB₀_𝔪/(1-ζ) at an ordinary point
ModularCurve.FullLevel.Diamond.isReduced_adicCompletion_quotient_span_one_sub_of_pow_eq_one_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_rigidDataH1Pow1,165 below · cited by 1 · depth 34 - Reduced generic fibre of the H₁ moduli ring, with its rank
ModularCurve.FullLevel.Diamond.isReduced_and_finrank_fractionRing_tensorProduct_levelModuliPackageAbs_eq_rigidDataH1Pow309 below · cited by 2 · depth 34 - q-expansion criterion at a supersingular point, H₁ level
ModularCurve.FullLevel.Diamond.mem_of_forall_coeff_mem_maximalIdeal_of_isMaximal_of_mem_ssJSet_chartAlgFin_rigidDataGamma1Pow_of_isPrimitiveRoot_mul2,299 below · cited by 1 · depth 34 - Unramified model for an ordinary completed local ring
ModularCurve.FullLevel.Diamond.nonempty_ringEquiv_adicCompletion_quotient_adicCompletion_unramified_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_rigidDataH1Pow1,468 below · cited by 1 · depth 34 - Supersingular completed local ring descends to an unramified base
ModularCurve.FullLevel.Diamond.nonempty_ringEquiv_adicCompletion_quotient_adicCompletion_unramified_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_rigidDataH1Pow1,469 below · cited by 1 · depth 34 - Trivial-diamond Γ₀(M')-relabelling fixes supersingular Γ₁(ℓ_g)-points
ModularCurve.FullLevel.Diamond.quotMk_eq_of_relabel_of_apply_eq_one_of_forall_smul_eq_zero_rigidDataGamma1Pow7 below · cited by 1 · depth 34 - Diamond action on the toric point of the Tate curve
ModularCurve.FullLevel.Diamond.toPoint_levelAut_eq_zsmul_toPoint_of_map_eq_tateToricPoint_rigidDataH1Pow141 below · cited by 1 · depth 34 - Orbit of Γ₁(ℓ_g)-level structures has at least 2[SL₂(ℤ):±Γ_{H_1}] elements
ModularCurve.FullLevel.Diamond.two_mul_index_gammaH_sup_zpowers_neg_one_le_ncard_orbit_gamma113 below · cited by 1 · depth 34 - Level automorphism relabels the Tate cusp pair by γ
ModularCurve.FullLevel.Diamond.zsmul_toPoint_add_zsmul_toPoint_eq_toPoint_levelAut_of_map_eq_cuspData_rigidDataH1Pow146 below · cited by 1 · depth 34 - Gauss subring of K lies over that of K₀ (q=3)
ModularCurve.FullLevel.algebraMap_mem_gauss_iff_of_levelH_of_eq_three0 below · cited by 1 · depth 34 - Gauss ring of K meets K₀ in O₀ (q=2)
ModularCurve.FullLevel.algebraMap_mem_gauss_iff_of_levelH_of_eq_two0 below · cited by 1 · depth 34 - Diamond action free at the cusp ∞, case q=3
ModularCurve.FullLevel.eq_refl_of_forall_apply_eq_of_forall_coeff_zero_mem_iff_chartAlgInf_xH_of_eq_three262 below · cited by 1 · depth 34 - Freeness of diamond automorphisms at the cusp, q=2
ModularCurve.FullLevel.eq_refl_of_forall_apply_eq_of_forall_coeff_zero_mem_iff_chartAlgInf_xH_of_eq_two262 below · cited by 1 · depth 34 - Reading admissible level-M' functions gives a κ_A-embedding
ModularCurve.FullLevel.exists_algHom_modularFunctionFieldFullC_forall_apply_residue_eq_ringHom_of_transcendental_of_tatePoint863 below · cited by 1 · depth 34 - Constants of A=A₀[ζ_A] lie in the image of `classify`
ModularCurve.FullLevel.exists_classify_eq_algebraMap_of_adjoin_eq_top_rigidDataH1Pow_of_finite_residueField_of_isPrimitiveRoot_mul_of_dvd1,428 below · cited by 2 · depth 34 - Minimal primes of the full-level ring are cyclotomic pins
ModularCurve.FullLevel.exists_eq_span_sub_algebraMap_of_mem_minimalPrimes_gamma0Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint2,119 below · cited by 1 · depth 34 - Coefficientwise Galois conjugation of level automorphisms at ξ
ModularCurve.FullLevel.exists_isLevelAutAt_semiconj_of_coeffMap_of_isLevelAutAt22 below · cited by 1 · depth 34 - Primitive qℓ-th root of unity on each component
ModularCurve.FullLevel.exists_isPrimitiveRoot_quotient_mk_of_mem_minimalPrimes_levelModuliPackageAbs_of_maximalIdeal_eq_span_natCast_gamma0Pow1,414 below · cited by 3 · depth 34 - Integral chart function vanishing at ∞ with unit diamond translates
ModularCurve.FullLevel.exists_mem_chartAlgInf_coeff_zero_eq_zero_forall_algEquiv_coeff_zero_ne_xH261 below · cited by 1 · depth 34 - Frobenius twist and cyclic-quotient j at a Tate point
ModularCurve.FullLevel.exists_place_curve_reduction_eq_map_frobenius_cyclicQuotientJ_eq_of_levelAut_of_originChart_of_tatePoint2,251 below · cited by 1 · depth 34 - Supersingular branch with second Drinfeld section at the origin
ModularCurve.FullLevel.exists_place_ringHom_chartAlgFin_residue_eq_originChart_levelAut_of_forall_nsmul_eq_zero_of_tatePoint2,365 below · cited by 1 · depth 34 - A K-point of `rigidDataH1Pow` with j-invariant j(mathsf q^q)
ModularCurve.FullLevel.exists_pt_laurentBaseChange_jOf_eq_jqNModC_rigidDataH1Pow_of_algebra_of_isPrimitiveRoot_mul_of_dvd301 below · cited by 1 · depth 34 - Restriction of full-level points along A₀ → A
ModularCurve.FullLevel.exists_pt_restrictScalars_jOf_eq_classify_comp_eq_gamma0Pow0 below · cited by 2 · depth 34 - Tate model and division values over the full-level q-expansion field
ModularCurve.FullLevel.exists_variableChange_tateBase_mem_laurentBaseChange_and_cuspData_mem124 below · cited by 1 · depth 34 - Degree of the Γ_H level field over L(j(q^{qℓ}))
ModularCurve.FullLevel.finrank_adjoin_jqNModC_laurentBaseChange_xHFunctionField_levelH_eq241 below · cited by 1 · depth 34 - Lying over the Gauss ring: j(τ)-floor versus j(qτ)-floor
ModularCurve.FullLevel.forall_algebraMap_mem_iff_iff_forall_qExpand_mem_iff_of_levelH951 below · cited by 1 · depth 34 - Classifying map image equals the j-integral chart algebra
ModularCurve.FullLevel.forall_classify_mem_chartAlgFin_and_forall_exists_classify_eq_of_jOf_eq_jqNModC_rigidDataH1Pow_of_isScalarTower_of_isPrimitiveRoot_mul_of_dvd2,134 below · cited by 2 · depth 34 - Cusp of the ∞-branch lies over the X₀(qM') Gauss ring, q=3
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_jInvChartInf_mem_xH_of_isAlgebraic_of_eq_three959 below · cited by 1 · depth 34 - Gauss ring at an ∞-branch cusp descends to level qM' (q=2)
ModularCurve.FullLevel.forall_mem_iff_mem_gauss_gamma0_mul_of_forall_mem_nonunits_of_jInvChartInf_mem_xH_of_isAlgebraic_of_eq_two959 below · cited by 1 · depth 34 - Completed stalks away from q are integrally closed domains
ModularCurve.FullLevel.isDomain_and_isIntegrallyClosed_adicCompletion_of_not_mem_levelModuliPackageAbs_gamma0Pow274 below · cited by 1 · depth 34 - Ordinary completed stalks of the full-level moduli ring are normal domains
ModularCurve.FullLevel.isDomain_and_isIntegrallyClosed_adicCompletion_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_gamma0Pow1,163 below · cited by 2 · depth 34 - Supersingular completed stalks of full-level moduli are normal domains
ModularCurve.FullLevel.isDomain_and_isIntegrallyClosed_adicCompletion_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_gamma0Pow1,016 below · cited by 2 · depth 34 - Igusa-level q-expansion field is separable over level-M' field, q=3
ModularCurve.FullLevel.isSeparable_modularFunctionFieldC_of_mem_xHFunctionFieldC_levelH_of_charP_of_eq_three316 below · cited by 1 · depth 34 - Separability over κ(̄ j,̄ j_{M'}) at q=2
ModularCurve.FullLevel.isSeparable_modularFunctionFieldC_of_mem_xHFunctionFieldC_levelH_of_charP_of_eq_two316 below · cited by 1 · depth 34 - Crossing points of the special fibre are not cusps (pole chart)
ModularCurve.FullLevel.jInvChartInf_not_mem_of_exists_two_minimalPrimes_span_le_chartAlgInf_laurentBaseChange_gamma0_mul956 below · cited by 3 · depth 34 - j(mathsf q^N) lies in the base-changed Γ_H expansion field
ModularCurve.FullLevel.jqNModC_mem_laurentBaseChange_xHFunctionField_levelH_of_dvd188 below · cited by 2 · depth 34 - Kernel of the Tate point classifier lies in every prime power
ModularCurve.FullLevel.ker_classify_le_pow_of_isPrime_of_jOf_eq_jqNModC_rigidDataH1Pow_of_adjoin_eq_top_of_isPrimitiveRoot_mul_of_dvd1,991 below · cited by 2 · depth 34 - Surjectivity of full-level points along nilpotent thickenings
ModularCurve.FullLevel.map_surjective_of_surjective_of_ker_pow_eq_bot_of_isUnit_of_ne_two_gamma0Pow212 below · cited by 2 · depth 34 - Integrality and cusp-regularity of j(qᵈ) for d ∣ M'
ModularCurve.FullLevel.mem_integers_and_cuspRegular_qExpand_jq_of_dvd44 below · cited by 6 · depth 34 - Gauss prime lies in every supersingular maximal ideal
ModularCurve.FullLevel.mem_of_coe_mem_nonunits_of_isMaximal_of_mem_ssJSet_chartAlgFin2,337 below · cited by 3 · depth 34 - Generic point count for the rigidified full-level problem
ModularCurve.FullLevel.natCard_algHom_apply_jOf_univ_eq_of_transcendental_gamma0Pow233 below · cited by 1 · depth 34 - No minimal prime of the generic fibre is maximal
ModularCurve.FullLevel.not_isMaximal_of_mem_minimalPrimes_tensorProduct_gamma0Pow230 below · cited by 1 · depth 34 - q-substitution carries Γ₀(M')-functions to level Γ_H(q²M')
ModularCurve.FullLevel.qExpand_mem_laurentBaseChange_xHFunctionField_levelH_of_mem_gamma07 below · cited by 4 · depth 34 - Unramifiedness over the X₀(M') floor at ∞-branch cusps, q=3
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_jInvChartInf_pos_of_forall_ord_pos_mem_of_forall_mem_nonunits_gauss_xH_of_eq_three662 below · cited by 1 · depth 34 - Cusps on the ∞-branch are unramified over X₀(M'), q=2
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_eq_one_of_ord_jInvChartInf_pos_of_forall_ord_pos_mem_of_forall_mem_nonunits_gauss_xH_of_eq_two662 below · cited by 1 · depth 34 - Igusa-branch cusps are unramified over the Γ₀(q²M') floor
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_gamma0_sq_mul_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_xH277 below · cited by 1 · depth 34 - No first-order deformations over a transcendental j-value
ModularCurve.FullLevel.snd_apply_eq_zero_of_apply_jOf_univ_eq_dualNumber_gamma0Pow212 below · cited by 1 · depth 34 - Ogg's unit is fixed by the Γ(q)∩Γ₀(M') level automorphisms
ModularCurve.FullLevel.AuxLevel.apply_eq_self_of_isLevelAutAt_of_mem_Gamma_of_coe_eq_qExpand_modularUnitSeries247 below · cited by 1 · depth 35 - Ogg's modular unit lies in the j-finite chart algebra
ModularCurve.FullLevel.AuxLevel.exists_coe_eq_qExpand_modularUnitSeries_mem_chartAlgFin_mul_eq_pow211 below · cited by 1 · depth 35 - Igusa-type valuation ring centred at y avoiding a blow-up end
ModularCurve.FullLevel.AuxLevel.exists_igusaType_valuationSubring_not_le_end_of_end_blowupChart318 below · cited by 1 · depth 35 - End of the blown-up chart labels one Igusa component prime
ModularCurve.FullLevel.AuxLevel.exists_isPrime_forall_igusaType_mem_maximalIdeal_iff_of_end_blowupChart634 below · cited by 1 · depth 35 - Relabelled Drinfeld pair passes through explicit cusp points
ModularCurve.FullLevel.AuxLevel.exists_isSectionThrough_relabel_coe_eq_cuspData_of_dvd179 below · cited by 1 · depth 35 - From an Ogg translate profile to a branch-labelling unit
ModularCurve.FullLevel.AuxLevel.exists_not_mem_forall_mem_of_oggProfile_of_drinfeldChartWitness87 below · cited by 1 · depth 35 - Translates of the modular unit: coefficients in mathfrak m_A iff q∤δ₁₀
ModularCurve.FullLevel.AuxLevel.forall_coeff_mem_maximalIdeal_iff_not_dvd_of_isLevelAutAt_of_coe_eq_qExpand_modularUnitSeries108 below · cited by 1 · depth 35 - Drinfeld chart image of y and comaps of its powers
ModularCurve.FullLevel.AuxLevel.map_eq_span_and_comap_pow_drinfeldChartWitness_chartAlgFin163 below · cited by 1 · depth 35 - Powers of the weighted centre contract from the blow-up chart
ModularCurve.FullLevel.AuxLevel.mem_weightedCentre_pow_of_eq_pow_mul_mem_blowupChart_of_drinfeldChartWitness_linked171 below · cited by 1 · depth 35 - Transfer of a blow-up chart to the local Drinfeld model
ModularCurve.FullLevel.AuxLevelOne.exists_blowupChart_ringHom_away_eq_div_add_sum_of_eq_add_sum_of_drinfeldChartWitness_linked_of_dvd132 below · cited by 1 · depth 35 - Chart element with toric mathsf q-expansion and level automorphism
ModularCurve.FullLevel.AuxLevelOne.exists_isLevelAutAt_mem_chartAlgFin_coe_eq_tateToricPoint_sub_and_apply_eq_of_isPrimitiveRoot_mul_of_dvd2,195 below · cited by 1 · depth 35 - A label unit at an end of the blown-up supersingular chart
ModularCurve.FullLevel.AuxLevelOne.exists_labelUnit_chartAlgFin_of_end_blowupChart_of_dvd461 below · cited by 1 · depth 35 - Weighted centre J survives every Igusa-type valuation of K
ModularCurve.FullLevel.AuxLevelOne.exists_mem_centre_not_mem_maximalIdeal_of_igusaType_blowupChart_of_dvd133 below · cited by 1 · depth 35 - Label units: uᵃ ∈ Jⁿ with exact exceptional order
ModularCurve.FullLevel.AuxLevelOne.exists_pow_mem_centre_pow_of_labelUnit_of_end_blowupChart_of_dvd296 below · cited by 1 · depth 35 - Completed blow-up chart C[J/a] at P in the Drinfeld model
ModularCurve.FullLevel.AuxLevelOne.exists_ringHom_ringEquiv_adicCompletion_blowupChartAt_localChart_of_drinfeldChartWitness_linked_of_dvd136 below · cited by 1 · depth 35 - Stalk facts at a supersingular point of the j-finite chart
ModularCurve.FullLevel.AuxLevelOne.isMaximal_and_mem_and_isNoetherianRing_stalk_and_dense_and_mem_iff_of_mem_ssJSet_of_isPrimitiveRoot_mul_of_dvd125 below · cited by 1 · depth 35 - Noetherianity and local data of the end blow-up chart C[J/a]
ModularCurve.FullLevel.AuxLevelOne.isNoetherianRing_and_mem_iff_and_exists_sub_of_end_blowupChart_of_drinfeldChartWitness_linked_of_dvd199 below · cited by 1 · depth 35 - Specialisation of the H₁ chart yields a κ(A)-algebra homomorphism
ModularCurve.FullLevel.Diamond.exists_algHom_forall_apply_residue_eq_apply_of_eq_map_classify_rigidDataH1Pow859 below · cited by 1 · depth 35 - Existence of a moduli place for the Frobenius-twisted Γ₀(M')-class
ModularCurve.FullLevel.Diamond.exists_isModuliPlaceOf_map_frobenius_of_forall_evalAt_eq_of_eq_map_classify_rigidDataH1Pow_of_tatePoint_pinGamma12,817 below · cited by 1 · depth 35 - A primitive q-th root of unity on each component of the H₁ moduli ring
ModularCurve.FullLevel.Diamond.exists_isPrimitiveRoot_quotient_mk_of_mem_minimalPrimes_levelModuliPackageAbs_of_maximalIdeal_eq_span_natCast_rigidDataH1Pow1,425 below · cited by 3 · depth 35 - Tate Γ₁(ℓ_g) point identifies level automorphisms with relabelling, sub-base edition
ModularCurve.FullLevel.Diamond.exists_pt_forall_isLevelAutAt_map_eq_act_of_exists_ringHom_rigidDataH1Pow_of_tate_pinGamma1_of_isScalarTower352 below · cited by 1 · depth 35 - Completions away from q are integrally closed domains
ModularCurve.FullLevel.Diamond.isDomain_and_isIntegrallyClosed_adicCompletion_of_not_mem_levelModuliPackageAbs_rigidDataH1Pow286 below · cited by 1 · depth 35 - Completed local ring at an ordinary point is normal
ModularCurve.FullLevel.Diamond.isDomain_and_isIntegrallyClosed_adicCompletion_of_nthSeries_eq_mul_X_pow_levelModuliPackageAbs_rigidDataH1Pow1,165 below · cited by 2 · depth 35 - Normality of completed local rings at supersingular points
ModularCurve.FullLevel.Diamond.isDomain_and_isIntegrallyClosed_adicCompletion_of_nthSeries_eq_mul_X_pow_mul_levelModuliPackageAbs_rigidDataH1Pow1,029 below · cited by 2 · depth 35 - Level automorphism transports level-q cusp coordinates up to μ
ModularCurve.FullLevel.Diamond.levelAut_apply_eq_unit_pow_mul_of_coe_eq_cuspPoint_variableChange130 below · cited by 1 · depth 35 - Level automorphism moves toric ℓ_g-torsion coordinates by c↦ c^{γ₀₀}
ModularCurve.FullLevel.Diamond.levelAut_apply_eq_unit_pow_mul_of_coe_eq_tateToricPoint_variableChange130 below · cited by 1 · depth 35 - Gauss nonunits lie in the supersingular maximal ideal
ModularCurve.FullLevel.Diamond.mem_of_coe_mem_nonunits_of_isMaximal_of_mem_ssJSet_chartAlgFin2,297 below · cited by 3 · depth 35 - Count of H₁-moduli points above a transcendental j
ModularCurve.FullLevel.Diamond.natCard_algHom_apply_jOf_univ_eq_of_transcendental_rigidDataH1Pow245 below · cited by 1 · depth 35 - No minimal prime of the generic fibre is maximal
ModularCurve.FullLevel.Diamond.not_isMaximal_of_mem_minimalPrimes_tensorProduct_rigidDataH1Pow244 below · cited by 1 · depth 35 - Smoothness of the generic fibre of the H₁ moduli ring
ModularCurve.FullLevel.Diamond.smooth_tensorProduct_levelModuliPackageAbs_rigidDataH1Pow_of_isFractionRing231 below · cited by 3 · depth 35 - Dual-number points with constant transcendental j are constant
ModularCurve.FullLevel.Diamond.snd_apply_eq_zero_of_apply_jOf_univ_eq_dualNumber_rigidDataH1Pow220 below · cited by 1 · depth 35 - Index of ±Γ_H(q²M') in SL₂(ℤ)
ModularCurve.FullLevel.Diamond.two_mul_index_gammaH_levelH_inf_ker_sup_zpowers_neg_one_eq6 below · cited by 1 · depth 35 - Specialisation of j(q^{dℓ'}) as the q-th power of a cyclic-quotient j
ModularCurve.FullLevel.apply_eq_cyclicQuotientJ_pow_of_levelAut_of_originChart_of_forall_nsmul_eq_zero_of_tatePoint2,242 below · cited by 1 · depth 35 - Integrality of j(mathsf q) over A[j(mathsf q^q)] in the level field
ModularCurve.FullLevel.coeffEmb_jq_mem_chartAlgFin_qExpand_laurentBaseChange_xHFunctionField191 below · cited by 1 · depth 35 - Supersingular points of the full-level chart are separated by prime-to-q level
ModularCurve.FullLevel.eq_of_isMaximal_of_mem_ssJSet_of_forall_coe_eq_qExpand_iff_chartAlgFin2,252 below · cited by 1 · depth 35 - A universal primitive ℓ-th root in the fine moduli ring
ModularCurve.FullLevel.exists_aeval_cyclotomic_eq_zero_forall_nontrivial_quotient_forall_isPrimitiveRoot_classify_gamma0Pow_of_maximalIdeal_eq_span282 below · cited by 1 · depth 35 - Every Ω-point of the full-level moduli ring has a tangent vector
ModularCurve.FullLevel.exists_algHom_dualNumber_fst_eq_snd_ne_zero_gamma0Pow221 below · cited by 1 · depth 35 - Extending an admissible valuation to FullC level M'
ModularCurve.FullLevel.exists_algHom_modularFunctionFieldFullC_of_ringHom_admissible309 below · cited by 1 · depth 35 - Dividing a cusp-regular function with zero reduction by q
ModularCurve.FullLevel.exists_eq_smul_of_residue_eq_zero_of_mem_integers_of_cuspRegular136 below · cited by 2 · depth 35 - Minimal primes of the level moduli ring are principal pins
ModularCurve.FullLevel.exists_eq_span_sub_algebraMap_of_mem_minimalPrimes_rigidDataH1Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint_of_isPrimitiveRoot_mul_of_dvd1,987 below · cited by 1 · depth 35 - Integral modular unit on Γ_H(q²M') vanishing at ∞
ModularCurve.FullLevel.exists_integralForms_levelH_coeff_zero_eq_zero_isIntegralElem_slash_isUnit24 below · cited by 3 · depth 35 - Semiconjugating level automorphisms by a coefficientwise field automorphism
ModularCurve.FullLevel.exists_isLevelAutAt_semiconj_of_coeffMap_of_isLevelAutAt_of_eq_levelH_inf_ker22 below · cited by 1 · depth 35 - Gauss prime of the j-chart algebra above q
ModularCurve.FullLevel.exists_isPrime_mem_of_forall_coeff_mem_maximalIdeal_chartAlgFin112 below · cited by 1 · depth 35 - Pole-chart function with unit constant terms, case q=3
ModularCurve.FullLevel.exists_mem_chartAlgInf_coeff_zero_eq_zero_forall_algEquiv_coeff_zero_ne_xH_of_eq_three261 below · cited by 1 · depth 35 - Pole-chart unit witness at ∞ on X_H(q²M'), q=2
ModularCurve.FullLevel.exists_mem_chartAlgInf_coeff_zero_eq_zero_forall_algEquiv_coeff_zero_ne_xH_of_eq_two261 below · cited by 1 · depth 35 - Weight-four forms with Tate cusp-point and c₄ expansions at level q
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_weight_four_qExpansion_eq_cuspPoint_sq_and_cFour110 below · cited by 1 · depth 35 - Weight-six form with q-expansion c₆ of Tate(q^{ q})
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_weight_six_qExpansion_eq_cSix_tateBase64 below · cited by 1 · depth 35 - Weight-three forms realising the level-q Tate cusp points
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_weight_three_qExpansion_eq_cuspPoint72 below · cited by 1 · depth 35 - Diamond operators on the function field of X_H(q²M')
ModularCurve.FullLevel.exists_monoidHom_gamma0_algEquiv_slash_floor_enum_ker_xH240 below · cited by 1 · depth 35 - First Drinfeld section is the origin at the Gauss place
ModularCurve.FullLevel.exists_originChart_fst_of_forall_ringHom_eq_zero_iff_mem_nonunits_of_tatePoint23 below · cited by 1 · depth 35 - Lifting full-level points along a scalar restriction
ModularCurve.FullLevel.exists_pt_restrictScalars_jOf_eq_classify_comp_eq_rigidDataH1Pow_of_isPrimitiveRoot_mul_of_dvd0 below · cited by 2 · depth 35 - Weight-one Tate model and cusp points over K
ModularCurve.FullLevel.exists_variableChange_weightOne_tateBase_mem_laurentBaseChange_and_cuspData_mem123 below · cited by 1 · depth 35 - Formal smoothness of the full-level moduli ring away from q
ModularCurve.FullLevel.formallySmooth_localization_atPrime_of_not_mem_levelModuliPackageAbs_gamma0Pow213 below · cited by 1 · depth 35 - Index of H₁ in (ℤ/q²M')^×, and -1notin H₁
ModularCurve.FullLevel.index_levelH_inf_ker_unitsMap_eq_and_neg_one_notMem0 below · cited by 1 · depth 35 - Integrality of the quotient B₀/(ξ_B-a)
ModularCurve.FullLevel.isDomain_quotient_span_sub_algebraMap_of_forall_nontrivial_of_forall_isPrimitiveRoot_classify_gamma0Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top2,115 below · cited by 1 · depth 35 - Unique minimal prime over π₀ in the j-finite chart
ModularCurve.FullLevel.le_of_mem_minimalPrimes_span_of_isPrime_chartAlgFin_of_coprime_level319 below · cited by 1 · depth 35 - Substitution mathsf q↦mathsf q^q raises level ℓ' to qℓ'
ModularCurve.FullLevel.qExpand_mem_laurentBaseChange_xHFunctionField_levelH_mul_of_mem7 below · cited by 2 · depth 35 - Unramifiedness over the X₀(q²M') floor at an Igusa cusp, q=3
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_gamma0_sq_mul_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_xH_of_eq_three277 below · cited by 1 · depth 35 - Unramifiedness over X₀(q²M') at an ∞-Igusa cusp place, q=2
ModularCurve.FullLevel.ramificationIndexAlong_inclusion_gamma0_sq_mul_eq_one_of_ord_jInvChartInf_pos_of_forall_mem_nonunits_gauss_xH_of_eq_two277 below · cited by 1 · depth 35 - Krull dimension at most one away from q
ModularCurve.FullLevel.ringKrullDim_localization_atPrime_le_one_of_not_mem_levelModuliPackageAbs_gamma0Pow105 below · cited by 1 · depth 35 - Index of ±Γ_H((qℓ)²M') in SL₂(ℤ)
ModularCurve.FullLevel.two_mul_index_gammaH_levelH_mul_sup_zpowers_neg_one_eq6 below · cited by 1 · depth 35 - Twice the j-fibre equals the level data on W₀
ModularCurve.FullLevel.two_mul_natCard_pt_jOf_eq_eq_natCard_isLevel_rigidDataPow_of_isAlgClosed4 below · cited by 1 · depth 35 - Two residues modulo qℓ with distinct reduced orders
ModularCurve.FullLevel.val_neg_natCast_pos_and_min_ne_of_isCoprime0 below · cited by 1 · depth 35 - Level-ℓ' function field inside level-qℓ', and contains j
ModularCurve.FullLevel.xHFunctionField_levelH_le_of_prime_and_jq_mem3 below · cited by 1 · depth 35 - Ogg's modular unit is fixed when q ∣ δ₁₀
ModularCurve.FullLevel.AuxLevel.apply_eq_self_of_isLevelAutAt_of_dvd_of_coe_eq_qExpand_modularUnitSeries65 below · cited by 1 · depth 36 - Fricke-type relation τ(x) w=q¹² in the Borel case
ModularCurve.FullLevel.AuxLevel.apply_mul_eq_pow_twelve_of_isLevelAutAt_of_dvd_apply_zero_zero_of_coe_eq_qExpand_modularUnitSeries65 below · cited by 1 · depth 36 - Contraction commutes with powers of the Drinfeld chart centre
ModularCurve.FullLevel.AuxLevel.comap_span_pow_eq_comap_span_pow_of_drinfeldChartWitness_linked165 below · cited by 1 · depth 36 - Inverse unipotent level automorphism scales Laurent coefficients by uⁿ
ModularCurve.FullLevel.AuxLevel.exists_isUnit_forall_coeff_apply_eq_pow_mul_coeff_of_isLevelAutAt_T_zpow_inv0 below · cited by 1 · depth 36 - Powers of the weighted centre as intersections of translated powers
ModularCurve.FullLevel.AuxLevel.weightedCentre_pow_eq_sInf_comap_levelAut_comap_span_pow_of_drinfeldChartWitness_linked165 below · cited by 1 · depth 36 - Invariance of the modular unit under Γ(q)∩Γ₀(M') level automorphisms
ModularCurve.FullLevel.AuxLevelOne.apply_eq_self_of_isLevelAutAt_of_mem_Gamma_of_coe_eq_qExpand_modularUnitSeries_of_dvd246 below · cited by 1 · depth 36 - Ogg's modular unit in the j-finite chart algebra
ModularCurve.FullLevel.AuxLevelOne.exists_coe_eq_qExpand_modularUnitSeries_mem_chartAlgFin_mul_eq_pow_of_dvd211 below · cited by 1 · depth 36 - An Igusa-type valuation ring avoiding a prescribed blow-up end
ModularCurve.FullLevel.AuxLevelOne.exists_igusaType_valuationSubring_not_le_end_of_end_blowupChart_of_dvd287 below · cited by 1 · depth 36 - End of blown-up supersingular chart labels an Igusa prime
ModularCurve.FullLevel.AuxLevelOne.exists_isPrime_forall_igusaType_mem_maximalIdeal_iff_of_end_blowupChart_of_dvd408 below · cited by 1 · depth 36 - Label unit isolating one branch through a supersingular point
ModularCurve.FullLevel.AuxLevelOne.exists_not_mem_forall_mem_of_oggProfile_of_drinfeldChartWitness_of_dvd55 below · cited by 1 · depth 36 - Coefficients of τ x lie in mathfrak m_A iff q∤δ₁₀
ModularCurve.FullLevel.AuxLevelOne.forall_coeff_mem_maximalIdeal_iff_not_dvd_of_isLevelAutAt_of_coe_eq_qExpand_modularUnitSeries_of_dvd108 below · cited by 1 · depth 36 - Drinfeld chart: Ψ(y) spans mathfrak m_S, and Ψ⁻¹(mathfrak m_Sⁿ)=yⁿ
ModularCurve.FullLevel.AuxLevelOne.map_eq_span_and_comap_pow_drinfeldChartWitness_chartAlgFin_of_dvd132 below · cited by 1 · depth 36 - Powers of the weighted centre contract from the varpiₜ-chart
ModularCurve.FullLevel.AuxLevelOne.mem_weightedCentre_pow_of_eq_pow_mul_mem_blowupChart_of_drinfeldChartWitness_linked_of_dvd140 below · cited by 1 · depth 36 - Supersingular maximal ideals agreeing on q-substituted functions coincide
ModularCurve.FullLevel.Diamond.eq_of_isMaximal_of_mem_ssJSet_of_forall_coe_eq_qExpand_iff_chartAlgFin2,214 below · cited by 1 · depth 36 - Tangent vectors at geometric points of the H₁ moduli ring
ModularCurve.FullLevel.Diamond.exists_algHom_dualNumber_fst_eq_snd_ne_zero_rigidDataH1Pow237 below · cited by 1 · depth 36 - Branch reading gives an embedding of the full level-M' field
ModularCurve.FullLevel.Diamond.exists_algHom_modularFunctionFieldFullC_forall_apply_residue_eq_ringHom_of_transcendental_rigidDataH1Pow_of_tatePoint_pinGamma1864 below · cited by 1 · depth 36 - Relabelled Drinfeld pair at the Tate point, level H₁
ModularCurve.FullLevel.Diamond.exists_isSectionThrough_relabel_coe_eq_cuspData_of_dvd_rigidDataH1Pow179 below · cited by 1 · depth 36 - Frobenius twist of the H₁ branch after place extension
ModularCurve.FullLevel.Diamond.exists_place_curve_reduction_eq_map_frobenius_cyclicQuotientJ_eq_of_levelAut_of_originChart_rigidDataH1Pow_of_tatePoint_pinGamma12,187 below · cited by 1 · depth 36 - Branch place at a supersingular point of the H₁ chart
ModularCurve.FullLevel.Diamond.exists_place_ringHom_chartAlgFin_residue_eq_originChart_levelAut_of_forall_nsmul_eq_zero_rigidDataH1Pow_of_tatePoint_pinGamma12,343 below · cited by 1 · depth 36 - Formal smoothness of the H₁ moduli local rings away from q
ModularCurve.FullLevel.Diamond.formallySmooth_localization_atPrime_of_not_mem_levelModuliPackageAbs_rigidDataH1Pow231 below · cited by 1 · depth 36 - Lifting H₁ moduli points along nilpotent-kernel surjections
ModularCurve.FullLevel.Diamond.map_surjective_of_surjective_of_ker_pow_eq_bot_of_isUnit_rigidDataH1Pow230 below · cited by 2 · depth 36 - Counting linked Γ₀(M')–Γ₁(ℓ_g)–Γ(q) level structures
ModularCurve.FullLevel.Diamond.natCard_isLevel_rigidDataH1Pow_eq_of_isAlgClosed238 below · cited by 1 · depth 36 - Degeneracy mathsf q↦mathsf q^q into the H₁-level function field
ModularCurve.FullLevel.Diamond.qExpand_mem_laurentBaseChange_xHFunctionField_of_mem_ker7 below · cited by 2 · depth 36 - Dimension ≤ 1 at q-invertible maximal ideals of B₀
ModularCurve.FullLevel.Diamond.ringKrullDim_localization_atPrime_le_one_of_not_mem_levelModuliPackageAbs_rigidDataH1Pow101 below · cited by 1 · depth 36 - Doubled point count of the rigid H₁ problem at j=t
ModularCurve.FullLevel.Diamond.two_mul_natCard_pt_jOf_eq_eq_natCard_isLevel_rigidDataH1Pow_of_isAlgClosed9 below · cited by 1 · depth 36 - Inclusion of q-expansion function fields and j at level H₁
ModularCurve.FullLevel.Diamond.xHFunctionField_ker_le_and_jq_mem3 below · cited by 2 · depth 36 - Unique minimal prime containing the Weil pin ξ_B - a
ModularCurve.FullLevel.existsUnique_mem_minimalPrimes_sub_algebraMap_mem_gamma0Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint2,110 below · cited by 1 · depth 36 - Non-constant dual-number point yields a non-zero tangent vector
ModularCurve.FullLevel.exists_algHom_dualNumber_fst_eq_snd_ne_zero_of_exists_pt_dualNumber_gamma0Pow0 below · cited by 1 · depth 36 - Preimage in B₀ of j(mathsf q^{qℓ'd}) as cyclic quotient j-invariant
ModularCurve.FullLevel.exists_classify_preimage_forall_apply_eq_cyclicQuotientJ_etale_of_tatePoint_gamma0Pow2,227 below · cited by 1 · depth 36 - Finite-type fine moduli package for Γ₀(N)×Γ(ℓ) level data
ModularCurve.FullLevel.exists_levelModuliPackageAbs_trivial_of_isUnit_two_three_gamma0Pow44 below · cited by 2 · depth 36 - Weight-four form with q-expansion c₄ of Tate(q^q)
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_weight_four_qExpansion_eq_cFour_tateBase63 below · cited by 1 · depth 36 - Diamond operators on the function field of X_H(q²M'), q=3
ModularCurve.FullLevel.exists_monoidHom_gamma0_algEquiv_slash_floor_enum_ker_xH_of_eq_three240 below · cited by 1 · depth 36 - Diamond operators on the function field of X_H(q²M'), q=2
ModularCurve.FullLevel.exists_monoidHom_gamma0_algEquiv_slash_floor_enum_ker_xH_of_eq_two240 below · cited by 1 · depth 36 - Non-trivial first-order deformations of full-level Weierstrass moduli points
ModularCurve.FullLevel.exists_pt_dualNumber_map_fstHom_eq_ne_map_inlAlgHom_gamma0Pow219 below · cited by 1 · depth 36 - Étale part of the pinned Tate point descends under qmapstoq^q
ModularCurve.FullLevel.exists_raw_etale_map_eq_map_qExpand_of_tatePoint169 below · cited by 1 · depth 36 - Unramified Gauss-type valuation ring on the prime-to-q level field
ModularCurve.FullLevel.exists_valuationSubring_gaussType_unramified_of_coprime_level3 below · cited by 1 · depth 36 - Finite type of the full-level fine moduli ring over A₀
ModularCurve.FullLevel.finiteType_levelModuliPackageAbs_gamma0Pow_of_maximalIdeal_eq_span140 below · cited by 1 · depth 36 - Integrality of the H₁ moduli ring over unramified constants
ModularCurve.FullLevel.isDomain_levelModuliPackageAbs_rigidDataH1Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint_of_isPrimitiveRoot_mul_of_dvd1,986 below · cited by 1 · depth 36 - Admissible residues are integral over κ_A[jmath̄]
ModularCurve.FullLevel.isIntegral_adjoin_residue_jq_residue_of_mem_admissible293 below · cited by 1 · depth 36 - Reducedness of the full-level moduli ring over an unramified base
ModularCurve.FullLevel.isReduced_levelModuliPackageAbs_gamma0Pow_of_maximalIdeal_eq_span1,387 below · cited by 1 · depth 36 - Anti-diagonal level automorphism shifts the j-readings
ModularCurve.FullLevel.levelAut_apply_qExpand_jq_eq_jqNModC_of_antidiagonal_of_ringHom261 below · cited by 1 · depth 36 - Relations among reductions are respected by admissible evaluations
ModularCurve.FullLevel.sum_algebraMap_mul_apply_eq_zero_of_sum_smul_residue_eq_zero137 below · cited by 1 · depth 36 - Uniqueness of Gauss-type valuations at level prime to q
ModularCurve.FullLevel.valuationSubring_eq_of_gaussType_of_coprime_level308 below · cited by 1 · depth 36 - Rigidity: reduction of a full-level change of variables is trivial
ModularCurve.FullLevel.variableChange_map_eq_one_of_eq_act_of_map_residue_eq_gamma0Pow3 below · cited by 2 · depth 36 - Ogg's modular unit is fixed by level automorphisms over Γ₀(q)
ModularCurve.FullLevel.AuxLevelOne.apply_eq_self_of_isLevelAutAt_of_dvd_of_coe_eq_qExpand_modularUnitSeries_of_dvd65 below · cited by 1 · depth 37 - Fricke relation for the modular unit at width q
ModularCurve.FullLevel.AuxLevelOne.apply_mul_eq_pow_twelve_of_isLevelAutAt_of_dvd_apply_zero_zero_of_coe_eq_qExpand_modularUnitSeries_unstretched_of_dvd65 below · cited by 1 · depth 37 - Contraction of powers of the Drinfeld-chart centre ideal
ModularCurve.FullLevel.AuxLevelOne.comap_span_pow_eq_comap_span_pow_of_drinfeldChartWitness_linked_of_dvd134 below · cited by 1 · depth 37 - Level automorphism at (T^s)⁻¹ scales Laurent coefficients geometrically
ModularCurve.FullLevel.AuxLevelOne.exists_isUnit_forall_coeff_apply_eq_pow_mul_coeff_of_isLevelAutAt_T_zpow_inv_of_dvd0 below · cited by 1 · depth 37 - Weighted centre powers are intersections of translated local centres
ModularCurve.FullLevel.AuxLevelOne.weightedCentre_pow_eq_sInf_comap_levelAut_comap_span_pow_of_drinfeldChartWitness_linked_of_dvd134 below · cited by 1 · depth 37 - Specialisation of j(qᵈ) as q-th power of cyclic-quotient invariant
ModularCurve.FullLevel.Diamond.apply_eq_cyclicQuotientJ_pow_of_levelAut_of_originChart_of_forall_nsmul_eq_zero_rigidDataH1Pow_of_tatePoint_pinGamma12,178 below · cited by 1 · depth 37 - Non-constant dual-number point gives non-zero tangent vector
ModularCurve.FullLevel.Diamond.exists_algHom_dualNumber_fst_eq_snd_ne_zero_of_exists_pt_dualNumber_rigidDataH1Pow0 below · cited by 1 · depth 37 - Gauss prime on the j-chart of X_{H_1}(q²M')
ModularCurve.FullLevel.Diamond.exists_isPrime_mem_of_forall_coeff_mem_maximalIdeal_chartAlgFin_of_eq_levelH_inf_ker112 below · cited by 1 · depth 37 - Fine moduli package for Γ₀(N)-kernels and a linked Γ₁(ℓ)-point
ModularCurve.FullLevel.Diamond.exists_levelModuliPackageAbs_trivial_rigidDataH1Pow38 below · cited by 2 · depth 37 - First Drinfeld section is the origin on the Gauss branch
ModularCurve.FullLevel.Diamond.exists_originChart_fst_of_forall_ringHom_eq_zero_iff_mem_nonunits_rigidDataH1Pow_of_tatePoint_pinGamma123 below · cited by 1 · depth 37 - Non-constant first-order deformations of H₁-moduli points
ModularCurve.FullLevel.Diamond.exists_pt_dualNumber_map_fstHom_eq_ne_map_inlAlgHom_rigidDataH1Pow235 below · cited by 1 · depth 37 - Étale part of the pinned H₁ Tate point as a q-expansion
ModularCurve.FullLevel.Diamond.exists_raw_etale_map_eq_map_qExpand_of_tatePoint_pinGamma1150 below · cited by 1 · depth 37 - Tate point: j(mathsf q^{qℓ'd}) as a cyclic quotient j-invariant
ModularCurve.FullLevel.algebraMap_jqNModC_eq_cyclicQuotientJ_of_eq_map_tatePoint_gamma0Pow153 below · cited by 1 · depth 37 - Specialising the Tate reading of j(mathsf q^{qℓ'd}) to cyclic quotients
ModularCurve.FullLevel.apply_eq_cyclicQuotientJ_of_classify_eq_jqNModC_of_tatePoint_gamma0Pow92 below · cited by 1 · depth 37 - Admissible evaluation extends to the reduced full modular function field
ModularCurve.FullLevel.exists_algHom_modularFunctionFieldFullC_of_ringHom_admissible_of_prime309 below · cited by 1 · depth 37 - Surjectivity of the Tate-point classifying map onto the j-chart algebra
ModularCurve.FullLevel.exists_clC_eq_of_mem_chartAlgFin_of_tatePoint_gamma0Pow2,211 below · cited by 1 · depth 37 - Tangent line at a point of the Γ₀(M')Γ(ℓ) Weierstrass moduli problem
ModularCurve.FullLevel.exists_forall_eq_map_dualNumber_smul_of_trivial_gamma0Pow20 below · cited by 2 · depth 37 - Raw Γ₀(M')×Γ(ℓ) structure on the twisted Tate curve
ModularCurve.FullLevel.exists_variableChange_raw_etale_tate_weightOne_level_fst_gamma0Pow167 below · cited by 1 · depth 37 - At most one minimal prime of the H₁ moduli ring
ModularCurve.FullLevel.finite_minimalPrimes_and_ncard_le_one_rigidDataH1Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint_of_isPrimitiveRoot_mul_of_dvd1,984 below · cited by 1 · depth 37 - At most ℓ-1 minimal primes of the full-level moduli ring
ModularCurve.FullLevel.finite_minimalPrimes_and_ncard_le_sub_one_gamma0Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_of_jOf_eq2,109 below · cited by 1 · depth 37 - Reducedness of the H₁-level fine moduli ring over A₀
ModularCurve.FullLevel.isReduced_levelModuliPackageAbs_rigidDataH1Pow_of_maximalIdeal_eq_span_of_adjoin_eq_top_tatePoint_of_isPrimitiveRoot_mul_of_dvd1,402 below · cited by 1 · depth 37 - Residual triviality of variable changes at level Γ₁(ℓ_g)
ModularCurve.FullLevel.variableChange_map_eq_one_of_eq_act_of_map_residue_eq_rigidDataH1Pow8 below · cited by 2 · depth 37 - Classifying preimage of j(mathsf q^{qd}) computing quotient j-invariants
ModularCurve.FullLevel.Diamond.exists_classify_preimage_forall_apply_eq_cyclicQuotientJ_etale_rigidDataH1Pow_of_tatePoint_pinGamma12,170 below · cited by 1 · depth 38 - Tangent line of the Γ₀(M')∩Γ₁(ℓ) Weierstrass moduli problem
ModularCurve.FullLevel.Diamond.exists_forall_eq_map_dualNumber_smul_of_trivial_rigidDataH1Pow29 below · cited by 2 · depth 38 - Points of the rigid H₁ datum force μ_q ⊂ F
ModularCurve.FullLevel.Diamond.exists_isPrimitiveRoot_of_pt_rigidDataH1Pow_of_perfectField204 below · cited by 1 · depth 38 - Representability of the raw Γ₀(N)–Γ₁(ℓ) Weierstrass functor
ModularCurve.FullLevel.Diamond.exists_represents_raw_trivial_rigidDataH1Pow4 below · cited by 1 · depth 38 - Raw étale Γ₀(M')∩Γ₁(ℓ_g)-structure on the twisted Tate curve
ModularCurve.FullLevel.Diamond.exists_variableChange_raw_etale_tate_weightOne_level_fst_level_snd_fst_of_ker148 below · cited by 1 · depth 38 - Exactly q-1 minimal primes in the H₁ moduli ring
ModularCurve.FullLevel.Diamond.finite_minimalPrimes_and_ncard_eq_sub_one_of_jOf_eq_jqNModC_rigidDataH1Pow1,973 below · cited by 1 · depth 38 - Integrality over A[j₀] for the Γ₀(N)×Γ₁(ℓ) moduli ring
ModularCurve.FullLevel.Diamond.isIntegral_adjoin_j0_levelModuliPackageAbs_trivial_rigidDataH1Pow30 below · cited by 1 · depth 38 - j(q^N) lies in the H₁-level field for N ∣ q²M'
ModularCurve.FullLevel.Diamond.jqNModC_mem_laurentBaseChange_xHFunctionField_of_dvd_of_eq_levelH_inf_ker188 below · cited by 1 · depth 38 - Level automorphism sends j(qᵈ) to j(q^q²d)
ModularCurve.FullLevel.Diamond.levelAut_apply_qExpand_jq_eq_jqNModC_of_antidiagonal_of_ringHom_of_eq_levelH_inf_ker6 below · cited by 1 · depth 38 - Infinitesimal lifting for the H₁ rigid Weierstrass moduli problem
ModularCurve.FullLevel.Diamond.map_surjective_of_surjective_of_ker_pow_eq_bot_of_isUnit_trivial_rigidDataH1Pow26 below · cited by 1 · depth 38 - Coefficients of the transported μ_{p^k}-kernel lie in the level field
ModularCurve.FullLevel.coeff_kernelVariableChangeDeg_mem_range_of_variableChange_cuspData_xP_mem_range_gamma0Pow_level_fst58 below · cited by 1 · depth 38 - A point of the rigid full-level problem forces μ_q ⊂ F
ModularCurve.FullLevel.exists_isPrimitiveRoot_of_pt_gamma0Pow_of_perfectField204 below · cited by 1 · depth 38 - Weight-one twist of Tate(mathsf q^ℓ) rational over level ℓ²M'
ModularCurve.FullLevel.exists_variableChange_weightOne_tateBase_mem_laurentBaseChange_and_cuspData_mem_of_prime_level114 below · cited by 1 · depth 38 - Full-level moduli ring has (ℓ-1)(q-1) minimal primes
ModularCurve.FullLevel.finite_minimalPrimes_and_ncard_eq_mul_of_jOf_eq_jqNModC_gamma0Pow2,099 below · cited by 1 · depth 38 - Residues of admissible functions are integral over κ_A[jmath̄ ]
ModularCurve.FullLevel.isIntegral_adjoin_residue_jq_residue_of_mem_admissible_of_prime293 below · cited by 1 · depth 38 - Surjectivity of Γ₀(M')×Γ(ℓ) Weierstrass points along nilpotent thickenings
ModularCurve.FullLevel.map_surjective_of_surjective_of_ker_pow_eq_bot_of_isUnit_trivial_gamma0Pow17 below · cited by 1 · depth 38 - Reduction-linear relations are respected by admissible readings
ModularCurve.FullLevel.sum_algebraMap_mul_apply_eq_zero_of_sum_smul_residue_eq_zero_of_prime137 below · cited by 1 · depth 38 - Tate point of the H₁ problem reads j(q^{qd})
ModularCurve.FullLevel.Diamond.algebraMap_jqNModC_eq_cyclicQuotientJ_of_eq_map_rigidDataH1Pow_of_tatePoint_pinGamma1153 below · cited by 1 · depth 39 - Classify-preimage of j(mathsf q^{qd}) specialises to cyclic-quotient j
ModularCurve.FullLevel.Diamond.apply_eq_cyclicQuotientJ_of_classify_eq_jqNModC_rigidDataH1Pow_of_tatePoint_pinGamma192 below · cited by 1 · depth 39 - K''-rationality of transported p^k-kernel polynomial coefficients
ModularCurve.FullLevel.Diamond.coeff_kernelVariableChangeDeg_mem_range_of_variableChange_tateToricPoint_one_fst_mem_range_of_ker58 below · cited by 1 · depth 39 - Surjectivity of the rigid H₁ classifying map at the Tate point
ModularCurve.FullLevel.Diamond.exists_clC_eq_of_mem_chartAlgFin_rigidDataH1Pow_of_tatePoint_pinGamma12,132 below · cited by 1 · depth 39 - Rationality of the weight-one twisted Tate model at diamond level
ModularCurve.FullLevel.Diamond.exists_variableChange_weightOne_tateBase_one_mem_laurentBaseChange_and_tateToricPoint_mem_of_ker114 below · cited by 1 · depth 39 - Weight-2d forms on Γ_H(N²M) with toric q-expansions
ModularCurve.FullLevel.exists_modularForm_gammaH_levelH_qExpansion_eq_smul_prod_toricPoint_add_gamma0Pow4 below · cited by 1 · depth 39 - Weight 2d forms on Γ_{H^flat}(M') with toric expansions
ModularCurve.FullLevel.Diamond.exists_modularForm_gammaH_qExpansion_eq_smul_prod_toricPoint_one_sub_of_ker4 below · cited by 1 · depth 40
ModularCurve.Gamma0Pair 1
- The quotient j-invariant is a class function on Y₀(N)
ModularCurve.Gamma0Pair.cyclicQuotientJ_zmultiples_smul_gen_eq_of_mk_eq_mk3 below · cited by 4 · depth 38
ModularCurve.HahnSpecialise 3
- Bijectivity of specialisation on cyclic subgroups of order N
ModularCurve.HahnSpecialise.specialiseCycSub_bijective6 below · cited by 1 · depth 16 - Specialisation is injective on cyclic subgroups of order N
ModularCurve.HahnSpecialise.specialiseCycSub_injective0 below · cited by 1 · depth 17 - Specialisation is bijective on N-torsion
ModularCurve.HahnSpecialise.specialise_bijOn_torsion4 below · cited by 2 · depth 17
ModularCurve.HpoolLevelRing 16
- Finite étale level rings of j on the X₀(p) model
ModularCurve.HpoolLevelRing.exists_finite_etale_levelRing_jChartFin828 below · cited by 1 · depth 17 - Finite étale level rings of the modular unit on X₀(p)
ModularCurve.HpoolLevelRing.exists_finite_etale_levelRing_self444 below · cited by 1 · depth 17 - Fibres of the level ring as base changes: étaleness and rank
ModularCurve.HpoolLevelRing.etale_fiber_levelRing_and_finrank_eq_of_tensorProduct_quotient0 below · cited by 2 · depth 18 - Spreading out étaleness of a level ring over ℤ[1/f]
ModularCurve.HpoolLevelRing.exists_forall_etale_levelRing_of_etale_fiber1 below · cited by 1 · depth 18 - Generic étaleness of j-fibres on the level-p modular curve
ModularCurve.HpoolLevelRing.exists_forall_etale_rat_tensorProduct_quotient_span_aeval_jChartFin122 below · cited by 1 · depth 18 - Étale j-level sets on the mod ℓ j-chart of X₀(p)
ModularCurve.HpoolLevelRing.exists_forall_etale_zmod_tensorProduct_quotient_span_aeval_jChartFin815 below · cited by 1 · depth 18 - Unramifiedness of Ogg's unit off a polynomial divisor
ModularCurve.HpoolLevelRing.exists_forall_isUnramifiedAt_polynomial_of_aeval_notMem123 below · cited by 1 · depth 18 - Ogg's unit and the two minimal primes over p
ModularCurve.HpoolLevelRing.exists_minimalPrimes_pair_modularUnitSeries222 below · cited by 2 · depth 18 - Characteristic-p fibre dictionary for the modular unit
ModularCurve.HpoolLevelRing.exists_pFibre_dictionary353 below · cited by 2 · depth 18 - Characteristic-q Laurent realisation of the finite j-chart of level p
ModularCurve.HpoolLevelRing.exists_ringHom_laurentSeries_ker_eq_span_natCast807 below · cited by 1 · depth 18 - Finiteness of level rings of the modular unit Δ(q)/Δ(qᵖ)
ModularCurve.HpoolLevelRing.finite_levelRing273 below · cited by 1 · depth 18 - Rank of (ℚ⊗ A)/(g(j)) equals (p+1)deg g
ModularCurve.HpoolLevelRing.finrank_rat_tensorProduct_quotient_span_aeval_jChartFin111 below · cited by 1 · depth 18 - Torsion-freeness of the level rings over ℤ[1/f]
ModularCurve.HpoolLevelRing.noZeroSMulDivisors_levelRing_of_forall_isPrime0 below · cited by 1 · depth 18 - Constant stalk rank (p-1)deg g of the level ring
ModularCurve.HpoolLevelRing.rankAtStalk_levelRing_eq248 below · cited by 1 · depth 18 - Geometric fibre at ℓ≠ p of the j-chart of X₀(p)
ModularCurve.HpoolLevelRing.exists_algEquiv_residueField_tensor_quotient_span_natCast_chartRing803 below · cited by 1 · depth 19 - Degree of a level quotient for Ogg's unit on X₀(p)
ModularCurve.HpoolLevelRing.finrank_rat_tensorProduct_quotient_span_aeval246 below · cited by 1 · depth 19
ModularCurve.IgusaCover 1
- Ramification in the Igusa cover: Kummer dichotomy at a place
ModularCurve.IgusaCover.ramificationIndexAlong_incl_eq_of_ord_hasseRootFn_pow_igusaFunctionFieldX1C1,048 below · cited by 1 · depth 23
ModularCurve.IgusaScheme 132
- Base change of Igusa chart rings to a place over ℓ ∤ N
ModularCurve.IgusaScheme.exists_algHom_tensor_chartAlg_injective_isIntegrallyClosed180 below · cited by 5 · depth 13 - Chart-pinned curve model of the Igusa scheme's geometric generic fibre
ModularCurve.IgusaScheme.exists_curveModel_iso_genericFibre_galoisCompat_chartPin144 below · cited by 1 · depth 13 - Igusa chart algebras inside a fibre model with cusp chart
ModularCurve.IgusaScheme.exists_fibreModel_cuspChart_of_chartAlg743 below · cited by 7 · depth 13 - Igusa's model of X₀(N₀) over ℤ₍ₚ₎, pinned
ModularCurve.IgusaScheme.exists_finiteMapData_ratCurveModel_igusaTo1,159 below · cited by 2 · depth 13 - Finite type of the two Igusa chart algebras over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.finiteType_chartAlgFin_and_chartAlgInf120 below · cited by 37 · depth 13 - Flatness of the two-chart Igusa scheme over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.flat_igusaTo1 below · cited by 4 · depth 13 - Integrality of the two-chart Igusa scheme
ModularCurve.IgusaScheme.isIntegral0 below · cited by 6 · depth 13 - Normality of the Igusa two-chart model on affine opens
ModularCurve.IgusaScheme.isIntegrallyClosed_sections_of_isAffineOpen5 below · cited by 1 · depth 13 - Properness of the two-chart Igusa scheme over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.isProper_igusaTo1 below · cited by 12 · depth 13 - Geometric reducedness of the fibres of the Igusa scheme at level Np
ModularCurve.IgusaScheme.isReduced_pullback_igusaTo_specMap_of_not_dvd134 below · cited by 4 · depth 13 - Igusa's two-chart model is locally of finite presentation
ModularCurve.IgusaScheme.locallyOfFinitePresentation_igusaTo122 below · cited by 2 · depth 13 - Generic fibre of the Igusa scheme is smooth and geometrically integral
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_and_geometricallyIntegral_pullback_snd_igusaTo_rat859 below · cited by 2 · depth 13 - Centre pins for the chart-pinned generic fibre of the Igusa scheme
ModularCurve.IgusaScheme.coeffEmb_sub_mem_nonunits_pointEquivPlace_ofGenerator_of_chartPin0 below · cited by 5 · depth 14 - ℚ-fibre of the ℤ_{(ℓ)}-chart algebra of the Igusa model
ModularCurve.IgusaScheme.exists_algEquiv_rat_tensor_chartAlg_chartRing0 below · cited by 9 · depth 14 - Base change to ℚ̄ of the two Igusa chart algebras
ModularCurve.IgusaScheme.exists_algEquiv_tensor_chartAlg_chartRing1 below · cited by 16 · depth 14 - Cuspidal sections and cusp coordinate j(qᵖ)/jᵖ for X₀(Np)
ModularCurve.IgusaScheme.exists_algHom_chartAlgInf_coeff_zero_and_mem_nonunits_of_not_dvd83 below · cited by 6 · depth 14 - The cusp ∞ as a ℤ_{(ℓ)}-point of the pole chart
ModularCurve.IgusaScheme.exists_algHom_chartAlgInf_eq_coeff_zero2 below · cited by 2 · depth 14 - Galois-compatible generic fibre isomorphism for the Igusa scheme
ModularCurve.IgusaScheme.exists_genericFibreIso_chartPin_and_galoisCompat0 below · cited by 4 · depth 14 - Generic fibre of the Igusa scheme is the curve model
ModularCurve.IgusaScheme.exists_genericFibreIso_chartPin_and_galoisCompat_of_algEquiv_chartAlg_chartRing0 below · cited by 1 · depth 14 - Generic fibres of the Igusa model: chart pins, Galois and place compatibility
ModularCurve.IgusaScheme.exists_genericFibreIso_chartPin_galoisCompat_and_ratPlaceCompat5 below · cited by 3 · depth 14 - Igusa scheme as base change of the two-chart ℤ-model
ModularCurve.IgusaScheme.exists_isPullback_twoChartIntegralModel_int_and_iso_pullback_and_iotaFin_comp_eq7 below · cited by 3 · depth 14 - Atkin–Lehner involution wₚ on Igusa's model of X₀(Np)
ModularCurve.IgusaScheme.exists_iso_involutive_iotaFin_comp_eq_atkinLehner_of_not_dvd143 below · cited by 1 · depth 14 - Chart-pinned degeneracy pair between Igusa models of X₀(Mℓ) and X₀(M)
ModularCurve.IgusaScheme.exists_pinned_degeneracyPair_inf153 below · cited by 1 · depth 14 - Finite-map data of arbitrarily large degree on the Igusa scheme
ModularCurve.IgusaScheme.exists_schemeHomOver_finiteMapData_levelSetsGenericallyEtale1,105 below · cited by 2 · depth 14 - Maximal smooth locus of the Igusa model contains the cusps
ModularCurve.IgusaScheme.exists_smoothLocus_maximal_and_section_mem885 below · cited by 1 · depth 14 - Centre pins on special fibres of the Igusa scheme
ModularCurve.IgusaScheme.exists_spBase_and_cuspChart_centrePin_of_genericFibre_iso_ofGenerator815 below · cited by 5 · depth 14 - The Igusa scheme has a two-affine open cover by its charts
ModularCurve.IgusaScheme.exists_twoAffineOpenCover_U0_eq_chartFinOpen1 below · cited by 3 · depth 14 - Two geometric components of the j-finite chart mod p
ModularCurve.IgusaScheme.finite_minimalPrimes_tensor_chartAlgFin_mul_and_ncard_eq_two_of_not_dvd136 below · cited by 7 · depth 14 - Geometric integrality of the Igusa scheme over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.geometricallyIntegral_igusaTo848 below · cited by 5 · depth 14 - Integrality of the geometric fibre charts of the Igusa scheme
ModularCurve.IgusaScheme.isDomain_tensor_chartAlgFin_and_chartAlgInf_of_isAlgClosed805 below · cited by 11 · depth 14 - Integral closedness of K ⊗_ℤ_{(ℓ)} chartAlgFin in characteristic zero
ModularCurve.IgusaScheme.isIntegrallyClosed_tensor_chartAlgFin_of_charZero81 below · cited by 2 · depth 14 - Integral closedness of R'⊗_ℤ_{(ℓ)}chartAlgFin for a DVR base
ModularCurve.IgusaScheme.isIntegrallyClosed_tensor_chartAlgFin_of_isDiscreteValuationRing168 below · cited by 1 · depth 14 - Constant-field extension of the pole chart stays a normal domain
ModularCurve.IgusaScheme.isIntegrallyClosed_tensor_chartAlgInf_of_charZero81 below · cited by 1 · depth 14 - Integral closedness of the pole chart over a DVR
ModularCurve.IgusaScheme.isIntegrallyClosed_tensor_chartAlgInf_of_isDiscreteValuationRing168 below · cited by 1 · depth 14 - Igusa: the two-chart model of X₀(N) over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.isProper_and_smooth_and_geometricallyIntegral858 below · cited by 13 · depth 14 - Each chart of X₀(Np) has reduced fibre with two components
ModularCurve.IgusaScheme.isReduced_quotient_and_ncard_minimalPrimes_span_natCast_of_not_dvd129 below · cited by 15 · depth 14 - j(q^N) and j are mutually integral
ModularCurve.IgusaScheme.jqN_mem_chartAlgFin_and_jFull_mem_chartAlg_jqN80 below · cited by 4 · depth 14 - Supersingular points of Y₀(N)_κ lie on the second copy
ModularCurve.IgusaScheme.ker_comp_atkinLehner_le_comap_retraction_of_mem_ssJSet_of_not_dvd940 below · cited by 2 · depth 14 - Reduction of Igusa-scheme points matches the fibre model's specialisation of places
ModularCurve.IgusaScheme.pointReduction_eq_congr_spPlace_of_cuspChart_centrePin191 below · cited by 5 · depth 14 - Frobenius on the second retraction of X₀(Np) mod p
ModularCurve.IgusaScheme.retraction_one_tmul_iota_eq_pow_of_not_dvd825 below · cited by 4 · depth 14 - Ogg's unit on the two components of X₀(Np) mod p
ModularCurve.IgusaScheme.retraction_one_tmul_modularUnit_eq_prod_ssJSet_of_not_dvd936 below · cited by 6 · depth 14 - Smoothness of the Igusa model's fibre at ℓ ∤ N
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_residue825 below · cited by 7 · depth 14 - Smoothness of characteristic-zero fibres of the integral model
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_snd_toBase_int_of_charZero128 below · cited by 6 · depth 14 - Igusa chart algebras as localisations with unchanged reduction mod ℓ
ModularCurve.IgusaScheme.chartAlg_eq_and_mem_iff_and_exists_ringEquiv_quotient_span_natCast2 below · cited by 5 · depth 15 - Geometric chart rings spanned by the integral chart algebras
ModularCurve.IgusaScheme.chartRing_le_span_coeffEmb_chartAlg0 below · cited by 5 · depth 15 - Fibres of the Igusa two-chart model are connected
ModularCurve.IgusaScheme.connectedSpace_pullback_igusaTo_specMap130 below · cited by 3 · depth 15 - Two q-expansions jointly inject κ⊗𝒪 into κ((q))
ModularCurve.IgusaScheme.eq_zero_of_forall_laurentLift_apply_eq_zero_of_not_dvd130 below · cited by 1 · depth 15 - Atkin–Lehner involution preserves the ℤ₍ₚ₎[j]-chart of X₀(Np)
ModularCurve.IgusaScheme.exists_algEquiv_chartAlgFin_mul_eq_atkinLehnerInvolutionFull79 below · cited by 5 · depth 15 - Special fibres of the two Igusa chart algebras
ModularCurve.IgusaScheme.exists_algEquiv_residueField_tensor_chartAlg_chartRing799 below · cited by 7 · depth 15 - Special fibres of the Igusa charts as characteristic-ℓ chart rings
ModularCurve.IgusaScheme.exists_algEquiv_residueField_tensor_chartAlg_chartRing_apply_tmul799 below · cited by 2 · depth 15 - Special-fibre chart identifications of the Igusa scheme, compatible on overlaps
ModularCurve.IgusaScheme.exists_algEquiv_residueField_tensor_chartAlg_chartRing_compat802 below · cited by 1 · depth 15 - Constant term gives a ℤ-point of the pole chart
ModularCurve.IgusaScheme.exists_algHom_int_chartAlgInf_eq_coeff_zero1 below · cited by 3 · depth 15 - Two-chart datum for the Igusa scheme: overlap is a basic open
ModularCurve.IgusaScheme.exists_chartFinOpen_inf_chartInfOpen_eq_basicOpen_and_mul_eq_one0 below · cited by 2 · depth 15 - Geometric generic fibre of the Igusa scheme as a curve model
ModularCurve.IgusaScheme.exists_curveModel_genericFibre_iso_and_galoisCompat152 below · cited by 1 · depth 15 - Local-ring points of the Igusa scheme: finite chart or pole of j
ModularCurve.IgusaScheme.exists_eq_spec_map_comp_iotaFin_or_iotaInf_of_mem_maximalIdeal0 below · cited by 2 · depth 15 - Igusa chart rings inside a cusp-chart fibre model
ModularCurve.IgusaScheme.exists_fibreModel_cuspChart_of_chartAlg_of_lift743 below · cited by 2 · depth 15 - Chart-pinned generic fibre of the Igusa model over ℚ
ModularCurve.IgusaScheme.exists_genericFibreIso_rat_chartPin0 below · cited by 1 · depth 15 - The K-fibre of the Igusa scheme as a glued two-chart curve
ModularCurve.IgusaScheme.exists_iso_glued_pullback_igusaTo_of_algEquiv_chartAlg_chartRing0 below · cited by 2 · depth 15 - Smoothness of Igusa's model at the cusp ∞ modulo p
ModularCurve.IgusaScheme.exists_mem_and_smooth_of_section_cuspInf_of_asIdeal_ne_bot147 below · cited by 1 · depth 15 - Denominators over ℤ_{(ℓ)}[j] in the modular function field
ModularCurve.IgusaScheme.exists_mul_mem_adjoin_jFull_jqN73 below · cited by 6 · depth 15 - Two minimal primes in the mod p chart of X₀(Np)
ModularCurve.IgusaScheme.exists_retraction_pair_residueField_tensor_chartAlgFin_mul_of_not_dvd815 below · cited by 4 · depth 15 - Minimal primes over p as kernels of q-expansion reductions
ModularCurve.IgusaScheme.exists_ringHom_laurentSeries_ker_eq_of_mem_minimalPrimes_of_not_dvd129 below · cited by 2 · depth 15 - Two components of the j-chart of X₀(Np) modulo p
ModularCurve.IgusaScheme.exists_ringHom_laurentSeries_pair_chartAlgFin_mul_frobenius_of_not_dvd824 below · cited by 5 · depth 15 - Finite type of the two integral chart algebras over ℤ
ModularCurve.IgusaScheme.finiteType_int_chartAlgFin_and_chartAlgInf119 below · cited by 9 · depth 15 - The two Igusa charts meet exactly where j, resp. 1/j, is invertible
ModularCurve.IgusaScheme.iotaInf_preimage_chartFinOpen_and_iotaFin_preimage_chartInfOpen0 below · cited by 4 · depth 15 - Pinned Igusa morphism: finite, surjective generic fibre and degree
ModularCurve.IgusaScheme.isFinite_and_surjective_curveChange_specMap_rat_and_exists_functionField_of_iotaFin_comp_eq_of_isFinite874 below · cited by 3 · depth 15 - The generic fibre of the Igusa scheme is Dedekind
ModularCurve.IgusaScheme.isIntegral_and_isLocallyNoetherian_and_forall_stalk_pullback_igusaTo_specMap_rat864 below · cited by 4 · depth 15 - Integrality of the characteristic-ℓ fibres of the Igusa scheme
ModularCurve.IgusaScheme.isIntegral_pullback_igusaTo_of_charP838 below · cited by 12 · depth 15 - Characteristic-zero fibres of the Igusa scheme are integral
ModularCurve.IgusaScheme.isIntegral_pullback_igusaTo_of_charZero144 below · cited by 3 · depth 15 - Integrality of the k-fibre of the two-chart model
ModularCurve.IgusaScheme.isIntegral_pullback_toBase_int_of_isUnit_natCast854 below · cited by 4 · depth 15 - The j-finite Igusa chart ring is integrally closed
ModularCurve.IgusaScheme.isIntegrallyClosed_chartAlgFin1 below · cited by 5 · depth 15 - Integral closedness of the chart ring at the j-pole
ModularCurve.IgusaScheme.isIntegrallyClosed_chartAlgInf1 below · cited by 5 · depth 15 - Properness over ℤ of the two-chart integral model
ModularCurve.IgusaScheme.isProper_toBase_int121 below · cited by 4 · depth 15 - Reducedness of k ⊗_ℤ_{(ℓ)} chartAlgFin in characteristic ℓ
ModularCurve.IgusaScheme.isReduced_chartAlgFin_tensor164 below · cited by 1 · depth 15 - Reducedness of k ⊗_ℤ_{(ℓ)} chartAlgInf at ℓ ∤ N
ModularCurve.IgusaScheme.isReduced_chartAlgInf_tensor164 below · cited by 1 · depth 15 - Regularity at the cusp ∞ of the pole chart over ℤ₍ₚ₎
ModularCurve.IgusaScheme.isRegularLocalRing_of_isLocalization_atPrime_chartAlgInf_cuspInfty139 below · cited by 2 · depth 15 - Retraction kernel detects the ∞-component: wₚj-jᵖ∈𝔭
ModularCurve.IgusaScheme.map_le_ker_retraction_iff_mem_of_mem_minimalPrimes_of_not_dvd825 below · cited by 3 · depth 15 - Igusa scheme as two-chart integral model over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.nonempty_iso_twoChartIntegralModel0 below · cited by 3 · depth 15 - A ℤ_{(ℓ)}-point of the Igusa scheme
ModularCurve.IgusaScheme.nonempty_schemeHomOver_id_igusaTo3 below · cited by 4 · depth 15 - Places restrict along chart-pinned morphisms of Igusa schemes
ModularCurve.IgusaScheme.pointEquivPlace_eq_restrictAlong_of_chart_pin4 below · cited by 1 · depth 15 - Mutual integrality of j and j(qᵈ) at level M
ModularCurve.IgusaScheme.qExpand_jq_mem_chartAlgFin_and_jFull_mem_chartAlg81 below · cited by 10 · depth 15 - Place compatibility of the ℚ̄- and ℚ-level Igusa chart models
ModularCurve.IgusaScheme.ratPlaceCompat_of_chartPins2 below · cited by 1 · depth 15 - Krull dimension one for the special fibre of the j-finite Igusa chart
ModularCurve.IgusaScheme.ringKrullDim_localization_chartAlgFin_tensor137 below · cited by 3 · depth 15 - Krull dimension one for the pole chart of the Igusa model
ModularCurve.IgusaScheme.ringKrullDim_localization_chartAlgInf_tensor137 below · cited by 3 · depth 15 - Relative dimension one for the Igusa scheme over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_igusaTo_of_smooth_fiber1 below · cited by 1 · depth 15 - Smoothness of the j-finite Igusa chart over k
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_chartFin_residue820 below · cited by 1 · depth 15 - Smoothness of the Igusa pole chart over k
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_chartInf_residue820 below · cited by 1 · depth 15 - Smoothness of the Igusa scheme over characteristic-ℓ fields
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_of_charP827 below · cited by 6 · depth 15 - Characteristic-zero fibres of the Igusa scheme are smooth curves
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_of_charZero120 below · cited by 2 · depth 15 - Smoothness of the Igusa fibre from its two charts
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_of_chartFin_of_chartInf0 below · cited by 2 · depth 15 - Smooth fibres in characteristic ℓ ∤ N of the integral model
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_snd_toBase_int_of_charP834 below · cited by 3 · depth 15 - Transport of fibrewise smoothness from the Igusa scheme to the ℤ-model
ModularCurve.IgusaScheme.smoothOfRelativeDimension_one_pullback_snd_toBase_int_of_pullback_snd_igusaTo6 below · cited by 2 · depth 15 - Smoothness of the integral model of X₀(p) over ℤ[1/p]
ModularCurve.IgusaScheme.smooth_pullback_snd_toBase_int_localizationAway839 below · cited by 1 · depth 15 - Smoothness of the integral model after inverting N
ModularCurve.IgusaScheme.smooth_pullback_snd_toBase_int_of_isUnit_natCast838 below · cited by 5 · depth 15 - Distinct minimal primes over p together with 1/j generate the unit ideal
ModularCurve.IgusaScheme.sup_sup_span_jInvChartInf_eq_top_of_mem_minimalPrimes_of_not_dvd244 below · cited by 2 · depth 15 - Minimal primes over ℓ avoid P(y)
ModularCurve.IgusaScheme.aeval_notMem_of_mem_minimalPrimes_span_natCast3 below · cited by 1 · depth 16 - Constant term of q-expansions as an R-point of the pole chart
ModularCurve.IgusaScheme.exists_algHom_chartAlgInf_algebraMap_eq_coeff_zero0 below · cited by 1 · depth 16 - Functions integral on both Igusa charts are constants in ℤ_{(ℓ)}
ModularCurve.IgusaScheme.exists_eq_algebraMap_of_mem_chartAlgFin_of_mem_chartAlgInf3 below · cited by 1 · depth 16 - Galois-compatible generic fibre of the Igusa scheme at ̄ j
ModularCurve.IgusaScheme.exists_genericFibre_iso_ofGenerator_jBar_and_galoisCompat3 below · cited by 1 · depth 16 - Igusa scheme as base change of the integral two-chart model
ModularCurve.IgusaScheme.exists_isPullback_twoChartIntegralModel_int_and_iso_pullback7 below · cited by 2 · depth 16 - Generic fibre of the Igusa scheme as rational two-chart model
ModularCurve.IgusaScheme.exists_iso_pullback_igusaTo_rat_twoChartIntegralModel_and_iotaFin10 below · cited by 1 · depth 16 - Maximal ideal at the cusp ∞ generated by p and 1/j
ModularCurve.IgusaScheme.exists_mul_eq_natCast_mul_add_jInvChartInf_mul_of_coeff_zero_mem90 below · cited by 2 · depth 16 - Pinned degeneracy pair between Igusa schemes mathfrak X_{Mℓ}rightrightarrowsmathfrak X_M
ModularCurve.IgusaScheme.exists_pinned_degeneracyPair153 below · cited by 1 · depth 16 - Pole-chart inclusion X₀(Nq)→ X₀(q) separates components mod q
ModularCurve.IgusaScheme.exists_ringHom_chartAlgInf_comap_minimalPrimes_ne_of_not_dvd205 below · cited by 1 · depth 16 - Rank of a pinned flat degeneracy map of Igusa schemes
ModularCurve.IgusaScheme.finrank_eq_of_pinned_of_flat_morphismRestrict892 below · cited by 1 · depth 16 - Finite surjections between Igusa schemes of good reduction are flat
ModularCurve.IgusaScheme.flat_and_locallyOfFinitePresentation_of_isFinite_of_not_dvd891 below · cited by 1 · depth 16 - Fibres of the Igusa scheme are geometrically connected
ModularCurve.IgusaScheme.geometricallyConnected_pullback_snd_igusaTo131 below · cited by 1 · depth 16 - Geometric connectedness of the Igusa scheme fibres for ℓ ∤ N
ModularCurve.IgusaScheme.geometricallyConnected_pullback_snd_igusaTo_of_not_dvd806 below · cited by 1 · depth 16 - Geometric connectedness over ℤ of the two-chart model of X₀(N)
ModularCurve.IgusaScheme.geometricallyConnected_toBase_int137 below · cited by 5 · depth 16 - Igusa regularity: j-chart of the special fibre is regular of dimension 1
ModularCurve.IgusaScheme.isRegularLocalRing_localization_chartAlgFin_tensor819 below · cited by 1 · depth 16 - Regularity of the pole chart of the Igusa special fibre
ModularCurve.IgusaScheme.isRegularLocalRing_localization_chartAlgInf_tensor819 below · cited by 1 · depth 16 - Finiteness of the two-chart Čech H¹ over ℤ_{(ℓ)}
ModularCurve.IgusaScheme.moduleFinite_chartAlgMid_quotient_range_inclFin_sup_range_inclInf121 below · cited by 1 · depth 16 - A ℤ_{(ℓ)}-point of the Igusa pole chart
ModularCurve.IgusaScheme.nonempty_algHom_chartAlgInf2 below · cited by 1 · depth 16 - Reductions of the Igusa chart algebra span the characteristic-ℓ chart ring
ModularCurve.IgusaScheme.piFin_image_spans_chartAlg182 below · cited by 1 · depth 16 - Pole chart ring spanned by reductions of the integral chart algebra
ModularCurve.IgusaScheme.piInf_image_spans_chartAlg182 below · cited by 1 · depth 16 - Galois-compatible generic fibre from the chart-ring identifications
ModularCurve.IgusaScheme.exists_genericFibreIso_galoisCompat_of_algEquiv_chartAlg_chartRing0 below · cited by 1 · depth 17 - Freeness at regular points for finite surjections of Igusa schemes
ModularCurve.IgusaScheme.free_localizedModule_sections_of_isRegularLocalRing_stalk_of_isFinite877 below · cited by 2 · depth 17 - Geometric connectedness passes from the Igusa scheme to the ℤ-model
ModularCurve.IgusaScheme.geometricallyConnected_pullback_snd_toBase_int_of_pullback_snd_igusaTo6 below · cited by 1 · depth 17 - Finiteness of the j-chart degeneracy map for Igusa schemes
ModularCurve.IgusaScheme.isFinite_specMap_chartAlgFin_of_coe_eq131 below · cited by 1 · depth 17 - Geometric fibres of the Igusa scheme have no isolated points
ModularCurve.IgusaScheme.not_isOpen_singleton_pullback_igusaTo_of_not_dvd862 below · cited by 1 · depth 17 - Stalks of the Igusa scheme have Krull dimension at most two
ModularCurve.IgusaScheme.ringKrullDim_stalk_le_two1 below · cited by 1 · depth 18 - Centre of a valuation ring passes between the Igusa charts
ModularCurve.IgusaScheme.forall_mem_asIdeal_iff_mem_nonunits_of_iotaFin_eq_of_iotaInf_eq0 below · cited by 1 · depth 21 - Finite chart and base generate the function field
ModularCurve.IgusaScheme.subfieldClosure_range_germToFunctionField_union_range_eq_top0 below · cited by 1 · depth 21 - Base change of the j-finite chart ring to a ramified DVR
ModularCurve.IgusaScheme.exists_algEquiv_tensor_chartAlgFin_mul_chartAlgFin_laurentBaseChange_of_not_dvd233 below · cited by 2 · depth 27 - Supersingular primes contain all minimal primes of the special fibre
ModularCurve.IgusaScheme.forall_minimalPrimes_le_of_mem_ssJSet_tensor_chartAlgFin_mul_of_not_dvd943 below · cited by 2 · depth 27 - Supersingular points of the j-chart meet every component mod p
ModularCurve.IgusaScheme.forall_minimalPrimes_span_natCast_le_of_forall_apply_jChartFin_mem_ssJSet_of_not_dvd10 below · cited by 2 · depth 27 - Crossing primes on X₀(Np) have supersingular j-invariant
ModularCurve.IgusaScheme.exists_mem_ssJSet_tmul_sub_mem_of_ker_le_of_ker_comp_le_tensor_chartAlgFin_mul_of_not_dvd937 below · cited by 2 · depth 28 - Crossing points of the finite chart have supersingular j
ModularCurve.IgusaScheme.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_natCast_le_of_forall_of_not_dvd159 below · cited by 2 · depth 28 - Base change of the j-finite Igusa chart to a DVR
ModularCurve.IgusaScheme.exists_algEquiv_tensor_chartAlgFin_mul_chartAlgFin_laurentBaseChange_of_charZero_of_not_dvd894 below · cited by 1 · depth 33 - Pole chart of the X₀(Np) model base-changes to a DVR
ModularCurve.IgusaScheme.exists_algEquiv_tensor_chartAlgInf_mul_chartAlgInf_laurentBaseChange_of_charZero_of_not_dvd894 below · cited by 1 · depth 35 - Two minimal primes in the geometric pole chart at level Np
ModularCurve.IgusaScheme.finite_minimalPrimes_tensor_chartAlgInf_mul_and_ncard_eq_two_of_not_dvd136 below · cited by 1 · depth 35 - Two q-expansions jointly detect zero after base change to κ
ModularCurve.IgusaScheme.eq_zero_of_forall_laurentLift_apply_eq_zero_chartAlgInf_of_not_dvd130 below · cited by 1 · depth 36
ModularCurve.InLine 1
- Points on the line of P₀ lie in ℤP₀
ModularCurve.InLine.some_mem_zmultiples_some_of_nonsingular3 below · cited by 3 · depth 29
ModularCurve.IsDiamondPullbackModL 4
- Kernel of a diamond pull-back action is ±Γ_H(M)
ModularCurve.IsDiamondPullbackModL.apply_eq_one_iff_gamma0Units_mem3 below · cited by 5 · depth 21 - Diamond pullback action fixes the Γ₀(N) q-expansion field
ModularCurve.IsDiamondPullbackModL.apply_eq_self_of_coe_mem_qExpFunctionFieldC_gamma00 below · cited by 4 · depth 26 - Uniqueness of a diamond pull-back action on ̄ F(Γ_H(M))
ModularCurve.IsDiamondPullbackModL.unique103 below · cited by 2 · depth 26 - Pull-back formula for the reduced diamond action at level M
ModularCurve.IsDiamondPullbackModL.coe_apply_eq_of_mem_Gamma0_of_level_mul1,242 below · cited by 1 · depth 29
ModularCurve.IsFrickeAutFull 1
- Uniqueness of the Fricke automorphism of F_N^{full}
ModularCurve.IsFrickeAutFull.eq_frickeInvolutionFull1 below · cited by 2 · depth 16
ModularCurve.IsGamma0PowAt 7
- Invariance of the (p,k)-kernel predicate under variable change
ModularCurve.IsGamma0PowAt.variableChange5 below · cited by 31 · depth 28 - Module-finite algebra representing Γ₀(M') kernel tuples
ModularCurve.IsGamma0PowAt.exists_moduleFinite_represents_tuple2 below · cited by 4 · depth 31 - Kernel polynomial at p^k cuts out a cyclic subgroup of order p^k
ModularCurve.IsGamma0PowAt.isAddCyclic_closure_and_natCard_eq_pow12 below · cited by 2 · depth 33 - Unique lifting of Γ₀(M') kernel tuples along nilpotent thickenings
ModularCurve.IsGamma0PowAt.existsUnique_tuple_map_eq_of_surjective_of_ker_pow_eq_bot12 below · cited by 8 · depth 35 - Uniqueness of the cyclic subgroup cut out by a Γ₀(p^k)-kernel
ModularCurve.IsGamma0PowAt.zmultiples_eq_zmultiples_of_isRoot_of_addOrderOf_eq3 below · cited by 4 · depth 35 - Unique lifting of Γ₀(p^k) kernel polynomials along nilpotent surjections
ModularCurve.IsGamma0PowAt.existsUnique_map_eq_of_surjective_of_ker_pow_eq_bot11 below · cited by 7 · depth 36 - Splitting a Γ₀(p^k) kernel polynomial over a faithfully flat extension
ModularCurve.IsGamma0PowAt.exists_faithfullyFlat_map_eq_prod_X_sub_C_of_ne_two4 below · cited by 1 · depth 38
ModularCurve.IsGamma1Link 2
- Linked Γ₁(ℓ)-point is ℓ^{k-1} times a kernel root
ModularCurve.IsGamma1Link.exists_root_toPoint_eq_pow_smul_toPoint_of_isAlgClosed16 below · cited by 4 · depth 31 - Lifting the Γ₁-link along a nilpotent thickening
ModularCurve.IsGamma1Link.of_map_of_surjective_of_ker_pow_eq_bot18 below · cited by 4 · depth 37
ModularCurve.IsGamma1Point 4
- Unique Tate normal form for a Γ₁(ℓ)-point
ModularCurve.IsGamma1Point.existsUnique_variableChange_isNormalForm6 below · cited by 4 · depth 32 - Invariance of Γ₁(ℓ)-point data under Weierstrass coordinate changes
ModularCurve.IsGamma1Point.variableChange0 below · cited by 14 · depth 32 - Rigidity of Γ₁(ℓ)-points under variable changes
ModularCurve.IsGamma1Point.variableChange_eq_one_of_smul_eq_of_variableChange_eq7 below · cited by 5 · depth 32 - Unique lifting of Γ₁(ℓ)-points along nilpotent thickenings
ModularCurve.IsGamma1Point.existsUnique_map_eq_of_surjective_of_ker_pow_eq_bot5 below · cited by 7 · depth 36
ModularCurve.IsInfReductionMap 9
- Kernel of the ∞-reduction map is killed by Uₚ
ModularCurve.IsInfReductionMap.baseChange_genU_self_apply_eq_zero_of_apply_eq_zero1,383 below · cited by 1 · depth 27 - Reduction to the infinity component intertwines ⟨ d⟩
ModularCurve.IsInfReductionMap.comp_baseChange_genDia_eq_genDiffModL_comp1,255 below · cited by 6 · depth 27 - Reduction at infinity intertwines T_ℓ with differential Hecke operator
ModularCurve.IsInfReductionMap.comp_baseChange_genT_eq_genDiffModL_comp1,292 below · cited by 2 · depth 27 - Reduction at infinity intertwines U_q for q ≠ p
ModularCurve.IsInfReductionMap.comp_baseChange_genU_eq_genDiffModL_comp_of_ne608 below · cited by 2 · depth 27 - Reduction maps at infinity carry Uₚ to Frobenius push-forward
ModularCurve.IsInfReductionMap.comp_baseChange_genU_self_eq_genDiffModL_comp138 below · cited by 2 · depth 27 - Kernel of an ∞-reduction map is spanned by p-divisible classes
ModularCurve.IsInfReductionMap.mem_span_tmul_intTwoCuspReduce_of_apply_eq_zero1 below · cited by 2 · depth 28 - Transposed Hecke correspondence at q ≠ p after mod p reduction
ModularCurve.IsInfReductionMap.exists_smul_correspondence_heckeAlphaModLH_heckeBetaModLH_apply_eq_of_ne911 below · cited by 1 · depth 33 - Fricke involution pair exchanging the mod p degeneracy maps
ModularCurve.IsInfReductionMap.exists_algEquiv_pair_intertwines_heckeAlphaModLH_heckeBetaModLH_and_diffQExp_pullbackAlong_eq639 below · cited by 1 · depth 34 - q-expansion of the Fricke pullback of ρ^∞(̄ f)
ModularCurve.IsInfReductionMap.smul_diffQExp_pullbackAlong_eq_ofPowerSeries_map_of_reduction_slash_fricke244 below · cited by 1 · depth 35
ModularCurve.IsLevelPStructure 13
- Level-p data transport along Weierstrass variable changes
ModularCurve.IsLevelPStructure.variableChange0 below · cited by 27 · depth 18 - Two level-ℓ structures differ by an invertible matrix
ModularCurve.IsLevelPStructure.exists_eq_nsmul_add_nsmul_of_isLevelPStructure4 below · cited by 5 · depth 29 - Unique normal form of a level-ℓ structure
ModularCurve.IsLevelPStructure.existsUnique_variableChange_isNormalForm6 below · cited by 2 · depth 30 - Rigidity of Katz full level-ℓ structures
ModularCurve.IsLevelPStructure.variableChange_eq_one_of_smul_eq_of_variableChange_eq_of_prime2 below · cited by 6 · depth 30 - Level-ℓ data gives an independent pair of ℓ-torsion points
ModularCurve.IsLevelPStructure.exists_nsmul_eq_zero_and_dvd_of_zsmul_add_zsmul_eq_zero4 below · cited by 6 · depth 32 - Relabelling a level-ℓ structure by a matrix invertible mod ℓ
ModularCurve.IsLevelPStructure.relabel_of_isUnit_det4 below · cited by 6 · depth 32 - Relabelling a level-ℓ structure is a right GL₂(ℤ/ℓ)-action
ModularCurve.IsLevelPStructure.relabel_relabel_and_relabel_one_and_relabel_eq_of_map_eq4 below · cited by 6 · depth 32 - Relabelling level-ℓ data commutes with Weierstrass coordinate changes
ModularCurve.IsLevelPStructure.relabel_variableChange5 below · cited by 1 · depth 32 - Relabelling rigidity for Katz level-ℓ structures over a field
ModularCurve.IsLevelPStructure.map_eq_of_relabel_variableChange_eq4 below · cited by 1 · depth 33 - Level-p data yield an injective map (ℤ/p)² → W(F)
ModularCurve.IsLevelPStructure.exists_injective_addMonoidHom_zmod_prod3 below · cited by 1 · depth 34 - Unique lifting of level-ℓ structures along nilpotent surjections
ModularCurve.IsLevelPStructure.existsUnique_map_eq_of_surjective_of_ker_pow_eq_bot7 below · cited by 8 · depth 35 - Distinct x-coordinates in a level-ℓ datum for ℓ ≥ 3
ModularCurve.IsLevelPStructure.isUnit_xP_sub_xQ0 below · cited by 1 · depth 35 - Relabelling acts freely on level-ℓ structures
ModularCurve.IsLevelPStructure.map_eq_one_of_relabel_eq4 below · cited by 1 · depth 37
ModularCurve.IsModPFormFn 2
- Base change of mod-p modular functions along a field map
ModularCurve.IsModPFormFn.coeffMap0 below · cited by 6 · depth 17 - Descent of the mod-p weight condition along algebraic extensions
ModularCurve.IsModPFormFn.of_coeffMap_algebraMap0 below · cited by 4 · depth 17
ModularCurve.IsModuliPlaceOf 2
- Moduli places of a Γ₀(N)-class share their centre
ModularCurve.IsModuliPlaceOf.mem_nonunits_iff_of_isIntegral_jModElt399 below · cited by 1 · depth 18 - Moduli places contain everything integral over K[̃ j]
ModularCurve.IsModuliPlaceOf.mem_toValuationSubring_of_isIntegral_jModElt0 below · cited by 1 · depth 18
ModularCurve.IsPlaceReductionModL 2
- Reduction mod ℓ acts coordinatewise on j and j_N
ModularCurve.IsPlaceReductionModL.coordinate_clauses268 below · cited by 9 · depth 13 - Reduction mod ℓ carries the cusp ∞ to the q-adic place
ModularCurve.IsPlaceReductionModL.apply_cuspInftyBar_eq_and_eq_cuspInftyBar_of_apply_eq_of_ord_ne_zero276 below · cited by 1 · depth 29
ModularCurve.JH 14
- Atkin–Lehner relation for Uₚ on Pic⁰ of modular curves
ModularCurve.JH.heckeOperatorHAlong_pullbackAlongHom_add_pullbackAlongHom_atkinLehner_smul_eq_pullbackAlongHom_comp_heckeBetaHBar_pushforwardAlongHom233 below · cited by 1 · depth 13 - Finite flat model for twisted pⁿ-torsion of J_H(M)
ModularCurve.JH.exists_finiteFlat_prolongation_pi_torsion_diamondTwist_of_not_dvd_of_galoisFactorsThroughFiniteLevel1,639 below · cited by 1 · depth 16 - Inertia away from Mp acts trivially on TₚJ_H
ModularCurve.JH.tateGaloisRep_eq_one_of_mem_inertiaSubgroupIn969 below · cited by 1 · depth 17 - n-torsion of J_H is fixed by a number field's Galois group
ModularCurve.JH.exists_finiteDimensional_smul_eq_self_of_torsion301 below · cited by 6 · depth 18 - Inertia-fixed ℓ-adic vectors on J_H(M) are monodromy plus p-old
ModularCurve.JH.exists_pow_smul_mem_span_inertia_sub_sup_old_of_rep_eq_self_tateModule_of_dvd_of_not_sq_dvd3,585 below · cited by 1 · depth 22 - Pull-back and push-forward between J_H(M) and J₁(M)
ModularCurve.JH.exists_pullback_pushforward_jOne_galois_and_comp_eq_nsmul_and_sum_diamondOneBar_eq223 below · cited by 5 · depth 22 - Freeness and rank 2g of T_ℓ J_H
ModularCurve.JH.finite_and_free_and_finrank_tateModule_eq_two_mul_genusFF853 below · cited by 7 · depth 22 - Inertia displacements at p ‖ M: ⟨ d₁⟩ Frobₚ = p Uₚ
ModularCurve.JH.genOpH_dia_galois_smul_sub_eq_natCast_smul_genOpH_U_of_isFrobeniusAt_of_mem_inertia2,844 below · cited by 1 · depth 22 - Forgetful pull-backs commute with the two degeneracy pull-backs
ModularCurve.JH.pullbackAlongHom_pullbackAlongHom_eq_degeneracyPullbackPair_pullbackAlongHom3 below · cited by 4 · depth 22 - Pull-back J_H(M)→ J₁(M) commutes with T_ℓ and ⟨ d⟩
ModularCurve.JH.pullbackAlongHom_heckeOperatorHAlong_eq_heckeOperatorOneBar_and_pullbackAlongHom_diamondHBar_eq_diamondOneBar277 below · cited by 3 · depth 23 - Fricke stability of a toric lattice in Tₚ J_H(M), up to p-powers
ModularCurve.JH.exists_pow_smul_tateEnd_fricke_mem_toricLattice_of_degeneracySwap0 below · cited by 1 · depth 25 - Injectivity of the degeneracy Gram operator on Tate modules
ModularCurve.JH.tateModule_eq_zero_of_forall_pushforwardAlongHom_degeneracy_eq_zero903 below · cited by 1 · depth 25 - Projection formula for the p-adic Weil pairing along X₁(M)→ X_H(M)
ModularCurve.JH.weilPairing_tateModule_jOne_pull_pull_eq_natCast_mul_of_pushforward_pullback_eq_nsmul120 below · cited by 1 · depth 25 - Surjectivity of the level-n map TₚJ_H → J_H[pⁿ]
ModularCurve.JH.exists_tateModule_proj_eq_of_mem_torsionBy264 below · cited by 2 · depth 32
ModularCurve.JHNeronObjectAtP 153
- Representing Pic⁰ makes the level datum an abelian scheme
ModularCurve.JHNeronObjectAtP.LevelData.abelianSchemePropertyBundle_of_nonempty_representsRelSubPic1,567 below · cited by 18 · depth 11 - Néron object for J_H(M) at p ∥ M with torus coordinates
ModularCurve.JHNeronObjectAtP.exists_levelData_representsRelSubPic_dictionary_of_xHDRModelAtP_torusCoords2,647 below · cited by 12 · depth 11 - Toric-by-finite filtration of Tₚ J_H(M) at p ∥ M
ModularCurve.JHNeronObjectAtP.exists_toricFiniteFiltration_tateModule_jH_self70 below · cited by 3 · depth 11 - Uₚ + wₚ^* equals β^*α_* on J_H(M)
ModularCurve.JHNeronObjectAtP.genOpH_U_add_ofAlgAut_smul_eq_pull_degPts_of_coe_eq_qExpand361 below · cited by 8 · depth 11 - Frobenius acts as Uₚ on toric ℓ^k-torsion
ModularCurve.JHNeronObjectAtP.genOpH_U_smul_eq_cyclotomicCharacter_toZModPow_smul_of_mem_toricPts123 below · cited by 1 · depth 11 - Frobenius on node units: p-th power twisted by the crossing permutation
ModularCurve.JHNeronObjectAtP.ptsSp_symm_eq_nodeUnit_pow_comp_frobPerm_of_isFrobeniusAt100 below · cited by 4 · depth 11 - Uₚ permutes node units of the glued Picard group by σ
ModularCurve.JHNeronObjectAtP.ptsSp_symm_hecke_U_nodeUnit_eq_nodeUnit_comp62 below · cited by 4 · depth 11 - Toric points are m-torsion in Pic⁰
ModularCurve.JHNeronObjectAtP.toricPts_le_torsion1 below · cited by 5 · depth 11 - Rigidity of A-morphisms μ_m^t → G_A on the special fibre
ModularCurve.JHNeronObjectAtP.eq_of_muBaseChange_residue_comp_eq39 below · cited by 6 · depth 12 - Prime-to-p toric points as Hom(ℤ[SS]⁰,μ_m)
ModularCurve.JHNeronObjectAtP.exists_addEquiv_toricPts_characterLattice_hom_of_ptsSp_nodeUnit6 below · cited by 2 · depth 12 - Hecke action on the special fibre is additive
ModularCurve.JHNeronObjectAtP.exists_addMonoidHom_apply_eq_ptsSp_symm_schemeHomOverComp_hecke0 below · cited by 4 · depth 12 - Endomorphisms act on toric lifts through M₀ mod m
ModularCurve.JHNeronObjectAtP.exists_comp_toricLift_fibreRestrictAlong_eq_toricLift_comp_mapDomainAlgHom40 below · cited by 3 · depth 12 - Endomorphisms act on the special-fibre torus through a lattice map
ModularCurve.JHNeronObjectAtP.exists_mapDomain_comp_torusFibre_eq_torusFibre_comp_fibreRestrictAlong22 below · cited by 3 · depth 12 - ψ-twist of the toric fibre differs by an integral map
ModularCurve.JHNeronObjectAtP.exists_mapRingHom_comp_torusFibre_eq_mapDomain_comp_torusFibre_comp_baseTwist22 below · cited by 2 · depth 12 - Toric characters read as node-unit classes in the special fibre
ModularCurve.JHNeronObjectAtP.exists_nodeUnit_eq_residue_toricLift_and_mul_and_eq_one0 below · cited by 6 · depth 12 - Frobenius acting on toric points via the reduced Frobenius matrix
ModularCurve.JHNeronObjectAtP.exists_smul_toricPoint_eq_toricPoint_galoisValues_comp_mapDomainAlgHom40 below · cited by 2 · depth 12 - Toric lifts μ_m^t → G_A over a place above p
ModularCurve.JHNeronObjectAtP.exists_toricLift_of_torusFibre43 below · cited by 1 · depth 12 - Frobenius and Uₚ torus matrices are mutually inverse
ModularCurve.JHNeronObjectAtP.frobMatrix_comp_torusMatrix_eq_id_of_hecke_U3 below · cited by 2 · depth 12 - Uₚ plus Atkin–Lehner equals degeneracy pull-push on J_H(M)
ModularCurve.JHNeronObjectAtP.genOpH_U_add_smul_eq_pull_degPts_of_roof234 below · cited by 1 · depth 12 - Hecke stability of the toric points of the Néron object
ModularCurve.JHNeronObjectAtP.genOpH_mem_toricPts64 below · cited by 5 · depth 12 - Stability of finite and toric points under an endomorphism over A
ModularCurve.JHNeronObjectAtP.mem_finPts_and_mem_toricPts_of_schemeHomOver_baseChange_pts62 below · cited by 3 · depth 12 - Finite points are the A-extendable m-torsion classes
ModularCurve.JHNeronObjectAtP.mem_finPts_iff_and_isTorsionPoint_section_and_specialPt0 below · cited by 11 · depth 12 - Frobenius equivariance of the special-fibre dictionary for J_H(M)
ModularCurve.JHNeronObjectAtP.ptsSp_symm_frobeniusTwist_eq_glueMap_of_pointReduction99 below · cited by 3 · depth 12 - Special fibre of the second degeneracy pull-back on Pic⁰
ModularCurve.JHNeronObjectAtP.ptsSp_symm_schemeHomOverComp_ptsSp_degPull_one_eq_mk_of_forall_apply_eq_zero_of_pullbackAlong986 below · cited by 3 · depth 12 - Toric point map: injective homomorphism, image the toric m-torsion
ModularCurve.JHNeronObjectAtP.toricPoint_convMul_and_injective_and_mem_toricPts_iff_and_natCard0 below · cited by 13 · depth 12 - Toric lifts reduce to torus characters on the special fibre
ModularCurve.JHNeronObjectAtP.exists_torusPt_residue_toricLift_and_torusFibre_injective0 below · cited by 1 · depth 13 - Principal divisors, constants and rational places of ̄ F
ModularCurve.JHNeronObjectAtP.hasPrincipalDivisors_and_constantsAreBase_and_surjective_residueField_fbar75 below · cited by 34 · depth 13 - Galois action on toric points of a μ_m^t-lift
ModularCurve.JHNeronObjectAtP.inertia_smul_eq_and_exists_decomposition_smul_eq_of_muLift42 below · cited by 1 · depth 13 - Multiplication by m on a base-changed level-Γ_H(M) Néron object
ModularCurve.JHNeronObjectAtP.locallyQuasiFinite_quasiCompact_flat_schemeNsmul_baseChange12 below · cited by 4 · depth 13 - Specialisation of A-integral points of a J_H(M) Néron object
ModularCurve.JHNeronObjectAtP.exists_addSubgroup_extendsToPlace_addMonoidHom_gluedPic0_eq_ptsSp_symm34 below · cited by 7 · depth 19 - Special-fibre dictionary respects multiples, identity and torsion
ModularCurve.JHNeronObjectAtP.ptsSp_nsmul_and_ptsSp_zero_and_smul_eq_zero_iff_isTorsionPoint0 below · cited by 4 · depth 20 - Surjectivity of the degeneracy push-forward on Pic⁰
ModularCurve.JHNeronObjectAtP.degPts_zero_surjective_of_pushforwardAlong140 below · cited by 4 · depth 23 - Inertia-invariant torsion of J_H(M) bounded by finite part
ModularCurve.JHNeronObjectAtP.exists_forall_natCard_torsion_inf_inertiaInvariants_le_natCard_finPts_mul_of_abelJacobiPin_of_wgen2,584 below · cited by 2 · depth 23 - Reduction of the finite part of T_ℓ J_H(M) and Uₚ
ModularCurve.JHNeronObjectAtP.exists_linearMap_finiteSubmodule_tateModule_jH_toPic0Pair_of_ne121 below · cited by 1 · depth 23 - p-old lattice in T_ℓ J_H(M) for p ∥ M
ModularCurve.JHNeronObjectAtP.exists_oldLattice_inf_toricLattice_eq_bot_and_finiteLattice_le_sup_tateModule_jH_of_ne1,445 below · cited by 1 · depth 23 - The p-old lattice in Tₚ J_H(M) at p ∥ M
ModularCurve.JHNeronObjectAtP.exists_oldLattice_inf_toricLattice_eq_bot_and_finiteLattice_le_sup_tateModule_jH_self1,489 below · cited by 2 · depth 23 - Toric Tate vectors as inertia coboundaries up to bounded ℓ-power
ModularCurve.JHNeronObjectAtP.exists_pow_smul_mem_span_inertia_sub_of_mem_toricLattice_tateModule_jH_of_abelJacobiPin_of_atkinLehner3,373 below · cited by 1 · depth 23 - Uₚ preserves the reduction domain and shifts node units by σ
ModularCurve.JHNeronObjectAtP.genOpH_U_mem_and_sp_genOpH_U_eq_nodeUnit_comp63 below · cited by 1 · depth 23 - Additivity of the level-M/p point dictionaries Λ
ModularCurve.JHNeronObjectAtP.levelData_pts_add_and_ptsSp_add_of_surjective_degPts0 below · cited by 4 · depth 23 - Diamond ⟨ d⟩ acts on the glued special fibre by glueMap
ModularCurve.JHNeronObjectAtP.ptsSp_symm_schemeHomOverComp_hecke_dia_eq_glueMap61 below · cited by 3 · depth 23 - Inertia fixes the finite m-torsion when p ∤ m
ModularCurve.JHNeronObjectAtP.smul_eq_self_of_mem_inertiaSubgroupIn_of_mem_finPts_of_coprime_of_representsRelSubPic5 below · cited by 1 · depth 23 - Inertia moves prime-to-p torsion into the toric subgroup
ModularCurve.JHNeronObjectAtP.smul_sub_mem_toricPts_of_mem_inertia_of_representsRelSubPic_of_atkinLehner2,802 below · cited by 2 · depth 23 - Uₚ and Frobenius on the toric lattice at ℓ=p
ModularCurve.JHNeronObjectAtP.tateGenOpH_U_comp_tateGaloisRep_frobenius_eq_cyclotomicCharacter_smul_of_mem_toricLattice_of_eq71 below · cited by 2 · depth 23 - Block form of Uₚ on the glued Pic⁰ pair
ModularCurve.JHNeronObjectAtP.toPic0Pair_ptsSp_symm_hecke_U_eq_blockOp61 below · cited by 5 · depth 23 - Good reduction identifies ℓ-adic Tate modules for ℓ ≠ p
ModularCurve.JHNeronObjectAtP.LevelData.exists_linearEquiv_tateModule_proj_eq_ptsSp_symm_section_of_ne27 below · cited by 1 · depth 24 - The base point of a level datum is Spec of a ring map
ModularCurve.JHNeronObjectAtP.LevelData.exists_ringHom_comp_eq_algebraMap_and_sigmaA_eq_specMap0 below · cited by 4 · depth 24 - Degeneracy maps vanish on toric p-power points
ModularCurve.JHNeronObjectAtP.degPts_eq_zero_of_mem_toricPts185 below · cited by 4 · depth 24 - Transport between two Néron objects for J_H(M) at p ∥ M
ModularCurve.JHNeronObjectAtP.exists_addEquiv_galois_map_toricPts_eq_map_finPts_eq_of_representsRelSubPic_of_abelianScheme73 below · cited by 2 · depth 24 - Inertia reaches the toric part of J_H(M)
ModularCurve.JHNeronObjectAtP.exists_forall_mem_toricPts_exists_smul_sub_eq_of_coprime_of_abelJacobiPin_of_atkinLehner3,370 below · cited by 1 · depth 24 - Abel–Jacobi-pinned Néron object for J_H(M) at p ∥ M
ModularCurve.JHNeronObjectAtP.exists_levelData_representsRelSubPic_level_abelJacobiPin_of_xHDRModelAtP_of_atkinLehner2,648 below · cited by 2 · depth 24 - Bounded exponent for the degeneracy push–pull kernel on torsion
ModularCurve.JHNeronObjectAtP.exists_nsmul_eq_zero_of_forall_degPts_pull_add_pull_eq_zero1,247 below · cited by 1 · depth 24 - Good inertia-invariant classes of J_H(M) extend over A
ModularCurve.JHNeronObjectAtP.extendsToPlace_pts_of_isGoodClass_of_abelJacobiPin_offDiag1,228 below · cited by 1 · depth 24 - Toric m-torsion via A-sections with trivial abelian-quotient reduction
ModularCurve.JHNeronObjectAtP.mem_toricPts_iff_exists_fibreMap_abqFibre_eq_one37 below · cited by 2 · depth 24 - Toric part lies in finite part of pⁿ-torsion, with bound
ModularCurve.JHNeronObjectAtP.toricPts_le_finPts_and_finite_and_natCard_finPts_le769 below · cited by 1 · depth 24 - Finiteness and order of the m-torsion of the special fibre
ModularCurve.JHNeronObjectAtP.LevelData.isFinite_schemeKerStr_special_and_finrank_eq_natCard_torsion736 below · cited by 1 · depth 25 - Rigidity of homomorphic μ^t_m-points over a henselian place
ModularCurve.JHNeronObjectAtP.eq_of_muBaseChange_residue_comp_eq_levelData26 below · cited by 1 · depth 25 - Galois-equivariant transport between two Néron objects at p
ModularCurve.JHNeronObjectAtP.exists_addEquiv_galois_map_toricPts_eq_map_finPts_eq_of_representsRelSubPic_of_ptsLaw_of_abelianScheme71 below · cited by 1 · depth 25 - Transport of the torus along an isomorphism of Néron objects
ModularCurve.JHNeronObjectAtP.exists_baseChange_comp_fst_eq_and_torusFibre_comp_eq_mapDomain_of_iso_of_representsRelSubPic_of_abelianScheme69 below · cited by 1 · depth 25 - Inertia displacements in TₚJ_H(M) lift to identity-reducing points
ModularCurve.JHNeronObjectAtP.exists_eq_tateGaloisRep_sub_self_and_reduction_of_mem_inertiaSubgroupIn_of_reflects_of_period0 below · cited by 1 · depth 25 - Divisibility between toric subgroups of the Néron object at p
ModularCurve.JHNeronObjectAtP.exists_mem_toricPts_mul_nsmul_eq0 below · cited by 1 · depth 25 - A power of Uₚ as Frobenius convolved with Verschiebung
ModularCurve.JHNeronObjectAtP.exists_pow_cartierDual_reduction_U_eq_frobenius_conv_verschiebung_of_finPtsWitness_of_isDiscreteValuationRing_of_bridge2,707 below · cited by 2 · depth 25 - Orthogonal of the toric lattice in Tₚ J_H(M)
ModularCurve.JHNeronObjectAtP.exists_pow_smul_mem_toricLattice_sup_oldLattice_of_forall_weilPairing_eq_zero469 below · cited by 1 · depth 25 - Inertia differences specialise into node units on J_H(M)
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_pts_smul_sub_eq_and_ptsSp_symm_mem_range_nodeUnit_of_mem_inertia_of_abelJacobiPins_of_representsRelSubPic1,915 below · cited by 1 · depth 25 - Special m-kernel: finiteness and order m^t·(dim A_κ[m])²
ModularCurve.JHNeronObjectAtP.isFinite_schemeKerStr_special_and_finrank_eq_mul_sq13 below · cited by 3 · depth 25 - Quasi-finiteness, quasi-compactness and flatness of [p^k] after base change
ModularCurve.JHNeronObjectAtP.locallyQuasiFinite_quasiCompact_flat_schemeNsmul_pow_baseChange_levelData144 below · cited by 1 · depth 25 - Finite part equals A-sections of the m-kernel scheme
ModularCurve.JHNeronObjectAtP.natCard_finPts_eq_natCard_sections_schemeKer1 below · cited by 2 · depth 25 - Tame monodromy bound for ℓ^k-torsion of J_H(M)
ModularCurve.JHNeronObjectAtP.natCard_torsion_le_natCard_image_smul_sub_mul_natCard_inertiaInvariants_of_forall_smul_sub_mem_toricPts3 below · cited by 1 · depth 25 - Two relative group laws with equal unit agree at genPt
ModularCurve.JHNeronObjectAtP.relativeGroupLaw_mul_eq_mul_genPt_of_one_eq19 below · cited by 1 · depth 25 - Degeneracy homomorphisms kill the toric point of the special fibre
ModularCurve.JHNeronObjectAtP.schemeHomOverComp_torusFibre_degeneracyHom_eq_one0 below · cited by 1 · depth 25 - Finiteness, flatness and fibres of the m-kernel over A
ModularCurve.JHNeronObjectAtP.schemeKerStr_baseChange_props16 below · cited by 2 · depth 25 - Inertia fixes the prime-to-p toric points at level Γ_H(M)
ModularCurve.JHNeronObjectAtP.smul_eq_self_of_mem_inertiaSubgroupIn_of_mem_toricPts3 below · cited by 1 · depth 25 - Inertia sends prime-to-p torsion into the toric subgroup
ModularCurve.JHNeronObjectAtP.smul_sub_mem_toricPts_of_mem_inertia_of_abelJacobiPin_of_wgen2,802 below · cited by 1 · depth 25 - Toric and finite parts of J_H(M)[m] multiply to #J_H(M)[m]
ModularCurve.JHNeronObjectAtP.toricPts_le_and_finPts_le_and_natCard_toricPts_mul_natCard_finPts_eq_of_coprime1,969 below · cited by 1 · depth 25 - Toric points of coprime orders: ab lies in a join b
ModularCurve.JHNeronObjectAtP.toricPts_mul_le_sup_of_coprime0 below · cited by 1 · depth 25 - Transport of toric lifts along an isomorphism of Néron objects
ModularCurve.JHNeronObjectAtP.exists_equiv_forall_toricLift_comp_eq_of_iso_of_representsRelSubPic_of_abelianScheme69 below · cited by 1 · depth 26 - fppf-local sections of m-torsion over the abelian-quotient square
ModularCurve.JHNeronObjectAtP.exists_fppfCover_section_schemeKer_of_abqFibre5 below · cited by 1 · depth 26 - Shear isomorphism for m-torsion over the abelian-quotient kernel
ModularCurve.JHNeronObjectAtP.exists_iso_pullback_schemeKer_torus_of_abqFibre0 below · cited by 1 · depth 26 - Special-fibre torus as joint kernel of the abelian-quotient pair
ModularCurve.JHNeronObjectAtP.exists_iso_torus_kerPair_abqFibre2 below · cited by 3 · depth 26 - Cartier transpose of Uₚ⟨ d₀⟩ is Frobenius (ordinary part)
ModularCurve.JHNeronObjectAtP.exists_units_forall_point_comp_cartierTranspose_U_comp_diamond_valuation_sub_pow_lt_one_of_ordinaryIdempotent_of_bridge1,358 below · cited by 1 · depth 26 - Node-unit Poincaré bundle puts a special-fibre class in the toric part
ModularCurve.JHNeronObjectAtP.ptsSp_symm_schemeHomOverComp_mem_range_nodeUnit_of_isNodeUnitModule_poincare_pullbackAlong1,672 below · cited by 1 · depth 26 - Verschiebung on the special fibre of the level-(M/p) abelian scheme
ModularCurve.JHNeronObjectAtP.LevelData.exists_verschiebung_comp_frobenius_eq_schemeNsmul36 below · cited by 2 · depth 27 - Frobenius and Verschiebung on the p-divisible levels mod p
ModularCurve.JHNeronObjectAtP.LevelData.restrict_frobenius_eq_pow_and_cartierDual_map_restrict_verschiebung_eq_pow_of_abelianSchemePropertyBundle9 below · cited by 1 · depth 27 - Descent of the abelian-quotient fibre maps to 𝔽ₚ
ModularCurve.JHNeronObjectAtP.exists_abqFibre_descent_zmodp7 below · cited by 1 · depth 27 - Descended diamond and Uₚ as D_Λ F on the p-fibre
ModularCurve.JHNeronObjectAtP.exists_descent_diamond_both_and_comp_hecke_U_eq_and_eq_verschiebung_of_blockOp_of_frobPullback_of_not_sq_dvd117 below · cited by 2 · depth 27 - Verschiebung equals Uₚ⟨ d₀⟩ on the connected part
ModularCurve.JHNeronObjectAtP.exists_units_forall_qc_comp_baseChange_U_comp_diamond_comp_eq_qc_comp_verschiebung_of_ordinaryIdempotent_of_bridge1,315 below · cited by 1 · depth 27 - Rank of the finite part of p^v-torsion of J_H(M)
ModularCurve.JHNeronObjectAtP.finrank_finitePart_schemeKer_baseChange_eq_pow_of_representsRelSubPic1,956 below · cited by 1 · depth 27 - Uₚ as Frobenius pull-back on glued special-fibre classes
ModularCurve.JHNeronObjectAtP.ptsSp_symm_hecke_U_mk_eq_mk_frobPullback_and_exists_mk_eq_of_snd_eq_zero63 below · cited by 3 · depth 27 - Order of the special p^v-kernel from representability
ModularCurve.JHNeronObjectAtP.LevelData.isFinite_schemeKerStr_special_and_finrank_eq_pow_two_mul_genusFF_of_representsRelSubPic1,940 below · cited by 3 · depth 28 - Frobenius, Uₚ and a diamond give [p] on ker abq₁
ModularCurve.JHNeronObjectAtP.exists_units_pullbackFst_abqFibre_comp_relFrobenius_comp_hecke_U_comp_hecke_dia_eq_comp_schemeNsmul121 below · cited by 1 · depth 28 - Frobenius equivariance of the special-fibre dictionary Λ.ptsSp
ModularCurve.JHNeronObjectAtP.levelData_ptsSp_frobeniusPushforward_eq_schemeHomOverComp_frobenius_ptsSp_of_hsp109 below · cited by 2 · depth 28 - Connected ordinary part of G[p^v] lies in ker(abq₁)
ModularCurve.JHNeronObjectAtP.mono_lift_and_exists_specMap_qc_comp_baseChange_comp_lift_eq_comp_pullbackFst_abqFibre_of_ordinaryIdempotent_of_bridge1,288 below · cited by 1 · depth 28 - Uₚ plus cross map equals degeneracy composite on glued Pic⁰
ModularCurve.JHNeronObjectAtP.ptsSp_symm_hecke_U_add_crossMap_eq_ptsSp_symm_degeneracyHom_degPull62 below · cited by 1 · depth 28 - Special m-kernel of the level-(M/p,H') abelian scheme has degree m^{2g'}
ModularCurve.JHNeronObjectAtP.LevelData.isFinite_schemeKerStr_special_and_finrank_eq_pow_two_mul_genusFF_of_abelianSchemePropertyBundle1,708 below · cited by 1 · depth 29 - Frobenius factorisation of Uₚ on the abelian-quotient coordinate
ModularCurve.JHNeronObjectAtP.exists_abqFibre_one_comp_baseChange_hecke_U_eq_comp_relFrobenius_comp_abqFibre_one_of_not_sq_dvd1,283 below · cited by 1 · depth 29 - Reducedness of the unit fibre of `abqFibre 1`
ModularCurve.JHNeronObjectAtP.isReduced_pullback_abqFibre_one_baseChange_one7 below · cited by 1 · depth 29 - The maps ιᵥ intertwine [n]^* with scheme-level [n]
ModularCurve.JHNeronObjectAtP.specMap_nsmulAlgHom_comp_eq_comp_schemeNsmul_of_forall_point_mul0 below · cited by 1 · depth 29 - Homomorphic level maps factor through the embedded p^v-torsion
ModularCurve.JHNeronObjectAtP.LevelData.exists_bialgHom_specMap_comp_eq_of_isHom_baseChange_level0 below · cited by 2 · depth 30 - Finite flat closed subgroup with toric points is the toric lift
ModularCurve.JHNeronObjectAtP.exists_bialgEquiv_comp_toricLift_eq_of_isClosedImmersion_of_flat_of_forall_mem_toricPts_iff14 below · cited by 3 · depth 30 - A bound p^{vt} for the joint kernel on finite levels
ModularCurve.JHNeronObjectAtP.finrank_quotient_sup_map_ker_counit_le_pow_toricRank_of_specMap_comp_eq4 below · cited by 1 · depth 30 - Uₚ and Frobenius on κ̄-points of the Néron fibre
ModularCurve.JHNeronObjectAtP.forall_point_comp_hecke_U_comp_abqFibre_one_eq_comp_abqFibre_one_comp_relFrobenius_comp_degPull_comp_hecke_dia_comp_abqFibre_zero_of_not_sq_dvd1,279 below · cited by 1 · depth 30 - Inertia-cyclotomic p-torsion classes lie in the finite part
ModularCurve.JHNeronObjectAtP.mem_finPts_of_inertia_cyclotomic2 below · cited by 3 · depth 30 - Ordinary corner: reduction to identity iff inertia acts cyclotomically
ModularCurve.JHNeronObjectAtP.reducesToOne_iff_inertia_cyclotomic_of_mem_corner_of_mem_finPts_of_ordinary_of_abelJacobiPin_of_inertF_of_levelData_of_algEquiv3,537 below · cited by 1 · depth 30 - Counting identity in the ordinary corner of J_H(M)[p]
ModularCurve.JHNeronObjectAtP.ncard_corner_finPts_mul_toricPts_eq_ncard_reducesToOne_mul_cyclotomic_of_abelJacobiPin_of_levelData_of_algEquiv3,191 below · cited by 1 · depth 31 - Toric–finite splitting on the ordinary corner of J_H(M)[p]
ModularCurve.JHNeronObjectAtP.ncard_corner_inter_toricPts_mul_ncard_corner_inter_finPts_eq_of_abelJacobiPin_of_representsRelSubPicLevel_of_levelData_of_algEquiv3,169 below · cited by 1 · depth 31 - Cyclotomic inertia forces finite-part points to reduce to one
ModularCurve.JHNeronObjectAtP.reducesToOne_of_inertia_cyclotomic_of_mem_finPts21 below · cited by 1 · depth 31 - Pairing annihilator of identity-reducing corner points is inertia-cyclotomic
ModularCurve.JHNeronObjectAtP.adjointCorner_finPts_forall_reducesToOne_pairing_eq_one_iff_inertia_cyclotomic_of_pairing_of_abelJacobiPin2,803 below · cited by 1 · depth 32 - Toric ⊆ identity-reducing ⊆ finite part; finiteness of J[p]^f
ModularCurve.JHNeronObjectAtP.corner_toricPts_subset_reducesToOne_subset_finPts_addSubgroup_cyclotomic_finPts_finite_of_abelJacobiPin2,789 below · cited by 1 · depth 32 - Idempotent and μₚ-pairing between corner and adjoint corner
ModularCurve.JHNeronObjectAtP.exists_idempotent_pairing_corner_adjointCorner_perfect_galois_radical_ncard_toric_cyclotomic_eq_of_abelJacobiPin_of_levelData_of_algEquiv3,182 below · cited by 1 · depth 32 - Transport of q-expansion fields and places along κ(P)→ K
ModularCurve.JHNeronObjectAtP.exists_ringHom_placeMap_injective_ord_eq_ssPlaces_qExpFunctionFieldC_of_ringHom81 below · cited by 1 · depth 32 - Constant-field extension of q-expansion function fields to K
ModularCurve.JHNeronObjectAtP.exists_ringHom_ringHom_placeMap_ord_eq_qExpFunctionFieldC_of_isAlgClosed58 below · cited by 2 · depth 32 - Membership in `finPts p` via supersingular orders of g
ModularCurve.JHNeronObjectAtP.mem_finPts_iff_forall_ssPlacesQExp_dvd_ord_of_rootFunction_smul_of_coe_eq_coeffMap_residue_of_abelJacobiPin_of_algEquiv2,604 below · cited by 1 · depth 32 - Toric points in a Hecke corner and its Weil annihilator
ModularCurve.JHNeronObjectAtP.ncard_corner_inter_toricPts_eq_ncard_weilAnnihilator_inter_toricPts_of_abelJacobiPin_of_representsRelSubPicLevel_of_algEquiv840 below · cited by 2 · depth 32 - Vanishing of the first glued component of the reduced w_*x
ModularCurve.JHNeronObjectAtP.toPic0Pair_ptsSp_symm_atkinLehner_fst_eq_zero_iff_exists_point_reducesToOne_of_mem_corner_of_mem_finPts_bridgePins1,324 below · cited by 1 · depth 32 - Toric and finite p-torsion: product of orders equals #J[p]
ModularCurve.JHNeronObjectAtP.toricPts_le_torsion_and_finPts_le_torsion_and_natCard_mul_natCard_eq_of_representsRelSubPicLevel3,049 below · cited by 2 · depth 32 - p-divisibility of the reduced root function's divisor
ModularCurve.JHNeronObjectAtP.dvd_ord_of_mem_finPts_of_coe_eq_coeffMap_residue_tauFree524 below · cited by 2 · depth 33 - Configured representative of a p-torsion class with Néron section
ModularCurve.JHNeronObjectAtP.exists_configured_rep_ord_mul_pow_eq_of_extendsToPlace_pts_of_smul_eq_zero1,151 below · cited by 2 · depth 33 - Fricke endomorphism over A of the J_H(M) Néron object inducing w_M
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_baseChange_pts_ofAlgAut_fricke_of_atkinLehnerComplement_placePin_of_representsRelSubPic_abelJacobi234 below · cited by 2 · depth 33 - Reduced Néron section of a finite p-torsion class
ModularCurve.JHNeronObjectAtP.exists_section_toPic0Pair_eq_mk_of_mem_finPts_of_forall_dvd_ord_tauFree1,164 below · cited by 2 · depth 33 - Divisibility of residue orders implies the class extends over A
ModularCurve.JHNeronObjectAtP.extendsToPlace_pts_of_forall_dvd_ord_residue_of_abelJacobiPin_offDiag_of_wgen1,989 below · cited by 1 · depth 33 - Finite part at level Γ_H: membership criterion
ModularCurve.JHNeronObjectAtP.mem_finPts_iff0 below · cited by 3 · depth 33 - Order of the finite p^v-torsion of J_H(M) at p ∥ M
ModularCurve.JHNeronObjectAtP.natCard_finPts_eq_pow_of_representsRelSubPic1,964 below · cited by 2 · depth 33 - Supersingular places of the level M/p q-expansion field count toricRank+1
ModularCurve.JHNeronObjectAtP.natCard_ssPlacesQExp_eq_toricRank_add_one_of_charP82 below · cited by 1 · depth 33 - Supersingular places over any algebraically closed field of characteristic p
ModularCurve.JHNeronObjectAtP.natCard_ssPlacesQExp_eq_toricRank_add_one_univ82 below · cited by 1 · depth 33 - Cyclotomic points of a Hecke corner and its Weil annihilator
ModularCurve.JHNeronObjectAtP.ncard_corner_inertiaCyclotomic_eq_ncard_weilAnnihilator_inertiaCyclotomic_of_abelJacobiPin_of_representsRelSubPicLevel520 below · cited by 1 · depth 33 - Trivial reduction iff the level-one layer point reduces to the identity
ModularCurve.JHNeronObjectAtP.ptsSp_symm_section_eq_zero_iff_exists_point_reducesToOne_of_mem_finPts_of_closedImmersion3 below · cited by 1 · depth 33 - Second coordinate detects vanishing of reductions on the ordinary corner
ModularCurve.JHNeronObjectAtP.ptsSp_symm_section_eq_zero_of_toPic0Pair_snd_eq_zero_of_mem_corner_of_mem_finPts_bridgePins63 below · cited by 1 · depth 33 - Atkin–Lehner swaps the two component coordinates of reduction
ModularCurve.JHNeronObjectAtP.toPic0Pair_ptsSp_symm_section_atkinLehner_fst_eq_zero_iff_snd_eq_zero_of_mem_finPts1,307 below · cited by 1 · depth 33 - Toric and finite p-torsion as mutual annihilators
ModularCurve.JHNeronObjectAtP.toricPts_finPts_mutual_annihilator_weilDatum_pairing_residueChar_of_abelJacobiPin_of_degeneracy3,116 below · cited by 2 · depth 33 - p-divisibility of the reduced divisor of a p-th root
ModularCurve.JHNeronObjectAtP.dvd_ord_of_iterate_mul_eq_one_of_barPt_comp_eq_pts_of_coe_eq_coeffMap_residue512 below · cited by 1 · depth 34 - Configured representative of a p-torsion class extending at P
ModularCurve.JHNeronObjectAtP.exists_configured_rep_and_isUnit_mul_pow_of_extendsToPlace_pts_of_smul_eq_zero1,134 below · cited by 1 · depth 34 - Configured representative of a finite p-torsion class at p
ModularCurve.JHNeronObjectAtP.exists_configured_rep_pic0Mk_eq_toPic0Pair_mk_of_mem_finPts_of_forall_dvd_ord_tauFree1,161 below · cited by 1 · depth 34 - Special-fibre pⁿ-torsion in abelian-quotient coordinates at p ∥ M
ModularCurve.JHNeronObjectAtP.exists_mem_finPts_toPic0Pair_ptsSp_symm_eq_and_eq_zero_iff_and_of_mem_toricPts_of_not_sq_dvd114 below · cited by 1 · depth 34 - Torsion Néron point extending over a place: its m-fold multiple is the unit
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_barPt_comp_eq_pts_and_iterate_mul_eq_one_of_extendsToPlace_of_nsmul_eq_zero1 below · cited by 2 · depth 34 - Composing group-law endomorphisms inducing a composite map on J_H
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_baseChange_pts_comp_of_forall_pts_eq0 below · cited by 1 · depth 34 - Model automorphism over A induces an endomorphism of G_A
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_baseChange_pts_ofAlgAut_of_baseChangeModelAut_of_relativeGroupLaw_eq_of_representsRelSubPic_baseChange_abelJacobi163 below · cited by 1 · depth 34 - Model automorphism gives a homomorphic endomorphism over A
ModularCurve.JHNeronObjectAtP.exists_schemeHomOver_baseChange_pts_ofAlgAut_of_modelAut_of_relativeGroupLaw_eq_of_representsRelSubPic_abelJacobi163 below · cited by 1 · depth 34 - Generation of `GluedPic0` by liftable point-pair differences
ModularCurve.JHNeronObjectAtP.mem_closure_gluedPic0_mk_configuredPair78 below · cited by 1 · depth 34 - Reduction of bidegree-(0,0) divisors of configured points
ModularCurve.JHNeronObjectAtP.ptsSp_symm_eq_mk_sum_of_pts_sum_configured945 below · cited by 1 · depth 34 - Reduction of an Atkin–Lehner translate depends only on the reduction
ModularCurve.JHNeronObjectAtP.resPt_comp_eq_of_resPt_comp_eq_of_pts_smul_atkinLehner_of_abelJacobiPin95 below · cited by 1 · depth 34 - Stability of the finite part under the Atkin–Lehner translate
ModularCurve.JHNeronObjectAtP.wbar_mem_finPts_of_mem_finPts_of_abelJacobiPin_tauFree60 below · cited by 3 · depth 34 - Toric p-torsion pairs trivially with finite p-torsion
ModularCurve.JHNeronObjectAtP.weilDatum_pairing_eq_one_of_mem_toricPts_of_mem_finPts_of_abelJacobiPin_of_degeneracy_of_representsRelSubPicLevel3,100 below · cited by 1 · depth 34 - Points over A: θ-twist equals composition with N
ModularCurve.JHNeronObjectAtP.baseChangePointOfBase_pts_ofAlgAut_smul_eq_comp_of_classifies_rigidify_pullback_curveChange_baseChange_of_abelJacobi158 below · cited by 1 · depth 35 - Generic divisor of a presentation of σ^*Poincaré on the Pl-model
ModularCurve.JHNeronObjectAtP.exists_divisor_ord_presentation_poincare_pullbackAlong_eq_of_barPt_comp_eq_pts125 below · cited by 2 · depth 35 - Finite p-torsion splits off the two degeneracy pull-backs
ModularCurve.JHNeronObjectAtP.exists_eq_add_pull_add_pull_of_mem_finPts_of_abelJacobiPin771 below · cited by 1 · depth 35 - Toric pⁿ-classes specialise to zero
ModularCurve.JHNeronObjectAtP.ptsSp_symm_eq_zero_of_mem_toricPts_pow2 below · cited by 1 · depth 35 - Model automorphisms act on J_H-points via θ
ModularCurve.JHNeronObjectAtP.pts_ofAlgAut_smul_eq_pts_comp_of_classifies_rigidify_pullback_curveChange_of_abelJacobi158 below · cited by 1 · depth 35 - Toric p-torsion pairs trivially with old p-torsion
ModularCurve.JHNeronObjectAtP.weilDatum_pairing_eq_one_of_mem_toricPts_of_eq_mk_pullbackAlong_of_abelJacobiPin232 below · cited by 1 · depth 35 - Toric p-torsion pairs trivially with identity-reduction classes
ModularCurve.JHNeronObjectAtP.weilDatum_pairing_eq_one_of_mem_toricPts_of_resPt_eq_one_of_abelJacobiPin_of_representsRelSubPicLevel3,083 below · cited by 1 · depth 35 - Lifting p^k-torsion along a level-(M/p) Néron datum
ModularCurve.JHNeronObjectAtP.LevelData.exists_pts_eq_barPt_comp_and_ptsSp_symm_eq_of_smul_eq_zero_of_abelianScheme737 below · cited by 1 · depth 36 - Presentation divisor of σ^*P is D' up to principal divisors
ModularCurve.JHNeronObjectAtP.exists_forall_divisor_congrRingEquiv_eq_add_ord_of_range_eq_lSpaceOn_restrict_poincare_pullbackAlong_of_barPt_comp_eq_pts118 below · cited by 1 · depth 36 - Group law, reduction and rigidity for A-sections of G
ModularCurve.JHNeronObjectAtP.exists_section_mul_inv_one_and_ptsSp_symm_eq0 below · cited by 1 · depth 36 - Inertia acts on toric m-points through the cyclotomic character
ModularCurve.JHNeronObjectAtP.smul_eq_nsmul_of_mem_toricPts_of_mem_inertiaSubgroupIn0 below · cited by 1 · depth 36 - The q-expansion of j lies in the fibre function field
ModularCurve.JHNeronObjectAtP.exists_coe_eq_jqModC_fbar6 below · cited by 1 · depth 39
ModularCurve.JHPlaceSpecialization 115
- Node-value law from regularity law at supersingular nodes
ModularCurve.JHPlaceSpecialization.ProlongationDatum.nodeValueLaw_of_regularityLaw_of_typeDichotomy3 below · cited by 1 · depth 25 - Second residue of a lower-level function is Frobenius of first
ModularCurve.JHPlaceSpecialization.ProlongationDatum.residueSnd_alpha_eq_qExpFrobeniusModL_residueFst_of_qExpand144 below · cited by 1 · depth 25 - Component map and glued specialization for X_H(M) at p ∥ M
ModularCurve.JHPlaceSpecialization.exists_componentMap_gluedSpecialization_of_isModel_of_coe_of_unit_of_cusp_of_attachedAnnulus_of_slope_of_fixReg1,958 below · cited by 1 · depth 25 - Existence of a prolongation datum with Gauss characterisation of R₁
ModularCurve.JHPlaceSpecialization.exists_prolongationDatum_mem_integers_iff_gauss137 below · cited by 1 · depth 25 - Gauss prolongation and place specialization for X_{H'}(M/p)
ModularCurve.JHPlaceSpecialization.exists_regularProlongation_sp_gauss_res_qexp_mapDomain_unique_surjective858 below · cited by 1 · depth 25 - Gauss specialisation inhabits the J_H place-specialisation structure
ModularCurve.JHPlaceSpecialization.exists_sp_eq_of_gauss54 below · cited by 1 · depth 25 - Finiteness of the diamond–Frobenius fixed locus on places
ModularCurve.JHPlaceSpecialization.finite_setOf_fixed_of_eq_gammaLift272 below · cited by 12 · depth 25 - Affine places descend along the Frobenius on places
ModularCurve.JHPlaceSpecialization.isAffinePlace_of_isAffinePlace_qExpFrobeniusPlaceModL52 below · cited by 8 · depth 25 - Affine places are stable under q-Frobenius and diamonds
ModularCurve.JHPlaceSpecialization.isAffinePlace_qExpFrobeniusPlaceModL_and_isAffinePlace_smul_diamondActionModL53 below · cited by 19 · depth 25 - Galois behaviour of the Gauss specialisation of places
ModularCurve.JHPlaceSpecialization.sp_smul_eq_of_mem_inertiaSubgroupIn_and_sp_smul_eq_qExpFrobeniusPlaceModL_of_isFrobeniusAt3 below · cited by 1 · depth 25 - End-slope law at both ends of a node annulus
ModularCurve.JHPlaceSpecialization.ProlongationDatum.annulus_ord_residue_eq_one_and_endSlope_both_ends_of_forall_isUnit_evalAt_mem_integers0 below · cited by 1 · depth 26 - An integral spanning set with jointly surjective residues
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_finset_isIntegral_span_residue_surjective319 below · cited by 1 · depth 26 - Level-M/p functions fill the first residue field
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_residue_alpha_eq1 below · cited by 3 · depth 26 - Specialization carries div(v) to div of its R₁-residue
ModularCurve.JHPlaceSpecialization.ProlongationDatum.mapDomain_sp_eq_ord_residue_alpha_full190 below · cited by 1 · depth 26 - First prolongation equals the Gauss ring of q-expansions
ModularCurve.JHPlaceSpecialization.ProlongationDatum.mem_integers_iff_gauss143 below · cited by 7 · depth 26 - Second residue vanishes when the first vanishes at a node
ModularCurve.JHPlaceSpecialization.ProlongationDatum.residue_snd_eq_zero_of_residue_fst_hasValue_zero_of_nodeValueLaw0 below · cited by 3 · depth 26 - δ injective; cuspidal places reduce to non-affine places
ModularCurve.JHPlaceSpecialization.delta_injective_and_not_isAffinePlace_reduce_of_isCuspidal_isZeroSide605 below · cited by 7 · depth 26 - Glued specialisation on the inertia invariants of J_H(M)
ModularCurve.JHPlaceSpecialization.exists_addMonoidHom_isGluedSpecialization_of_isModel_of_coe_of_unit_of_cusp_of_orient1,755 below · cited by 1 · depth 26 - Surjective component map, good representatives, principal good divisor
ModularCurve.JHPlaceSpecialization.exists_comp_sndDegLaw_surjective_repOfKer_principalGood_of_isModel_of_coe_of_unit_of_cusp_of_attachedAnnulus_of_slope_of_fixReg1,618 below · cited by 1 · depth 26 - Gauss prolongation to the level-M modular function field
ModularCurve.JHPlaceSpecialization.exists_regularProlongation_mem_integers_iff_gauss_and_residue_coeffMap135 below · cited by 5 · depth 26 - Supersingular places are fixed by Frobenius–diamond–Frobenius
ModularCurve.JHPlaceSpecialization.fixed_of_mem_ssPlacesQExp1,241 below · cited by 10 · depth 26 - Places with nonzero order at Δ(q)/Δ(qᵖ) are cuspidal
ModularCurve.JHPlaceSpecialization.isCuspidal_of_ord_ne_zero_of_coe_eq_coeffEmb_modularUnitSeries46 below · cited by 5 · depth 26 - Vanishing component map forces good classes at level Γ_H
ModularCurve.JHPlaceSpecialization.isGoodClass_of_comp_eq_zero_of_exists_isGoodDiv55 below · cited by 1 · depth 26 - Cuspidal places lie on the ∞-side or the 0-side
ModularCurve.JHPlaceSpecialization.isInftySide_or_isZeroSide_of_isCuspidal271 below · cited by 17 · depth 26 - Zero side and infinity side of the cusps are disjoint
ModularCurve.JHPlaceSpecialization.not_isInftySide_of_isZeroSide138 below · cited by 14 · depth 26 - Cusp orders of the residue of the modular unit Δ(q)/Δ(qᵖ)
ModularCurve.JHPlaceSpecialization.ProlongationDatum.ord_residue_eq_mul_ord_of_coe_eq_modularUnitSeries_of_not_isAffinePlace67 below · cited by 1 · depth 27 - Component group of J_H(M) at p ∥ M from annulus depths
ModularCurve.JHPlaceSpecialization.exists_depth_comp_depthCompLaw_annulusDepthLaw_sndDegLaw_surjective_repOfKer_principalGood_of_coe_of_unit_of_cusp_of_attachedAnnulus_of_slope_of_fixReg1,617 below · cited by 1 · depth 27 - Divisibility of Pic⁰ of the reduced modular function field
ModularCurve.JHPlaceSpecialization.exists_zsmul_eq_pic0_fbar617 below · cited by 1 · depth 27 - Cuspidality for j from cuspidality for j(qᵖ)
ModularCurve.JHPlaceSpecialization.isCuspidal_of_isCuspidalPrime123 below · cited by 5 · depth 27 - Glued principality of good principal gluing data
ModularCurve.JHPlaceSpecialization.isGluedPrincipal_glueData_of_forall_apply_eq_ord_of_isModel_of_coe_of_unit_of_cusp_of_orient1,257 below · cited by 1 · depth 27 - Specialisation of the zero and polar divisors of j
ModularCurve.JHPlaceSpecialization.mapDomain_sp_zeros_sub_algebraMap_eq_and_mapDomain_sp_poles_eq_of_coe_eq_jqModC592 below · cited by 3 · depth 27 - Order of Ogg's unit Δ(q)/Δ(qᵖ) at ∞-side places
ModularCurve.JHPlaceSpecialization.ord_eq_mul_ord_of_coe_eq_coeffEmb_modularUnitSeries_of_isInftySide165 below · cited by 2 · depth 27 - Zeros of j-a specialise under place-specialisation packets
ModularCurve.JHPlaceSpecialization.ord_pos_sp_sub_algebraMap_of_ord_pos593 below · cited by 10 · depth 27 - Non-integral j at w forces a pole at sp(w)
ModularCurve.JHPlaceSpecialization.ord_sp_neg_of_forall_ord_sub_algebraMap_le594 below · cited by 10 · depth 27 - Sum of ramification weights of ∞-side places equals one
ModularCurve.JHPlaceSpecialization.sum_ramificationIndexAlong_filter_isInftySide_fiberAlong_eq_one_of_forall_ord_sub_nonpos348 below · cited by 2 · depth 27 - Common normalisation of a good function for both prolongations
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_smul_mem_integers_residue_ne_zero_of_isGoodDiv_of_admissible_of_unit_of_cusp1,255 below · cited by 2 · depth 28 - First-component chart-local membership from regularity on the ∞-side
ModularCurve.JHPlaceSpecialization.ProlongationDatum.mem_chartLocalSetFst_of_isCuspChartFstAt296 below · cited by 1 · depth 28 - One-sided divisor and cusp laws from a model and a unit
ModularCurve.JHPlaceSpecialization.ProlongationDatum.oneSidedDivisorLaw_and_oneSidedCuspLaw_of_isModel_of_unit0 below · cited by 7 · depth 28 - Surjectivity of the depth component map for J_H(M) at p ∥ M
ModularCurve.JHPlaceSpecialization.comp_surjective_of_depthCompLaw_of_annulusInf58 below · cited by 1 · depth 28 - Vanishing depth component class of a principal divisor
ModularCurve.JHPlaceSpecialization.componentGroupProj_depthDual_add_degree_sndDiv_smul_eq_zero_of_div_of_annulusInf_of_fixReadAffine269 below · cited by 1 · depth 28 - A depth component map for J_H(M) at p ∥ M
ModularCurve.JHPlaceSpecialization.exists_comp_depthCompLaw_of_principalLaw_of_annulusInf0 below · cited by 1 · depth 28 - A depth function obeying the annulus depth law at every node
ModularCurve.JHPlaceSpecialization.exists_depth_forall_annulusDepthLaw16 below · cited by 1 · depth 28 - Inertia-fixed strict places of both kinds avoiding a finite set
ModularCurve.JHPlaceSpecialization.exists_families_isStrictFst_isStrictSnd_notMem_forall_inertia_smul_eq_of_gammaLift_ed2406 below · cited by 7 · depth 28 - Kernel classes of the component reading admit good representatives
ModularCurve.JHPlaceSpecialization.exists_isGoodDiv_pic0Mk_eq_of_comp_eq_zero_of_depthCompLaw_of_annulusInf_of_verticalSlope_of_fixReg1,483 below · cited by 1 · depth 28 - Principal good divisor of bidegree (m(e),-m(e)) from vertical slopes
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_isGoodDiv_degree_fstDiv_eq_sum_lcm_div_of_annulus_of_verticalSlope194 below · cited by 1 · depth 28 - Inertia-fixed representatives of inertia-invariant classes in J_H(M)
ModularCurve.JHPlaceSpecialization.exists_rep_inertiaFixed_support_strict_or_node_of_mem_inertiaInvariants_of_annulus_of_fixReg1,596 below · cited by 1 · depth 28 - Existence of an ∞-side place above b
ModularCurve.JHPlaceSpecialization.exists_restrictAlong_eq_and_isInftySide_of_forall_ord_sub_nonpos338 below · cited by 3 · depth 28 - Glued principality of the gluing datum of a common unit
ModularCurve.JHPlaceSpecialization.isGluedPrincipal_glueData_of_forall_apply_eq_ord_of_mem_integers_of_residue_ne_zero_of_isModel_of_unit_of_cusp_of_orient1,245 below · cited by 1 · depth 28 - Non-∞-side ramification above b sums to at least p
ModularCurve.JHPlaceSpecialization.le_sum_ramificationIndexAlong_filter_not_isInftySide_fiberAlong346 below · cited by 1 · depth 28 - Depth at an inertia-fixed place is chart-independent
ModularCurve.JHPlaceSpecialization.valuation_evalAt_param_eq_of_annulus_of_annulus0 below · cited by 1 · depth 28 - Cusp chart at infinity: integrality and regularity over a cusp
ModularCurve.JHPlaceSpecialization.ProlongationDatum.mem_integers_and_residue_mem_and_mem_of_mem_cuspChartSetInf147 below · cited by 2 · depth 29 - Inertia-invariant rational positions on the supersingular annuli
ModularCurve.JHPlaceSpecialization.exists_annulusPositionLaw_inertiaInvariant_exists_fixed_of_annulus9 below · cited by 2 · depth 29 - Twist-type divisors: inertia-fixed strict part plus glued-trivial good part
ModularCurve.JHPlaceSpecialization.exists_inertiaFixed_isStrict_add_isGoodDiv_gluedMk_eq_zero_add_principal_of_isTwistType_of_inertiaStable_of_annulus_of_fixRead1,476 below · cited by 2 · depth 29 - Twist type after subtracting an inertia-fixed divisor
ModularCurve.JHPlaceSpecialization.exists_inertiaFixed_isTwistType_sub_of_inertiaStable_of_annulus417 below · cited by 1 · depth 29 - Inertia-stable representatives with strict or nodal support
ModularCurve.JHPlaceSpecialization.exists_inertiaStable_pic0Mk_eq_support_strict_or_node_of_inertiaStable1,545 below · cited by 1 · depth 29 - Good function with node residue orders -lcm(e)/e(s)
ModularCurve.JHPlaceSpecialization.exists_isGoodDiv_ord_residue_eq_neg_lcm_div_of_annulus_of_verticalSlope0 below · cited by 1 · depth 29 - Good representative of an inertia-fixed class with vanishing component reading
ModularCurve.JHPlaceSpecialization.exists_isGoodDiv_pic0Mk_eq_of_forall_componentGroupProj_depthDual_add_eq_zero_of_annulusInf_of_verticalSlope_of_fixRead1,482 below · cited by 1 · depth 29 - Inertia-fixed admissible representative of an inertia-stable divisor
ModularCurve.JHPlaceSpecialization.exists_principal_degZero_forall_support_sub_inertia_smul_eq_of_splitting1,483 below · cited by 1 · depth 29 - A simple zero after specialisation is attained at one place
ModularCurve.JHPlaceSpecialization.ord_eq_one_and_forall_ord_eq_zero_of_forall_sp_eq_imp_ord_nonneg_of_ord_eq_one0 below · cited by 1 · depth 29 - Two-sided representative of a glued-trivial divisor class on X_H(M)
ModularCurve.JHPlaceSpecialization.ProlongationDatum.IsModel.exists_isStrictFst_isStrictSnd_reduceFst_eq_reduceSnd_eq_pic0Mk_eq_of_isGoodDiv_of_gammaLift_of_unit_of_cusp1,258 below · cited by 1 · depth 30 - Rigidity of divisors with equal first-kind and second-kind reductions
ModularCurve.JHPlaceSpecialization.ProlongationDatum.IsModel.sum_single_add_sum_single_eq_of_ord_eq_nsmul_sub_of_gammaLift_of_discLaw_of_unit_of_cusp1,269 below · cited by 1 · depth 30 - Inertia-equivariant function with prescribed simple zero on X_H(M)
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_ord_eq_one_forall_isStrict_reduceFst_reduceSnd_notMem_forall_inertia_smul_eq1,543 below · cited by 1 · depth 30 - Gauss lemma at the cusp: integrality over A[x'⁻¹]
ModularCurve.JHPlaceSpecialization.ProlongationDatum.mem_integralOverPoleChart_of_mem_integers_of_forall_inv_mem_imp_mem338 below · cited by 1 · depth 30 - Tent-weighted circle degrees of inertia-stable divisors are integers
ModularCurve.JHPlaceSpecialization.den_twistCircleDeg_eq_one_of_inertiaStable_of_annulus54 below · cited by 4 · depth 30 - Glued Picard classes from inertia-fixed strict divisors on X_H(M)
ModularCurve.JHPlaceSpecialization.exists_inertiaFixed_isStrict_gluedMk_glueData_eq_of_annulus427 below · cited by 1 · depth 30 - Good representative with vanishing glued class for a twisted divisor
ModularCurve.JHPlaceSpecialization.exists_isGoodDiv_gluedMk_eq_zero_pic0Mk_eq_of_isTwistOf_of_gluedMk_twistSpData_eq_zero_of_inertiaStable_of_annulus1,463 below · cited by 1 · depth 30 - Strict places in general position on X_H(M)
ModularCurve.JHPlaceSpecialization.exists_isStrictFst_isStrictSnd_generalPosition_disjoint_forall_inertia_smul_eq_of_gammaLift_of_unit409 below · cited by 2 · depth 30 - Existence of a twisted fibre datum from annulus data
ModularCurve.JHPlaceSpecialization.exists_twistedFibreDatum_laws_of_annulus77 below · cited by 1 · depth 30 - Subtracting a strict degree-zero divisor preserves twist type
ModularCurve.JHPlaceSpecialization.isTwistOf_sub_and_twistSpData_sub_eq_of_forall_isStrict0 below · cited by 1 · depth 30 - Vanishing depth reading implies twist type
ModularCurve.JHPlaceSpecialization.isTwistType_of_componentGroupProj_depthDual_eq_zero_of_inertiaStable_of_annulus237 below · cited by 1 · depth 30 - Inertia-invariance of the two place readings on X_H(M)
ModularCurve.JHPlaceSpecialization.reduceFst_smul_eq_and_reduceSnd_smul_eq_of_mem_inertiaSubgroupIn0 below · cited by 2 · depth 30 - Admissibility of the twisted gluing datum at p ‖ M
ModularCurve.JHPlaceSpecialization.twistSpData_mem_admissible_of_isTwistOf191 below · cited by 1 · depth 30 - A-integral value at a strict place of the first kind
ModularCurve.JHPlaceSpecialization.ProlongationDatum.IsModel.exists_hasValue_and_hasValue_residue_reduceFst_of_isStrictFst_of_forall_ord_nonneg_of_unit_of_cusp48 below · cited by 2 · depth 31 - Galois-equivariant bi-integral lift of node-compatible residue pairs
ModularCurve.JHPlaceSpecialization.ProlongationDatum.IsModel.exists_mem_riemannRochSpace_residue_eq_forall_arithmeticGalois_smul_eq_of_isGoodDiv1,494 below · cited by 1 · depth 31 - Inertia-equivariant common unit with prescribed simple zero and order tables
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_commonUnit_ord_eq_one_orderTables_of_realisation_forall_inertia_smul_eq267 below · cited by 1 · depth 31 - Frobenius-compatible prolongations restrict to the same valuation ring
ModularCurve.JHPlaceSpecialization.ProlongationDatum.integers_comap_eq_integers_comap_of_residue_eq_qExpFrobeniusModL0 below · cited by 2 · depth 31 - R₁ ∩ ℚ̄ = A, units Q(j), and level-lowering residues
ModularCurve.JHPlaceSpecialization.ProlongationDatum.integers_fst_isGeneric_and_forall_exists_valuation_sub_alpha_lt_one3 below · cited by 2 · depth 31 - Section-pair bounds on the two components at a node configuration
ModularCurve.JHPlaceSpecialization.ProlongationDatum.sectionPair_bounds_of_regularityLaw_of_isModel_of_unit_of_cusp1,257 below · cited by 1 · depth 31 - Rigidity of the base divisor for normalised bi-integral functions
ModularCurve.JHPlaceSpecialization.ProlongationDatum.sum_single_add_sum_single_eq_of_ord_eq_nsmul_sub_of_residue_eq_one_of_hasValue_one_of_discLaw_of_cusp1,258 below · cited by 1 · depth 31 - Integrality of annulus position moments of inertia-stable divisors
ModularCurve.JHPlaceSpecialization.den_twistPosMoment_eq_one_of_inertiaStable_of_annulus54 below · cited by 1 · depth 31 - Inertia-fixed strict places with prescribed reduction, off finitely many
ModularCurve.JHPlaceSpecialization.exists_finset_forall_exists_isStrictFst_reduceFst_eq_and_isStrictSnd_reduceSnd_eq_forall_inertia_smul_eq_of_gammaLift412 below · cited by 1 · depth 31 - Places with non-affine first reduction are cuspidal
ModularCurve.JHPlaceSpecialization.isCuspidal_of_not_isAffinePlace_reduceFst350 below · cited by 1 · depth 31 - Pinned chart: strict parts of E push to the base points
ModularCurve.JHPlaceSpecialization.mapDomain_fstDiv_eq_and_mapDomain_sndDiv_eq_of_twistSp_eq_zero_of_pin341 below · cited by 1 · depth 31 - Residue boxes and node values for a good divisor
ModularCurve.JHPlaceSpecialization.ProlongationDatum.IsModel.residue_mem_riemannRochSpace_mapDomain_and_node_hasValue_of_isGoodDiv682 below · cited by 1 · depth 32 - Bi-integral S-fixed family with independent residue pairs
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_linearIndependent_residuePair_forall_arithmeticGalois_smul_eq_of_finiteDimensional0 below · cited by 1 · depth 32 - Unit U with S-fixed bounded-denominator basis of U· L(D)
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_unit_smul_riemannRochSpace_basis_coeffMap_eq_smul_forall_arithmeticGalois_smul_eq353 below · cited by 1 · depth 32 - Sum of two residue degrees equals the degree of j
ModularCurve.JHPlaceSpecialization.ProlongationDatum.finrank_adjoin_residue_add_finrank_adjoin_residue_eq_finrank_adjoin584 below · cited by 1 · depth 32 - Incomparability of the two prolongations of a Γ_H datum
ModularCurve.JHPlaceSpecialization.ProlongationDatum.not_integers_le_integers_and_not_integers_le_integers2 below · cited by 1 · depth 32 - Chord bounds and rigidity of coupled sheet scalings at supersingular nodes
ModularCurve.JHPlaceSpecialization.exists_endOrder_ineq_and_coupledScalings_hasValue_of_isTwistOf_of_twistSp_eq_zero_of_annulus70 below · cited by 1 · depth 32 - S-invariant common unit moving L(D)-poles to integral j
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_commonUnit_forall_pole_integral_forall_arithmeticGalois_smul_eq_of_riemannRochSpace345 below · cited by 1 · depth 33 - Residue bounds and node values for bi-integral Riemann–Roch sections
ModularCurve.JHPlaceSpecialization.ProlongationDatum.residue_fst_box_and_residue_snd_box_and_node_hasValue_of_mem_riemannRochSpace678 below · cited by 4 · depth 33 - Chord inequality and rigidity along supersingular annuli
ModularCurve.JHPlaceSpecialization.exists_chord_le_endOrders_and_rigid_of_isTwistOf_of_twistSp_eq_zero_of_annulus60 below · cited by 1 · depth 33 - Node telescoping identity for coupled sheet scalings
ModularCurve.JHPlaceSpecialization.exists_hasValue_residue_div_pow_and_div_eq_twistAngFactor_of_coupled_of_inertiaStable1 below · cited by 1 · depth 33 - Good divisors with a common unit n-th root give admissible gluing data
ModularCurve.JHPlaceSpecialization.glueData_mem_admissible_of_isGoodDiv_of_forall_mul_eq_ord_of_residue_ne_zero_of_isModel1,248 below · cited by 1 · depth 35 - Good effective divisors avoiding a finite set of fibre places
ModularCurve.JHPlaceSpecialization.exists_isGoodDiv_reduce_notMem_isPrincipal_sub_of_smul_eq600 below · cited by 1 · depth 36 - Non-strict places lie over supersingular node pairs
ModularCurve.JHPlaceSpecialization.exists_mem_reduceFst_eq_reduceSnd_eq_of_not_isStrictFst_of_not_isStrictSnd0 below · cited by 1 · depth 36 - Moving lemma for divisor classes on J_H(M) at p ∥ M
ModularCurve.JHPlaceSpecialization.exists_rep_reduce_notMem_of_moving_of_disjoint_of_isModel_lawBlock_of_orient1,552 below · cited by 1 · depth 36 - Representatives of J_H classes with non-supersingular j-values on supports
ModularCurve.JHPlaceSpecialization.exists_rep_forall_exists_ord_sub_pos_residue_notMem_of_isModel_of_regularityLaw_of_orderLawFixed_of_coe_of_unit_of_cusp_of_orient1,551 below · cited by 1 · depth 37 - First reading of a place: cuspidality and j-values
ModularCurve.JHPlaceSpecialization.isCuspidal_iff_not_isAffinePlace_reduceFst_and_hasValue_reduceFst_of_ord_pos602 below · cited by 2 · depth 37 - Zero-side places of X_H(M) are cuspidal
ModularCurve.JHPlaceSpecialization.isCuspidal_of_isZeroSide1 below · cited by 2 · depth 37 - Bi-integral basis of a Riemann–Roch space with independent residues
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_basis_mem_integers_riemannRochSpace_linearIndependent_residue588 below · cited by 4 · depth 38 - Removing one bad place by a principal divisor
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_apply_eq_neg_one_forall_support_good_of_not_good_of_coe_of_unit_of_cusp_of_orient1,549 below · cited by 1 · depth 38 - Removing a bad place with fixed affine first reduction
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_apply_eq_neg_one_forall_support_good_of_fixed_of_isAffinePlace1,515 below · cited by 1 · depth 39 - Removability of a bad place on either branch
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_apply_eq_neg_one_forall_support_good_of_isInftySide_or_isStrictFst_or_isZeroSide_or_isStrictSnd1,537 below · cited by 1 · depth 39 - Typology of places of X_H(M) above p with p ‖ M
ModularCurve.JHPlaceSpecialization.isCuspidal_or_fixed_and_isAffinePlace_or_isStrictFst_or_isStrictSnd598 below · cited by 3 · depth 39 - Common unit with a simple pole at a fixed place
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_commonUnit_pole_of_reduceFst_fixed_of_isAffinePlace_of_regularityLaw_of_coe_of_unit_of_cusp_of_orient1,510 below · cited by 1 · depth 40 - Constant shift at a fixed affine pole avoiding a bad set
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_sub_algebraMap_forall_ord_pos_reduceFst_notMem_of_fixed_of_isAffinePlace675 below · cited by 1 · depth 40 - Removing a bad place of the first kind on X_H(M)
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_apply_eq_neg_one_forall_support_good_of_isInftySide_or_isStrictFst1,534 below · cited by 1 · depth 40 - Removing a bad place of the second kind on X_H(M)
ModularCurve.JHPlaceSpecialization.exists_isPrincipal_apply_eq_neg_one_forall_support_good_of_isZeroSide_or_isStrictSnd1,534 below · cited by 1 · depth 40 - Common unit with a simple pole at V₀
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_commonUnit_pole_reduceFst_of_regularityLaw_of_coe_of_unit_of_cusp_of_orient1,529 below · cited by 1 · depth 41 - Common unit with prescribed simple pole on the second sheet
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_commonUnit_pole_reduceSnd_of_regularityLaw_of_coe_of_unit_of_cusp_of_orient1,529 below · cited by 1 · depth 41 - Effective good divisor of large degree in general position
ModularCurve.JHPlaceSpecialization.ProlongationDatum.exists_isGoodDiv_forall_mem_support_good_le_degree_fstDiv_sndDiv730 below · cited by 3 · depth 41 - Lifting a good j-value along the first degeneracy map
ModularCurve.JHPlaceSpecialization.exists_ord_sub_algebraMap_pos_residue_notMem_of_isAffinePlace_reduceFst597 below · cited by 2 · depth 41 - Constant shift preserving a simple pole and both Gauss residues
ModularCurve.JHPlaceSpecialization.exists_sub_algebraMap_mem_integers_residue_ne_zero_ord_eq_neg_one_of_ord_eq_neg_one_of_forall_ord_neg47 below · cited by 2 · depth 41 - Second reading at a φ-fixed place equals δ(φ(r₁))
ModularCurve.JHPlaceSpecialization.reduceSnd_eq_apply_qExpFrobeniusPlaceModL_reduceFst_of_fixed_of_typeDichotomy3 below · cited by 1 · depth 41
ModularCurve.JOne 20
- Inertia at q acts unipotently on diamond-norms of prime-to-q torsion
ModularCurve.JOne.smul_smul_sub_self_eq_of_mem_inertiaSubgroupIn_of_eq_sum_diamondOneBar2,768 below · cited by 1 · depth 17 - Torsion of J₁(M) fixed by a number field's stabiliser
ModularCurve.JOne.exists_finiteDimensional_smul_eq_self_of_torsion245 below · cited by 4 · depth 18 - Galois action on J₁(M) commutes with the Hecke action
ModularCurve.JOne.galois_smul_heckeAlgOne_smul10 below · cited by 4 · depth 19 - Frobenius at q acting as q U_q on J₁(M₀q)
ModularCurve.JOne.diamondOneBar_smul_smul_sub_self_eq_smul_heckeOperatorOneBar_of_isFrobeniusAt_of_eq_sum_diamondOneBar2,861 below · cited by 1 · depth 20 - Finite-index inertia subgroups fixing μ_q and J₁(M)[m]
ModularCurve.JOne.exists_le_inertiaSubgroupIn_finiteIndex_forall_apply_eq_self_of_pow_eq_one_forall_smul_eq_self_of_torsion246 below · cited by 7 · depth 20 - Frobenius–Hecke relation at q for Γ₁(M₀)∩Γ₀(q) classes
ModularCurve.JOne.diamondOneBar_smul_pullbackAlongHom_smul_sub_self_eq_smul_heckeOperatorOneBar_of_isFrobeniusAt2,856 below · cited by 1 · depth 21 - Inertia-invariant diamond-norm vectors in TₚJ₁(M) modulo monodromy and old parts
ModularCurve.JOne.exists_pow_smul_mem_span_inertia_sub_sup_range_of_rep_eq_self_of_mem_range_diamondNorm_tateModule_of_dvd_of_not_sq_dvd3,611 below · cited by 1 · depth 21 - Degeneracy pull-back inputs hold whenever Nt ∣ N'
ModularCurve.JOne.degeneracyPullbackInputs33 below · cited by 6 · depth 22 - Degeneracy pull-backs commute with T_q and ⟨ d⟩
ModularCurve.JOne.degeneracyPullbackPair_comm_heckeOperatorOneBar_diamondOneBar247 below · cited by 4 · depth 22 - Transport of the Γ_H-level relation to J₁(M₀q)
ModularCurve.JOne.diamondOneBar_smul_pullbackAlongHom_smul_sub_self_eq_smul_heckeOperatorOneBar_of_genOpH280 below · cited by 1 · depth 22 - Galois equivariance of Hecke and diamond operators on J₁(M)
ModularCurve.JOne.smul_heckeOperatorOneBar_and_smul_diamondOneBar274 below · cited by 1 · depth 22 - Degeneracy pull-backs commute with ⟨ d⟩ for ℓ∤ d
ModularCurve.JOne.degeneracyPullbackPair_comm_diamondOneBar7 below · cited by 1 · depth 23 - Degeneracy pull-backs commute with T_q for q≠ℓ
ModularCurve.JOne.degeneracyPullbackPair_comm_heckeOperatorOneBar241 below · cited by 1 · depth 23 - Galois equivariance of the two degeneracy pull-backs on J₁(N)
ModularCurve.JOne.degeneracyPullbackPair_galois_smul5 below · cited by 2 · depth 23 - Descent of an H-invariant point of J₁(Mp) to the fixed field
ModularCurve.JOne.exists_ringHom_spec_fixedField_comp_eq_gpts_of_forall_smul_eq_self137 below · cited by 1 · depth 25 - Injectivity of the degeneracy Gram operator on TₚJ₁(N)
ModularCurve.JOne.tateModule_eq_zero_of_forall_pushforwardAlongHom_x1LevelInclBar_x1LevelSubstBar_eq_zero898 below · cited by 1 · depth 26 - Diamond twist of mixed push–pulls on J₁(N)
ModularCurve.JOne.diamondOneBar_pushforwardAlongHom_x1LevelSubstBar_pullbackAlongHom_x1LevelInclBar_eq294 below · cited by 1 · depth 27 - Push–pull along the first degeneracy map is multiplication by degree
ModularCurve.JOne.pushforwardAlongHom_pullbackAlongHom_x1LevelInclBar_eq_finrankAlong_smul9 below · cited by 1 · depth 27 - Push–pull along the degeneracy map β₁ is degree multiplication
ModularCurve.JOne.pushforwardAlongHom_pullbackAlongHom_x1LevelSubstBar_eq_finrankAlong_smul9 below · cited by 1 · depth 27 - α_{1,*}β₁^* equals deg(j)· Tₚ on J₁(N)
ModularCurve.JOne.pushforwardAlongHom_x1LevelInclBar_pullbackAlongHom_x1LevelSubstBar_eq_finrankAlong_smul_heckeOperatorOneBar77 below · cited by 1 · depth 27
ModularCurve.JOneES 2
- Base change of the q-expansion function field is a function field
ModularCurve.JOneES.exists_transcendental_finiteDimensional_laurentBaseChange1 below · cited by 124 · depth 12 - The q-expansion function field over ℚ is a one-variable function field
ModularCurve.JOneES.exists_transcendental_finiteDimensional_qExpFunctionFieldC0 below · cited by 13 · depth 13
ModularCurve.JZero 111
- Finite generation of J₀(N)(ℚ) for prime N≥ 5
ModularCurve.JZero.addGroup_fg_invariants_rat_of_prime_of_five_le2,179 below · cited by 3 · depth 8 - Divisibility of the class group J₀(N) over ℚ̄
ModularCurve.JZero.divisible754 below · cited by 29 · depth 8 - Mordell–Weil for J₀(N), prime level N≥ 5
ModularCurve.JZero.addGroup_fg_invariants_of_prime_of_five_le2,178 below · cited by 1 · depth 9 - p-power torsion of J₀(N) has order p^{2gn}
ModularCurve.JZero.exists_abelJacobiCard712 below · cited by 4 · depth 10 - Descent height data on J₀(N)(K) for prime N ≥ 5
ModularCurve.JZero.exists_descent_height_two_invariants_of_prime_of_five_le2,177 below · cited by 1 · depth 10 - Finite definition field for the n-torsion of J₀(N)
ModularCurve.JZero.exists_finiteDimensional_smul_eq_self_of_torsion462 below · cited by 8 · depth 10 - 2-torsion of J₀(N) is fixed by a finite extension
ModularCurve.JZero.exists_finiteDimensional_torsion_two_le_invariants715 below · cited by 2 · depth 10 - Weak Mordell–Weil at 2 for J₀(N)
ModularCurve.JZero.finiteIndex_range_nsmul_two_invariants1,003 below · cited by 2 · depth 10 - Monotonicity of J₀(N)-invariants in the base field
ModularCurve.JZero.invariants_le_invariants_of_le0 below · cited by 2 · depth 10 - Open stabilisers for the Galois action on J₀(N)
ModularCurve.JZero.isOpen_stabilizer78 below · cited by 7 · depth 10 - Torsion classes in J₀(M) have open stabilisers
ModularCurve.JZero.torsion_fixed_by_open79 below · cited by 1 · depth 10 - Points of J₀(N) over ℚ̄ have open stabilisers
ModularCurve.JZero.exists_finiteDimensional_fixingSubgroup_smul_eq76 below · cited by 9 · depth 11 - Inertia fixes prime-to-ℓ Kummer classes on J₀(N)
ModularCurve.JZero.exists_finset_inertiaSubgroupIn_smul_eq_of_prime_smul_sub_eq_zero979 below · cited by 1 · depth 11 - Descent inequalities for the naive height on J₀(N)
ModularCurve.JZero.naiveHeight_descent_of_prime_of_five_le1,969 below · cited by 1 · depth 11 - Northcott property of the naive height on J₀(N)(K)
ModularCurve.JZero.naiveHeight_northcott210 below · cited by 1 · depth 11 - Galois-stable effective representative of a fixed class on J₀(N)
ModularCurve.JZero.exists_galoisStable_rep182 below · cited by 5 · depth 12 - Subadditivity of the naive height on J₀(N)
ModularCurve.JZero.naiveHeight_add_le209 below · cited by 1 · depth 12 - Height growth under 2^k on J₀(N), prime level ≥ 5
ModularCurve.JZero.naiveHeight_growth_of_prime_of_five_le1,961 below · cited by 1 · depth 12 - Degree reduction for the naive height on J₀(N)
ModularCurve.JZero.naiveHeight_reduce225 below · cited by 1 · depth 12 - pⁿ-torsion of J₀(N) has p^{2gn} points
ModularCurve.JZero.cardinalityAJ_genusFF460 below · cited by 12 · depth 13 - p-adic Weil pairing on the Tate module of J₀(M)
ModularCurve.JZero.exists_tateModule_pairing_rep_eq_cyclotomicCharacter_mul907 below · cited by 2 · depth 13 - Finiteness of p^k-torsion of J₀(N) from the point count
ModularCurve.JZero.finite_torsion_pow_of_cardinalityAJ0 below · cited by 11 · depth 13 - Height form bounded above by naive height of representatives
ModularCurve.JZero.heightForm_le215 below · cited by 1 · depth 13 - Lower bound for the height form on near-minimal representatives
ModularCurve.JZero.heightForm_lower_of_prime_of_five_le1,956 below · cited by 1 · depth 13 - Quasi-invariance of the J₀(N) height form under linear equivalence
ModularCurve.JZero.heightForm_quasiInvariant_of_prime_of_five_le1,829 below · cited by 1 · depth 13 - Height form bounds the naive height from below
ModularCurve.JZero.exists_isRepOf_heightForm_lower1,040 below · cited by 1 · depth 14 - Quasi-invariance of the height form along a class, prime level ≥ 5
ModularCurve.JZero.heightForm_quasiInvariant_eps_of_prime_of_five_le1,829 below · cited by 1 · depth 14 - Height pairing against principal divisors is almost zero
ModularCurve.JZero.pairing_principal_le_of_prime_of_five_le1,823 below · cited by 2 · depth 14 - n^{2g} divides the n-torsion count of J₀(N)
ModularCurve.JZero.pow_two_mul_genusFF_le_card_torsion746 below · cited by 3 · depth 14 - Point heights of a representative bounded by its naive height
ModularCurve.JZero.ptsum_pointHt_le_divNaiveHeight214 below · cited by 3 · depth 14 - Chord height at a non-cuspidal place equals the value-tuple height
ModularCurve.JZero.absLogHeight_chordVec_eq_evalAt_of_ne146 below · cited by 2 · depth 15 - Chord-height comparison bounded by divisor mass, prime level
ModularCurve.JZero.chordLine_core_of_prime_of_five_le1,805 below · cited by 1 · depth 15 - Naive height of a representative bounded by base mass
ModularCurve.JZero.divNaiveHeight_le_baseMass_of_isRepOf299 below · cited by 1 · depth 15 - Height form dominates base mass on suitable representatives
ModularCurve.JZero.exists_isRepOf_baseMass_le_heightForm1,032 below · cited by 1 · depth 15 - Point heights bounded linearly by the j-coordinate height
ModularCurve.JZero.exists_pointHt_le_absLogHeight_jCoord210 below · cited by 1 · depth 15 - Point height bounded by (2g+1) times base height
ModularCurve.JZero.exists_pointHt_le_mul_baseHt273 below · cited by 7 · depth 15 - Bounded-mass principal divisors with small self-pairing at every place
ModularCurve.JZero.pairing_chord_self_le_of_prime_of_five_le1,812 below · cited by 1 · depth 15 - The base mass ignores the cusp at infinity
ModularCurve.JZero.baseMass_add_single_cuspInftyBar0 below · cited by 2 · depth 16 - Base mass of a one-point divisor
ModularCurve.JZero.baseMass_single0 below · cited by 2 · depth 16 - Chord-line height identity for sections on X₀(N), N≥ 5 prime
ModularCurve.JZero.chordLine_section_ledger_of_prime_of_five_le1,803 below · cited by 1 · depth 16 - Effective divisors of off-cusp mass at most one
ModularCurve.JZero.eq_single_add_single_cuspInftyBar_of_offBaseMass_le_one0 below · cited by 2 · depth 16 - Height of j(v) bounded linearly by the model point height
ModularCurve.JZero.exists_absLogHeight_jCoord_le_pointHt294 below · cited by 1 · depth 16 - Height of the chord vector at v equals h_D(v)+h_{D+K}(v)+O(1)
ModularCurve.JZero.exists_abs_absLogHeight_regVal_sub_pointHt_add_pointHt_le297 below · cited by 2 · depth 16 - Additivity up to O(1) of model heights on X₀(N)
ModularCurve.JZero.exists_abs_pointHt_sub_add_pointHt_le273 below · cited by 2 · depth 16 - Equal-degree linear systems: heights agree up to ε h + C
ModularCurve.JZero.exists_abs_pointHt_sub_pointHt_le_mul_add_of_degree_eq273 below · cited by 9 · depth 16 - Point heights of a base-point-free subfamily differ boundedly
ModularCurve.JZero.exists_abs_pointHt_sub_pointHt_le_of_forall_exists_ord_add_eq_zero263 below · cited by 14 · depth 16 - Level transport along 1· p=p for places and J₀
ModularCurve.JZero.exists_addEquiv_placeEquiv_oneMul0 below · cited by 1 · depth 16 - Height form dominates base mass at confluent divisors
ModularCurve.JZero.exists_baseMass_le_heightForm_of_exists_two_le1,012 below · cited by 1 · depth 16 - Existence of a cusp-maximal representative of a class in J₀(N)
ModularCurve.JZero.exists_isRepOf_forall_apply_cuspInftyBar_le221 below · cited by 1 · depth 16 - Pair-height sums bounded by base heights for reduced divisors
ModularCurve.JZero.exists_sum_pairHt_le_of_forall_le_one1,005 below · cited by 1 · depth 16 - Height form equals genus times base mass when off-cusp mass ≤ 1
ModularCurve.JZero.heightForm_eq_genusFF_mul_baseMass_of_offBaseMass_le_one4 below · cited by 1 · depth 16 - Height form minus base mass, grouped by points
ModularCurve.JZero.heightForm_sub_baseMass_eq0 below · cited by 2 · depth 16 - Tate module of J₀(N) free of rank 2g
ModularCurve.JZero.nonempty_tateModule_basis_of_abelJacobiCard0 below · cited by 1 · depth 16 - The off-cusp mass ignores the cusp ∞̄
ModularCurve.JZero.offBaseMass_add_single_cuspInftyBar0 below · cited by 1 · depth 16 - Off-cusp mass is at most the genus when L(D^∘-∞)=0
ModularCurve.JZero.offBaseMass_le_genusFF_of_riemannRochSpace_eq_bot254 below · cited by 1 · depth 16 - Off-cusp mass of a one-point divisor
ModularCurve.JZero.offBaseMass_single0 below · cited by 1 · depth 16 - Quotients of sections of k· E with bounded k
ModularCurve.JZero.quot_rep231 below · cited by 1 · depth 16 - Two-line height identity at non-cuspidal base places, prime level ≥ 5
ModularCurve.JZero.sum_pairHt_twoLine_ledger_of_nonCuspidal_of_prime_of_five_le1,798 below · cited by 1 · depth 16 - Chord functions at w are a scalar multiple of the chord vector
ModularCurve.JZero.chordFun_evalAt_eq_smul_chordVec149 below · cited by 1 · depth 17 - Embedding bases separate places: nonvanishing chord vectors
ModularCurve.JZero.chordVec_ne_zero_of_ne242 below · cited by 11 · depth 17 - Degree of the off-cusp part equals off-cusp mass
ModularCurve.JZero.degree_erase_cuspInftyBar144 below · cited by 2 · depth 17 - Wronskians of a spanning family as a base-point-free subsystem of |2D+K|
ModularCurve.JZero.diffCoeff_wronskian_mem_riemannRochSpace_and_exists_ord_add_eq_zero285 below · cited by 1 · depth 17 - Tangent datum height exceeds 2h(P) by (2g-2+ε)t(P)+C
ModularCurve.JZero.exists_absLogHeight_regVal_sub_two_mul_pointHt_le305 below · cited by 1 · depth 17 - Homogeneous certificate s_k^{M+1}=sumⱼ qⱼ(s)uⱼ on X₀(N)
ModularCurve.JZero.exists_isHomogeneous_sum_aeval_mul_eq_pow193 below · cited by 1 · depth 17 - Lower bound for pair heights against a fixed place
ModularCurve.JZero.exists_sub_mul_baseHt_le_pairHt276 below · cited by 2 · depth 17 - Many-point determinantal Jensen inequality on X₀(N)
ModularCurve.JZero.exists_sum_pairHt_le_of_det_evalAt_ne_zero986 below · cited by 1 · depth 17 - Index of speciality of the off-cusp part: ℓ(K-D^∘)+deg D^∘=g
ModularCurve.JZero.finrank_riemannRochSpace_canonicalDivisorOf_sub_erase_add_offBaseMass263 below · cited by 2 · depth 17 - Height form of a one-point divisor
ModularCurve.JZero.heightForm_single0 below · cited by 1 · depth 17 - Two representatives of a class of J₀(N) differ by a principal divisor
ModularCurve.JZero.isPrincipal_sub_of_isRepOf0 below · cited by 1 · depth 17 - Archimedean Jensen bound at infinite places of number fields
ModularCurve.JZero.jensen_arch470 below · cited by 1 · depth 17 - One-sided archimedean Jensen bound at non-cuspidal base places
ModularCurve.JZero.jensen_arch_at_le_of_nonCuspidal475 below · cited by 2 · depth 17 - Archimedean Jensen line at ∞̄ and non-cuspidal places
ModularCurve.JZero.jensen_arch_at_of_nonCuspidal472 below · cited by 1 · depth 17 - One-sided Jensen inequality at bad finite places
ModularCurve.JZero.jensen_bad_at_le402 below · cited by 2 · depth 17 - Bad-place regularised Jensen inequality on X₀(N), N prime ≥ 5
ModularCurve.JZero.jensen_bad_at_of_prime_of_five_le1,666 below · cited by 2 · depth 17 - Jensen bound at bad primes, prime level at least five
ModularCurve.JZero.jensen_bad_primes_of_prime_of_five_le1,671 below · cited by 1 · depth 17 - A ν-adic Jensen identity uniform in the base place
ModularCurve.JZero.jensen_good_at752 below · cited by 2 · depth 17 - Explicit one-sided Jensen inequality at good finite places
ModularCurve.JZero.jensen_good_at_le753 below · cited by 2 · depth 17 - Non-archimedean Jensen formula at good places, cusp included
ModularCurve.JZero.jensen_good_primes753 below · cited by 1 · depth 17 - Quadratic normality of L((2g+1)∞̄) on X₀(N)
ModularCurve.JZero.riemannRochSpace_embDivisor_mul_self256 below · cited by 4 · depth 17 - Confluent many-point Jensen inequality for a section frame
ModularCurve.JZero.sum_pairHt_le_of_isUnit_det_jetMatrix277 below · cited by 2 · depth 17 - Heights of 2× 2 minors of two tuples
ModularCurve.JZero.exists_absLogHeight_minors_le1 below · cited by 1 · depth 18 - Pair height against a fixed place, up to ε h
ModularCurve.JZero.exists_abs_pairHt_sub_pointHt_div_le274 below · cited by 2 · depth 18 - Uniform non-archimedean disc charts at pivots on X₀(N)
ModularCurve.JZero.exists_chart_of_isPivot391 below · cited by 1 · depth 18 - Upper bound h_f ≤ deg A (1+ε) t + C off the cusp
ModularCurve.JZero.exists_pointHt_le_degree_mul_baseHt_of_mem_riemannRochSpace278 below · cited by 1 · depth 18 - Non-vanishing chord datum for an embedding basis
ModularCurve.JZero.exists_regVal_chord_ne_zero263 below · cited by 4 · depth 18 - Approach to a base point: limits of archimedean chordal terms
ModularCurve.JZero.exists_seq_tendsto_place318 below · cited by 2 · depth 18 - Archimedean Jensen comparison at all complex embeddings
ModularCurve.JZero.jensen_arch_embedding469 below · cited by 3 · depth 18 - One-sided archimedean proximity bound with explicit normalisation
ModularCurve.JZero.prox_sum_le_of_forall_log_secVal_le471 below · cited by 1 · depth 18 - Projective normality of the |(2g+1)∞| model of X₀(N)
ModularCurve.JZero.riemannRochSpace_embDivisor_mul_eq259 below · cited by 3 · depth 18 - Chow reciprocity for sections of the embedding divisor
ModularCurve.JZero.chowReciprocity_embedding239 below · cited by 1 · depth 19 - Archimedean Chow-side estimate for embedding sections on X₀(N)
ModularCurve.JZero.chowSide_arch_embedding435 below · cited by 1 · depth 19 - Finite chart data covering places where sᵢ has least order
ModularCurve.JZero.exists_forall_exists_ord_sub_evalAt_eq_one_and_derivative_evalEval_ne_zero294 below · cited by 1 · depth 19 - At each place a normalised coordinate is a local parameter
ModularCurve.JZero.exists_ord_div_sub_evalAt_eq_one264 below · cited by 3 · depth 19 - Chordal proximity sums agree with the Chow side up to O(k)
ModularCurve.JZero.prox_sum_chowSide258 below · cited by 3 · depth 19 - Hyperplane sections of the embedding system have degree 2g+1
ModularCurve.JZero.sum_toNat_hyperplaneSection_eq_embDegree176 below · cited by 5 · depth 19 - Total multiplicity of a section cycle equals kcdotembDegree
ModularCurve.JZero.sum_toNat_sectionCycle_eq_mul_embDegree176 below · cited by 4 · depth 19 - Chow-side archimedean comparison off the support of B
ModularCurve.JZero.chowSide_arch_embedding_off_support416 below · cited by 1 · depth 20 - Chow side at the cusp from off-support bounds
ModularCurve.JZero.chowSide_cusp_of_off_support324 below · cited by 1 · depth 20 - Local constancy of μ(h) on small chordal balls, uniformly in μ
ModularCurve.JZero.exists_abv_evalAt_eq_abv_evalAt_of_le_prox283 below · cited by 1 · depth 20 - Separating form vanishing on the model of X₀(N)
ModularCurve.JZero.exists_isHomogeneous_aeval_eq_zero_and_eval_ne_zero277 below · cited by 1 · depth 20 - Nonzero plane relation between s_l/sᵢ and a function h
ModularCurve.JZero.exists_ne_zero_eval_zero_ne_zero_evalEval_div_eq_zero114 below · cited by 1 · depth 20 - A function is large at places chordally near one of its poles
ModularCurve.JZero.exists_one_le_abv_evalAt_of_le_prox283 below · cited by 1 · depth 20 - Good chart datum at a prescribed place of X₀(N)
ModularCurve.JZero.exists_ord_sub_evalAt_eq_one_and_derivative_evalEval_ne_zero293 below · cited by 1 · depth 20 - Places approaching the base cusp: proximities and section sizes
ModularCurve.JZero.exists_seq_tendsto_cuspInftyBar307 below · cited by 1 · depth 21 - Two-sided pencil bound for the Chow-side comparison
ModularCurve.JZero.pencil_secProd_chowForm_two_sided413 below · cited by 1 · depth 21 - Good hyperplane sections through a given place on X₀(N)
ModularCurve.JZero.exists_hyperplaneSection_sum_log_secVal_ge404 below · cited by 1 · depth 22 - Upper bound for normalised section values on X₀(N)
ModularCurve.JZero.exists_log_secVal_sub_le309 below · cited by 1 · depth 22 - Boundedness of the hyperplane-section Chow cocycle
ModularCurve.JZero.hyperplaneSection_cocycle_bounded326 below · cited by 1 · depth 22 - Auxiliary hyperplane section with bounded defect on X₀(N)
ModularCurve.JZero.exists_hyperplaneSection_defect_le323 below · cited by 1 · depth 23 - Cocycle identity for the hyperplane-section defect
ModularCurve.JZero.hyperplaneSection_cocycle257 below · cited by 1 · depth 23 - Pivot value of a linear section equals the Chow form at e
ModularCurve.JZero.secProd_one_linSec_eq_eval_chowForm256 below · cited by 1 · depth 24
ModularCurve.JZeroGoodReductionSpecialization 1
- Specialization is surjective on q-primary torsion, q≠ℓ
ModularCurve.JZeroGoodReductionSpecialization.torsBijFor_of_charP_of_not_dvd1,781 below · cited by 1 · depth 9
ModularCurve.JZeroNeronIdentityComponent 13
- Localised Néron sections embed with finite index into Eisenstein quotient
ModularCurve.JZeroNeronIdentityComponent.exists_addMonoidHom_localizedModule_sections_eisensteinQuotientRationalLocalized_of_sectionsEquiv240 below · cited by 2 · depth 12 - Eisenstein-primary Hopf orders and an H¹ comparison
ModularCurve.JZeroNeronIdentityComponent.exists_jZeroTorsionHopfOrder_forall_nonempty_localizedModule_fppfCohomology_kernel_addEquiv758 below · cited by 1 · depth 12 - Hecke action on the fppf points sheaf of J⁰
ModularCurve.JZeroNeronIdentityComponent.exists_ringHom_heckeAlg_end_of_sectionsEquiv2 below · cited by 2 · depth 12 - Multiplication by n on J⁰ is locally of finite presentation
ModularCurve.JZeroNeronIdentityComponent.locallyOfFinitePresentation_schemeNsmul1 below · cited by 2 · depth 12 - Smallness of H¹_{fppf}(Specℤ,G[n])
ModularCurve.JZeroNeronIdentityComponent.small_fppfCohomology_one_kernel_zsmul2 below · cited by 2 · depth 12 - Eisenstein idempotents on the q^m-torsion kernel schemes
ModularCurve.JZeroNeronIdentityComponent.exists_heckeAlg_tower_idempotent_schemeKer_of_ringHom_of_sectionsEquiv11 below · cited by 2 · depth 13 - Hopf-algebra tower of Eisenstein-primary q-torsion on J₀(p)
ModularCurve.JZeroNeronIdentityComponent.exists_hopfAlgebra_tower_pointsSheaf_levelMap_of_idempotent9 below · cited by 2 · depth 13 - Eisenstein part of G[q^m] as Hecke-stable retract
ModularCurve.JZeroNeronIdentityComponent.exists_retract_kernel_zsmul_pointsSheaf_of_eisensteinProjector16 below · cited by 2 · depth 13 - Multiplication by n>0 on J⁰ is locally quasi-finite
ModularCurve.JZeroNeronIdentityComponent.locallyQuasiFinite_schemeNsmul10 below · cited by 2 · depth 13 - Eisenstein q-primary Néron torsion core with point embedding
ModularCurve.JZeroNeronIdentityComponent.exists_jZeroNeronPrimaryTorsionCore_forall_exists_points_embedding822 below · cited by 1 · depth 14 - A Hecke element outside the Eisenstein ideal acting as t_m
ModularCurve.JZeroNeronIdentityComponent.exists_notMem_forall_zsmul_eq_zero_imp_app_eq7 below · cited by 1 · depth 14 - Reducedness of the q^m-torsion of the Néron identity component
ModularCurve.JZeroNeronIdentityComponent.isReduced_schemeKer_pow3 below · cited by 3 · depth 14 - Sheaf endomorphism acting by φ makes φ additive on points
ModularCurve.JZeroNeronIdentityComponent.schemeHomOverComp_mul_eq_mul_of_sectionsEquiv_end0 below · cited by 1 · depth 14
ModularCurve.JZeroNeronIdentityComponentGood 1
- Mod-2 residue dictionary for the Eisenstein torsion core
ModularCurve.JZeroNeronIdentityComponentGood.exists_jZeroNeronPrimaryTorsionCore_two_residue_iff_reductionModL824 below · cited by 1 · depth 13
ModularCurve.JZeroNeronObjectAtP 121
- Uₚ acts as an involution on toric points
ModularCurve.JZeroNeronObjectAtP.heckeGen_smul_heckeGen_smul_eq_self_of_mem_toricPts481 below · cited by 1 · depth 11 - Rigidity of μ_m^t-homomorphisms over a henselian valuation base
ModularCurve.JZeroNeronObjectAtP.eq_of_muBaseChange_residue_comp_eq39 below · cited by 5 · depth 12 - Reduction mod m of a toric matrix across an isomorphism
ModularCurve.JZeroNeronObjectAtP.exists_forall_muPt_comp_eq_comp_of_torusMatrix_of_rigid0 below · cited by 1 · depth 12 - Existence of the level-N₀ Jacobian model at p
ModularCurve.JZeroNeronObjectAtP.exists_levelModel_isJacobian1,827 below · cited by 6 · depth 12 - The generating set of `finPts` is already a subgroup
ModularCurve.JZeroNeronObjectAtP.mem_finPts_iff0 below · cited by 17 · depth 12 - Multiplicativity on torsion torus points of the special fibre
ModularCurve.JZeroNeronObjectAtP.schemeHomOverComp_mul_torusPt_fibreRestrictAlong_of_torsion1 below · cited by 1 · depth 12 - Hecke stability of the toric m-torsion of J₀(N₀p)
ModularCurve.JZeroNeronObjectAtP.smul_mem_toricPts65 below · cited by 3 · depth 12 - Toric points lie in the kernel of both degeneracy push-forwards
ModularCurve.JZeroNeronObjectAtP.toricPts_le_ker_degeneracyPushforwardPair304 below · cited by 5 · depth 12 - Extendability over a valuation ring is closed under inversion
ModularCurve.JZeroNeronObjectAtP.ExtendsToPlace.inv0 below · cited by 2 · depth 13 - Extendable points are closed under the relative group law
ModularCurve.JZeroNeronObjectAtP.ExtendsToPlace.mul0 below · cited by 2 · depth 13 - Unit point of a relative group law extends to a place
ModularCurve.JZeroNeronObjectAtP.ExtendsToPlace.one0 below · cited by 2 · depth 13 - Endomorphisms act on the special-fibre torus by a character matrix
ModularCurve.JZeroNeronObjectAtP.exists_mapDomain_comp_torusFibre_eq_torusFibre_comp_fibreRestrictAlong22 below · cited by 4 · depth 13 - Base-changed endomorphisms preserve the toric lift on ℚ̄-points
ModularCurve.JZeroNeronObjectAtP.exists_muPt_comp_toricLift_eq_comp_fibreRestrictAlong63 below · cited by 1 · depth 13 - Toricity of p-new finite points up to a bounded multiple
ModularCurve.JZeroNeronObjectAtP.exists_nsmul_mem_toricPts_of_mem_finPts1,736 below · cited by 5 · depth 13 - Valuative criterion: extending a ℚ̄-point over a place
ModularCurve.JZeroNeronObjectAtP.exists_schemeHomOver_barPt_comp_eq_of_isProper0 below · cited by 7 · depth 13 - Toric lifts μ_m^t of the special-fibre torus over a place
ModularCurve.JZeroNeronObjectAtP.exists_toricLift_of_torusFibre43 below · cited by 1 · depth 13 - Multiplication by m is quasi-finite, quasi-compact and flat
ModularCurve.JZeroNeronObjectAtP.locallyQuasiFinite_quasiCompact_flat_schemeNsmul_baseChange12 below · cited by 2 · depth 13 - Degeneracy morphisms kill the ℚ̄-points of the toric lift
ModularCurve.JZeroNeronObjectAtP.muPt_toricLift_degeneracyHom_eq_one303 below · cited by 1 · depth 13 - Hecke stability of the extendable m-torsion subgroup
ModularCurve.JZeroNeronObjectAtP.smul_mem_finPts0 below · cited by 7 · depth 13 - Prime-to-p inertia differences lie in the toric part
ModularCurve.JZeroNeronObjectAtP.smul_sub_self_mem_toricPts_of_isGluedSpecialization2,620 below · cited by 1 · depth 13 - Toric m-torsion subgroup as closure of toric points
ModularCurve.JZeroNeronObjectAtP.toricPts_of_pos0 below · cited by 6 · depth 13 - Agreement of places for two q-expansion-pinned models of X₀(N₀)
ModularCurve.JZeroNeronObjectAtP.LevelModel.pointEquivPlace_ofGenerator_eq_of_comp_eeta0_of_chartPin106 below · cited by 3 · depth 14 - Norm along φ_κ acts as Frobenius pushforward on Pic⁰
ModularCurve.JZeroNeronObjectAtP.LevelModel.symm_fibreMap_frobeniusNormHom_eq_frobeniusPushforwardModL_symm1,181 below · cited by 1 · depth 14 - Unique extension of a geometric generic point over a valuation ring
ModularCurve.JZeroNeronObjectAtP.existsUnique_schemeHomOver_barPt_comp_eq_of_isProper0 below · cited by 4 · depth 14 - Bialgebra comorphism of the toric lift through the m-torsion
ModularCurve.JZeroNeronObjectAtP.exists_bialgHom_muCoord_forall_torsionPoint_comp_fst_eq11 below · cited by 1 · depth 14 - Degeneracy maps on Pic⁰ commute with base twists
ModularCurve.JZeroNeronObjectAtP.fibreMap_abq_schemeHomOverComp_eq_of_pullbackHom_pin858 below · cited by 1 · depth 14 - Counting A-integral m-torsion in the joint degeneracy kernel
ModularCurve.JZeroNeronObjectAtP.finite_and_card_le_of_kernel_coset_representatives1,013 below · cited by 1 · depth 14 - Finiteness of the fixed points of frobSp²
ModularCurve.JZeroNeronObjectAtP.finite_fixedPoints_frobSp_comp_self971 below · cited by 7 · depth 14 - Galois action on toric μ_m-points: inertia and decomposition
ModularCurve.JZeroNeronObjectAtP.inertia_smul_eq_and_exists_decomposition_smul_eq_of_muLift42 below · cited by 1 · depth 14 - Toric torsion as integral points reducing into the torus
ModularCurve.JZeroNeronObjectAtP.mem_toricPts_iff_exists_fibreMap_abqFibre_eq_one1,637 below · cited by 1 · depth 14 - Degeneracy maps kill the toric lift over the residue field
ModularCurve.JZeroNeronObjectAtP.muBaseChange_toricLift_degeneracyHom_eq_one21 below · cited by 1 · depth 14 - Lower bound m^t for the toric m-torsion subgroup
ModularCurve.JZeroNeronObjectAtP.pow_toricRank_le_card_toricPts1,632 below · cited by 3 · depth 14 - Toric points lie in the finite part
ModularCurve.JZeroNeronObjectAtP.toricPts_le_finPts1 below · cited by 11 · depth 14 - Finiteness of n-torsion in the special fibre of J₀(N₀)
ModularCurve.JZeroNeronObjectAtP.LevelData.finite_torsionSubset_special1,600 below · cited by 1 · depth 15 - Frobenius conjugation and fibre points of the Igusa model
ModularCurve.JZeroNeronObjectAtP.LevelModel.fibrePt_eq_fibrePt_comp_frobenius_of_isFrobeniusAt86 below · cited by 1 · depth 15 - Base-changed Abel–Jacobi classifies 𝒪(y)⊗𝒪(-ε₀)
ModularCurve.JZeroNeronObjectAtP.LevelModel.nonempty_poincare_pullbackAlong_ajZero_baseChange_iso_ofPoint874 below · cited by 1 · depth 15 - Special fibre of an A-point equals reduction mod λ
ModularCurve.JZeroNeronObjectAtP.LevelModel.ptsSp_symm_schemeHomOverComp_resPt_eq_reductionModL741 below · cited by 2 · depth 15 - The Néron object's `frobSp` is Frobenius push-forward mod p
ModularCurve.JZeroNeronObjectAtP.frobSp_eq_frobeniusPushforwardModL923 below · cited by 3 · depth 15 - Finiteness and rank bound for m-torsion of the joint kernel
ModularCurve.JZeroNeronObjectAtP.isFinite_schemeKerStr_kerPairLaw_special_and_finrank_le5 below · cited by 2 · depth 15 - Local quasi-finiteness of m-torsion in the degeneracy kernel
ModularCurve.JZeroNeronObjectAtP.locallyQuasiFinite_schemeKerStr_kerPairLaw1,000 below · cited by 2 · depth 15 - Order of the toric m-torsion subgroup: m^{toricRank}
ModularCurve.JZeroNeronObjectAtP.natCard_toricPts1,633 below · cited by 6 · depth 15 - Special m-kernel of the J₀(N₀) datum has order m^{2g}
ModularCurve.JZeroNeronObjectAtP.LevelData.isFinite_schemeKerStr_special_and_finrank_eq_pow_two_mul_genusFF1,598 below · cited by 2 · depth 16 - Degree-zero twists by A-sections come from D₀
ModularCurve.JZeroNeronObjectAtP.LevelModel.exists_schemeHomOver_poincare_pullbackAlong_iso_rigidify_sectionTwist_of_sum_eq_zero281 below · cited by 1 · depth 16 - Reduction of a point classifying a rigidified section twist
ModularCurve.JZeroNeronObjectAtP.LevelModel.nonempty_poincare_baseChange_pullbackAlong_iso_pointTwist_of_iso_rigidify_sectionTwist29 below · cited by 1 · depth 16 - Rigidified section twists restrict to the geometric generic fibre
ModularCurve.JZeroNeronObjectAtP.LevelModel.nonempty_poincare_pullbackAlong_iso_pointTwist_of_iso_rigidify_sectionTwist29 below · cited by 1 · depth 16 - Poincaré pullback at a degree-zero class is the point twist
ModularCurve.JZeroNeronObjectAtP.LevelModel.nonempty_poincare_pullbackAlong_pts_pic0Mk_iso_pointTwist15 below · cited by 1 · depth 16 - Assembling the at-p Néron datum from a Néron object
ModularCurve.JZeroNeronObjectAtP.exists_jZeroNeronAtPDataOrdV22_toric_eq_fin_eq_of_forall_smul_sub_mem5,559 below · cited by 1 · depth 16 - Prime-to-p abelian quotient family for the level-N₀p Néron object
ModularCurve.JZeroNeronObjectAtP.exists_abq_family_of_coprime2,209 below · cited by 1 · depth 17 - Assembly of the at-p Néron datum from a Néron object
ModularCurve.JZeroNeronObjectAtP.exists_jZeroNeronAtPDataOrdV22_of_children2,106 below · cited by 1 · depth 17 - Non-toric 𝔪-torsion at p forces lower-level torsion
ModularCurve.JZeroNeronObjectAtP.hasLowerLevelTorsion_of_mem_finPts_of_not_mem_toricPts2,138 below · cited by 1 · depth 17 - Non-toric 𝔪-torsion at p descends to level N₀
ModularCurve.JZeroNeronObjectAtP.heckeTorsion_ne_bot_of_mem_finPts_of_not_mem_toricPts2,161 below · cited by 1 · depth 17 - Cardinality of the A-extendable m-torsion: m^{t+4g₀}
ModularCurve.JZeroNeronObjectAtP.natCard_finPts1,619 below · cited by 4 · depth 17 - Frobenius acts as p Tₚ on toric torsion
ModularCurve.JZeroNeronObjectAtP.smul_eq_hecke_of_isFrobeniusAt_of_mem_toricPts_of_forall_smul_sub_mem_toricPts5,239 below · cited by 1 · depth 17 - Frobenius squared acts by p² on prime-to-p toric torsion
ModularCurve.JZeroNeronObjectAtP.smul_smul_eq_of_isFrobeniusAt_of_mem_toricPts_of_forall_smul_sub_mem_toricPts3,616 below · cited by 1 · depth 17 - Prime-to-p toric points lie in the monodromy toric part
ModularCurve.JZeroNeronObjectAtP.toricPts_le_toricMonodromyPart_of_forall_smul_sub_mem_toricPts1,702 below · cited by 3 · depth 17 - Kernel of an abelian-quotient family equals the toric points
ModularCurve.JZeroNeronObjectAtP.abq_eq_zero_iff_mem_toricPts_of_forall_reductionModL_eq989 below · cited by 2 · depth 18 - Hecke equivariance of an abelian-quotient family at p
ModularCurve.JZeroNeronObjectAtP.abq_heckeGen_smul_of_forall_reductionModL_eq1,225 below · cited by 1 · depth 18 - Decomposition group equivariance of the abelian quotient family
ModularCurve.JZeroNeronObjectAtP.abq_smul_of_mem_decompositionSubgroup_of_forall_reductionModL_eq980 below · cited by 1 · depth 18 - A prime-to-p abelian-quotient family on the Néron object
ModularCurve.JZeroNeronObjectAtP.exists_abq_family_forall_reductionModL_eq1,823 below · cited by 1 · depth 18 - From Néron object and extension to a v2.2 at-p datum
ModularCurve.JZeroNeronObjectAtP.exists_jZeroNeronAtPDataOrdV22_of_children_of_neronExtension2,057 below · cited by 1 · depth 18 - Divisibility of the toric point subgroups
ModularCurve.JZeroNeronObjectAtP.exists_mem_toricPts_mul_nsmul_eq0 below · cited by 1 · depth 18 - Lower-level 𝔪-torsion from a non-zero abelian-quotient coordinate
ModularCurve.JZeroNeronObjectAtP.hasLowerLevelTorsion_of_ptsSp_symm_fibreMap_abqFibre_ne_zero1,897 below · cited by 1 · depth 18 - Transport of 𝔪-torsion from level N₀p to level N₀
ModularCurve.JZeroNeronObjectAtP.heckeTorsion_ne_bot_of_ptsSp_symm_fibreMap_abqFibre_ne_zero1,920 below · cited by 1 · depth 18 - Order of the special m-kernel: m^{t+4g₀}
ModularCurve.JZeroNeronObjectAtP.isFinite_schemeKerStr_special_and_finrank_eq1,611 below · cited by 2 · depth 18 - Finite-part m-torsion counted by A-sections of the m-kernel
ModularCurve.JZeroNeronObjectAtP.natCard_finPts_eq_natCard_sections_schemeKer1 below · cited by 1 · depth 18 - Inertia-invariant n-torsion bounded by finite part times component group
ModularCurve.JZeroNeronObjectAtP.natCard_jZeroTorsion_inf_inertiaInvariants_le1,627 below · cited by 1 · depth 18 - Inertia displacement count for ℓ^k-torsion of J₀(N₀p)
ModularCurve.JZeroNeronObjectAtP.natCard_jZeroTorsion_le_mul_of_prime_pow6 below · cited by 1 · depth 18 - Existence of a Néron extension of an at-p Néron object
ModularCurve.JZeroNeronObjectAtP.nonempty_neronExtension84 below · cited by 1 · depth 18 - Newform eigenplane not inside the finite part of Tₚ J₀(N₀p)
ModularCurve.JZeroNeronObjectAtP.not_eigenPlane_le_span_tateModule_finPts_of_isNewform_of_inertia_smul_sub_mem_finPts2,099 below · cited by 1 · depth 18 - Non-toric finite p-points have non-zero abelian-quotient coordinates
ModularCurve.JZeroNeronObjectAtP.ptsSp_symm_fibreMap_abqFibre_ne_zero_of_mem_finPts_of_not_mem_toricPts1,741 below · cited by 2 · depth 18 - Range of the abelian-quotient maps is the m-torsion
ModularCurve.JZeroNeronObjectAtP.range_abq_eq_torsionBy_of_forall_reductionModL_eq1,761 below · cited by 1 · depth 18 - Kernel of [m] on the Néron object over a place A
ModularCurve.JZeroNeronObjectAtP.schemeKerStr_baseChange_props1,616 below · cited by 1 · depth 18 - Inertia fixes the prime-to-p toric points
ModularCurve.JZeroNeronObjectAtP.smul_eq_self_of_mem_inertiaSubgroupIn_of_mem_toricPts3 below · cited by 2 · depth 18 - Toric ab-torsion lies in toric a- plus b-torsion
ModularCurve.JZeroNeronObjectAtP.toricPts_mul_le_sup_of_coprime0 below · cited by 1 · depth 18 - Reduction mod p intertwines Hecke action on the special fibre
ModularCurve.JZeroNeronObjectAtP.LevelData.reductionModL_smul_eq_ptsSp_symm_schemeHomOverComp742 below · cited by 4 · depth 19 - Hecke operators extend to the Néron model over O_A
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_hecke_endomorphism7 below · cited by 1 · depth 19 - Decomposition-group stability of A-extendable points
ModularCurve.JZeroNeronObjectAtP.NeronExtension.extN_galois_smul7 below · cited by 1 · depth 19 - Inertia-invariant points extend to sections of the Néron extension
ModularCurve.JZeroNeronObjectAtP.NeronExtension.inertiaInvariants_le_extPts7 below · cited by 1 · depth 19 - Hopf-flat p-torsion subgroups lie in the finite part
ModularCurve.JZeroNeronObjectAtP.NeronExtension.le_finPts_of_hopf_of_forall_smul_sub_mem181 below · cited by 1 · depth 19 - Degeneracy morphisms intertwine Hecke endomorphisms over the base
ModularCurve.JZeroNeronObjectAtP.comp_degeneracyHom_eq_degeneracyHom_comp5 below · cited by 2 · depth 19 - The abelian-quotient class map on finite-part m-torsion
ModularCurve.JZeroNeronObjectAtP.exists_addMonoidHom_finPts_eq_ptsSp_symm_fibreMap_abqFibre2 below · cited by 2 · depth 19 - Endomorphisms act on toric lifts through M₀ mod m
ModularCurve.JZeroNeronObjectAtP.exists_comp_toricLift_fibreRestrictAlong_eq_toricLift_comp_mapDomainAlgHom40 below · cited by 3 · depth 19 - Semilinear twist of the toric part of the special fibre
ModularCurve.JZeroNeronObjectAtP.exists_mapRingHom_comp_torusFibre_eq_mapDomain_comp_torusFibre_comp_baseTwist22 below · cited by 1 · depth 19 - Prime-to-p division modulo points extending to the place
ModularCurve.JZeroNeronObjectAtP.exists_mem_inertiaInvariants_nsmul_eq_zero_sub_extendsToPlace21 below · cited by 1 · depth 19 - Glued full Néron model of J₀(N₀p) over O_A
ModularCurve.JZeroNeronObjectAtP.exists_neronGlue35 below · cited by 1 · depth 19 - Divisibility of A-points of the Néron identity component
ModularCurve.JZeroNeronObjectAtP.exists_nsmul_eq22 below · cited by 1 · depth 19 - Frobenius action on toric points via a reduced Frobenius matrix
ModularCurve.JZeroNeronObjectAtP.exists_smul_toricPoint_eq_toricPoint_galoisValues_comp_mapDomainAlgHom40 below · cited by 2 · depth 19 - Extendable prime-to-p torsion on J₀(N₀p) is inertia-invariant
ModularCurve.JZeroNeronObjectAtP.finPts_le_inertiaInvariantTorsion5 below · cited by 1 · depth 19 - Counting extendable m-torsion points with trivial abelian-quotient reduction
ModularCurve.JZeroNeronObjectAtP.finite_and_natCard_le_pow_toricRank_of_ptsSp_symm_fibreMap_abqFibre_eq_zero1,016 below · cited by 1 · depth 19 - Frobenius and Uₚ torus matrices are mutually inverse
ModularCurve.JZeroNeronObjectAtP.frobMatrix_comp_torusMatrix_eq_id_of_forall_prime_pow_smul_toricPoint45 below · cited by 1 · depth 19 - Order of the special m-kernel: m^t times a square
ModularCurve.JZeroNeronObjectAtP.isFinite_schemeKerStr_special_and_finrank_eq_mul_sq13 below · cited by 1 · depth 19 - Toric 𝔪-torsion bound at p∈𝔪
ModularCurve.JZeroNeronObjectAtP.natCard_toricPts_inf_heckeTorsion_le1,205 below · cited by 1 · depth 19 - Uₚ on the two special-fibre coordinates at p
ModularCurve.JZeroNeronObjectAtP.ptsSp_symm_fibreMap_abqFibre_comp_eq_of_degeneracyHom_heckeGen_self1,015 below · cited by 1 · depth 19 - Frobenius commutes with the transported Hecke operator on the reduction
ModularCurve.JZeroNeronObjectAtP.ptsSp_symm_schemeHomOverComp_frobSp960 below · cited by 2 · depth 19 - Rigidity: morphisms out of G are determined on ℚ̄-points
ModularCurve.JZeroNeronObjectAtP.schemeHomOver_ext_of_forall_pts_comp_eq5 below · cited by 1 · depth 19 - Toric points: a character group mapping isomorphically onto T̃[m]
ModularCurve.JZeroNeronObjectAtP.toricPoint_convMul_and_injective_and_mem_toricPts_iff_and_natCard0 below · cited by 3 · depth 19 - Saturation of the toric torsion tower of J₀(N₀p)
ModularCurve.JZeroNeronObjectAtP.toricPts_eq_inf1,634 below · cited by 1 · depth 19 - Prolongation of p-torsion realised by a finite flat Hopf algebra
ModularCurve.JZeroNeronObjectAtP.NeronExtension.extN_of_mem_of_hopf_of_forall_smul_sub_mem180 below · cited by 1 · depth 20 - Generic fibre of the Néron extension open immersion is an isomorphism
ModularCurve.JZeroNeronObjectAtP.NeronExtension.isIso_genericFibreRestrict_openImm6 below · cited by 2 · depth 20 - Membership in the finite part of a Néron extension
ModularCurve.JZeroNeronObjectAtP.NeronExtension.mem_finPts_iff0 below · cited by 2 · depth 20 - Vanishing of the component map and sections over the inertia-invariant base
ModularCurve.JZeroNeronObjectAtP.comp_eq_zero_iff_exists_schemeHomOver_shGenLift_eq16 below · cited by 1 · depth 20 - fppf-local sections of the m-torsion comparison map
ModularCurve.JZeroNeronObjectAtP.exists_fppfCover_section_schemeKer_of_abqFibre5 below · cited by 1 · depth 20 - Inertia-invariant points as K_A-points of the base change
ModularCurve.JZeroNeronObjectAtP.exists_inertiaInvariants_to_schemeHomOver_specGenericFibreInclusion_bijective1 below · cited by 2 · depth 20 - Split torus open in the joint degeneracy kernel
ModularCurve.JZeroNeronObjectAtP.exists_isOpenImmersion_torus_kerPair_degeneracyHom974 below · cited by 1 · depth 20 - Shear isomorphism for m-torsion over the abelian-quotient kernel pair
ModularCurve.JZeroNeronObjectAtP.exists_iso_pullback_schemeKer_torus_of_abqFibre0 below · cited by 1 · depth 20 - Toric part as joint kernel of the abelian-quotient pair
ModularCurve.JZeroNeronObjectAtP.exists_iso_torus_kerPair_abqFibre2 below · cited by 2 · depth 20 - Divisibility by m coprime to p on A-points
ModularCurve.JZeroNeronObjectAtP.exists_nsmul_eq_of_coprime17 below · cited by 1 · depth 20 - Toric points lift to A-sections with torus special point
ModularCurve.JZeroNeronObjectAtP.exists_section_and_torusPt_of_mem_toricPts1 below · cited by 1 · depth 20 - Finite flat prolongation of a Hopf-realised subgroup of J₀(N₀p)[p]
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_isFinite_forall_ptsN_comp_eq_of_hopf178 below · cited by 1 · depth 21 - Preconnectedness of the Néron object base-changed to A
ModularCurve.JZeroNeronObjectAtP.preconnectedSpace_pullback_g_sigmaA7 below · cited by 1 · depth 21 - Raynaud prolongation of the connected part into the Néron model
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_forall_mul_eq_specMap_forall_specMap_comp_eq_ptsN_of_bialgHom_of_isLocalRing154 below · cited by 1 · depth 22 - Inertia-invariant points give sections of the Néron extension
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_ptsN_eq_comp_of_mem_inertiaInvariants7 below · cited by 1 · depth 22 - Finite part of the m-torsion of the Néron model
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_hopfAlgebra_finPts_equiv_forall_specMap_comp_eq_ptsN67 below · cited by 1 · depth 23 - Identity-component finite part lies inside the Néron finite part
ModularCurve.JZeroNeronObjectAtP.NeronExtension.finPts_le_finPts0 below · cited by 1 · depth 23 - Néron extension criterion: closure meets the special fibre
ModularCurve.JZeroNeronObjectAtP.NeronExtension.extN_iff_closure_inter_preimage_closedPoint_nonempty18 below · cited by 1 · depth 24 - Multiplication by m on a Néron extension is quasi-finite and flat
ModularCurve.JZeroNeronObjectAtP.NeronExtension.locallyQuasiFinite_quasiCompact_flat_schemeNsmul36 below · cited by 1 · depth 24 - Néron model as union of component-group translates
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_sections_forall_exists_mem_range_openImm_comp_mul22 below · cited by 1 · depth 25 - Flatness and local quasi-finiteness of [m] after base change
ModularCurve.JZeroNeronObjectAtP.locallyQuasiFinite_quasiCompact_flat_schemeNsmul_baseChange_shStr12 below · cited by 1 · depth 25 - Every component group element is hit by a global section
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_section_specN_eq8 below · cited by 1 · depth 26 - Points over the closed point specialise to section images
ModularCurve.JZeroNeronObjectAtP.NeronExtension.exists_specializes_section_base_closedPoint_eq16 below · cited by 1 · depth 26 - Points off the closed fibre lie in the open immersion's image
ModularCurve.JZeroNeronObjectAtP.NeronExtension.mem_range_openImm_of_base_ne_closedPoint5 below · cited by 1 · depth 26
ModularCurve.JZeroNeronPrimaryTorsionCore 4
- Points of H_m over 𝔽̄₂ number a power of two
ModularCurve.JZeroNeronPrimaryTorsionCore.exists_natCard_algHom_H_algebraicClosure_zmod_two_eq_pow0 below · cited by 2 · depth 12 - Fppf H¹ of a Néron core sheaf has q-power order
ModularCurve.JZeroNeronPrimaryTorsionCore.exists_natCard_fppfCohomology_one_eq_pow1,484 below · cited by 1 · depth 12 - Global sections of mathcal J_m have q-power order
ModularCurve.JZeroNeronPrimaryTorsionCore.exists_natCard_fppfCohomology_zero_eq_pow715 below · cited by 1 · depth 12 - Multiplication by q^m annihilates the sheaves mathcal J_m
ModularCurve.JZeroNeronPrimaryTorsionCore.pow_nsmul_sections_eq_zero7 below · cited by 1 · depth 13
ModularCurve.JZeroNeronPrimaryTorsionFFModels 1
- Mod-q fibre of the finite flat model has q-power point count
ModularCurve.JZeroNeronPrimaryTorsionFFModels.exists_natCard_withConv_hffBarQ_algHom_eq_pow2 below · cited by 1 · depth 12
ModularCurve.JZeroNeronPrimaryTorsionFlag 18
- Admissible chain from a primary Néron torsion flag
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_admissibleChain_filtAlpha_eq0 below · cited by 1 · depth 11 - Layer inequality l₁-l₀+[const]≤ d_g-dₜ for odd q
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_cokernel_h1_sub_h0_add_ite_kind_le_dg_sub_dt_of_ne_two1,207 below · cited by 1 · depth 11 - Mult-kind layer bound l₁+dₜ≤ l₀+1 at odd q
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_cokernel_h1_add_dt_le_h0_add_one_of_kind_eq_mult_of_ne_two1,182 below · cited by 1 · depth 12 - Const-kind layer: l₁ + dₜ ≤ l₀ at odd q
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_cokernel_h1_add_dt_le_h0_of_kind_eq_const_of_ne_two1,156 below · cited by 1 · depth 12 - Per-layer cohomology inequality l₁-l₀+dₐ≤ d_g-dₜ at 2
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_cokernel_h1_sub_h0_add_da_le_dg_sub_dt_two1,165 below · cited by 1 · depth 13 - Point counts along a flag step multiply by a power of 2
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_natCard_algHom_succ_eq_pow_mul_natCard_algHom_castSucc_two1 below · cited by 3 · depth 13 - Multiplicative layers: trivial H⁰ and μ_q-kernel bound
ModularCurve.JZeroNeronPrimaryTorsionFlag.natCard_fppfCohomology_zero_cokernel_eq_one_and_exists_ker_natCard_eq_pow_of_kind_eq_mult_of_ne_two1,163 below · cited by 1 · depth 13 - Constant-kind layer: H⁰ count and H¹ injection into ℤ/q
ModularCurve.JZeroNeronPrimaryTorsionFlag.natCard_fppfCohomology_zero_cokernel_eq_pow_and_exists_hom_restriction_constantZMod_of_kind_eq_const_of_ne_two1,133 below · cited by 1 · depth 13 - Cohomology bounds for a layer of the 2-primary flag
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_cokernel_dt_le_h0_and_h1_add_da_le_one_two1,164 below · cited by 1 · depth 14 - Hopf-algebra layer representing a flag step quotient sheaf
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_hopfAlgebra_range_eq_hopfKer_sectionsEquiv37 below · cited by 6 · depth 14 - Finiteness of fppf H¹ of a flag layer at q=2
ModularCurve.JZeroNeronPrimaryTorsionFlag.finite_fppfCohomology_one_layer_two1,163 below · cited by 1 · depth 14 - Point counts multiply across a Hopf–Galois flag step
ModularCurve.JZeroNeronPrimaryTorsionFlag.natCard_algHom_succ_eq_mul_natCard_algHom_hopfKer1,076 below · cited by 6 · depth 14 - Multiplicative flag layers: Galois acts by n_σ-th convolution power
ModularCurve.JZeroNeronPrimaryTorsionFlag.ringEquiv_apply_algHom_eq_convPow_of_range_eq_hopfKer_of_kind_eq_mult1 below · cited by 2 · depth 14 - Constant-kind flag layers have Galois-invariant ℚ̄-points
ModularCurve.JZeroNeronPrimaryTorsionFlag.ringEquiv_apply_algHom_eq_of_range_eq_hopfKer_of_kind_eq_const0 below · cited by 2 · depth 14 - Flag projections are ℤ-bialgebra maps
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_bialgHom_toAlgHom_eq_pi0 below · cited by 1 · depth 15 - Hopf-algebra model of a flag-layer cokernel at 2
ModularCurve.JZeroNeronPrimaryTorsionFlag.exists_hopfAlgebra_cokernel_sectionsEquiv_natCard_two1,163 below · cited by 1 · depth 15 - Finiteness of H¹_{fppf} at a constant flag layer, q odd
ModularCurve.JZeroNeronPrimaryTorsionFlag.finite_fppfCohomology_one_cokernel_of_kind_eq_const_of_ne_two1,154 below · cited by 1 · depth 15 - Finiteness of H¹_{fppf} for multiplicative flag layers
ModularCurve.JZeroNeronPrimaryTorsionFlag.finite_fppfCohomology_one_cokernel_of_kind_eq_mult_of_ne_two1,180 below · cited by 1 · depth 15
ModularCurve.JZeroNeronPrimaryTorsionSheaf 2
- Mazur's Proposition I.1.7 at the prime 2
ModularCurve.JZeroNeronPrimaryTorsionSheaf.prop17_of_forall_nonempty_jZeroNeronPrimaryTorsionFlag_two1,168 below · cited by 1 · depth 11 - Two-adic bound h¹+a≤ h⁰+δ from non-empty flags
ModularCurve.JZeroNeronPrimaryTorsionSheaf.h1_add_le_h0_add_delta_of_forall_nonempty_jZeroNeronPrimaryTorsionFlag_two1,167 below · cited by 1 · depth 12
ModularCurve.KatzGamma0Form 2
- Level reduction for even-weight Katz forms on Γ₀(p)
ModularCurve.KatzGamma0Form.exists_pullbackLevelP_eq_of_qTwist_qExpansion_eq_of_even_of_five_le413 below · cited by 2 · depth 17 - Level reduction for Katz Γ₀(p) forms, p≥ 5
ModularCurve.KatzGamma0Form.exists_pullbackLevelP_eq_of_qTwist_qExpansion_eq_of_five_le412 below · cited by 1 · depth 18
ModularCurve.KatzLevelPForm 8
- q-expansion principle for Γ₀(p)-type Katz forms
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnSndLine_of_evalCusp_eq_zero328 below · cited by 1 · depth 18 - q-expansion principle at split Cartan level p
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnLines_of_forall_evalCusp_eq_zero328 below · cited by 1 · depth 19 - q-expansion principle for Γ₀(p)-type Katz level-p forms
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnSndLine_of_evalCusp_eq_zero_of_field313 below · cited by 1 · depth 19 - Descent of the q-expansion principle from fields
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnSndLine_of_evalCusp_eq_zero_of_forall_field18 below · cited by 1 · depth 19 - Descent of swap-invariant level-p Katz forms to level one
ModularCurve.KatzLevelPForm.existsUnique_pullbackLevelP_eq_of_swapInvariant13 below · cited by 1 · depth 19 - Vanishing at one cusp kills line-dependent Katz level-p forms
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnLines_of_evalCusp_eq_zero_of_field318 below · cited by 1 · depth 20 - Vanishing of Katz level-p forms: fields to rings
ModularCurve.KatzLevelPForm.eq_zero_of_dependsOnlyOnLines_of_evalCusp_eq_zero_of_forall_field17 below · cited by 1 · depth 20 - Generic-curve vanishing of a level-p Katz form
ModularCurve.KatzLevelPForm.eq_zero_of_forall_toFun_genericCurve_eq_zero4 below · cited by 2 · depth 20
ModularCurve.LambdaModularPolynomialData 4
- Fricke twist u ↦ 1/16 - u of the λ-modular equation
ModularCurve.LambdaModularPolynomialData.eval2_sixteenth_sub_eq_zero185 below · cited by 2 · depth 22 - Every Y-coefficient of Ψ has X-degree at most q+1
ModularCurve.LambdaModularPolynomialData.natDegree_coeff_le180 below · cited by 2 · depth 22 - Reciprocal symmetry of the modular polynomial Ψ_q
ModularCurve.LambdaModularPolynomialData.psi_reciprocal196 below · cited by 1 · depth 22 - Swapped λ-modular equation via the Atkin–Lehner involution
ModularCurve.LambdaModularPolynomialData.eval2_swap_eq_zero176 below · cited by 2 · depth 23
ModularCurve.LambdaNodeLocalized 29
- Primes containing the λ-node relations coincide
ModularCurve.LambdaNodeLocalized.eq_of_isPrime_of_forall_lambdaEval_mem342 below · cited by 1 · depth 20 - Anharmonic group of order six acting on the λ-field
ModularCurve.LambdaNodeLocalized.exists_anharmonic_mulSemiringAction_lambdaFieldOver213 below · cited by 2 · depth 20 - Maximal ideals of the λ-extension carry a level-two value
ModularCurve.LambdaNodeLocalized.exists_forall_lambdaEval_mem_of_isMaximal228 below · cited by 1 · depth 20 - Node-ring expansion of j(q²) over j = 0, 1728
ModularCurve.LambdaNodeLocalized.exists_qExpand_two_jq_sub_eq_unit_mul_pow_jWidth_of_eq_zero_or_eq_1728206 below · cited by 2 · depth 20 - An anharmonic automorphism transporting level-two nodes over j=0,1728
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_lambdaFieldOver_forall_map_lambdaEval_mem214 below · cited by 1 · depth 20 - Stabiliser of a λ-node over j=0,1728 acts tangentially by (ζ,ζ^q)
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_sub_smul_mem_sq_sup_of_stabilizer_of_eq_zero_or_eq_1728208 below · cited by 1 · depth 20 - Fixed ring of a node automorphism is a crossing model
ModularCurve.LambdaNodeLocalized.exists_ringHom_uvCrossingModel_pow_jWidth_range_eq_fixedPoints_adicCompletion361 below · cited by 1 · depth 20 - Branch pins of the crossing model at j ∈ {0,1728}
ModularCurve.LambdaNodeLocalized.exists_span_pair_eq_of_uvCrossingModel_apply_eq_qExpand_two_jq_sub_of_kroneckerCongruence9 below · cited by 1 · depth 20 - Invariant subring of the level-two ring with completion the ̂ g-fixed ring
ModularCurve.LambdaNodeLocalized.exists_subring_adicCompletion_ringEquiv_eqLocus_of_stabilizer_of_eq_zero_or_eq_1728377 below · cited by 1 · depth 20 - Crossing-chart expansion of J-x and J_q-x^q at a wide node
ModularCurve.LambdaNodeLocalized.exists_units_uvCrossingModel_apply_eq_qExpand_two_jq_sub_of_range_eq_fixedPoints347 below · cited by 1 · depth 20 - Noetherian local λ-node ring of dimension 2 at (l,l^q)
ModularCurve.LambdaNodeLocalized.isNoetherianRing_isLocalRing_lambdaLocalizedAtPoint_coeffSubring213 below · cited by 7 · depth 20 - Level-one q-expansions at q² lie in the λ-field
ModularCurve.LambdaNodeLocalized.qExpand_two_mem_lambdaFieldOver_of_mem_fieldOver27 below · cited by 5 · depth 20 - Maximal ideals of the level-two ring are λ-nodes
ModularCurve.LambdaNodeLocalized.exists_eq_comap_maximalIdeal_lambdaLocalizedAtPoint_of_isMaximal358 below · cited by 1 · depth 21 - Every element of λ-field over K is a quotient in the localised node ring
ModularCurve.LambdaNodeLocalized.exists_mul_eq_of_mem_lambdaFieldOver0 below · cited by 2 · depth 21 - Nonzero kernel element of `lambdaEval` from modular polynomial data
ModularCurve.LambdaNodeLocalized.exists_ne_zero_lambdaEval_eq_zero0 below · cited by 1 · depth 21 - Completed λ-node ring at a supersingular point is a crossing
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_adicCompletion_lambdaLocalizedAtPoint_uvCrossingModel345 below · cited by 2 · depth 21 - An involution μ↦ 1/(256μ) of the λ-field of level q
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_lambdaFieldOver_map_eq_inv211 below · cited by 2 · depth 21 - Anharmonic involution μ↦ 1/16-μ on the λ-field
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_lambdaFieldOver_map_eq_sixteenth_sub207 below · cited by 2 · depth 21 - Every element of the localized node ring is congruent to a constant
ModularCurve.LambdaNodeLocalized.exists_sub_const_mem_maximalIdeal_lambdaLocalizedAtPoint208 below · cited by 1 · depth 21 - Level-two node ring as a localisation of an integral closure
ModularCurve.LambdaNodeLocalized.isLocalization_atPrime_lambdaLocalizedAtPoint_of_isIntegralElem355 below · cited by 3 · depth 21 - Branch ideals are prime in the localised λ-node ring
ModularCurve.LambdaNodeLocalized.isPrime_span_uniformizer_branches_lambdaLocalizedAtPoint211 below · cited by 2 · depth 21 - Module-finiteness of the λ-level integral closure over the node local ring
ModularCurve.LambdaNodeLocalized.moduleFinite_of_forall_mem_iff_isIntegralElem_qExpand_modularLocalizedAtPoint179 below · cited by 1 · depth 21 - Relations between the λ-series vanish at (l,l^q) mod q
ModularCurve.LambdaNodeLocalized.pointEval_eq_zero_of_lambdaEval_eq_zero_of_ne_two205 below · cited by 10 · depth 21 - Maximal ideal Q contracted from the λ-localisation at (l',l'^q)
ModularCurve.LambdaNodeLocalized.eq_comap_maximalIdeal_lambdaLocalizedAtPoint_of_sub_const_mem356 below · cited by 1 · depth 22 - Branch form of the λ-Kronecker congruence mod q
ModularCurve.LambdaNodeLocalized.eval2_branch_eq_zero_of_lambdaEval_eq_zero191 below · cited by 1 · depth 22 - Finite spanning set over the descended node ring at level two
ModularCurve.LambdaNodeLocalized.exists_finset_forall_isIntegralElem_eq_sum_mul_of_mem_lambdaFieldOver178 below · cited by 1 · depth 22 - Liftable level-two value at a maximal ideal of B
ModularCurve.LambdaNodeLocalized.exists_level_two_value_sub_const_mem_of_isMaximal356 below · cited by 1 · depth 22 - An involutive root of Ψ extends to a λ-field automorphism
ModularCurve.LambdaNodeLocalized.exists_ringEquiv_lambdaFieldOver_of_involutive_subst174 below · cited by 2 · depth 22 - Fricke symmetry of the λ-pair at level two
ModularCurve.LambdaNodeLocalized.lambdaEval_aeval_sixteenth_sub_swap_eq_zero185 below · cited by 1 · depth 23
ModularCurve.LevelComponent 4
- Universal Katz level-ℓ Weil pairing as ℓ-th root of unity
ModularCurve.LevelComponent.exists_pow_eq_one_and_forall_weilPairing0_toPoint_mapRing_eq_of_mk_eq_univ39 below · cited by 3 · depth 30 - Local constancy of the Weil pairing of a level-ℓ basis
ModularCurve.LevelComponent.exists_not_mem_and_exists_pow_eq_one_forall_weilPairing0_toPoint_mapRing_localizationAway_eq33 below · cited by 1 · depth 31 - Infinitesimal rigidity of Γ₀(M') and level-ℓ structures
ModularCurve.LevelComponent.act_eq_of_mapRing_fstHom_eq_of_map_fstHom_eq_one_of_smul_curve_eq_gamma0Pow16 below · cited by 1 · depth 38 - Infinitesimal rigidity of the Γ₀(M')∩Γ₁(ℓ_g) level datum
ModularCurve.LevelComponent.act_eq_of_mapRing_fstHom_eq_of_map_fstHom_eq_one_of_smul_curve_eq_rigidDataH1Pow14 below · cited by 1 · depth 38
ModularCurve.LevelModuliPackageAbs 76
- Points lifting along ι are classified through ι
ModularCurve.LevelModuliPackageAbs.apply_mem_range_of_map_eq_map_univ0 below · cited by 4 · depth 32 - Uniqueness of an abstract representing package of a level-moduli datum
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_map_univ_eq0 below · cited by 11 · depth 32 - Regular two-dimensional complete local ring at a supersingular point
ModularCurve.LevelModuliPackageAbs.exists_isDrinfeldBasisAdic_isRegularLocalRing_hasseParam_adicCompletion_of_forall_mem_ssJSet_gamma0Pow1,037 below · cited by 1 · depth 32 - Invertible moduli-problem automorphisms act on the fine moduli ring
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_apply_jOf_univ_eq_classify_act_eq0 below · cited by 4 · depth 33 - Hasse parameter and j-invariant at a supersingular point, Γ₀-tuple level
ModularCurve.LevelModuliPackageAbs.exists_coeff_nthSeries_sub_mul_mem_span_and_map_j0_sub_algebraMap_eq_mul_pow_of_factorsThrough_of_five_le_gamma0Pow73 below · cited by 2 · depth 33 - Coefficient ring and factorisation for completed fine moduli rings
ModularCurve.LevelModuliPackageAbs.exists_coefficientRing_factorsThrough_adicCompletion_of_levelModuliDatum11 below · cited by 12 · depth 33 - Regular complete local ring at a supersingular Drinfeld-level point
ModularCurve.LevelModuliPackageAbs.exists_isDrinfeldBasisAdic_isRegularLocalRing_hasseParam_adicCompletion_of_forall_mem_ssJSet_rigidDataH1Pow1,053 below · cited by 1 · depth 33 - Complete local ring at a supersingular point, with level relabelling
ModularCurve.LevelModuliPackageAbs.exists_isDrinfeldBasisAdic_isRegularLocalRing_hasseParam_problemAut_linearPart_adicCompletion_of_forall_mem_ssJSet_gamma0Pow1,093 below · cited by 1 · depth 33 - Universal formal Drinfeld basis at a supersingular point
ModularCurve.LevelModuliPackageAbs.exists_reducesToOrigin_isDrinfeldBasisAdic_universal_of_factorsThrough_of_ne_two_gamma0Pow960 below · cited by 4 · depth 33 - Uniqueness of the classifying W₀-algebra map, Γ₀-power level
ModularCurve.LevelModuliPackageAbs.algHom_eq_of_isBaseChange_lawIso_appAdic_eq_gamma0Pow78 below · cited by 1 · depth 34 - Existence of a classifying W₀-algebra map for Drinfeld bases
ModularCurve.LevelModuliPackageAbs.exists_algHom_isBaseChange_lawIso_appAdic_eq_gamma0Pow936 below · cited by 1 · depth 34 - Hasse parameter and j at a supersingular Drinfeld point
ModularCurve.LevelModuliPackageAbs.exists_coeff_nthSeries_sub_mul_mem_span_and_map_j0_sub_algebraMap_eq_mul_eval_of_factorsThrough_rigidDataH1Pow78 below · cited by 2 · depth 34 - Supersingular completion: Drinfeld basis, Hasse parameter, relabelling linear part
ModularCurve.LevelModuliPackageAbs.exists_isDrinfeldBasisAdic_isRegularLocalRing_hasseParam_problemAut_linearPart_adicCompletion_of_forall_mem_ssJSet_rigidDataH1Pow1,117 below · cited by 1 · depth 34 - Rigidified universality of the H₁ moduli ring at a supersingular point
ModularCurve.LevelModuliPackageAbs.exists_reducesToOrigin_isDrinfeldBasisAdic_universal_of_factorsThrough_rigidDataH1Pow972 below · cited by 4 · depth 34 - Level relabelling on the deformation ring: linear part cγ̄
ModularCurve.LevelModuliPackageAbs.exists_ringEquiv_originParam_linearPart_of_problemAut_relabel_of_reducesToOrigin_universal_gamma0Pow_of_mem_ssJSet189 below · cited by 1 · depth 34 - Residue base change of the universal formal group and Drinfeld-basis transport
ModularCurve.LevelModuliPackageAbs.isBaseChange_and_isDrinfeldBasisAdic_residue_of_toPowerSeries_eq_gamma0Pow3 below · cited by 1 · depth 34 - Reducedness of R/(1-ζ) at an ordinary Drinfeld point
ModularCurve.LevelModuliPackageAbs.isReduced_quotient_span_one_sub_of_pow_eq_one_of_factorsThrough_of_nthSeries_eq_mul_X_pow_gamma0Pow1,155 below · cited by 1 · depth 34 - Equal classifying kernels give equal q-Weil pairing values
ModularCurve.LevelModuliPackageAbs.weilPairing0_drinfeld_mapRing_eq_of_ker_classify_eq_rigidDataPow254 below · cited by 1 · depth 34 - Weil pairing determined by the kernel of `classify`
ModularCurve.LevelModuliPackageAbs.weilPairing0_mapRing_eq_of_ker_classify_eq_rigidDataPow41 below · cited by 1 · depth 34 - Uniqueness of the classifying map at level H₁
ModularCurve.LevelModuliPackageAbs.algHom_eq_of_isBaseChange_lawIso_appAdic_eq_rigidDataH1Pow77 below · cited by 1 · depth 35 - Igusa presentation of the ordinary completed local ring
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_adjoinRoot_powerSeries_of_factorsThrough_of_nthSeries_eq_mul_X_pow_of_five_le_gamma0Pow1,151 below · cited by 2 · depth 35 - Ordinary H₁ local ring as W₀[[t]][X]/(g), g Eisenstein-like
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_adjoinRoot_powerSeries_of_factorsThrough_of_nthSeries_eq_mul_X_pow_rigidDataH1Pow1,153 below · cited by 2 · depth 35 - Pinned relabelling lifts to a residue-trivial automorphism of R
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_comp_eq_classify_act_of_problemAut_relabel_of_factorsThrough_gamma0Pow132 below · cited by 1 · depth 35 - Existence of the classifying W₀-algebra map on formal deformations
ModularCurve.LevelModuliPackageAbs.exists_algHom_isBaseChange_lawIso_appAdic_eq_rigidDataH1Pow947 below · cited by 1 · depth 35 - Raw Drinfeld points over T come from R
ModularCurve.LevelModuliPackageAbs.exists_algHom_of_raw_lawIso_appAdic_eq_gamma0Pow21 below · cited by 1 · depth 35 - Equal classifying kernels yield a common model
ModularCurve.LevelModuliPackageAbs.exists_eq_act_mapRing_of_ker_classify_eq2 below · cited by 2 · depth 35 - Linear part of the relabelling endomorphism on Drinfeld parameters
ModularCurve.LevelModuliPackageAbs.exists_originParam_linearPart_of_algHom_comp_eq_classify_act_of_problemAut_relabel_gamma0Pow_of_mem_ssJSet182 below · cited by 1 · depth 35 - Lifting a Drinfeld basis to a raw Γ₀(M')-level datum
ModularCurve.LevelModuliPackageAbs.exists_raw_lawIso_appAdic_eq_of_isDrinfeldBasisAdic_gamma0Pow924 below · cited by 1 · depth 35 - Relabelling automorphisms act linearly on Drinfeld origin parameters
ModularCurve.LevelModuliPackageAbs.exists_ringEquiv_originParam_linearPart_of_problemAut_relabel_of_reducesToOrigin_universal_rigidDataH1Pow_of_mem_ssJSet218 below · cited by 1 · depth 35 - Residue base change and Drinfeld basis of the reduced law
ModularCurve.LevelModuliPackageAbs.isBaseChange_and_isDrinfeldBasisAdic_residue_of_toPowerSeries_eq_rigidDataH1Pow3 below · cited by 1 · depth 35 - Two rigidified lifts induce the same moduli point
ModularCurve.LevelModuliPackageAbs.map_univ_eq_of_isBaseChange_lawIso_appAdic_eq_gamma0Pow77 below · cited by 1 · depth 35 - Fibres of j on a representing object count moduli points
ModularCurve.LevelModuliPackageAbs.natCard_algHom_apply_jOf_univ_eq_natCard_pt_jOf_eq0 below · cited by 2 · depth 35 - Equal `classify` kernels give equal Drinfeld Weil pairing values
ModularCurve.LevelModuliPackageAbs.weilPairing0_drinfeld_mapRing_eq_of_ker_classify_eq_rigidDataH1Pow254 below · cited by 1 · depth 35 - First-order action of level relabelling on Drinfeld origin parameters
ModularCurve.LevelModuliPackageAbs.apply_originParam_sub_inv_u_mul_mem_sq_of_act_mapRing_eq_relabel_gamma0Pow118 below · cited by 1 · depth 36 - Igusa presentation of the ordinary local ring, normal position
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_adjoinRoot_powerSeries_of_nthSeries_eq_mul_X_pow_of_eq_one_of_ne_one_rigidDataH1Pow1,139 below · cited by 1 · depth 36 - Igusa presentation of the completed ordinary stalk in normal position
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_adjoinRoot_powerSeries_of_nthSeries_eq_mul_X_pow_of_five_le_of_eq_one_of_ne_one_gamma0Pow1,137 below · cited by 1 · depth 36 - Relabelling automorphism of the rigidified ring R
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_comp_eq_classify_act_of_problemAut_relabel_of_factorsThrough_rigidDataH1Pow183 below · cited by 1 · depth 36 - Artinian points with prescribed reduction arise from R → T
ModularCurve.LevelModuliPackageAbs.exists_algHom_of_raw_lawIso_appAdic_eq_rigidDataH1Pow26 below · cited by 1 · depth 36 - Linear part of a Γ₀(M')-relabelling on Drinfeld origin parameters
ModularCurve.LevelModuliPackageAbs.exists_originParam_linearPart_of_algHom_comp_eq_classify_act_of_problemAut_relabel_rigidDataH1Pow_of_mem_ssJSet214 below · cited by 1 · depth 36 - Artinian lift of a Drinfeld basis to a raw H₁ datum
ModularCurve.LevelModuliPackageAbs.exists_raw_lawIso_appAdic_eq_of_isDrinfeldBasisAdic_rigidDataH1Pow934 below · cited by 1 · depth 36 - GL₂ relabelling of the Drinfeld pair realised by an automorphism
ModularCurve.LevelModuliPackageAbs.exists_raw_linComb_algEquiv_map_univ_eq_gamma0Pow113 below · cited by 1 · depth 36 - Basis relabelling by γ is induced by an automorphism of B₀
ModularCurve.LevelModuliPackageAbs.exists_raw_linComb_algEquiv_map_univ_eq_rigidDataH1Pow113 below · cited by 1 · depth 36 - Relabelling automorphism pulls back to γ-relabelled datum over a domain
ModularCurve.LevelModuliPackageAbs.exists_variableChange_act_mapRing_classify_act_univ_eq_relabel_of_isDomain_gamma0Pow_of_isUnit123 below · cited by 2 · depth 36 - Lifts of the universal curve differ by a trivial variable change
ModularCurve.LevelModuliPackageAbs.exists_variableChange_map_eq_one_smul_map_eq_map_of_lawIso_gamma0Pow46 below · cited by 1 · depth 36 - Uniqueness of Γ₀(M') kernel data under a trivial variable change
ModularCurve.LevelModuliPackageAbs.kernel_map_eq_kernelVariableChangeDeg_of_smul_map_eq_gamma0Pow12 below · cited by 1 · depth 36 - Level-ℓ data of two lifts differ by the variable change C
ModularCurve.LevelModuliPackageAbs.levelPData_map_eq_variableChange_of_smul_map_eq_gamma0Pow8 below · cited by 1 · depth 36 - Drinfeld pairs transported by a variable change reducing to one
ModularCurve.LevelModuliPackageAbs.levelTransport_map_eq_act_map_of_smul_map_eq_gamma0Pow23 below · cited by 1 · depth 36 - Equality of moduli points from matching Drinfeld basis data
ModularCurve.LevelModuliPackageAbs.map_univ_eq_of_isBaseChange_lawIso_appAdic_eq_rigidDataH1Pow76 below · cited by 1 · depth 36 - Residue compatibility of the classifying map at Γ₀-power level
ModularCurve.LevelModuliPackageAbs.residue_classify_eq_of_map_residue_eq_gamma0Pow4 below · cited by 1 · depth 36 - Scaling of a γ-relabelling is a (q+1)-st root of unity mod 𝔪
ModularCurve.LevelModuliPackageAbs.u_pow_sub_one_mem_and_of_act_mapRing_eq_relabel_gamma0Pow_of_mem_ssJSet45 below · cited by 1 · depth 36 - First-order action of a relabelling on Drinfeld origin parameters
ModularCurve.LevelModuliPackageAbs.apply_originParam_sub_inv_u_mul_mem_sq_of_act_mapRing_eq_relabel_rigidDataH1Pow118 below · cited by 1 · depth 37 - Inverse problem automorphisms come from an algebra automorphism
ModularCurve.LevelModuliPackageAbs.exists_algEquiv_map_univ_eq_act0 below · cited by 2 · depth 37 - One-dimensional tangent space at an ordinary q-torsion point
ModularCurve.LevelModuliPackageAbs.exists_algHom_dualNumber_of_represents_nsmul_eq_one_of_nthSeries_eq_mul_X_pow_gamma0Pow939 below · cited by 1 · depth 37 - First-order deformations with q-torsion section form a line (H₁ level)
ModularCurve.LevelModuliPackageAbs.exists_algHom_dualNumber_of_represents_nsmul_eq_one_of_nthSeries_eq_mul_X_pow_rigidDataH1Pow937 below · cited by 1 · depth 37 - Relabelling automorphism pulls back by a variable change
ModularCurve.LevelModuliPackageAbs.exists_variableChange_act_mapRing_classify_act_univ_eq_relabel_toPoint_of_mem_gamma0_of_isDomain_rigidDataH1Pow_of_isUnit180 below · cited by 2 · depth 37 - Two deformations with isomorphic formal groups differ by a trivial variable change
ModularCurve.LevelModuliPackageAbs.exists_variableChange_map_eq_one_smul_map_eq_map_of_lawIso_rigidDataH1Pow47 below · cited by 1 · depth 37 - Kernel data of two lifts differ by the variable change
ModularCurve.LevelModuliPackageAbs.kernel_map_eq_kernelVariableChangeDeg_of_smul_map_eq_rigidDataH1Pow12 below · cited by 1 · depth 37 - Equality of Γ₁(ℓ_g)-data under an infinitesimal variable change
ModularCurve.LevelModuliPackageAbs.levelPData_map_eq_variableChange_of_smul_map_eq_rigidDataH1Pow6 below · cited by 1 · depth 37 - Transport of Drinfeld pairs under a residually trivial variable change
ModularCurve.LevelModuliPackageAbs.levelTransport_map_eq_act_map_of_smul_map_eq_rigidDataH1Pow23 below · cited by 1 · depth 37 - Full-level moduli deformation ring is the Igusa root ring S[X]/(g)
ModularCurve.LevelModuliPackageAbs.nonempty_algEquiv_adjoinRoot_of_factorsThrough_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_gamma0Pow964 below · cited by 1 · depth 37 - Completed local ring at level H₁ is S[X]/(g)
ModularCurve.LevelModuliPackageAbs.nonempty_algEquiv_adjoinRoot_of_factorsThrough_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_rigidDataH1Pow968 below · cited by 1 · depth 37 - Completed stalk of the Γ₀(M')×Γ(ℓ) moduli ring is W₀[[t]]
ModularCurve.LevelModuliPackageAbs.nonempty_algEquiv_powerSeries_of_factorsThrough_trivial_gamma0Pow30 below · cited by 1 · depth 37 - Completed stalk of the linked Γ₀(M')×Γ₁(ℓ) package is W₀[[t]]
ModularCurve.LevelModuliPackageAbs.nonempty_algEquiv_powerSeries_of_factorsThrough_trivial_rigidDataH1Pow39 below · cited by 1 · depth 37 - Residue compatibility of the classifying map for `rigidDataH1Pow`
ModularCurve.LevelModuliPackageAbs.residue_classify_eq_of_map_residue_eq_rigidDataH1Pow4 below · cited by 1 · depth 37 - Rigidity of the scalar of a relabelling automorphism at supersingular j
ModularCurve.LevelModuliPackageAbs.u_pow_sub_one_mem_and_of_act_mapRing_eq_relabel_rigidDataH1Pow_of_mem_ssJSet182 below · cited by 1 · depth 37 - Pro-representability and lifting for deformations in a fine level-moduli problem
ModularCurve.LevelModuliPackageAbs.existsUnique_map_comp_univ_eq_and_exists_comp_eq_of_factorsThrough0 below · cited by 2 · depth 38 - R-points as pairs: an S-point and an Igusa root
ModularCurve.LevelModuliPackageAbs.exists_algHom_equiv_subtype_eval_map_eq_zero_natural_of_factorsThrough_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_gamma0Pow961 below · cited by 1 · depth 38 - Artinian W₀-points of R as Igusa root pairs, naturally
ModularCurve.LevelModuliPackageAbs.exists_algHom_equiv_subtype_eval_map_eq_zero_natural_of_factorsThrough_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_rigidDataH1Pow965 below · cited by 1 · depth 38 - Deformation ring points as normalised Drinfeld triples, naturally
ModularCurve.LevelModuliPackageAbs.exists_algHom_equiv_isDrinfeldBasisOver_natural_of_factorsThrough_gamma0Pow39 below · cited by 1 · depth 39 - Natural normal form for W₀-algebra maps out of R
ModularCurve.LevelModuliPackageAbs.exists_algHom_equiv_isDrinfeldBasisOver_natural_of_factorsThrough_rigidDataH1Pow44 below · cited by 1 · depth 39 - Torsion-lift ring S represents q-torsion lifts of ̄ Q_k
ModularCurve.LevelModuliPackageAbs.exists_equiv_algHom_symm_apply_eq_of_represents_nsmul_eq_one_of_factorsThrough_gamma0Pow1 below · cited by 1 · depth 39 - Torsion-lift ring S represents pinned q-torsion points at level H₁
ModularCurve.LevelModuliPackageAbs.exists_equiv_algHom_symm_apply_eq_of_represents_nsmul_eq_one_of_factorsThrough_rigidDataH1Pow1 below · cited by 1 · depth 39 - Drinfeld partners of a q-torsion point are roots of the Igusa factor
ModularCurve.LevelModuliPackageAbs.exists_equiv_isDrinfeldBasisOver_subtype_eval_map_eq_zero_natural_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_gamma0Pow951 below · cited by 1 · depth 39 - Drinfeld bases lifting the origin as roots of g, naturally
ModularCurve.LevelModuliPackageAbs.exists_equiv_isDrinfeldBasisOver_subtype_eval_map_eq_zero_natural_of_nthSeries_eq_X_mul_mul_of_isDomain_adjoinRoot_rigidDataH1Pow951 below · cited by 1 · depth 39 - Lifting q-torsion points and Drinfeld partners to base change
ModularCurve.LevelModuliPackageAbs.exists_section_equiv_isDrinfeldBasisOver_isDrinfeldBasis_of_isCoefficientHom_gamma0Pow34 below · cited by 1 · depth 40 - Transport of q-torsion sections and Drinfeld bases at level H₁
ModularCurve.LevelModuliPackageAbs.exists_section_equiv_isDrinfeldBasisOver_isDrinfeldBasis_of_isCoefficientHom_rigidDataH1Pow34 below · cited by 1 · depth 40
ModularCurve.LevelN 18
- Regular differentials inject into weight-2 cusp forms for Γ(N)
ModularCurve.LevelN.exists_linearMap_regularDifferentials_cuspForm_injective10 below · cited by 1 · depth 19 - Galois structure of the level-N modular function field
ModularCurve.LevelN.exists_monoidHom_algEquiv_fixedField_eq_adjoin6 below · cited by 11 · depth 19 - Genus lower bound for the level-N modular function field
ModularCurve.LevelN.twelve_mul_add_mul_index_le_genusFF131 below · cited by 1 · depth 19 - Regular differentials of the level N function field as F(τ) dτ
ModularCurve.LevelN.exists_linearMap_regularDifferentials_mdifferentiable7 below · cited by 1 · depth 20 - Places over j = 1728, 0, ∞ with bounded order
ModularCurve.LevelN.exists_place_ord_jGen_le_two_three_level24 below · cited by 3 · depth 20 - A place with j-pole fixed by the translation automorphism
ModularCurve.LevelN.exists_place_ord_neg_forall_smul_eq6 below · cited by 5 · depth 20 - Place above j(τ₀) fixed by stabiliser automorphisms
ModularCurve.LevelN.exists_place_ord_sub_pos_forall_smul_eq6 below · cited by 3 · depth 20 - Regular differentials of level N vanish at every cusp
ModularCurve.LevelN.isZeroAtImInfty_slash_of_mem_regularDifferentials7 below · cited by 1 · depth 20 - Weight-two Γ(N)-invariance of (a/b) (c/e)'
ModularCurve.LevelN.slash_eq_self_of_mem_Gamma_of_mul_eq6 below · cited by 1 · depth 20 - A q_N-expansion algebra map on the level-N function field
ModularCurve.LevelN.exists_algHom_laurentSeries_qExpansion17 below · cited by 5 · depth 21 - Analytic order at τ₀ is e times the order at a place
ModularCurve.LevelN.exists_place_analyticOrderAt_eq_mul_ord6 below · cited by 2 · depth 21 - Level-M realisation of the q-expansion function field
ModularCurve.LevelN.exists_algHom_laurentBaseChange_apply_eq_qExpand20 below · cited by 2 · depth 23 - Integrality over the j-line at j=∞
ModularCurve.LevelN.exists_monic_eval_eq_zero_coeff_eq_aeval_inv_div_of_forall_valuation_le_one0 below · cited by 1 · depth 23 - The level-M modular function algebra is a domain
ModularCurve.LevelN.isDomain_ring6 below · cited by 2 · depth 23 - Valuation bound for E(σ_γ z) under a non-zero limit
ModularCurve.LevelN.valuation_apply_smul_le_one_of_tendsto_div_smul6 below · cited by 1 · depth 23 - Elements fixing Φ pointwise lie in ±Γ₁(M)
ModularCurve.LevelN.Descent.fixer_le225 below · cited by 1 · depth 25 - Ramification of X(M)→ X(1) at j=0 and j=1728
ModularCurve.LevelN.exists_place_ord_jGen_eq_three_two_and_stabilizer_subset_zpowers39 below · cited by 2 · depth 25 - Fixer of the Γ₀(M) function field lies in ±Γ₀(M)
ModularCurve.LevelN.Descent.fixer_le_gamma0190 below · cited by 2 · depth 33
ModularCurve.LevelOneFibre 3
- Genus of X₀(q) is less than the number of supersingular j-invariants
ModularCurve.LevelOneFibre.genusFF_lt_card_of_ssJSet499 below · cited by 3 · depth 19 - Supersingular j-invariant count equals genus of X₀(1· q) plus one
ModularCurve.LevelOneFibre.card_eq_genusFF_one_mul_add_one_of_ssJSet486 below · cited by 3 · depth 24 - Supersingular j-invariants number the genus of X₀(q) plus one
ModularCurve.LevelOneFibre.card_eq_genusFF_add_one_of_ssJSet485 below · cited by 1 · depth 25
ModularCurve.LevelP 18
- Étaleness of the level-n basis ring for nΔ invertible
ModularCurve.LevelP.BasisRing.etale4 below · cited by 6 · depth 18 - The level-p basis ring is flat over the base ring
ModularCurve.LevelP.BasisRing.flat8 below · cited by 2 · depth 18 - Flatness of the universal level-p basis, coordinate-change and Borel rings
ModularCurve.LevelP.flat_univBasisRing_vcRing_borelRing0 below · cited by 3 · depth 18 - Universal base ring ℤ[aᵢ][1/(pΔ)] is integrally closed domain
ModularCurve.LevelP.isDomain_and_isIntegrallyClosed_univBase0 below · cited by 2 · depth 18 - The universal Borel pair is a level-p structure
ModularCurve.LevelP.isLevelPStructure_borelDataPrime6 below · cited by 2 · depth 18 - Unit discriminant of the Vélu quotient by a p-torsion line
ModularCurve.LevelP.isUnit_discriminant_quotientByLine9 below · cited by 1 · depth 18 - Vélu's x-only quotient depends only on the line
ModularCurve.LevelP.quotientByLine_eq_of_inLine8 below · cited by 1 · depth 18 - Vélu quotient of Tate(qᵖ) by the q-point is Tate(q)
ModularCurve.LevelP.quotientByLine_tateBase_nonToricPoint_fst58 below · cited by 1 · depth 18 - Vélu's x-only quotient commutes with Weierstrass variable changes
ModularCurve.LevelP.quotientByLine_variableChange2 below · cited by 1 · depth 18 - Existence of the classifying map of a level-p structure
ModularCurve.LevelP.BasisRing.exists_ringHom_basisData_map_eq0 below · cited by 8 · depth 19 - Freeness and rank p²-1 of the p-torsion point ring
ModularCurve.LevelP.TorsionPointRing.free_and_finrank_eq0 below · cited by 3 · depth 19 - Vélu's x-only quotient equals the summing-set quotient
ModularCurve.LevelP.quotientByLine_eq_veluQuotient_oddOrderSummingSet2 below · cited by 2 · depth 19 - Faithful flatness of the level-p basis ring
ModularCurve.LevelP.BasisRing.faithfullyFlat8 below · cited by 1 · depth 20 - Classifying map out of the torsion point ring
ModularCurve.LevelP.TorsionPointRing.exists_ringHom_apply_torsionPt_eq0 below · cited by 4 · depth 20 - Ring maps out of the level-p basis ring are determined by basis data
ModularCurve.LevelP.BasisRing.ringHom_ext_of_basisData_map_eq0 below · cited by 3 · depth 31 - Universal relabelling of the level-ℓ basis exists
ModularCurve.LevelP.exists_levelPData_map_eq_relabel_univData5 below · cited by 1 · depth 32 - Reducedness of the universal level-ℓ basis ring
ModularCurve.LevelP.isReduced_univBasisRing11 below · cited by 2 · depth 32 - Reducedness of the universal p-torsion point ring
ModularCurve.LevelP.TorsionPointRing.isReduced_of_isUnit6 below · cited by 2 · depth 33
ModularCurve.LevelRelabelling 20
- Transitivity of Γ(N₀) on Weil-normalised level-ℓ structures
ModularCurve.LevelRelabelling.exists_mem_Gamma_relabel_eq_of_weilPairing0_eq45 below · cited by 3 · depth 28 - Relabelling action of Γ₀(M') on the rigidified moduli problem
ModularCurve.LevelRelabelling.exists_isModuliRelabelling_gamma0Pow129 below · cited by 9 · depth 29 - Relabelling by g then g' with gg' ≡ 1 (mod q)
ModularCurve.LevelRelabelling.RawDrinfeldPair.relabel_relabel_eq_self_of_mul_map_eq_one_of_isLevel104 below · cited by 3 · depth 30 - Relabelling problem automorphisms for the Γ₀(M')×Γ(ℓ)×Γ(q) datum
ModularCurve.LevelRelabelling.exists_problemAut_relabel_one_mul_of_isUnit_det_gamma0Pow128 below · cited by 1 · depth 30 - Relabelling a Drinfeld basis depends only on g mod q
ModularCurve.LevelRelabelling.RawDrinfeldPair.relabel_eq_relabel_of_map_eq_of_isLevel_of_two_le3 below · cited by 6 · depth 31 - Relabelling raw Drinfeld pairs is a right M₂(ℤ)-action
ModularCurve.LevelRelabelling.RawDrinfeldPair.relabel_relabel0 below · cited by 1 · depth 31 - Diamond relabelling by Γ₀(M') on `rigidDataH1Pow`
ModularCurve.LevelRelabelling.exists_isModuliRelabelling_rigidDataH1Pow180 below · cited by 7 · depth 31 - Natural relabelling of level-ℓ data by integral matrices
ModularCurve.LevelRelabelling.exists_natural_relabel_levelPData19 below · cited by 4 · depth 31 - Weil-normalised level-ℓ structures number #SL₂(ℤ/ℓ)
ModularCurve.LevelRelabelling.natCard_isLevelPStructure_weilPairing0_eq_eq_natCard_specialLinearGroup51 below · cited by 1 · depth 31 - Rigidity of Katz level-ℓ structures over algebraically closed fields
ModularCurve.LevelRelabelling.variableChange_eq_one_of_smul_eq_of_variableChange_eq_of_isLevelPStructure3 below · cited by 1 · depth 31 - Natural [a]-multiplication on Γ₁(ℓ)-data over A-algebras
ModularCurve.LevelRelabelling.exists_natural_zsmul_gamma1Point172 below · cited by 4 · depth 32 - Relabelled level data read back as the relabelled points
ModularCurve.LevelRelabelling.toPoint_relabel_eq_zsmul_add_zsmul0 below · cited by 6 · depth 32 - Relabelling by a matrix invertible mod qℓ gives problem automorphisms
ModularCurve.LevelRelabelling.exists_problemAut_relabel_of_isUnit_det_gamma0Pow128 below · cited by 3 · depth 33 - Relabelling commutes with a Weierstrass change of variables
ModularCurve.LevelRelabelling.relabel_smul_variableChange1 below · cited by 11 · depth 33 - Relabelling cusp data on the Tate curve is linear in (v,w)
ModularCurve.LevelRelabelling.relabel_tateBase_cuspData_eq_cuspData_zsmul_add_zsmul50 below · cited by 10 · depth 33 - Relabelling by g ≡ 1 mod n fixes n-torsion pairs
ModularCurve.LevelRelabelling.RawDrinfeldPair.relabel_eq_self_of_map_eq_one_of_isTorsionPoint0 below · cited by 2 · depth 34 - Relabelling the Drinfeld pair by g gives a moduli automorphism
ModularCurve.LevelRelabelling.exists_problemAut_relabel_drinfeld_of_isUnit_det_rigidDataH1Pow108 below · cited by 2 · depth 34 - Relabelling of level data commutes with base change
ModularCurve.LevelRelabelling.relabel_map_eq_map_relabel0 below · cited by 10 · depth 34 - Drinfeld-slot relabelling by g is a problem automorphism
ModularCurve.LevelRelabelling.exists_problemAut_act_mk_eq_mk_relabel_of_isUnit_det_gamma0Pow105 below · cited by 2 · depth 35 - Drinfeld slot relabelling by g is a problem automorphism
ModularCurve.LevelRelabelling.exists_problemAut_act_mk_eq_mk_relabel_of_isUnit_det_rigidDataH1Pow105 below · cited by 2 · depth 37
ModularCurve.MTorsionNeBot 1
- Nonzero 𝔪-torsion descends along surjections from finite modules
ModularCurve.MTorsionNeBot.of_surjective_of_finite0 below · cited by 2 · depth 19
ModularCurve.MazurII142 1
- Multiplicity one for J[𝔪] from an Oda packet
ModularCurve.MazurII142.OdaDictionaryNoBT1.finrank_eq_two_of_finrank_ker_frob_eq0 below · cited by 1 · depth 18
ModularCurve.ModularPolynomialData 58
- Symmetry of the prime-level modular polynomial
ModularCurve.ModularPolynomialData.evalSymm_of_prime44 below · cited by 21 · depth 9 - Modular equation Φ_ℓ(j(qᵈ),j(q^{dℓ}))=0 over any base ring
ModularCurve.ModularPolynomialData.eval_jqNModC_mul_eq_zero2 below · cited by 68 · depth 9 - Forward modular equation for odd Vélu quotients
ModularCurve.ModularPolynomialData.isRoot_map_j_veluQuotient_j_of_addOrderOf_eq96 below · cited by 3 · depth 9 - Separability of Φₚ(j₀,Y) for non-integral j₀
ModularCurve.ModularPolynomialData.separable_map_eval2_of_not_isIntegral233 below · cited by 1 · depth 9 - Separability of Φ̄_N over 𝔽̄_ℓ(X), prime level
ModularCurve.ModularPolynomialData.separable_map_ratFunc_of_prime_of_not_dvd47 below · cited by 2 · depth 9 - Uniqueness of prime-level modular polynomial data
ModularCurve.ModularPolynomialData.eq_of_prime42 below · cited by 4 · depth 10 - Bidegree bound deg cᵢ ≤ p(ψ(p)-i) at prime level
ModularCurve.ModularPolynomialData.natDegree_coeff_le_mul_dedekindPsi_sub48 below · cited by 9 · depth 10 - Separability of Φₚ(j₀,Y) for non-integral j₀
ModularCurve.ModularPolynomialData.separable_map_eval2_of_not_isIntegral_of_isAlgClosed232 below · cited by 1 · depth 10 - Separability of Φₚ(jmath̄(q),Y) over K((q))
ModularCurve.ModularPolynomialData.separable_map_jqModC_of_prime45 below · cited by 3 · depth 10 - Transpose of Φ monic of degree ψ(N) over ℚ(j)
ModularCurve.ModularPolynomialData.transposeToAdjoin_monic_of_qExpansion5 below · cited by 1 · depth 10 - Φ_N(j(Λ),j(Λ'))=0 for cyclic sublattices of index N
ModularCurve.ModularPolynomialData.eval_jLattice_eq_zero_of_isAddCyclic19 below · cited by 2 · depth 11 - Swapped modular equation Φ_ℓ(j(q^{dℓ}),j(qᵈ))=0 over any ring
ModularCurve.ModularPolynomialData.eval_jqNModC_of_mul_eq_zero2 below · cited by 22 · depth 11 - Roots of Φ_N(a+t,Y) are Laurent series for odd N
ModularCurve.ModularPolynomialData.hasRamBound_one_of_isRoot_off_zero_1728_of_odd209 below · cited by 1 · depth 11 - Roots of Φ_N(t,Y) over ℚ̄ have ramification bound 3 (N odd)
ModularCurve.ModularPolynomialData.hasRamBound_three_of_isRoot_at_zero_of_odd209 below · cited by 3 · depth 11 - Roots of Φ_N(1728+t,Y) have ramification bound 2 for odd N
ModularCurve.ModularPolynomialData.hasRamBound_two_of_isRoot_at_1728_of_odd209 below · cited by 3 · depth 11 - Vélu quotient j-invariant is a root of Φ₂ₙ₊₁(j(W), · )
ModularCurve.ModularPolynomialData.isRoot_map_j_veluQuotient_j_of_addOrderOf_eq_of_isAlgClosed95 below · cited by 2 · depth 11 - Degree bound p+1 for the coefficients of Φₚ
ModularCurve.ModularPolynomialData.natDegree_coeff_le46 below · cited by 10 · depth 11 - Separability of Φ₂(j₀,Y) for non-integral j₀
ModularCurve.ModularPolynomialData.separable_map_eval2_of_not_isIntegral_of_isAlgClosed_two67 below · cited by 1 · depth 11 - Coset representatives give roots of Φ_N(j(τ),·)
ModularCurve.ModularPolynomialData.eval_E4_cube_div_discriminant_coset_eq_zero15 below · cited by 2 · depth 12 - Coset conjugates as roots of Φ_N(j(q^N),Y)
ModularCurve.ModularPolynomialData.exists_isPrimitiveRoot_forall_isRoot_cosetConj_complex18 below · cited by 5 · depth 12 - Level-2 modular equation via Vélu quotients
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_veluQuotient2_j65 below · cited by 4 · depth 12 - Diagonal modular polynomial has unit leading coefficient
ModularCurve.ModularPolynomialData.isUnit_leadingCoeff_diag72 below · cited by 2 · depth 12 - Roots of Φ_N(j(E),Y) lie in L
ModularCurve.ModularPolynomialData.mem_of_isRoot_map_j_of_transcendental_of_odd203 below · cited by 4 · depth 12 - Separability of Φ_N over K(X) when N≠ 0 in K
ModularCurve.ModularPolynomialData.separable_map_ratFunc_of_natCast_ne_zero24 below · cited by 23 · depth 12 - Weighted support bounds for the level-p modular polynomial
ModularCurve.ModularPolynomialData.weighted_support_le74 below · cited by 6 · depth 12 - Uniqueness of modular polynomial data at every level
ModularCurve.ModularPolynomialData.eq_all70 below · cited by 8 · depth 13 - Symmetry of the modular polynomial at levels N>1
ModularCurve.ModularPolynomialData.evalSymm_of_one_lt77 below · cited by 30 · depth 13 - Modular equation on H: Φ_N(j(σ),j(Nσ))=0
ModularCurve.ModularPolynomialData.eval_E4_cube_div_discriminant_smul_eq_zero13 below · cited by 1 · depth 13 - Coset roots of the modular polynomial descend from ℂ
ModularCurve.ModularPolynomialData.forall_isRoot_cosetConj_jqModC_of_complex0 below · cited by 4 · depth 13 - Irreducibility of the modular polynomial over K(X)
ModularCurve.ModularPolynomialData.irreducible_map_ratFunc_of_natCast_ne_zero113 below · cited by 9 · depth 13 - j(q^N) is integral over ℚ(j(q))
ModularCurve.ModularPolynomialData.isIntegral_jqN0 below · cited by 1 · depth 13 - Unit leading coefficient of Φ_N(X,X) for non-square N
ModularCurve.ModularPolynomialData.isUnit_leadingCoeff_diag_of_not_isSquare71 below · cited by 3 · depth 13 - Roots of Φ_N(j(W),Y) lie in the N-torsion field
ModularCurve.ModularPolynomialData.mem_of_isRoot_map_j_of_transcendental213 below · cited by 2 · depth 13 - Uniqueness of the level-2 modular polynomial
ModularCurve.ModularPolynomialData.phi_eq_phiTwo55 below · cited by 3 · depth 13 - Evaluation symmetry of an irreducible modular polynomial datum
ModularCurve.ModularPolynomialData.evalSymm_of_irreducible5 below · cited by 2 · depth 14 - Transfer of the modular-fibre factorisation to arbitrary algebraically closed fields
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_fullKernelQuotient_j_of_transcendental10 below · cited by 3 · depth 15 - Modular polynomial at a transcendental j splits over Vélu quotients
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_fullKernelQuotient_j_of_transcendental_of_charZero222 below · cited by 3 · depth 15 - Coset factorisation of the modular equation over K((q))
ModularCurve.ModularPolynomialData.map_adjoin_jqNModC_eq_cosetTwoVarPoly24 below · cited by 4 · depth 15 - Separability of the modular polynomial over K(X) at prime level
ModularCurve.ModularPolynomialData.separable_map_ratFunc_of_prime47 below · cited by 2 · depth 15 - Symmetry of the modular polynomial at squarefree level
ModularCurve.ModularPolynomialData.evalSymm_of_squarefree77 below · cited by 2 · depth 16 - The reversed modular polynomial at prime level
ModularCurve.ModularPolynomialData.exists_reversed_eval2_inv_jq_inv_jqN_eq_zero46 below · cited by 1 · depth 16 - Roots of the level-2 fibre polynomial are Vélu quotient j-invariants
ModularCurve.ModularPolynomialData.exists_veluQuotient2_j_eq_of_mem_roots_fibrePoly66 below · cited by 1 · depth 16 - Every root of Φ_ℓ(j(W),Y) is a Vélu quotient j-invariant
ModularCurve.ModularPolynomialData.exists_veluQuotient_j_eq_of_mem_roots_fibrePoly208 below · cited by 1 · depth 16 - Modular equation of odd prime level via Vélu quotients
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_veluQuotient_j204 below · cited by 3 · depth 16 - Irreducibility of the modular polynomial over K(jqNModC_M)
ModularCurve.ModularPolynomialData.irreducible_map_adjoin_jqNModC105 below · cited by 2 · depth 16 - Irreducible Φ_N is the minimal polynomial of j(q^N)
ModularCurve.ModularPolynomialData.minpoly_jqN_eq1 below · cited by 2 · depth 16 - The level-3 modular polynomial datum is classical Φ₃
ModularCurve.ModularPolynomialData.phi_eq_phiThree55 below · cited by 2 · depth 16 - Separability of Φ_N modulo a prime ℓ∤ N
ModularCurve.ModularPolynomialData.separable_map_ratFunc_of_not_dvd25 below · cited by 2 · depth 16 - Modular equation as Vélu product at transcendental j
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_veluQuotient_j_of_transcendental_of_isAlgClosed196 below · cited by 1 · depth 17 - Divisibility of the composed modular polynomial by the resultant
ModularCurve.ModularPolynomialData.dvd_resultant_of_mul75 below · cited by 1 · depth 18 - Modular equation at transcendental j as product over Vélu quotients
ModularCurve.ModularPolynomialData.fibrePoly_j_eq_prod_veluQuotient_j_of_transcendental195 below · cited by 1 · depth 18 - Integrality of the modular equation over ℤ((q))
ModularCurve.ModularPolynomialData.eval_int_eq_zero0 below · cited by 1 · depth 20 - Reversed modular polynomial at a prime and its reduction
ModularCurve.ModularPolynomialData.exists_monic_eval2_inv_div_pow_eq_zero_and_map_eq_X_pow_mul_X_sub_one97 below · cited by 2 · depth 30 - Degree bound for the coefficients of Φ_N
ModularCurve.ModularPolynomialData.natDegree_coeff_le_mul_dedekindPsi_sub_all85 below · cited by 1 · depth 31 - Kronecker's congruence for the modular polynomial Φₚ
ModularCurve.ModularPolynomialData.map_map_intCast_eq_of_charP50 below · cited by 2 · depth 34 - Separability of Φ_N(j(E),Y) for transcendental j(E)
ModularCurve.ModularPolynomialData.separable_map_eval2RingHom_j_of_transcendental309 below · cited by 1 · depth 35 - Weighted degree bound for the level-N modular polynomial
ModularCurve.ModularPolynomialData.natDegree_coeff_le_level_mul_dedekindPsi_sub75 below · cited by 3 · depth 36 - Strict degree bounds for the modular polynomial near j=∞
ModularCurve.ModularPolynomialData.natDegree_coeff_lt_of_le_and_natDegree_coeff_sub_one_eq75 below · cited by 3 · depth 36
ModularCurve.ModuliPoint 2
- Frobenius squared fixes supersingular Γ₀(N)-moduli points
ModularCurve.ModuliPoint.map_frobenius_map_frobenius_eq_self_of_mem_ssLocus_univ6 below · cited by 1 · depth 13 - Equality of Γ₀(N) moduli points is a single step
ModularCurve.ModuliPoint.mk_eq_mk_iff_step1 below · cited by 6 · depth 18
ModularCurve.MultCovering 118
- Supersingular annuli with j ≠ 0, 1728 have modulus p
ModularCurve.MultCovering.AnnCtx.exists_isUnit_modulus_eq_mul_of_ssValue_ne0 below · cited by 1 · depth 23 - Annuli of the multiplicative covering are not circles
ModularCurve.MultCovering.AnnCtx.exists_mem_dom_abv_evalAt_param_ne0 below · cited by 18 · depth 23 - Divisibility of annulus moduli by p
ModularCurve.MultCovering.AnnCtx.exists_mem_modulus_eq_mul0 below · cited by 1 · depth 23 - Each annulus modulus divides p³
ModularCurve.MultCovering.AnnCtx.exists_mem_pow_modulusExp_eq_modulus_mul0 below · cited by 1 · depth 23 - Good family bounded by μ(p)^{nᵢ} on the ̄ 0-chart
ModularCurve.MultCovering.abv_evalAt_goodFamily_le_pow_hasseExp_of_mem_zeroChart_dom657 below · cited by 4 · depth 23 - Proximity comparison on the ∞̄ chart via a fibre coordinate
ModularCurve.MultCovering.chartComparison_infChart_of_fibreCoord525 below · cited by 1 · depth 23 - Chordal proximity comparison on the ̄ 0 chart of X₀(p)
ModularCurve.MultCovering.chartComparison_zeroChart_of_chartData_of_fibreCoord642 below · cited by 1 · depth 23 - Cross-piece chordal comparison from exhaustive charts and annuli
ModularCurve.MultCovering.crossComparison_of_forall_mem_chart_dom_or_mem_annIn_dom1,342 below · cited by 1 · depth 23 - The cusp ∞̄ lies in the domain of the ∞-chart
ModularCurve.MultCovering.cuspInftyBar_mem_infChart_dom0 below · cited by 5 · depth 23 - Embedding bases on X₀(p) have m(p)+1 members
ModularCurve.MultCovering.eq_mAnnuli_add_one_of_isEmbBasis496 below · cited by 18 · depth 23 - Embedding basis expanded in the good family
ModularCurve.MultCovering.eq_sum_linkMatrix_mul_goodFamily0 below · cited by 1 · depth 23 - Good-family values lie in mathfrak m_A on the inner annuli
ModularCurve.MultCovering.evalAt_goodFamily_mem_maximalIdeal_of_mem_annIn_dom636 below · cited by 1 · depth 23 - A Fricke-separated covering chart context for X₀(p)
ModularCurve.MultCovering.exists_chartCtx_separated_covering1,033 below · cited by 2 · depth 23 - Existence of a good-family context at prime level p
ModularCurve.MultCovering.exists_famCtx1,243 below · cited by 1 · depth 23 - A good-family context orthogonal at both cusps exists
ModularCurve.MultCovering.exists_famCtx_orth_linearIndependent_zeroChart_residue1,243 below · cited by 4 · depth 23 - Bi-filtered unimodular recombination of a good family
ModularCurve.MultCovering.exists_famCtx_toFamData_eq_of_bifiltered630 below · cited by 4 · depth 23 - Source lift of a supersingular j-value on the zero chart
ModularCurve.MultCovering.exists_lift_jF_sub_mem_chart_src_integers_and_ord_nodeSrc_pos137 below · cited by 1 · depth 23 - Unimodular recombination with attained node orders at p=11
ModularCurve.MultCovering.exists_unimodular_famData_ord_nodeSrc_zeroChart_residue_eq_neg_hasseExp_div_jWidth_of_eq_eleven968 below · cited by 1 · depth 23 - Unimodular recombination of the good family carrying wide-node certificates
ModularCurve.MultCovering.exists_unimodular_famData_wideCertificates983 below · cited by 3 · depth 23 - Hasse exponent equal to node width forces a simple pole
ModularCurve.MultCovering.forall_ord_goodFamily_eq_zero_and_ord_residue_goodFamilyZero_eq_neg_one_of_hasseExp_eq_jWidth634 below · cited by 5 · depth 23 - Hasse exponents and node orders for p = 11
ModularCurve.MultCovering.hasseExp_and_ord_node_residue_of_eq_eleven644 below · cited by 3 · depth 23 - Hasse exponent equals supersingular node width, genus zero
ModularCurve.MultCovering.hasseExp_eq_jWidth_of_genus_zero640 below · cited by 2 · depth 23 - Hasse exponents bounded by the modulus exponent 3
ModularCurve.MultCovering.hasseExp_le_modulusExp628 below · cited by 4 · depth 23 - The Hasse exponent vanishes at index zero
ModularCurve.MultCovering.hasseExp_zero628 below · cited by 7 · depth 23 - Uniform p^B-window making a finite family units of the ∞̄-chart
ModularCurve.MultCovering.infChart_chartData188 below · cited by 1 · depth 23 - Chart data for the good family on the ∞̄-chart
ModularCurve.MultCovering.infChart_chartData_goodFamily510 below · cited by 3 · depth 23 - q-expansion unit criterion on the ∞̄-chart
ModularCurve.MultCovering.infChart_mem_integers_residue_ne_zero_of_qCoeff132 below · cited by 1 · depth 23 - Residue of an A-integral q-expansion on the ∞̄-chart
ModularCurve.MultCovering.infChart_residue_coeffMap132 below · cited by 1 · depth 23 - Residue on the ∞̄-chart computes `modularRedLocHom`
ModularCurve.MultCovering.infChart_residue_eq_modularRedLocHom132 below · cited by 1 · depth 23 - Residue of j on the ∞̄-chart is jmath̄
ModularCurve.MultCovering.infChart_residue_jF133 below · cited by 1 · depth 23 - Residue of j(qᵖ) on the ∞̄-chart equals ̄ j^{ p}
ModularCurve.MultCovering.infChart_residue_jpF133 below · cited by 1 · depth 23 - Nonzero ∞̄-chart residue as quotient of A-integral expansions
ModularCurve.MultCovering.infChart_residue_ne_zero_iff_exists_quotient132 below · cited by 1 · depth 23 - j lies in the integers of the ∞̄-chart
ModularCurve.MultCovering.jF_mem_infChart_integers133 below · cited by 1 · depth 23 - j(qᵖ) lies in the ∞̄-chart's valuation ring
ModularCurve.MultCovering.jpF_mem_infChart_integers133 below · cited by 1 · depth 23 - Link budget makes the change-of-basis matrices p-integral
ModularCurve.MultCovering.linkBudget_spec2 below · cited by 12 · depth 23 - Left inverse property of the link matrix
ModularCurve.MultCovering.linkMatrixInv_mul0 below · cited by 12 · depth 23 - Link matrices satisfy M· M⁻¹=1
ModularCurve.MultCovering.linkMatrix_mul_inv0 below · cited by 12 · depth 23 - Inf- and zero-charts partition the non-supersingular places
ModularCurve.MultCovering.mem_infChart_dom_xor_mem_zeroChart_dom257 below · cited by 1 · depth 23 - Quotient criterion for the ∞̄-chart's valuation ring
ModularCurve.MultCovering.mem_infChart_integers_iff132 below · cited by 1 · depth 23 - ∞̄-chart integers are the localised modular ring
ModularCurve.MultCovering.mem_infChart_integers_iff_coe_mem_modularLocalized132 below · cited by 3 · depth 23 - A-integral q-expansions lie in the ∞̄-chart
ModularCurve.MultCovering.mem_infChart_integers_of_forall_coeff_mem132 below · cited by 2 · depth 23 - Nodes of the Fricke-transported chart: ̃ j = aᵖ supersingular
ModularCurve.MultCovering.mem_zeroChart_nodes_iff3 below · cited by 12 · depth 23 - At most simple poles of the rescaled family on the zero chart
ModularCurve.MultCovering.neg_one_le_ord_nodeSrc_zeroChart_residue_goodFamilyZero628 below · cited by 2 · depth 23 - Nonemptiness of the annulus context over a chart context
ModularCurve.MultCovering.nonempty_annCtx919 below · cited by 5 · depth 23 - No place of the ∞̄-chart domain is supersingular-centred
ModularCurve.MultCovering.not_isSSCentred_of_mem_infChart_dom132 below · cited by 1 · depth 23 - Places in the zero chart are never supersingularly centred
ModularCurve.MultCovering.not_isSSCentred_of_mem_zeroChart_dom86 below · cited by 1 · depth 23 - Hasse members of the good family have positive exponent
ModularCurve.MultCovering.one_le_hasseExp628 below · cited by 9 · depth 23 - Width-one tubes: constancy of rescaled good-family ratios
ModularCurve.MultCovering.residue_evalAt_goodFamilyZero_div_eq_evalAt_nodeSrc_of_forall_widthOne642 below · cited by 1 · depth 23 - Chart data on the ̄ 0-chart when all nodes have width one
ModularCurve.MultCovering.zeroChart_chartData_goodFamilyZero_of_forall_ssValue_ne259 below · cited by 1 · depth 23 - Chart data on the ̄0-chart from independent reductions
ModularCurve.MultCovering.zeroChart_chartData_goodFamilyZero_of_linearIndependent1,114 below · cited by 1 · depth 23 - Zero-chart data for the rescaled good family, 5≤ p<13
ModularCurve.MultCovering.zeroChart_chartData_goodFamilyZero_of_lt_thirteen1,128 below · cited by 1 · depth 23 - Rescaled good family consists of units on the ̄0-chart
ModularCurve.MultCovering.zeroChart_residue_goodFamilyZero_ne_zero628 below · cited by 19 · depth 23 - ̄ 0-chart integrality of a 0-orthogonal family
ModularCurve.MultCovering.FamData.goodFamilyZero_mem_zeroChart_integers628 below · cited by 2 · depth 24 - Coefficientwise p-integrality of p⁻¹wₚ t_l
ModularCurve.MultCovering.FamData.inf_h0_of_one_le_hasseExp628 below · cited by 2 · depth 24 - Linear independence of the zero-chart residues of a Gauss-orthogonal family
ModularCurve.MultCovering.FamData.linearIndependent_zeroChart_residue_goodFamilyZero2 below · cited by 2 · depth 24 - Hasse exponent at least one for nonzero indices
ModularCurve.MultCovering.FamData.one_le_hasseExp_of_orth628 below · cited by 2 · depth 24 - Smith identification on the ̄ 0-chart of X₀(p)⊗ k
ModularCurve.MultCovering.FamData.t_zeroChart_of_orth954 below · cited by 2 · depth 24 - Proximity bound at an ∞̄-chart place against a small place
ModularCurve.MultCovering.abs_prox_evalVec_le_of_mem_infChart_dom_of_forall_abv_evalAt_goodFamily_lt_one87 below · cited by 1 · depth 24 - Profile of a zero-free good-family member on an annulus
ModularCurve.MultCovering.abv_evalAt_goodFamily_eq_abv_evalAt_param_of_ord_residue_eq_one_of_forall_ord_eq_zero634 below · cited by 7 · depth 24 - Good-family values are small on supersingular annuli
ModularCurve.MultCovering.abv_evalAt_goodFamily_lt_one_of_mem_annIn_dom635 below · cited by 12 · depth 24 - Chordal separation across two distinct supersingular annuli
ModularCurve.MultCovering.crossComparison_annIn_annIn1,314 below · cited by 1 · depth 24 - Chordal separation between a supersingular annulus and the zero chart
ModularCurve.MultCovering.crossComparison_annIn_zeroChart1,321 below · cited by 1 · depth 24 - The cusp ∞̄ avoids the ̄ 0-chart of X₀(p)
ModularCurve.MultCovering.cuspInftyBar_not_mem_zeroChart_dom194 below · cited by 4 · depth 24 - The cusp 0 lies outside the ∞-component chart
ModularCurve.MultCovering.cuspZeroBar_not_mem_infChart_dom103 below · cited by 24 · depth 24 - Change of basis from an embedding basis to the good family
ModularCurve.MultCovering.eq_sum_linkMatrix_smul_goodFamily0 below · cited by 11 · depth 24 - Content-two members combine to the width-one node product
ModularCurve.MultCovering.exists_combination_hasseExp_two_eq_prod_widthOne869 below · cited by 4 · depth 24 - Recombining a good family by a bi-filtered digit matrix
ModularCurve.MultCovering.exists_famData_of_bifiltered_digits635 below · cited by 2 · depth 24 - Content-one witnesses at wide nodes on the zero chart
ModularCurve.MultCovering.exists_hasseExp_eq_one_unramified_and_separates960 below · cited by 1 · depth 24 - A uniform p-power budget for the link matrices
ModularCurve.MultCovering.exists_linkBudget1 below · cited by 1 · depth 24 - Infinity-side witnesses at supersingular nodes: non-vanishing and minimal multiplicity
ModularCurve.MultCovering.exists_rootMultiplicity_ssValue_minimal870 below · cited by 1 · depth 24 - Wide-node certificates for the good family at p = 11
ModularCurve.MultCovering.exists_unimodular_famData_wideCertificates_of_eq_eleven967 below · cited by 2 · depth 24 - Fricke involution at level 1· p sends j to j(qᵖ)
ModularCurve.MultCovering.frickeInvolutionBar_jF79 below · cited by 1 · depth 24 - Index-zero member of a good family equals 1
ModularCurve.MultCovering.goodFamily_zero_eq_one0 below · cited by 10 · depth 24 - Hasse exponents at most one when all widths are one
ModularCurve.MultCovering.hasseExp_le_one_of_forall_widthOne628 below · cited by 1 · depth 24 - The ∞̄-chart of a covering context is `chartFst`
ModularCurve.MultCovering.infChart_def0 below · cited by 1 · depth 24 - Good family members are units of the ∞̄-chart
ModularCurve.MultCovering.infChart_goodFamily_residue_ne_zero40 below · cited by 36 · depth 24 - Node data for the good family on the ∞̄-chart
ModularCurve.MultCovering.infChart_nodeData40 below · cited by 6 · depth 24 - Each Hasse member is simple at some node
ModularCurve.MultCovering.infChart_nodeData_exists_node_of_member40 below · cited by 24 · depth 24 - Cusp-chart residues of a p-adically orthogonal family on X₀(p)
ModularCurve.MultCovering.infChart_residue_eq_ssPolyBar_mul_of_orthogonal894 below · cited by 5 · depth 24 - Residues of j and j(qᵖ) on the ∞̄-chart
ModularCurve.MultCovering.infChart_residue_jF_jpF132 below · cited by 4 · depth 24 - The two spellings of j(qᵖ) at level 1· p agree
ModularCurve.MultCovering.jpF_eq_jqFun0 below · cited by 14 · depth 24 - Reduced good family spans the width-one polar system
ModularCurve.MultCovering.mem_span_zeroChart_residue_of_forall_ord_nodeSrc_ge869 below · cited by 3 · depth 24 - Pole bound at tube nodes for the rescaled good family
ModularCurve.MultCovering.neg_hasseExp_div_jWidth_le_ord_nodeSrc_zeroChart_residue_goodFamilyZero945 below · cited by 4 · depth 24 - Existence of a chart context for X₀(p) over A
ModularCurve.MultCovering.nonempty_chartCtx1,034 below · cited by 5 · depth 24 - Good-family members have no zeros on a width-one supersingular tube
ModularCurve.MultCovering.ord_goodFamily_eq_zero_of_ord_residue_eq_one_of_jWidth_eq_one634 below · cited by 4 · depth 24 - Equality case: zero-free supersingular annulus, opposite node orders
ModularCurve.MultCovering.ord_nodeSrc_zeroChart_residue_eq_neg_ord_nodeTgt_of_hasseExp_eq_jWidth_mul953 below · cited by 1 · depth 24 - Node order and separation for a digit recombination on the ̄0-chart
ModularCurve.MultCovering.ord_nodeSrc_zeroChart_residue_of_digits41 below · cited by 1 · depth 24 - Simple value at a width-three node after digit recombination
ModularCurve.MultCovering.ord_nodeSrc_zeroChart_residue_sub_algebraMap_eq_one_of_digits40 below · cited by 1 · depth 24 - Node orders of digit-recombined good family reductions
ModularCurve.MultCovering.ord_nodeTgt_infChart_residue_of_digits636 below · cited by 1 · depth 24 - Rescaled good family reduces to functions regular off the nodes
ModularCurve.MultCovering.zeroChart_residue_goodFamilyZero_ord_nonneg_of_not_mem_nodes628 below · cited by 6 · depth 24 - Hasse exponent at most one for doubly orthogonal bases
ModularCurve.MultCovering.FamData.hasseExp_le_one_of_orth945 below · cited by 1 · depth 25 - At most simple poles at nodes of ̄0-chart reductions
ModularCurve.MultCovering.FamData.neg_one_le_ord_nodeSrc_residue_goodFamilyZero628 below · cited by 1 · depth 25 - Regularity off the nodes of the rescaled family on the ̄ 0-chart
ModularCurve.MultCovering.FamData.residue_goodFamilyZero_ord_nonneg_of_not_mem_nodes628 below · cited by 1 · depth 25 - Independence of compConst from the good family
ModularCurve.MultCovering.compConst_eq_compConst2 below · cited by 3 · depth 25 - Cross-tube chordal comparison for X₀(11)
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_eq_eleven1,287 below · cited by 1 · depth 25 - Tube-to-tube chordal comparison for adapted good families
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_orth_of_linearIndependent1,009 below · cited by 1 · depth 25 - Chordal separation of supersingular tubes from the ̄0-chart
ModularCurve.MultCovering.crossComparison_annIn_zeroChart_of_adapted1,023 below · cited by 1 · depth 25 - Chordal separation of a supersingular annulus from the ̄0-chart
ModularCurve.MultCovering.crossComparison_annIn_zeroChart_of_ord_one_of_zeroFree673 below · cited by 2 · depth 25 - A simple zero at some supersingular node
ModularCurve.MultCovering.exists_node_ord_infChart_residue_eq_one_of_eq_ssPolyBar_mul39 below · cited by 3 · depth 25 - Non-negativity of the Hasse content of a good family
ModularCurve.MultCovering.hasseContent_nonneg628 below · cited by 2 · depth 25 - Hasse exponent vanishes for a member equal to 1
ModularCurve.MultCovering.hasseExp_eq_zero_of_t_eq_one628 below · cited by 2 · depth 25 - Hasse exponents of the good family are at most two for p≥ 13
ModularCurve.MultCovering.hasseExp_le_two_of_thirteen_le632 below · cited by 5 · depth 25 - Reduction of the good family on the ∞̄-chart
ModularCurve.MultCovering.infChart_residue_goodFamily0 below · cited by 6 · depth 25 - Members of Hasse content at most one span the narrow-node polar system
ModularCurve.MultCovering.mem_span_zeroChart_residue_hasseExp_le_one_of_forall_ord_nodeSrc_ge869 below · cited by 2 · depth 25 - At p=11 the enumerated supersingular values are 0 and 1728
ModularCurve.MultCovering.ssValue_eq_zero_or_eq_1728_of_eq_eleven20 below · cited by 2 · depth 25 - Smith form of the good family on the ̄ 0-chart
ModularCurve.MultCovering.zeroChart_residue_goodFamilyZero_smith0 below · cited by 1 · depth 25 - Invariance of the comparison constant under p-integral recombination
ModularCurve.MultCovering.compConst_eq_of_t_eq_sum5 below · cited by 2 · depth 26 - Chordal proximity bound across two distinct supersingular annuli
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_deep_of_zeroFree649 below · cited by 2 · depth 26 - Tube-against-tube proximity bound at width-one supersingular nodes
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_jWidth_eq_one879 below · cited by 1 · depth 26 - Cross-comparison of two wide tubes by leading terms
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_leadingTerms649 below · cited by 1 · depth 26 - Chordal cross-comparison of places in two distinct annuli
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_outer878 below · cited by 1 · depth 26 - Chordal proximity bound across two distinct supersingular annuli
ModularCurve.MultCovering.crossComparison_annIn_annIn_of_outer_of_lt_hasseExp648 below · cited by 1 · depth 26 - Chordal separation of a supersingular annulus from the ̄0-chart
ModularCurve.MultCovering.crossComparison_annIn_zeroChart_of_twoMembers677 below · cited by 1 · depth 26 - Two wide supersingular nodes force two Hasse exponents 2
ModularCurve.MultCovering.exists_ne_hasseExp_eq_two_of_jWidth_ne_one954 below · cited by 1 · depth 26 - Two-member certificate at a supersingular node with j=0
ModularCurve.MultCovering.exists_unimodular_famData_twoMembers_certificate_of_ssValue_eq_zero881 below · cited by 1 · depth 26 - Coefficients of an ∞̄-integral combination of a good family
ModularCurve.MultCovering.mem_of_eq_sum_smul_goodFamily1 below · cited by 1 · depth 26 - Non-vanishing tangent determinant for two content-2 family members
ModularCurve.MultCovering.tangentDet_ne_zero_of_hasseExp_two639 below · cited by 1 · depth 26 - Frobenius fixes the zero-chart reductions of the good family
ModularCurve.MultCovering.coeffMap_frobenius_zeroChart_residue_goodFamilyZero635 below · cited by 1 · depth 27
ModularCurve.NodeLocalized 69
- A ∩ K is all of K or a discrete valuation ring
ModularCurve.NodeLocalized.coeffSubring_eq_or_isDiscreteValuationRing1 below · cited by 59 · depth 15 - Inertia-fixed number field whose integers reduce onto 𝔽_{q²}
ModularCurve.NodeLocalized.exists_finiteDimensional_forall_inertia_apply_eq_and_mem_range_redRestrict0 below · cited by 15 · depth 15 - Uniformiser and ramification index for A∩ K
ModularCurve.NodeLocalized.exists_forall_redRestrict_eq_zero_iff_and_natCast_eq_pow_mul0 below · cited by 19 · depth 15 - Descent of an integral q-expansion to a subfield K
ModularCurve.NodeLocalized.exists_mem_fieldOver_coeffMap_eq_of_coeffMap_redRestrict_eq_of_isIntegral78 below · cited by 1 · depth 16 - A ∩ K is a discrete valuation ring
ModularCurve.NodeLocalized.isDiscreteValuationRing_coeffSubring0 below · cited by 20 · depth 16 - Noetherian local ring of dimension two on the plane model
ModularCurve.NodeLocalized.isNoetherianRing_isLocalRing_modularLocalizedAtPoint_coeffSubring168 below · cited by 33 · depth 16 - Branch ideals at a point of the special fibre are prime
ModularCurve.NodeLocalized.isPrime_span_uniformizer_branches_modularLocalizedAtPoint162 below · cited by 11 · depth 16 - Modular relations over A∩ K vanish at (a,a^q)
ModularCurve.NodeLocalized.pointEval_eq_zero_of_modularEval_eq_zero152 below · cited by 35 · depth 16 - Kronecker congruence: relations vanish on both branches
ModularCurve.NodeLocalized.eval2_branch_eq_zero_of_modularEval_eq_zero58 below · cited by 1 · depth 17 - j and j_q lie in the integral closure of A₀[j]
ModularCurve.NodeLocalized.jqModC_mem_jIntegralClosure_and_jqNModC_mem146 below · cited by 8 · depth 17 - Galois invariance of node-local elements with rational coefficients
ModularCurve.NodeLocalized.arithmeticGalois_smul_eq_self_of_mem_modularLocalizedAtPoint_coeffSubring_bot153 below · cited by 1 · depth 18 - Gauss coordinate at a supersingular node with j=1728
ModularCurve.NodeLocalized.exists_gaussCoordinate_of_crossingPresentation_ofNat1728176 below · cited by 3 · depth 18 - Gauss coordinate at a supersingular centre j=0
ModularCurve.NodeLocalized.exists_gaussCoordinate_of_crossingPresentation_zero176 below · cited by 3 · depth 18 - Fricke involution exchanges node rings at (a,a^q) and (a^q,a)
ModularCurve.NodeLocalized.exists_ringEquiv_modularLocalizedAtPoint_coe_eq_frickeInvolutionBar157 below · cited by 8 · depth 18 - Finiteness of the residue field of A∩ K
ModularCurve.NodeLocalized.finite_residueField_coeffSubring0 below · cited by 2 · depth 18 - Reduction kernel at the node is the branch ideal (varpi, j_q-j^{ q})
ModularCurve.NodeLocalized.modularRedLocHom_eq_zero_iff_mem_span_branchFst154 below · cited by 9 · depth 18 - Fricke twin: kernel along the branch j - j_q^{ q}
ModularCurve.NodeLocalized.modularRedLocHom_frickeInvolutionBar_eq_zero_iff_mem_span_branchSnd160 below · cited by 3 · depth 18 - Branch number at a node, unit form of ord = n
ModularCurve.NodeLocalized.ord_modularRedLocHom_eq_iff_exists_isUnit194 below · cited by 4 · depth 18 - Kernel of reduction on A∩ℚ is generated by q
ModularCurve.NodeLocalized.redRestrict_bot_eq_zero_iff_exists_eq_natCast_mul0 below · cited by 1 · depth 18 - Unique centred place with given value of j_q-j^q
ModularCurve.NodeLocalized.existsUnique_place_centred_hasValue_nodeCoord316 below · cited by 2 · depth 19 - Crossing parameter attains each admissible value once at j=1728
ModularCurve.NodeLocalized.existsUnique_place_centred_ofNat1728_hasValue_of_crossingPresentation257 below · cited by 2 · depth 19 - Unique place at the j=0 node with prescribed crossing value
ModularCurve.NodeLocalized.existsUnique_place_centred_zero_hasValue_of_crossingPresentation257 below · cited by 2 · depth 19 - Inert quadratic coefficient extension by a primitive cube root of unity
ModularCurve.NodeLocalized.exists_coeffSubring_inertQuadratic_cubeRoot1 below · cited by 2 · depth 19 - Node coordinate j_q-j^q has a value in the annulus
ModularCurve.NodeLocalized.exists_hasValue_nodeCoord_of_centred200 below · cited by 15 · depth 19 - Unit principle at the width-two supersingular node j = 1728
ModularCurve.NodeLocalized.exists_int_mul_pow_param_isUnit_of_forall_centred_ofNat1728_ord_eq_zero_of_crossingPresentation639 below · cited by 1 · depth 19 - Unit normalisation at the width-three supersingular node j=0
ModularCurve.NodeLocalized.exists_int_mul_pow_param_isUnit_of_forall_centred_zero_ord_eq_zero_of_crossingPresentation639 below · cited by 1 · depth 19 - K(j,j_q) lies in the fraction field of the node ring
ModularCurve.NodeLocalized.exists_mul_eq_of_mem_fieldOver0 below · cited by 13 · depth 19 - Places of K(j,j_q) lift to ℚ̄(X₀(q))
ModularCurve.NodeLocalized.exists_place_bar_restrict_fieldOver_eq3 below · cited by 1 · depth 19 - A height-one prime of the j-integral closure as a place over K
ModularCurve.NodeLocalized.exists_place_fieldOver_mem_iff_of_height_one160 below · cited by 1 · depth 19 - Height-one primes at the node admit centred ℚ̄-points
ModularCurve.NodeLocalized.exists_ringHom_ker_eq_centred_of_height_one_of_natCast_notMem158 below · cited by 1 · depth 19 - Elements of the node-localized ring take A-values at W
ModularCurve.NodeLocalized.exists_sub_algebraMap_mem_nonunits_of_mem_modularLocalizedAtPoint0 below · cited by 4 · depth 19 - Two-branch normalisation at a supersingular node of X₀(q)
ModularCurve.NodeLocalized.exists_twoBranchNormalisation_qpow_of_forall_centred_ord_eq_zero684 below · cited by 4 · depth 19 - Unit values at places centred on a supersingular node
ModularCurve.NodeLocalized.isUnit_evalAt_of_forall_centred_ord_eq_zero_of_gaussUnit562 below · cited by 5 · depth 19 - Regularity at a supersingular node gives membership in the local ring
ModularCurve.NodeLocalized.mem_modularLocalizedAtPoint_coeffSubring_of_isIntegral_of_mem_fieldOver_of_redRestrict_eq_of_forall_centred_ord_nonneg566 below · cited by 3 · depth 19 - Node coordinate minus its value is a uniformiser at W
ModularCurve.NodeLocalized.ord_nodeCoord_sub_eq_one_of_centred356 below · cited by 2 · depth 19 - Crossing parameter uniformises at centred places of the j=1728 tube
ModularCurve.NodeLocalized.ord_sub_eq_one_of_centred_ofNat1728_of_crossingPresentation276 below · cited by 1 · depth 19 - Crossing parameter uniformises at centred places, j=0
ModularCurve.NodeLocalized.ord_sub_eq_one_of_centred_zero_of_crossingPresentation276 below · cited by 1 · depth 19 - Section prime at the width-two supersingular node j = 1728
ModularCurve.NodeLocalized.exists_heightOnePrime_sectionOfCrossingParam_centred_ofNat1728199 below · cited by 2 · depth 20 - Height-one section prime for an admissible crossing value at j=0
ModularCurve.NodeLocalized.exists_heightOnePrime_sectionOfCrossingParam_centred_zero199 below · cited by 2 · depth 20 - Residue at a centred place over a supersingular node
ModularCurve.NodeLocalized.exists_mem_and_red_eq_of_hasValue_frobNodePair_of_centred_of_ssJSet_of_ne_zero_of_ne_1728426 below · cited by 1 · depth 20 - Scaling a modular function to a nonzero Gauss reduction
ModularCurve.NodeLocalized.exists_smul_gaussUnit376 below · cited by 2 · depth 20 - Surjection from W[[X₀,X₁]] onto the completed node ring
ModularCurve.NodeLocalized.exists_surjective_mvPowerSeries_adicCompletion_modularLocalizedAtPoint170 below · cited by 1 · depth 20 - Two-branch normalisation at the node j=1728: width divides the Fricke exponent
ModularCurve.NodeLocalized.exists_twoBranchNormalisation_qpow_ofNat1728_width_dvd594 below · cited by 1 · depth 20 - Two-branch normalisation at the node j=0: width divides the Fricke exponent
ModularCurve.NodeLocalized.exists_twoBranchNormalisation_qpow_zero_width_dvd594 below · cited by 1 · depth 20 - Crossing presentations force q-adically equal values at node places
ModularCurve.NodeLocalized.forall_natCast_pow_dvd_sub_of_hasValue_eq_of_crossingPresentation146 below · cited by 1 · depth 20 - Centred values at the node j=1728 agree q-adically
ModularCurve.NodeLocalized.forall_natCast_pow_dvd_sub_of_hasValue_eq_of_crossingPresentation_ofNat1728146 below · cited by 1 · depth 20 - Gauss units q/G and w_q(G) for the node coordinate G
ModularCurve.NodeLocalized.gaussData_nodeCoord202 below · cited by 6 · depth 20 - Unit values of a Gauss pair at nodes centred at 1728
ModularCurve.NodeLocalized.isUnit_evalAt_ofNat1728_of_gaussPair_of_isAlgClosed621 below · cited by 5 · depth 20 - Unit values of a Gauss pair at the node j=0
ModularCurve.NodeLocalized.isUnit_evalAt_zero_of_gaussPair_of_isAlgClosed621 below · cited by 5 · depth 20 - Non-vanishing of the near-branch node value at supersingular nodes
ModularCurve.NodeLocalized.ne_zero_of_hasValue_frobNodePair_of_forall_centred_ord_eq_zero449 below · cited by 2 · depth 20 - Order one at a centred place for a height-one uniformiser
ModularCurve.NodeLocalized.ord_generator_eq_one_of_heightOne_of_ringIff122 below · cited by 2 · depth 20 - Places of ℚ̄(X₀(q)) determined by values over a number field
ModularCurve.NodeLocalized.place_eq_of_forall_hasValue_iff_of_mem_fieldOver149 below · cited by 2 · depth 20 - Crossing presentations cut out the two branch primes
ModularCurve.NodeLocalized.span_uniformizer_pair_eq_branches_or_swap_of_maximalIdeal_eq_span163 below · cited by 1 · depth 20 - Zero or pole at a place centred at a supersingular node
ModularCurve.NodeLocalized.exists_centred_ord_ne_zero_of_not_isUnit_frobNodePair448 below · cited by 2 · depth 21 - Elements of order zero at a node are monomials
ModularCurve.NodeLocalized.exists_isUnit_and_eq_pow_mul_pow_mul_pow_mul_of_forall_centred_ord_eq_zero_of_crossingPresentation272 below · cited by 2 · depth 21 - Residue compatibility at a supersingular node of X₀(q)
ModularCurve.NodeLocalized.exists_mem_and_red_eq_of_hasValue_frobNodePair_of_centred_of_ssJSet569 below · cited by 2 · depth 21 - Membership in the node local ring at (a,a^q), a ≠ 0,1728
ModularCurve.NodeLocalized.mem_modularLocalizedAtPoint_coeffSubring_of_isIntegral_of_mem_fieldOver_of_ne_zero_of_ne_1728424 below · cited by 3 · depth 21 - Regularity at points (b,b^q) with b^{q^2}≠ b
ModularCurve.NodeLocalized.mem_modularLocalizedAtPoint_of_mem_modularLocalized_of_isIntegral179 below · cited by 2 · depth 21 - Nonvanishing of the near-branch node value at a supersingular centre
ModularCurve.NodeLocalized.ne_zero_of_hasValue_frobNodePair_of_forall_centred_ord_eq_zero_of_mem_ssJSet592 below · cited by 2 · depth 21 - Finite spanning set for the integral closure of A₀[j]
ModularCurve.NodeLocalized.exists_finset_forall_mem_jIntegralClosure_eq_sum_mul154 below · cited by 2 · depth 22 - Height-one prime containing p, avoiding q and the node
ModularCurve.NodeLocalized.exists_heightOne_mem_of_mul_eq_of_not_isUnit_frobNodePair405 below · cited by 2 · depth 22 - A prime of the j-integral closure through p=fs avoiding the node
ModularCurve.NodeLocalized.exists_isPrime_mem_of_mul_eq_of_not_isUnit_frobNodePair361 below · cited by 2 · depth 22 - Node-local functions as quotients integral over (A∩ K)[j]
ModularCurve.NodeLocalized.exists_mul_eq_mem_jIntegralClosure_of_not_isUnit_frobNodePair147 below · cited by 2 · depth 22 - Membership in the node-local ring over a number field
ModularCurve.NodeLocalized.mem_modularLocalizedAtPoint_coeffSubring_of_isIntegral_of_mem_fieldOver566 below · cited by 2 · depth 22 - Regularity at a supersingular node gives localised membership
ModularCurve.NodeLocalized.mem_modularLocalizedAtPoint_of_mem_modularLocalized_of_forall_centred_ord_eq_zero_of_ssJSet568 below · cited by 1 · depth 22 - Branch order at a node versus membership in (varpi,G)+Hⁿ
ModularCurve.NodeLocalized.natCast_le_ord_modularRedLocHom_iff_mem_sup_span_pow194 below · cited by 3 · depth 22 - Wide two-branch normalisation at supersingular nodes with j∈{0,1728}
ModularCurve.NodeLocalized.exists_twoBranchNormalisation_qpow_width_dvd_and_mul_ord_charLGeomPlaceOfPoint_eq_neg_of_eq_zero_or_eq_ofNat1728740 below · cited by 4 · depth 26 - Two-branch normalisation at a node, with node order -m
ModularCurve.NodeLocalized.exists_twoBranchNormalisation_qpow_and_ord_charLGeomPlaceOfPoint_eq_neg696 below · cited by 4 · depth 27 - Enlarging the coefficient field to lift a residue value
ModularCurve.NodeLocalized.exists_le_redRestrict_eq_and_forall_redRestrict_eq_zero_iff_and_eq_pow_mul_coeffSubring_of_liesOverPrime117 below · cited by 1 · depth 29
ModularCurve.PDPairing 7
- Γ(4) is a free group
ModularCurve.PDPairing.isFreeGroup_Gamma_four0 below · cited by 4 · depth 12 - Hecke self-adjointness of the integer pairing on parabolic classes
ModularCurve.PDPairing.pairZFun_heckeT0_comm0 below · cited by 2 · depth 12 - Degeneracy maps are adjoint for the integer parabolic pairing
ModularCurve.PDPairing.pairZFun_jDeg0_iDeg00 below · cited by 3 · depth 12 - Non-degeneracy of pairZ modulo a prime p≥ 5
ModularCurve.PDPairing.pairZ_nondegenerate_mod0 below · cited by 2 · depth 12 - Parabolic pairings as a fixed multiple of mod-3 perfect forms
ModularCurve.PDPairing.exists_forall_smul_eq_pairZ_and_perfect_mod_three2 below · cited by 2 · depth 13 - The two transfer Hecke operators agree at H=top
ModularCurve.PDPairing.heckeT0_apply_eq_heckeT_top_apply2 below · cited by 3 · depth 14 - The index of Γ(4) in SL₂(ℤ) is 48
ModularCurve.PDPairing.index_Gamma_four0 below · cited by 1 · depth 14
ModularCurve.Period 17
- Hecke operators commute with the character involution on Γ₀(N)
ModularCurve.Period.charInvolution_heckeOperatorHom0 below · cited by 5 · depth 10 - Equivariant holomorphic primitive of a weight-2 cusp form
ModularCurve.Period.CuspForm.exists_equivariantPrimitive_gamma00 below · cited by 13 · depth 11 - Nonzero weight-2 cusp forms have nonzero period character
ModularCurve.Period.CuspForm.periodHom_ne_zero_of_ne_zero2 below · cited by 1 · depth 11 - Periods of an equivariant primitive vanish on parabolic elements
ModularCurve.Period.IsEquivariantPrimitive.isParabolicHom_periodHom0 below · cited by 3 · depth 11 - Finite-dimensionality of Hom(Γ₀(N),ℂ)
ModularCurve.Period.moduleFinite_addMonoidHom_gamma0_complex0 below · cited by 2 · depth 11 - Equal derivatives give equal period homomorphisms
ModularCurve.Period.IsEquivariantPrimitive.periodHom_eq_of_hasDerivAt0 below · cited by 5 · depth 12 - An integral basis of parabolic characters survives base change
ModularCurve.Period.exists_basis_parabolicHoms_castAddHom_comp7 below · cited by 15 · depth 12 - Mod p parabolic eigenclass gives maximal ideal of T₂(N)
ModularCurve.Period.exists_ideal_heckeAlgebra_two_of_int_modp_eigenclass591 below · cited by 1 · depth 12 - Integral lifting of parabolic characters mod n vanishing on torsion
ModularCurve.Period.exists_parabolicHoms_int_castAddHom_comp_eq_of_forall_isOfFinOrder2 below · cited by 4 · depth 12 - Nonzero parabolic realisation of a normalised eigenform over k
ModularCurve.Period.exists_parabolicRealization30 below · cited by 1 · depth 12 - Hecke operators preserve parabolic homomorphisms on Γ₀(N)
ModularCurve.Period.heckeOperatorHom_preserves_parabolic0 below · cited by 8 · depth 12 - Signature form of the Eichler–Shimura Betti bound
ModularCurve.Period.six_mul_finrank_parabolicHoms_add_le_index0 below · cited by 1 · depth 13 - Integral parabolic characters base-change to torsion-free rings
ModularCurve.Period.exists_basis_parabolicHoms_of_isAddTorsionFree0 below · cited by 23 · depth 14 - Parabolic characters plus cusp count bounded by dimHom(Γ,K)+1
ModularCurve.Period.finrank_parabolicHoms_add_natCard_le_finrank_addMonoidHom_add_one3 below · cited by 2 · depth 14 - Period of the trace sum equals the transfer sum of periods
ModularCurve.Period.traceSum_period_eq0 below · cited by 1 · depth 14 - Unique parabolic extension of a character across -1
ModularCurve.Period.existsUnique_isParabolicHom_sup_zpowers_neg_one_apply_eq0 below · cited by 2 · depth 19 - Base change of a perfect pairing on integral parabolic homomorphisms
ModularCurve.Period.exists_perfectPairing_parabolicHoms_baseChange1 below · cited by 1 · depth 20
ModularCurve.PhiGen 32
- Integrality of the descended coefficients of Φ_ℓ
ModularCurve.PhiGen.PhiGenDescends.intCoeffs0 below · cited by 5 · depth 9 - Evaluation symmetry of a modular polynomial packet with descended coefficients
ModularCurve.PhiGen.evalSymm_of_coeff_evalAtJ_eq14 below · cited by 3 · depth 9 - Assembling Φ_ℓ from a descended integral coefficient family
ModularCurve.PhiGen.exists_modularPolynomialData_coeff_eq10 below · cited by 5 · depth 9 - Descent of the coefficients of Φ_ℓ to ℚ((q))
ModularCurve.PhiGen.exists_phiGenDescends1 below · cited by 5 · depth 9 - Coefficients of the generic level-ℓ modular polynomial lie in ℚ[j]
ModularCurve.PhiGen.mem_adjoin_jq_of_phiGenDescends16 below · cited by 5 · depth 9 - Twisted prime-level splitting of the modular polynomial Φₚ
ModularCurve.PhiGen.splits_prime_at_slot44 below · cited by 42 · depth 9 - Vanishing of descended coefficients above degree ℓ+1
ModularCurve.PhiGen.PhiGenDescends.c_eq_zero3 below · cited by 1 · depth 10 - Descended coefficient family has top coefficient 1
ModularCurve.PhiGen.PhiGenDescends.c_top3 below · cited by 2 · depth 10 - Formal descent data give q-expansions of the Hecke coset polynomial
ModularCurve.PhiGen.PhiGenDescends.hasSum_cosetPoly_coeff10 below · cited by 1 · depth 10 - Pole order at most ℓ+1 for descended coefficients
ModularCurve.PhiGen.PhiGenDescends.poleOrderLE3 below · cited by 3 · depth 10 - Descended coefficients annihilate j(q^ℓ)
ModularCurve.PhiGen.PhiGenDescends.sum_mul_jqN_pow_eq_zero3 below · cited by 2 · depth 10 - Integrality descent for polynomials in j
ModularCurve.PhiGen.aeval_jq_intCoeffs_descent0 below · cited by 2 · depth 10 - Injectivity of evaluation at j(q)
ModularCurve.PhiGen.evalAtJ_injective3 below · cited by 7 · depth 10 - Symmetry of Φ_ℓ from its splitting into conjugates
ModularCurve.PhiGen.evalSymm_of_splits6 below · cited by 1 · depth 10 - Pole bound for the non-constant coefficients of phiProd
ModularCurve.PhiGen.phiProd_conj_coeff_eq_zero_of_le0 below · cited by 1 · depth 10 - Leading t-coefficient of the constant term of Φ_ℓ
ModularCurve.PhiGen.phiProd_conj_coeff_zero_lead0 below · cited by 2 · depth 10 - Descended coefficients force Φ to split into conjugates
ModularCurve.PhiGen.splits_of_coeff_evalAtJ_eq0 below · cited by 3 · depth 10 - Prime-level modular polynomial splits over any field with ζₚ
ModularCurve.PhiGen.splits_of_prime43 below · cited by 6 · depth 10 - Irreducibility of Φ_ℓ(j,Y) over ℚ(j) from its conjugate factorisation
ModularCurve.PhiGen.phiIrreducible_of_splits6 below · cited by 1 · depth 11 - Summing q-twists over the ℓ-th roots of unity
ModularCurve.PhiGen.sum_qTwist_coeff72 below · cited by 5 · depth 11 - The ℓ+1 conjugate expansions of j are distinct
ModularCurve.PhiGen.conj_injective0 below · cited by 3 · depth 12 - A field endomorphism permutes the ℓ-th roots of unity
ModularCurve.PhiGen.exists_galoisPerm1 below · cited by 3 · depth 12 - Coefficientwise Galois descent for Laurent series
ModularCurve.PhiGen.mem_range_coeffEmb_of_forall_coeffMap_eq1 below · cited by 5 · depth 12 - Joint descent for q-expansions with rational coefficients
ModularCurve.PhiGen.mem_range_coeffEmb_qExpand_of_mem_inter1 below · cited by 4 · depth 12 - Prime-level splitting of Φₚ at the slot uq^e
ModularCurve.PhiGen.splits_prime_at_slot_of_isPrimitiveRoot48 below · cited by 9 · depth 13 - Pole order at t=0 is unchanged by coefficient extension
ModularCurve.PhiGen.tPoleOrderLE_coeffEmb_iff3 below · cited by 1 · depth 13 - Pole order descends along t ↦ t^N
ModularCurve.PhiGen.tPoleOrderLE_of_qExpand3 below · cited by 1 · depth 13 - Pole bound ℓ²+ℓ-1 for non-constant coefficients
ModularCurve.PhiGen.tPoleOrderLE_phiProd_conj_of_ne_zero0 below · cited by 1 · depth 13 - Splitting of Φₚ(j(qᵖ),Y) over K((q))
ModularCurve.PhiGen.splits_prime_of_isPrimitiveRoot48 below · cited by 2 · depth 14 - Descended symmetric functions of the conjugates lie in ℚ[j]
ModularCurve.PhiGen.mem_adjoin_jq_of_qExpand_descent_phiProd_modularUnit20 below · cited by 1 · depth 18 - Coefficient at ℓ n of ℓ² f(q^{ℓ^2})+sum_b f(ζᵇ q)
ModularCurve.PhiGen.weightTwo_coeff_sum_slots73 below · cited by 1 · depth 21 - Powers of j have integral q-expansions
ModularCurve.PhiGen.intCoeffs_jq_pow0 below · cited by 2 · depth 25
ModularCurve.PlaceSpecialization 585
- Weak cusp rule for a place-specialization packet on X₀(p)
ModularCurve.PlaceSpecialization.cuspRuleFor130 below · cited by 1 · depth 9 - Strong cusp rule for place-specialisation packets on X₀(p)
ModularCurve.PlaceSpecialization.cuspRuleStrongFor129 below · cited by 1 · depth 10 - Frobenius acts on J₀(N) as special-fibre Frobenius push-forward
ModularCurve.PlaceSpecialization.spPic0_frobenius_smul_eq0 below · cited by 6 · depth 10 - Inertia acts trivially on J₀(N) through a specialization packet
ModularCurve.PlaceSpecialization.spPic0_inertia_smul0 below · cited by 4 · depth 10 - Surjectivity of the class map of a place-specialisation packet
ModularCurve.PlaceSpecialization.spPic0_surjective162 below · cited by 9 · depth 10 - Decomposition-group stability of the kernel of the component map
ModularCurve.PlaceSpecialization.componentMap_decomposition_smul_eq_zero_of_eq_zero88 below · cited by 1 · depth 11 - Frobenius stability of the kernel of the component map
ModularCurve.PlaceSpecialization.componentMap_frobenius_smul_eq_zero_of_eq_zero4 below · cited by 5 · depth 11 - Hecke stability of the kernel of the component map at q
ModularCurve.PlaceSpecialization.componentMap_heckeAlg_smul_eq_zero_of_eq_zero_of_isModel965 below · cited by 5 · depth 11 - T_ℓ acts as ℓ+1 through the component map
ModularCurve.PlaceSpecialization.componentMap_heckeGen_smul_eq_add_one_smul_of_isModel2,484 below · cited by 3 · depth 11 - Injectivity of `spPic0` on prime-to-q torsion
ModularCurve.PlaceSpecialization.eq_zero_of_primeToTorsion_of_spPic0_eq_zero995 below · cited by 1 · depth 11 - Hecke-equivariant component map and toric monodromy detection
ModularCurve.PlaceSpecialization.exists_heckeModule_componentGroup_toricMonodromyPart_mem_of_isModel2,736 below · cited by 3 · depth 11 - Hecke-equivariance of the Pic⁰ specialisation map
ModularCurve.PlaceSpecialization.exists_heckeModule_pic0_spPic0_heckeAlg_smul999 below · cited by 2 · depth 11 - Prime-to-q torsion classes lift along `spPic0`
ModularCurve.PlaceSpecialization.exists_primeToTorsion_spPic0_eq_of_primeToTorsion1,772 below · cited by 1 · depth 11 - Lifting m-torsion from the component group to inertia invariants
ModularCurve.PlaceSpecialization.exists_torsion_preimage_componentMap_of_isModel1,311 below · cited by 3 · depth 11 - Lifting m-torsion through the glued specialization at q
ModularCurve.PlaceSpecialization.exists_torsion_preimage_gluedSpecialization_of_isModel1,137 below · cited by 3 · depth 11 - Widths, component map and glued specialisation over a place above q
ModularCurve.PlaceSpecialization.exists_widths_componentMap_gluedSpecialization_placeWidthChar_of_isModel1,892 below · cited by 5 · depth 11 - Injectivity on prime-to-q torsion of component and glued specialization maps
ModularCurve.PlaceSpecialization.gluedSpecialization_componentMap_injective_primeToTorsion_of_isModel1,053 below · cited by 7 · depth 11 - Frobenius law for the glued specialization
ModularCurve.PlaceSpecialization.gluedSpecialization_frobenius_smul_eq_glueMap4 below · cited by 5 · depth 11 - Hecke action at q on node units of the glued specialisation
ModularCurve.PlaceSpecialization.gluedSpecialization_nodeUnit_heckeGen_eq_nodePerm_symm_comp526 below · cited by 5 · depth 11 - Inertia differences on prime-to-q torsion are toric
ModularCurve.PlaceSpecialization.inertia_smul_sub_self_componentMap_eq_zero_toPic0Pair_eq_zero_of_isModel1,962 below · cited by 4 · depth 11 - Eichler–Shimura relation: sp^{Pic^0} intertwines T_ℓ with the geometric Hecke correspondence
ModularCurve.PlaceSpecialization.spPic0_heckeGen_ell_eq_heckeFibreGeom209 below · cited by 6 · depth 11 - Decomposition-group equivariance of the glued specialization's Pic⁰-pair
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_decomposition_smul_eq_spPic0_smul158 below · cited by 1 · depth 11 - Decomposition-group stability of the vanishing Pic⁰-pair
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_decomposition_smul_eq_zero_of_eq_zero88 below · cited by 1 · depth 11 - Hecke stability of the toric kernel of the glued specialization
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_heckeAlg_smul_eq_zero_of_eq_zero_of_isModel1,199 below · cited by 3 · depth 11 - Hecke equivariance of the projected glued specialization at q
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_heckeGen_equivariant_of_isModel957 below · cited by 5 · depth 11 - Hecke law for the glued specialization on Pic⁰ pairs
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_heckeGen_smul_eq_heckePic0Fibre_of_isModel1,173 below · cited by 1 · depth 11 - Good kernel classes of order prime to q vanish
ModularCurve.PlaceSpecialization.IsGluedSpecialization.eq_zero_of_isGoodClass_of_nsmul_eq_zero_of_not_dvd_of_isModel1,052 below · cited by 1 · depth 12 - Glued specialization is onto by good inertia-invariant classes
ModularCurve.PlaceSpecialization.IsGluedSpecialization.exists_isGoodClass_apply_eq482 below · cited by 2 · depth 12 - Good classes in the kernel of a glued specialization are m-divisible
ModularCurve.PlaceSpecialization.IsGluedSpecialization.exists_nsmul_eq_of_isGoodClass_of_apply_eq_zero_of_isModel1,125 below · cited by 2 · depth 12 - Good classes form a subgroup of the inertia invariants
ModularCurve.PlaceSpecialization.exists_addSubgroup_mem_iff_isGoodClass0 below · cited by 5 · depth 12 - Inertia displacement of prime-to-q torsion: good divisor with vanishing Pic⁰ pair
ModularCurve.PlaceSpecialization.exists_goodRep_toPic0Pair_eq_zero_smul_sub_self_of_isModel1,928 below · cited by 3 · depth 12 - Kind-respecting good admissible representative for T_ℓ on J₀(Nq)
ModularCurve.PlaceSpecialization.exists_good_admissible_rep_heckeDivBar_good_admissible_kindResp_of_isModel930 below · cited by 5 · depth 12 - Good admissible representatives stable under the q-correspondence
ModularCurve.PlaceSpecialization.exists_good_admissible_rep_heckeDivBar_self_good_admissible284 below · cited by 1 · depth 12 - Hecke descent family away from ℓ on the special fibre
ModularCurve.PlaceSpecialization.exists_heckeDescent_family_qne_ell965 below · cited by 5 · depth 12 - Glued classes killed by `toPic0Pair` lift to toric monodromy
ModularCurve.PlaceSpecialization.exists_mem_toricMonodromyPart_sp_eq_of_toPic0Pair_eq_zero_of_isModel2,694 below · cited by 1 · depth 12 - Existence of a place specialization carrying a model prolongation tuple
ModularCurve.PlaceSpecialization.exists_prolongationTuple_isModel_and_orderLawFixed1,658 below · cited by 5 · depth 12 - Component map and Ogg bidegree divisor at characteristic-q widths
ModularCurve.PlaceSpecialization.exists_widths_comp_sndDegLaw_surjective_repOfKer_principalGood_of_widthPinChar_of_isModel1,734 below · cited by 1 · depth 12 - Widths, component map, glued specialisation and Hecke matrices
ModularCurve.PlaceSpecialization.exists_widths_componentMap_gluedSpecialization_placeWidthChar_correspondence_heckeComponentAction_agree_of_isModel2,483 below · cited by 1 · depth 12 - Hecke equivariance of spPic⁰ at primes q≠ℓ
ModularCurve.PlaceSpecialization.heckePic0Fibre_spPic0_eq_spPic0_heckeGen_smul_of_ne_ell956 below · cited by 4 · depth 12 - Glued principality of the gluing datum of a principal divisor
ModularCurve.PlaceSpecialization.isGluedPrincipal_glueData_of_forall_apply_eq_ord_of_regularityLaw_of_nodeValueLaw_of_nonempty678 below · cited by 1 · depth 12 - Vanishing of the component map implies a good class
ModularCurve.PlaceSpecialization.isGoodClass_of_comp_eq_zero_of_exists_isGoodDiv163 below · cited by 3 · depth 12 - Place specialisation refines reduction mod ℓ on J₀(N)
ModularCurve.PlaceSpecialization.reductionModL_eq_zero_of_spPic0_eq_zero_and_isPlaceReductionModL_sp937 below · cited by 3 · depth 12 - Inertia-invariance of inertial displacements of prime-to-q torsion
ModularCurve.PlaceSpecialization.smul_sub_self_mem_inertiaInvariants_of_primeToTorsion_of_isModel1,961 below · cited by 2 · depth 12 - Specialization on Pic⁰ agrees with reduction mod ℓ
ModularCurve.PlaceSpecialization.spPic0_eq_reductionModL937 below · cited by 2 · depth 12 - Hecke propagation of glued vanishing away from q
ModularCurve.PlaceSpecialization.toPic0Pair_gluedSpecialization_heckeGen_dvd_smul_eq_zero_of_eq_zero_of_isModel1,171 below · cited by 1 · depth 12 - Strict two-sided representatives of good classes killed by sp
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictFst_isStrictSnd_reduceFst_eq_reduceSnd_eq_pic0Mk_eq720 below · cited by 2 · depth 13 - Multiplication by m on the residue polydisc over a base divisor
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_reduceFst_eq_reduceSnd_eq_ord_eq_nsmul_sub_sub_of_evalAt_pow_ne614 below · cited by 1 · depth 13 - Rigidity of a two-sided base divisor under n-torsion relations
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.sum_single_add_sum_single_eq_of_ord_eq_nsmul_sub_of_evalAt_ne754 below · cited by 4 · depth 13 - Cusp law on the infinity branch for prolongation tuples
ModularCurve.PlaceSpecialization.ProlongationTuple.cuspLawInfty_of_sp_eq_spPlace_of_cuspChart636 below · cited by 2 · depth 13 - One-sided first-copy cusp law at infinity-side cusps
ModularCurve.PlaceSpecialization.ProlongationTuple.cuspLawInfty_oneSided666 below · cited by 12 · depth 13 - One-sided zero-side cusp law at level N
ModularCurve.PlaceSpecialization.ProlongationTuple.cuspLawZero_oneSided668 below · cited by 10 · depth 13 - One-sided first-copy divisor law off Frob²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.divisorLawFst_oneSided666 below · cited by 14 · depth 13 - One-sided second divisor law for prolongation tuples
ModularCurve.PlaceSpecialization.ProlongationTuple.divisorLawSnd_oneSided668 below · cited by 11 · depth 13 - Moving good classes off a finite set of reductions
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_good_admissible_rep_reduce_notMem_of_isGoodClass_of_isModel925 below · cited by 4 · depth 13 - Values at places over a supersingular node reduce to branch residues
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidue_red_of_hasValue_of_sp_eq_spPlace862 below · cited by 3 · depth 13 - Atkin–Lehner symmetry: two laws suffice for a model
ModularCurve.PlaceSpecialization.ProlongationTuple.isModel_of_divisorLawFst_of_cuspLawInfty81 below · cited by 1 · depth 13 - Gauss jump law from model and one-sided regularity
ModularCurve.PlaceSpecialization.ProlongationTuple.jumpLaw_of_isModel_of_oneSidedRegularityLaw680 below · cited by 1 · depth 13 - Node-value law from the regularity law at supersingular places
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeValueLaw_of_regularityLaw385 below · cited by 10 · depth 13 - Norm reduction and order formula for a prolongation tuple
ModularCurve.PlaceSpecialization.ProlongationTuple.normReduction_of_not_dvd_of_surjective249 below · cited by 2 · depth 13 - One-sided regularity law at supersingular places for models
ModularCurve.PlaceSpecialization.ProlongationTuple.oneSidedRegularityLaw_of_isModel_of_not_dvd1,683 below · cited by 1 · depth 13 - Both residues regular at a Frobenius-square-fixed ordinary affine place
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_nonneg_and_ord_residueSnd_nonneg_of_fixed_of_isAffineGeomPlace_of_notMem_ssPlaces_of_sp_eq_spPlace420 below · cited by 1 · depth 13 - Hecke equivariance of the depth functional in the component group
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_eq_heckeComponentAction_of_heckeGen_smul2,247 below · cited by 1 · depth 13 - Component map of J₀(Nq) with second-copy degree law
ModularCurve.PlaceSpecialization.exists_comp_sndDegLaw_coordMem_repOfKer_widthChar_of_isModel1,733 below · cited by 1 · depth 13 - Depth function and surjective component map on inertia invariants
ModularCurve.PlaceSpecialization.exists_depth_comp_depthCompLaw_depthValueLaw_sndDegLaw_surjective_repOfKer_repOfInvariant_principalGood_of_widthPinChar_of_isModel1,732 below · cited by 3 · depth 13 - Inertia-invariant lifting of all but finitely many places
ModularCurve.PlaceSpecialization.exists_finset_forall_exists_sp_eq_forall_inertia_smul_eq426 below · cited by 9 · depth 13 - Inertial displacements at non-strict supersingular places give node units
ModularCurve.PlaceSpecialization.exists_gluedMk_eq_nodeUnit_of_isGoodDiv_of_admissible_of_pic0Mk_eq_smul_single_sub_self_of_isModel242 below · cited by 1 · depth 13 - Inertia displacements σ V-V admit good admissible representatives
ModularCurve.PlaceSpecialization.exists_goodRep_admissible_smul_single_sub_self_of_isModel1,474 below · cited by 2 · depth 13 - Inertial differences realise prescribed node units at one node pair
ModularCurve.PlaceSpecialization.exists_inertia_smul_sub_self_sp_eq_nodeUnit_of_isModel2,693 below · cited by 2 · depth 13 - Good effective divisors avoiding a finite set of reductions
ModularCurve.PlaceSpecialization.exists_isGoodDiv_reduce_notMem_isPrincipal_sub_of_smul_eq175 below · cited by 2 · depth 13 - Non-affine fibre places lift to both cuspidal sides
ModularCurve.PlaceSpecialization.exists_isInftySide_reduceFst_eq_and_isZeroSide_reduceSnd_eq_of_not_isAffineGeomPlace216 below · cited by 27 · depth 13 - Principal divisor on X₀(Nq) realising the width sum
ModularCurve.PlaceSpecialization.exists_isPrincipal_isGoodDiv_degree_fstDiv_eq_widthSum_widthChar_of_isModel1,395 below · cited by 2 · depth 13 - Inertia-fixed strict points in general position over the special fibre
ModularCurve.PlaceSpecialization.exists_isStrictFst_isStrictSnd_general_position_disjoint_forall_inertia_smul_eq469 below · cited by 15 · depth 13 - Inertia-fixed lift of the second kind over a given place
ModularCurve.PlaceSpecialization.exists_isStrictSnd_restrictAlong_eq_forall_inertia_smul_eq4 below · cited by 9 · depth 13 - Hecke transport of node units on the glued fibre at q
ModularCurve.PlaceSpecialization.exists_matrix_eq_correspondence_gluedSpecialization_nodeUnit_heckeGen_of_ne_of_isModel1,780 below · cited by 3 · depth 13 - A lawful prolongation tuple over a level-one place specialisation
ModularCurve.PlaceSpecialization.exists_prolongationTuple_isModel_and_orderLawFixed_level_one686 below · cited by 3 · depth 13 - Moving divisor classes off places outside the supersingular locus
ModularCurve.PlaceSpecialization.exists_rep_reduce_notMem_of_moving_of_disjoint_of_isModel959 below · cited by 1 · depth 13 - Charts at every place not fixed by φ²
ModularCurve.PlaceSpecialization.hasCharts_of_sp_eq_spPlace_of_not_dvd644 below · cited by 1 · depth 13 - Coordinates for specialisations arising from fibre models
ModularCurve.PlaceSpecialization.hasCoordinates_of_sp_eq_spPlace196 below · cited by 2 · depth 13 - Inertial displacements of strict divisors: goodness and vanishing glue datum
ModularCurve.PlaceSpecialization.isGoodDiv_and_glueData_smul_sub_self_eq_zero_of_forall_isStrict3 below · cited by 4 · depth 13 - U_q preserves good divisors and transports their gluing data
ModularCurve.PlaceSpecialization.isGoodDiv_heckeDivBar_self_and_glueData_mem_admissible281 below · cited by 1 · depth 13 - Local semicontinuity from reduced divisors, coordinates and charts
ModularCurve.PlaceSpecialization.localSemicontinuity_of_reducesDivisors_of_hasCoordinates_of_hasCharts247 below · cited by 1 · depth 13 - Strictness of the second reduction transfers to the first
ModularCurve.PlaceSpecialization.not_fixed_reduceFst_of_isStrictSnd0 below · cited by 18 · depth 13 - Prime-to-q torsion trivial on the glued reduction vanishes
ModularCurve.PlaceSpecialization.pic0Mk_eq_zero_of_isGoodDiv_of_mk_glueData_eq_zero_of_nsmul_eq_zero_of_isModel1,052 below · cited by 1 · depth 13 - Surjectivity of the reduction map of a level-N place specialisation
ModularCurve.PlaceSpecialization.red_surjective_of_level114 below · cited by 11 · depth 13 - Inertia invariance of the first level-N reduction
ModularCurve.PlaceSpecialization.reduceFst_arithmeticGalois_smul2 below · cited by 18 · depth 13 - Atkin–Lehner transport swaps the two reductions at q ∤ N
ModularCurve.PlaceSpecialization.reduceFst_atkinLehnerBar_smul76 below · cited by 31 · depth 13 - Surjectivity of the first reduction map on places
ModularCurve.PlaceSpecialization.reduceFst_surjective53 below · cited by 9 · depth 13 - Inertia invariance of the second level-N reduction
ModularCurve.PlaceSpecialization.reduceSnd_arithmeticGalois_smul3 below · cited by 13 · depth 13 - Atkin–Lehner transport exchanges the two degeneracy reductions
ModularCurve.PlaceSpecialization.reduceSnd_atkinLehnerBar_smul76 below · cited by 23 · depth 13 - Hecke stability of the specialisation kernel for q ≠ ℓ
ModularCurve.PlaceSpecialization.spPic0_heckeGen_smul_eq_zero_of_ne_ell963 below · cited by 1 · depth 13 - Model and order laws pin the place specialisation of X₀(N)
ModularCurve.PlaceSpecialization.sp_eq_spPlace_of_isModel_of_orderLawFixed342 below · cited by 5 · depth 13 - Inertial displacements of prime-to-q torsion are good toric classes
ModularCurve.PlaceSpecialization.IsGluedSpecialization.isGoodClass_and_toPic0Pair_apply_smul_sub_self_of_isModel1,962 below · cited by 2 · depth 14 - Existence of a model level-one prolongation pair at q
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_isModel341 below · cited by 8 · depth 14 - First prolongation integers are the localised modular ring
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.mem_integersFst_iff_coe_mem_modularLocalized124 below · cited by 51 · depth 14 - Second prolongation integers as Fricke pull-back of localised reduction
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.mem_integersSnd_iff_coe_frickeInvolutionBar_mem_modularLocalized124 below · cited by 5 · depth 14 - Order law at Frobenius-fixed places for prolongation pairs
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.orderLawFixed269 below · cited by 10 · depth 14 - Regularity law for level-one prolongation pairs of X₀(q)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.regularityLaw563 below · cited by 9 · depth 14 - Value law at a smooth point of the first copy
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_hasValue_of_mem_smoothLocalRingFst668 below · cited by 4 · depth 14 - Strict two-sided representative of a good degree-zero class
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictFst_isStrictSnd_reduceFst_eq_reduceSnd_eq_pic0Mk_eq_of_isGoodDiv720 below · cited by 2 · depth 14 - Incidence data representing m-division on the residue polydisc
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_mDivRepresents588 below · cited by 1 · depth 14 - Roots of the incidence system give m-division divisors
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_reduceFst_eq_reduceSnd_eq_ord_eq_of_mDivRepresents_of_forall_eval_eq_zero218 below · cited by 1 · depth 14 - Common normalisation of a good admissible divisor on both branches
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_smul_mem_integers_of_isGoodDiv_of_admissible680 below · cited by 1 · depth 14 - Unit Jacobian at the centre of the m-division system
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.isUnit_det_jacobian_centre_of_mDivRepresents124 below · cited by 1 · depth 14 - At most a simple pole for ε at affine base points
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.neg_one_le_ord_residueFst_of_eq_one_add_mul_of_evalAt_ne725 below · cited by 2 · depth 14 - Simple pole bound for the second residue of ε
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.neg_one_le_ord_residueSnd_of_eq_one_add_mul_of_evalAt_ne728 below · cited by 1 · depth 14 - Depth of an inertia-fixed place over a node is less than E
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.depth_lt_of_depthValueLaw_of_nodeEquation468 below · cited by 6 · depth 14 - Positivity of depth at inertia-fixed places over a node
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.depth_pos_of_depthValueLaw_of_nodeEquation468 below · cited by 5 · depth 14 - Node depth of inertia-fixed places is a power of v_A(q)
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.exists_yDepth_eq_pow_of_forall_inertia_smul_eq12 below · cited by 2 · depth 14 - Crossing exponent at supersingular nodes, all characteristics q∤ N
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidthChar_mul_of_orderLawFixed846 below · cited by 12 · depth 14 - Inertia-invariant annulus over a supersingular node of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_annulus_mem_dom_iff_reduceFst_eq_of_mem_ssPlaces1,333 below · cited by 1 · depth 14 - One-sided divisor laws for the modular unit Δ/Δ_q
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_divisor_oneSidedFst_laws_modularUnit659 below · cited by 3 · depth 14 - Envelope and local equation for an inertial displacement at a node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_envelope_localEquation_smul_single_sub_single947 below · cited by 1 · depth 14 - Inertia-fixed node presentation at supersingular places of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_range_redRestrict_forall_nodeCoordinates_presentation_of_orderLawFixed1,270 below · cited by 4 · depth 14 - Existence of an inertia-fixed strict place avoiding W
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_isStrictFst_and_smul_eq_self445 below · cited by 2 · depth 14 - Node integers have q-expansions over a number field
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_le_mem_nodeIntegersOver_of_mem_nodeIntegers115 below · cited by 2 · depth 14 - Modular unit Δ/Δ_q: R₁-integral, residue of order 1-q
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_integersFst_residue_ne_zero_of_coe_eq_modularUnitSeries_level32 below · cited by 18 · depth 14 - Riemann–Roch functions with prescribed residue orders on both prolongations
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_riemannRochSpace_ord_residue_eq_neg_of_splitDatum807 below · cited by 1 · depth 14 - One-point moving on X₀(Nq) into the strict locus
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_ord_eq_one_forall_isStrict_reduceFst_reduceSnd_notMem922 below · cited by 1 · depth 14 - A finite spanning order for the two prolongations at level Nq
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_orderSubalgebra_finite_span_eq_top204 below · cited by 1 · depth 14 - Correcting an exponent by powers of a non-unit
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_pow_mul_zpow_mem_integersSnd_residue_ne_zero4 below · cited by 3 · depth 14 - Inertia-invariant places at prescribed depth over a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_reduceFst_eq_and_smul_eq_self_and_yDepth_eq_pow_of_orderLawFixed1,313 below · cited by 2 · depth 14 - Tube equation for an inertial displacement on an annulus
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_tubeEquation_smul_sub_self_of_annulus9 below · cited by 1 · depth 14 - Good principal divisors yield admissible gluing data
ModularCurve.PlaceSpecialization.ProlongationTuple.glueData_mem_admissible_of_isGoodDiv_of_ord_eq_of_not_dvd231 below · cited by 1 · depth 14 - Value bridge at a supersingular node over number fields
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidue_red_of_hasValue_of_mem_nodeIntegersOver_of_sp_eq_spPlace860 below · cited by 3 · depth 14 - Non-affine first reduction forces cuspidality of V
ModularCurve.PlaceSpecialization.ProlongationTuple.isCuspidal_of_not_isAffineGeomPlace_reduceFst81 below · cited by 14 · depth 14 - Atkin–Lehner involution exchanges the ∞- and 0-sides
ModularCurve.PlaceSpecialization.ProlongationTuple.isInftySide_atkinLehnerBar_smul_iff76 below · cited by 14 · depth 14 - Cuspidal places of level Nq lie on the infinity or zero side
ModularCurve.PlaceSpecialization.ProlongationTuple.isInftySide_or_isZeroSide_of_isCuspidal256 below · cited by 15 · depth 14 - Atkin–Lehner at q swaps the zero and infinity sides
ModularCurve.PlaceSpecialization.ProlongationTuple.isZeroSide_atkinLehnerBar_smul_iff76 below · cited by 5 · depth 14 - Saturation of the node residue maps at a supersingular place
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidue_saturated_of_orderLawFixed817 below · cited by 6 · depth 14 - Cuspidal places reduce to non-affine places on the first copy
ModularCurve.PlaceSpecialization.ProlongationTuple.not_isAffineGeomPlace_reduceFst_of_isCuspidal0 below · cited by 20 · depth 14 - Zero side and infinity side of a place are disjoint
ModularCurve.PlaceSpecialization.ProlongationTuple.not_isInftySide_of_isZeroSide146 below · cited by 8 · depth 14 - Reduced modular unit has order zero at ordinary affine places
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_eq_zero_of_coe_eq_modularUnitSeries_of_notMem_ssPlaces239 below · cited by 7 · depth 14 - The 0-side Frobenius relation between the two reductions
ModularCurve.PlaceSpecialization.ProlongationTuple.reduceFst_eq_frobOnPlacesGeomLevel_reduceSnd_of_isZeroSide248 below · cited by 5 · depth 14 - Second residue of the level-q modular unit vanishes
ModularCurve.PlaceSpecialization.ProlongationTuple.residue_eq_zero_of_mem_integersSnd_of_coe_eq_modularUnitSeries100 below · cited by 5 · depth 14 - Divisor bounds for the reduced pair of a bi-integral section
ModularCurve.PlaceSpecialization.ProlongationTuple.sectionPair_bounds_of_regularityLaw_of_isModel678 below · cited by 2 · depth 14 - Split datum from a local equation at one supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.splitDatum_of_forall_reduceFst_eq_ord_eq673 below · cited by 1 · depth 14 - ∞-side places over a non-integral j-place have ramification sum 1
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ramificationIndexAlong_heckeAlphaBar_filter_isInftySide_fiberAlong_eq_one_of_forall_ord_jq_sub_nonpos245 below · cited by 1 · depth 14 - Value integrality at supersingular nodes from the fixed-place order law
ModularCurve.PlaceSpecialization.ProlongationTuple.valueIntegralityLaw_of_orderLawFixed453 below · cited by 23 · depth 14 - Principal divisors have trivial depth class in the component group
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_degree_sndDiv_smul_eq_zero_of_div1,545 below · cited by 1 · depth 14 - Hecke transport of the depth functional: annulus case, q≥ 5
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_eq_heckeComponentAction_of_heckeGen_smul_of_five_le_of_not_isGoodDiv2,172 below · cited by 1 · depth 14 - Hecke transport of depth functionals in component groups, q<5
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_eq_heckeComponentAction_of_heckeGen_smul_of_lt_five2,237 below · cited by 1 · depth 14 - Depth functional of a principal divisor lies in the Gram image
ModularCurve.PlaceSpecialization.depthDual_add_mem_range_gramMap_of_isPrincipal1,553 below · cited by 2 · depth 14 - Vanishing of the depth functional on good divisors
ModularCurve.PlaceSpecialization.depthDual_eq_zero_of_isGoodDiv0 below · cited by 4 · depth 14 - Arbitrarily many strict places of both kinds avoiding a finite set
ModularCurve.PlaceSpecialization.exists_families_isStrictFst_isStrictSnd_notMem230 below · cited by 4 · depth 14 - Inertia-fixed strict families of places at level Nq
ModularCurve.PlaceSpecialization.exists_families_isStrictFst_isStrictSnd_notMem_forall_inertia_smul_eq442 below · cited by 8 · depth 14 - Inertia-fixed representative with strict or supersingular support
ModularCurve.PlaceSpecialization.exists_inertiaFixedSupport_degZero_pic0Mk_eq_of_isModel1,581 below · cited by 1 · depth 14 - Charts at affine places not fixed by φ²
ModularCurve.PlaceSpecialization.exists_isChartAt_of_isAffineGeomPlace292 below · cited by 1 · depth 14 - Existence of charts at non-affine places off the φ²-fixed locus
ModularCurve.PlaceSpecialization.exists_isChartAt_of_not_isAffineGeomPlace642 below · cited by 1 · depth 14 - Genus-zero transfer of good-class and glued-specialization data
ModularCurve.PlaceSpecialization.exists_isGoodClass_iff_isGluedSpecialization_of_not_genusFF_pos1,681 below · cited by 1 · depth 14 - Good function with prescribed pole orders at supersingular nodes
ModularCurve.PlaceSpecialization.exists_isGoodDiv_ord_residueFst_eq_neg_lcm_div_widthChar_of_orderLawFixed1,394 below · cited by 1 · depth 14 - Depth-component kernel classes are classes of good divisors
ModularCurve.PlaceSpecialization.exists_isGoodDiv_pic0Mk_eq_of_comp_eq_zero_of_depthCompLaw_depthValueLaw_repOfInvariant_of_isModel1,541 below · cited by 1 · depth 14 - T_ℓ (ℓ≠ q) acts on node units by a correspondence matrix
ModularCurve.PlaceSpecialization.exists_matrix_eq_correspondence_gluedSpecialization_nodeUnit_heckeGen_of_ne_of_isModel_of_prolongation_of_regularityLaw_nodeValueLaw986 below · cited by 1 · depth 14 - An A-value for j(q^q) at places with affine second reduction
ModularCurve.PlaceSpecialization.exists_ord_jQFun_sub_pos_of_isAffineGeomPlace_reduceSnd2 below · cited by 11 · depth 14 - Model prolongation tuple with order law at genus-zero level
ModularCurve.PlaceSpecialization.exists_prolongationTuple_isModel_and_orderLawFixed_of_not_genusFF_pos1,659 below · cited by 5 · depth 14 - Model prolongation tuple with regularity, node-value and order laws
ModularCurve.PlaceSpecialization.exists_prolongationTuple_isModel_regularityLaw_nodeValueLaw1,666 below · cited by 8 · depth 14 - Existence of a lawful level-one prolongation tuple
ModularCurve.PlaceSpecialization.exists_prolongationTuple_isModel_regularityLaw_nodeValueLaw_level_one686 below · cited by 5 · depth 14 - Regular prolongation at level N reducing j and j_N
ModularCurve.PlaceSpecialization.exists_regularProlongation_sp_jq_jqN330 below · cited by 5 · depth 14 - Moving the strict part of a divisor off T₀, kind-respecting
ModularCurve.PlaceSpecialization.exists_rep_eq_off_strict_reduce_notMem_heckeDivBar_strictPart_good_kindResp_of_isModel929 below · cited by 4 · depth 14 - Moving lemma: representatives with j-residues avoiding S
ModularCurve.PlaceSpecialization.exists_rep_forall_exists_ord_sub_pos_residue_notMem_of_isModel_of_regularityLaw_of_orderLawFixed_of_ssPlaces956 below · cited by 1 · depth 14 - Frobenius squared fixes supersingular places under a place specialization
ModularCurve.PlaceSpecialization.frobOnPlacesGeomLevel_frobOnPlacesGeomLevel_eq_self_of_mem_ssPlaces381 below · cited by 17 · depth 14 - Atkin–Lehner at q swaps strictness of the first and second kinds
ModularCurve.PlaceSpecialization.isStrictFst_atkinLehnerBar_smul_iff78 below · cited by 6 · depth 14 - Model prolongation tuples determine the strict labels
ModularCurve.PlaceSpecialization.isStrictFst_iff_and_isStrictSnd_iff_of_isModel_of_isModel746 below · cited by 1 · depth 14 - Strictness of a place versus φ²-fixedness of its first reduction
ModularCurve.PlaceSpecialization.isStrictFst_or_isStrictSnd_iff0 below · cited by 12 · depth 14 - Local semicontinuity at first-kind places over a non-fixed place
ModularCurve.PlaceSpecialization.localSemicontinuityFst_of_reducesDivisors_of_hasCoordinates_of_hasCharts240 below · cited by 1 · depth 14 - Local semicontinuity for the second component from the first
ModularCurve.PlaceSpecialization.localSemicontinuitySnd_of_localSemicontinuityFst80 below · cited by 1 · depth 14 - First-component inclusion under a good/bad splitting over v
ModularCurve.PlaceSpecialization.mem_chartLocalSetFst_of_split237 below · cited by 2 · depth 14 - First reduction of the cusp ∞̄ is the j-line cusp
ModularCurve.PlaceSpecialization.redFst_cuspInftyBar12 below · cited by 28 · depth 14 - Places with no integral j-value reduce to the cusp
ModularCurve.PlaceSpecialization.redFst_eq_placeInfty_of_forall_ord_le_zero8 below · cited by 27 · depth 14 - Second reduction of the cusp ̄ 0 at q
ModularCurve.PlaceSpecialization.redSnd_cuspZeroBar95 below · cited by 6 · depth 14 - Inertia acts unipotently on prime-to-q torsion of J₀(q)
ModularCurve.PlaceSpecialization.smul_sub_self_mem_inertiaInvariants_of_primeToTorsion_of_isModel_levelOne1,494 below · cited by 1 · depth 14 - Cusp law at ∞ for level-one prolongation pairs
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.cuspLawInfty214 below · cited by 5 · depth 15 - Cusp law at 0 from the cusp law at ∞
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.cuspLawZero_of_cuspLawInfty89 below · cited by 5 · depth 15 - Divisor law on the first branch for level-one prolongation pairs
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.divisorLawFst313 below · cited by 5 · depth 15 - Fricke transport of the level-one divisor law
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.divisorLawSnd_of_divisorLawFst88 below · cited by 5 · depth 15 - Valuation rings over the Gauss ring of the j-line
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.integers_eq_or_eq_of_forall_mem_iff127 below · cited by 2 · depth 15 - Inertia-fixed correction making a stable divisor twistable
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_fixedGood_isTwistOf_sub_of_inertiaStable452 below · cited by 1 · depth 15 - Kernel reach at level N for twistable inertia-stable divisors
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_fixedStrict_add_kernelGood_of_isTwistOf_of_inertiaStable1,500 below · cited by 2 · depth 15 - Twist vector from a vanishing depth class in the component group
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_isTwistOf_of_componentGroupProj_depthDual_eq_zero_of_inertiaStable_of_laws1,284 below · cited by 1 · depth 15 - Chart data at a strict first-kind place over a fixed reduction
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_chartData_of_isStrictFst231 below · cited by 1 · depth 15 - Chart data at a strict place of the second kind
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_chartData_of_isStrictSnd231 below · cited by 1 · depth 15 - Simple zero of j-a above a strict affine place
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictFst_reduceFst_eq_ord_jFun_sub_eq_one115 below · cited by 3 · depth 15 - Simple zero of j_N-a on a strict first-kind disc
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictFst_reduceFst_eq_ord_jNFun_sub_eq_one114 below · cited by 2 · depth 15 - A simple zero of j(q^{Nq})-a on a strict second-kind disc
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictSnd_reduceSnd_eq_ord_jNQFun_sub_eq_one114 below · cited by 2 · depth 15 - Unique strict second-kind place where j(q^q)-a vanishes simply
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictSnd_reduceSnd_eq_ord_jQFun_sub_eq_one115 below · cited by 2 · depth 15 - Bi-integral family in L(E'+(m-1)E₀) with independent residue pairs
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_linearIndependent_residue_pair_riemannRochSpace_add_nsmul537 below · cited by 1 · depth 15 - Inertia-fixed lift of a residue pair to L(D)
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_mem_riemannRochSpace_residue_eq_forall_arithmeticGalois_smul_eq_of_isGoodDiv869 below · cited by 2 · depth 15 - Realising a node-compatible residue pair by a bi-integral section
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_mem_riemannRochSpace_residue_eq_of_isGoodDiv863 below · cited by 1 · depth 15 - Expansion homomorphism at a uniformiser of the residue disc
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_ringHom_tExpansion_of_ord_residue_eq_one673 below · cited by 1 · depth 15 - Order bound for a first residue from a t-expansion
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.residue_eq_zero_or_le_ord_residue_of_tExpansion_red_eq_zero_of_ord_residue_eq_one669 below · cited by 1 · depth 15 - Depth bound at a supersingular node: depth V < e'E
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.depth_lt_mul_of_yDepth_pow_eq468 below · cited by 2 · depth 15 - Depth dichotomy for the node coordinate y at a rational place
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.exists_hasDepth_or_depthBetween_y_of_nodeEquation_of_orderLawFixed459 below · cited by 3 · depth 15 - Relational depth equals value-group depth of y
ModularCurve.PlaceSpecialization.ProlongationTuple.NodeCoordinates.hasValuation_y_iff_yDepth_eq0 below · cited by 4 · depth 15 - Constants from A lie in the first smooth local ring
ModularCurve.PlaceSpecialization.ProlongationTuple.algebraMap_mem_smoothLocalRingFst0 below · cited by 3 · depth 15 - Crossing exponent at a supersingular node equals width times e_K
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidth_mul_of_orderLawFixed827 below · cited by 3 · depth 15 - Unique place over a supersingular node with prescribed value of y
ModularCurve.PlaceSpecialization.ProlongationTuple.existsUnique_reduceFst_eq_and_hasValue_y_of_orderLawFixed1,302 below · cited by 1 · depth 15 - Existence of a level-N annulus datum satisfying all its laws
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_annulusDatumLevel_laws1,347 below · cited by 2 · depth 15 - Node annulus at a supersingular place with parameter y
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_annulus_dom_iff_reduceFst_eq_and_param_eq_y_of_ringEquiv_uvCrossingModel587 below · cited by 2 · depth 15 - Common unit with simple zero and residue order tables
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_ord_eq_one_orderTables_of_realisation0 below · cited by 1 · depth 15 - Common unit with a simple pole above an ordinary fixed place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_of_reduceFst_fixed_ordinary_of_regularityLaw929 below · cited by 2 · depth 15 - Common unit with a simple pole at V₀, surviving the first residue
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_reduceFst_of_regularityLaw953 below · cited by 1 · depth 15 - Common unit with a simple pole at V₀
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_reduceSnd_of_regularityLaw953 below · cited by 1 · depth 15 - A coefficient complete DVR for the completed K-node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_completeDVR_ringHom_adicCompletion_nodeIntegersOver4 below · cited by 5 · depth 15 - Crossing presentation of the K-node ring at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_of_orderLawFixed_of_saturated843 below · cited by 8 · depth 15 - Completed node ring as crossing model W[[U,V]]/(UV-π^E)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_exponent_ringEquiv_adicCompletion_nodeIntegersOver_uvCrossingModel1,154 below · cited by 6 · depth 15 - Function realising prescribed node units and residue orders -n_w
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_forall_isUnit_mul_pow_nodeIntegers_and_ord_residueFst_eq_and_forall_ord_eq_zero1,063 below · cited by 1 · depth 15 - Value bridge at a supersingular node for j-integral elements
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_residue_red_of_mem_jIntegralClosure_of_sp_eq_spPlace834 below · cited by 1 · depth 15 - Node coordinates over an inertia-fixed field with uniformiser q
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_field_nonempty_nodeCoordinates776 below · cited by 3 · depth 15 - Uniform node presentation over one inertia-fixed number field
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_forall_nodeCoordinates_presentation_of_orderLawFixed1,271 below · cited by 1 · depth 15 - Inertia-fixed node presentation xy=q^eu at supersingular places
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_nodeCoordinates_presentation_of_orderLawFixed1,270 below · cited by 4 · depth 15 - Integral bases with k-independent residue pairs
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_integers_linearIndependent_residue_pair_of_finiteDimensional0 below · cited by 3 · depth 15 - Node integers are fractions with denominator a unit at the node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mul_eq_mem_jIntegralClosure_of_mem_nodeIntegersOver_of_sp_eq_spPlace286 below · cited by 2 · depth 15 - Elements of K(j_q,j_q^{Nq}) are node-ring fractions
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mul_eq_of_mem_fieldOver_nodeIntegersOver126 below · cited by 8 · depth 15 - Residue surjectivity of the node ring over K at a supersingular place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_not_isUnit_sub_nodeConst_of_evalAt_mem_range_redRestrict_of_orderLawFixed469 below · cited by 8 · depth 15 - Branch-adapted pair from a level-q crossing presentation
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_pair_nodeIntegersOver_ord_eq_placeRamificationJ_of_crossingPresentation389 below · cited by 1 · depth 15 - Seed datum from node coordinates with q-normalised node equation
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_seedDatum_of_nodeCoordinates_nodeEquation0 below · cited by 1 · depth 15 - Gauss-order function at a supersingular node: slope drops
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_slopeDrop_eq_sum_div_depth_of_sum_div_reduceFst_eq570 below · cited by 4 · depth 15 - Slope-drop function at a supersingular node on subdivided depths
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_slopeDrop_eq_sum_div_depth_of_yDepth_pow_eq570 below · cited by 2 · depth 15 - Node residue of the first prolongation equals reduced value at V
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidueFst_red_evalAt_of_orderLawFixed454 below · cited by 8 · depth 15 - Degeneracy pull-backs lie in the K-node ring, residues reduced
ModularCurve.PlaceSpecialization.ProlongationTuple.heckeAlphaBar_mem_nodeIntegersOver_and_nodeResidue_eq_coeffMap79 below · cited by 3 · depth 15 - Node ring over K at supersingular place is local noetherian
ModularCurve.PlaceSpecialization.ProlongationTuple.isLocalRing_and_isNoetherianRing_nodeIntegersOver_of_nodeCoordinates_of_orderLawFixed_of_range_redRestrict1,198 below · cited by 8 · depth 15 - Locality of the node ring over a number field
ModularCurve.PlaceSpecialization.ProlongationTuple.isLocalRing_nodeIntegersOver_of_orderLawFixed_of_regularityLaw400 below · cited by 4 · depth 15 - Non-vanishing node residue forces a unit in the local node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.isUnit_of_not_hasValue_nodeResidue_zero_of_isLocalRing411 below · cited by 3 · depth 15 - Crossing compatibility of residues against a local parameter
ModularCurve.PlaceSpecialization.ProlongationTuple.le_ord_residue_and_exists_hasValue_of_mul0 below · cited by 1 · depth 15 - Ramification mass off the ∞-side is at least q
ModularCurve.PlaceSpecialization.ProlongationTuple.le_sum_ramificationIndexAlong_heckeAlphaBar_filter_not_isInftySide_fiberAlong237 below · cited by 1 · depth 15 - Sheet-one divisor law with regularity at supersingular places
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceFst_filter_sheetOne_eq_ord_residueFst_of_regularityLaw1,120 below · cited by 3 · depth 15 - Second-sheet divisor law on X₀(Nq), ordinary fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceSnd_filter_sheetTwo_eq_ord_residueSnd_of_regularityLaw1,122 below · cited by 2 · depth 15 - Push-forward of the ∞-side divisor of the modular unit
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_restrictAlong_filter_isInftySide_divisor_modularUnit246 below · cited by 1 · depth 15 - Specialization pushes the polar divisor of j to ordᵥ(jmath̄)
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_sp_filter_neg_divisor_j_eq_ord_jqModC_of_isModel245 below · cited by 1 · depth 15 - Units of the first smooth local ring from vanishing orders
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_smoothLocalRingFst_and_inv_mem_of_forall_ord_eq_zero0 below · cited by 1 · depth 15 - Node depths below one partition the crossing exponent
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeDepths_lt_one_and_partition_of_nodeEquation_of_orderLawFixed457 below · cited by 12 · depth 15 - Order of the first-branch residue and the ideal (varpi,x,yⁿ)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidueFst_eq_zero_or_le_ord_iff_mem_span_of_orderLawFixed_of_range_redRestrict1,249 below · cited by 2 · depth 15 - Order on the second branch versus the ideal (varpi,y,xⁿ)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidueSnd_eq_zero_or_le_ord_iff_mem_span_of_orderLawFixed_of_range_redRestrict1,249 below · cited by 2 · depth 15 - Node residues are rational over red(A∩ K)(jmath̃,jmath̃_N)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidue_mem_closure_redRestrict85 below · cited by 3 · depth 15 - Residue of j-j₀ is a uniformiser downstairs
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_reduceFst_residue_jFun_sub_eq_one365 below · cited by 1 · depth 15 - Order of the reduced modular unit at non-affine places
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_modularUnit_eq_mul_ord_jqModC_of_not_isAffineGeomPlace82 below · cited by 1 · depth 15 - ∞-side places: second reduction is Frobenius of the first
ModularCurve.PlaceSpecialization.ProlongationTuple.reduceSnd_eq_frobOnPlacesGeomLevel_reduceFst_of_isInftySide242 below · cited by 4 · depth 15 - Residues of j-a and j(q^q)-a on both prolongations
ModularCurve.PlaceSpecialization.ProlongationTuple.residue_jFun_sub_jQFun_sub75 below · cited by 23 · depth 15 - Residues of j_N-a and j_{Nq}-a on both prolongations
ModularCurve.PlaceSpecialization.ProlongationTuple.residue_jNFun_sub_jNQFun_sub75 below · cited by 7 · depth 15 - Node order bound by branch orders of the two reductions
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_div_reduceFst_le_ord_residueFst_add_ord_residueSnd488 below · cited by 4 · depth 15 - Order of a principal divisor above a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_div_reduceFst_le_ord_residues488 below · cited by 2 · depth 15 - The ∞-side cusps over a cusp have ramification sum one
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ramificationIndexAlong_heckeAlphaBar_filter_isInftySide_fiberAlong_eq_one236 below · cited by 3 · depth 15 - Presentation-invariance of the node crossing valuation
ModularCurve.PlaceSpecialization.ProlongationTuple.valuation_pow_crossingExponent_eq0 below · cited by 3 · depth 15 - Node depths are independent of the chosen presentation
ModularCurve.PlaceSpecialization.ProlongationTuple.xDepth_eq_and_yDepth_eq_of_nodeCoordinates122 below · cited by 7 · depth 15 - Hecke equivariance of the depth class, good case
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_eq_heckeComponentAction_of_heckeGen_smul_of_isGoodDiv2,025 below · cited by 1 · depth 15 - Hecke transport of the depth functional: annulus case, q<5
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_eq_heckeComponentAction_of_heckeGen_smul_of_lt_five_of_not_isGoodDiv2,235 below · cited by 1 · depth 15 - Uniqueness of place specialisations when the special fibre has positive genus
ModularCurve.PlaceSpecialization.eq_of_genusFF_pos814 below · cited by 1 · depth 15 - Uniqueness of specialisation packets carrying a model prolongation tuple
ModularCurve.PlaceSpecialization.eq_of_isModel_of_orderLawFixed1,050 below · cited by 3 · depth 15 - Depths and component map over the residue field of A
ModularCurve.PlaceSpecialization.exists_depth_comp_depthCompLaw_depthValueLaw_sndDegLaw_surjective_repOfKer_repOfInvariant_principalGood_of_widthPinChar_of_isModel_residueField1,733 below · cited by 1 · depth 15 - Hecke neighbours inherit strictness and kind off a finite set
ModularCurve.PlaceSpecialization.exists_finset_isStrict_and_kind_of_mem_support_heckeDivBar_single_of_reduce_notMem223 below · cited by 1 · depth 15 - Good admissible representatives of inertia displacements at level q
ModularCurve.PlaceSpecialization.exists_goodRep_toPic0Pair_eq_zero_smul_sub_self_levelOne1,113 below · cited by 1 · depth 15 - Inertia-stable divisor classes with strict or supersingular support
ModularCurve.PlaceSpecialization.exists_inertiaStable_pic0Mk_eq_of_inertiaStable_of_isModel935 below · cited by 1 · depth 15 - Existence of a level-one prolongation pair
ModularCurve.PlaceSpecialization.exists_levelOneProlongationPair124 below · cited by 6 · depth 15 - A-value of j at places with affine first reduction
ModularCurve.PlaceSpecialization.exists_ord_jFun_sub_pos_of_isAffineGeomPlace_reduceFst2 below · cited by 8 · depth 15 - Integral value of j(q^N) at places with affine reduction
ModularCurve.PlaceSpecialization.exists_ord_jNFun_sub_pos_of_isAffineGeomPlace_reduceFst2 below · cited by 3 · depth 15 - Integral value of j(q^{Nq}) at affine second reductions
ModularCurve.PlaceSpecialization.exists_ord_jNQFun_sub_pos_of_isAffineGeomPlace_reduceSnd2 below · cited by 4 · depth 15 - Cross-power law for node depths under the ℓ-degeneracy maps
ModularCurve.PlaceSpecialization.exists_reduceFst_eq_and_yDepth_restrictAlong_heckeAlphaBar_pow_width_eq_of_ne_of_not_dvd1,950 below · cited by 1 · depth 15 - A level-one place specialisation forces k algebraically closed
ModularCurve.PlaceSpecialization.isAlgClosed10 below · cited by 7 · depth 15 - Algebraic closedness of the residue field at level prime to q
ModularCurve.PlaceSpecialization.isAlgClosed_of_level_of_not_dvd83 below · cited by 2 · depth 15 - Atkin–Lehner at q swaps strictness of the two kinds
ModularCurve.PlaceSpecialization.isStrictSnd_atkinLehnerBar_smul_iff78 below · cited by 5 · depth 15 - First reduction intertwines T_ℓ with the fibre correspondence
ModularCurve.PlaceSpecialization.mapDomain_reduceFst_heckeDivBar_eq_heckeDivFibre_mapDomain_reduceFst_of_ne_of_isModel_of_orderLawFixed842 below · cited by 2 · depth 15 - First reduction of the Hecke divisor of one place
ModularCurve.PlaceSpecialization.mapDomain_reduceFst_heckeDivBar_single_apply_eq_correspondence_of_ne346 below · cited by 2 · depth 15 - Zeros of j-j₀ descend along the first reduction
ModularCurve.PlaceSpecialization.ord_jGeomGen_sub_pos_of_ord_jFun_sub_pos1 below · cited by 3 · depth 15 - Prime-to-q torsion with trivial glued reduction vanishes, level one
ModularCurve.PlaceSpecialization.pic0Mk_eq_zero_of_isGoodDiv_of_mk_glueData_eq_zero_of_nsmul_eq_zero_of_isModel_levelOne1,092 below · cited by 1 · depth 15 - Strict first-kind places: unramified over α and unique
ModularCurve.PlaceSpecialization.ramificationIndexAlong_heckeAlphaBar_eq_one_and_eq_of_isChartAt_of_isStrictFst1 below · cited by 1 · depth 15 - First level-one reduction is the place j=b̄
ModularCurve.PlaceSpecialization.redFst_eq_charLGeomPlaceOfPoint_of_ord_pos5 below · cited by 20 · depth 15 - Supersingularity propagates along the ℓ-Hecke correspondence, ℓ ≠ q
ModularCurve.PlaceSpecialization.reduceFst_mem_ssPlaces_of_mem_support_heckeDivBar_single_of_ne345 below · cited by 3 · depth 15 - Places with non-integral j specialise to j=∞
ModularCurve.PlaceSpecialization.sp_eq_placeInfty_of_forall_ord_le_zero7 below · cited by 13 · depth 15 - Only two prolongations with transcendental j-residue
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.integers_eq_or_eq_of_transcendental126 below · cited by 1 · depth 16 - Order of the first residue at non-geometric places
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.ord_residue_fst_eq_zero_of_forall_ne11 below · cited by 5 · depth 16 - Residues of j and j_q on a level-one prolongation pair
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residue_jFun_jqFun85 below · cited by 13 · depth 16 - Summed divisor law for the pencil j+μ j_q
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.sum_ord_pencil_eq304 below · cited by 1 · depth 16 - Integrality of inertia-stable circle degrees and depth moments
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.den_circleDeg_eq_one_and_den_depthMoment_eq_one_of_inertiaStable159 below · cited by 4 · depth 16 - Good representative of an inertia-stable twistable class
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_isGoodDiv_pic0Mk_eq_of_isTwistOf_of_mk_spData_eq_zero_of_inertiaStable1,486 below · cited by 1 · depth 16 - Subtracting a balanced strict divisor preserves twist data
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.isTwistOf_sub_and_spData_sub_eq_of_forall_isStrict162 below · cited by 1 · depth 16 - Admissibility of the twisted gluing datum at level N
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.spData_mem_admissible444 below · cited by 1 · depth 16 - Local quotient form at a smooth strict first-kind point
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_evalBar_eq_mul_evalBar_of_mem_smoothLocalRingFst214 below · cited by 1 · depth 16 - Regular functions on a strict second-kind disc as p/s
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_evalBar_eq_mul_evalBar_of_mem_smoothLocalRingSnd219 below · cited by 1 · depth 16 - Strict two-sided representative of a good degree-zero class, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_isStrictFst_isStrictSnd_reduceFst_eq_reduceSnd_eq_pic0Mk_eq_of_isGoodDiv_levelOne769 below · cited by 2 · depth 16 - Integral t-expansions at a smooth point of the reduction
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_tExpansion_of_ord_residue_eq_one672 below · cited by 1 · depth 16 - Residues of good-divisor sections and their node values
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.residue_mem_riemannRochSpace_mapDomain_and_hasValue_of_isGoodDiv485 below · cited by 3 · depth 16 - Rigidity of a two-sided base divisor under a torsion relation
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.sum_single_add_sum_single_eq_of_ord_eq_nsmul_sub_of_evalAt_ne_levelOne768 below · cited by 2 · depth 16 - First residues of K-rational functions are Frobenius-fixed
ModularCurve.PlaceSpecialization.ProlongationTuple.arithFrobC_pow_smul_residueFst_eq_of_isFrobeniusAt_of_coe_mem_fieldOver8 below · cited by 2 · depth 16 - Crossing exponent at a ramified supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidth_mul_of_orderLawFixed_of_one_lt_placeRamificationJ824 below · cited by 1 · depth 16 - Crossing exponent at an unramified supersingular node equals jWidth· e_K
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidth_mul_of_orderLawFixed_of_placeRamificationJ_eq_one825 below · cited by 1 · depth 16 - Node parameter separates the places above a supersingular place
ModularCurve.PlaceSpecialization.ProlongationTuple.eq_of_evalAt_y_eq_of_reduceFst_eq_of_ringEquiv_uvCrossingModel219 below · cited by 2 · depth 16 - Common unit with simple pole over an ordinary fixed place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_of_reduceFst_fixed_ordinary_of_regularityLaw_univ929 below · cited by 1 · depth 16 - Surjective depth–component homomorphism on inertia invariants, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_comp_depthCompLaw_and_surjective_levelOne1,340 below · cited by 1 · depth 16 - Crossing presentation of the K-node ring, q ≥ 5
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_of_orderLawFixed_of_saturated_of_five_le834 below · cited by 1 · depth 16 - Crossing presentation at a supersingular node for q<5
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_of_orderLawFixed_of_saturated_of_lt_five504 below · cited by 1 · depth 16 - Inertia-invariant rational depth at supersingular nodes of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_depthQ_cleared_law_and_forall_inertia_smul_eq467 below · cited by 1 · depth 16 - Seed function for the vertical cycle at supersingular nodes
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_forall_isUnit_mul_pow_nodeIntegers_and_ord_residueFst_eq_neg818 below · cited by 1 · depth 16 - Places with nodal value law at w have first reduction w
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_forall_reduceFst_eq_of_forall_hasValue_of_sp_eq_spPlace221 below · cited by 1 · depth 16 - Node value law at a supersingular place of level Nq
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_iff_hasValue_residueFst_zero_jIntegralClosure_of_sp_eq_spPlace833 below · cited by 2 · depth 16 - Agreement of the two residue conditions at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_residueFst_iff_residueSnd_jIntegralClosure_of_sp_eq_spPlace832 below · cited by 2 · depth 16 - Node coordinates at a supersingular place over an inertia-fixed field
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_nonempty_nodeCoordinates771 below · cited by 1 · depth 16 - Surjectivity of the glued class map onto GluedPic⁰
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_strict_gluedPic0Mk_glueData_eq482 below · cited by 1 · depth 16 - Bi-integral basis of a Riemann–Roch space with independent residue pairs
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_linearIndependent_residuePair_of_riemannRochSpace188 below · cited by 1 · depth 16 - Separating supersingular places by functions integral over the j-ring
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_jIntegralClosure_ord_residues_pos_and_eq_zero_of_ne1,170 below · cited by 3 · depth 16 - Level-q node ring elements lift to level-Nq node integers
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_nodeIntegersOver_of_mem_modularLocalizedAtPoint77 below · cited by 8 · depth 16 - Denominators may be chosen non-vanishing on the first branch
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mul_eq_of_mem_integers_nodeResidueFst_ne_zero0 below · cited by 5 · depth 16 - R₂-integral quotients admit denominators with non-zero second residue
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mul_eq_of_mem_integers_nodeResidueSnd_ne_zero0 below · cited by 5 · depth 16 - Node units and uniformiser ratios take values in k^×
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_nodeUnit_lam_mu_hasValue_of_ord_eq_one2 below · cited by 1 · depth 16 - Moving lemma on X₀(Nq) with inertia equivariance
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_ord_eq_one_forall_isStrict_reduceFst_reduceSnd_notMem_forall_inertia_smul_eq_of_isModel927 below · cited by 2 · depth 16 - Place realising a height-one prime at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_place_forall_iff_mem_and_hasValue_of_height_one_of_natCast_notMem198 below · cited by 3 · depth 16 - Admissible values attained by y at places over a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_reduceFst_eq_and_evalAt_y_eq_of_ringEquiv_uvCrossingModel506 below · cited by 3 · depth 16 - Completed node ring is a crossing model with order laws
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_ringEquiv_adicCompletion_nodeIntegersOver_uvCrossingModel_of_isMaximal408 below · cited by 8 · depth 16 - Complete Nakayama surjection onto a completed node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_surjective_mvPowerSeries_adicCompletion_nodeIntegersOver4 below · cited by 5 · depth 16 - Unit principle at a supersingular node of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_zpow_unit_principle_evalAt_y_of_ringEquiv_uvCrossingModel416 below · cited by 1 · depth 16 - Gauss order at the first end adds the varpi-exponent
ModularCurve.PlaceSpecialization.ProlongationTuple.gaussOrder_fst_end_ringEquiv_adicCompletion_eq_add_of_eq_nodeConst_pow_mul2 below · cited by 4 · depth 16 - Depth-zero Gauss order of varpiᵈux' in the crossing model
ModularCurve.PlaceSpecialization.ProlongationTuple.gaussOrder_snd_end_ringEquiv_adicCompletion_eq_add_of_eq_nodeConst_pow_mul2 below · cited by 4 · depth 16 - Second node residue equals the reduction of g(V)
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidueSnd_red_evalAt_of_orderLawFixed454 below · cited by 5 · depth 16 - Values of node-ring elements specialise to the first residue
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_and_hasValue_nodeResidueFst_of_hasValue1 below · cited by 6 · depth 16 - Node integrality and regular residues at supersingular places
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_nodeIntegers_and_residue_mem_of_mem_jIntegralClosure82 below · cited by 8 · depth 16 - Node residue values at supersingular places are K-rational
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_range_redRestrict_of_hasValue_nodeResidueFst392 below · cited by 6 · depth 16 - Fibre-product and saturation laws for the K-node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeIntegersOver_fibreProduct_of_orderLawFixed_of_range_redRestrict1,247 below · cited by 5 · depth 16 - Kernel of the first node residue is (varpi, x)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidueFst_eq_zero_iff_mem_span_of_orderLawFixed_of_range_redRestrict1,248 below · cited by 1 · depth 16 - Kernel of the second node residue is (varpi,y)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidueSnd_eq_zero_iff_mem_span_of_orderLawFixed_of_range_redRestrict1,248 below · cited by 1 · depth 16 - Saturation of the two node residue maps at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidue_saturated_of_orderLawFixed_of_isNoetherianRing814 below · cited by 2 · depth 16 - Rational node coordinates from a simple zero of ̄ g-̄ g^{q^2}
ModularCurve.PlaceSpecialization.ProlongationTuple.nonempty_nodeCoordinates_bot_of_ord_sub_pow_sq_eq_one1 below · cited by 5 · depth 16 - Node integers have residues of non-negative order at w
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_nodeResidue_nonneg_of_regularityLaw0 below · cited by 4 · depth 16 - Unit first residue at ordinary affine φ²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_eq_zero_of_forall_reduceFst_eq_ord_eq_zero270 below · cited by 2 · depth 16 - First residue regular at an ordinary fixed place, Atkin–Lehner case
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_nonneg_of_not_hasValue_modularUnit597 below · cited by 2 · depth 16 - Regularity of the second residue at φ v via a modular unit
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueSnd_nonneg_of_hasValue_modularUnit124 below · cited by 2 · depth 16 - Node coordinate minus its value is a uniformiser
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_y_sub_algebraMap_evalAt_eq_one_of_ringEquiv_uvCrossingModel501 below · cited by 1 · depth 16 - Side identity at ∞-side places where j(q^q) is regular
ModularCurve.PlaceSpecialization.ProlongationTuple.reduceSnd_eq_frobOnPlacesGeomLevel_reduceFst_of_isInftySide_of_ne_of_ord_jQFun_nonneg219 below · cited by 1 · depth 16 - One-sided first-copy regularity away from the supersingular places
ModularCurve.PlaceSpecialization.ProlongationTuple.regularityLawFst_oneSided269 below · cited by 2 · depth 16 - Depth-wise zero count at a node equals crossing-model count
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ord_eq_finsum_rank_mul_length_of_total_eq199 below · cited by 2 · depth 16 - Node total over a supersingular place via the crossing model
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ord_eq_finsum_rank_mul_length_total_of_nodeResidue_ne_zero429 below · cited by 6 · depth 16 - Total order over a node bounded by horizontal crossing-model count
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ord_le_finsum_rank_mul_length_total196 below · cited by 2 · depth 16 - Krull dimension at least two for the completed node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.two_le_ringKrullDim_adicCompletion_nodeIntegersOver2 below · cited by 6 · depth 16 - Valuation of values less than one detects the maximal ideal
ModularCurve.PlaceSpecialization.ProlongationTuple.valuation_evalAt_lt_one_iff_mem_maximalIdeal126 below · cited by 7 · depth 16 - Depth dual of a principal divisor lies in the Gram image
ModularCurve.PlaceSpecialization.depthDual_add_mem_range_gramMap_of_isPrincipal_widthChar1,985 below · cited by 2 · depth 16 - Inertia-fixed admissible representatives of inertia-invariant classes in J₀(q)
ModularCurve.PlaceSpecialization.exists_degZero_mk_eq_and_forall_inertia_smul_eq_and_isStrictType_or_redFst_mem_residueField1,743 below · cited by 1 · depth 16 - Level-one glued class of an inertial displacement is a node unit
ModularCurve.PlaceSpecialization.exists_goodRep_gluedMk_eq_nodeUnit_smul_single_sub_self_levelOne1,086 below · cited by 1 · depth 16 - Level-one prolongation tuples give prolongation pairs
ModularCurve.PlaceSpecialization.exists_levelOneProlongationPair_of_prolongationTuple0 below · cited by 10 · depth 16 - Lifting a uniformiser at a supersingular place, with pole control
ModularCurve.PlaceSpecialization.exists_ord_sub_pow_sq_eq_one_of_mem_ssPlaces766 below · cited by 4 · depth 16 - Level-one prolongation tuple with node coordinates and depth
ModularCurve.PlaceSpecialization.exists_prolongationTuple_nodeCoordinates_depthValueLaw_levelOne1,069 below · cited by 1 · depth 16 - Cross-power law for node depths along the level-ℓ Hecke roof
ModularCurve.PlaceSpecialization.exists_reduceFst_eq_and_yDepth_restrictAlong_heckeAlphaBar_pow_placeWidthChar_eq_of_ne_of_not_dvd1,949 below · cited by 1 · depth 16 - Moving lemma for degree-zero classes at level 1· q
ModularCurve.PlaceSpecialization.exists_rep_reduce_notMem_of_moving_of_disjoint_levelOne932 below · cited by 1 · depth 16 - Reduction of places commutes with both degeneracy legs
ModularCurve.PlaceSpecialization.exists_spRoof_pullbackAlong_restrictAlong_compat_of_exists_placeMap_fullC_v2236 below · cited by 1 · depth 16 - Counting ∞-side zeros via the q-order drop under reduction
ModularCurve.PlaceSpecialization.exists_sum_ord_isInftySide_eq_order_sub_order202 below · cited by 2 · depth 16 - First reduction at q commutes with T_ℓ, ℓ ≠ q
ModularCurve.PlaceSpecialization.mapDomain_reduceFst_heckeDivBar_eq_heckeDivFibre_mapDomain_reduceFst_of_ne278 below · cited by 1 · depth 16 - Integrality over ℚ̄[j,j_N] gives regularity at affine places
ModularCurve.PlaceSpecialization.mem_valuationSubring_of_isIntegral_of_sp_isAffineGeomPlace0 below · cited by 2 · depth 16 - Infinity-side places have the same first reduction as ∞̄
ModularCurve.PlaceSpecialization.redFst_eq_redFst_cuspInftyBar_of_isInftySide11 below · cited by 1 · depth 16 - Second level-one reduction at a place with integral j_q
ModularCurve.PlaceSpecialization.redSnd_eq_charLGeomPlaceOfPoint_of_ord_pos5 below · cited by 13 · depth 16 - Place specialisation commutes with both degeneracy maps
ModularCurve.PlaceSpecialization.restrictAlong_heckeAlphaC_sp_and_restrictAlong_heckeBetaC_sp_eq_sp_restrictAlong_of_isModel933 below · cited by 6 · depth 16 - Class-level map determined by the place-level map
ModularCurve.PlaceSpecialization.spPic0_eq_of_sp_eq0 below · cited by 2 · depth 16 - Specialisation of a place where j-b vanishes
ModularCurve.PlaceSpecialization.sp_eq_charLGeomPlaceOfPoint_of_ord_pos5 below · cited by 6 · depth 16 - Depth dictionary gives the depth divisor and second-branch degree
ModularCurve.PlaceSpecialization.sum_height_mul_multidegree_comp_eq_depthDiv_and_apply_inl_one_eq_degree_sndDiv_level144 below · cited by 1 · depth 16 - Node depth along the degeneracy tower is a ramification power
ModularCurve.PlaceSpecialization.yDepth_restrictAlong_towerInclBar_eq_yDepth_pow_ramificationIndexAlong_heckeAlphaC1,296 below · cited by 1 · depth 16 - Depth along the ℓ-substitution leg is a ramification power
ModularCurve.PlaceSpecialization.yDepth_restrictAlong_towerSubstBar_eq_yDepth_pow_ramificationIndexAlong_heckeBetaC985 below · cited by 1 · depth 16 - Good admissible representative of σ V-V at a wide node
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_goodRep_admissible_smul_single_sub_self_of_eq_zero_or_eq1,025 below · cited by 1 · depth 17 - Good admissible representative of σ V-V at a supersingular node
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_goodRep_admissible_smul_single_sub_self_of_ne_zero_of_ne913 below · cited by 1 · depth 17 - Integrality of j where j+μ j_q is integral at a place
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_ord_jFun_sub_pos_of_ord_jFun_add_mul_jqFun_sub_pos170 below · cited by 1 · depth 17 - Moving representatives of J₀(q)-classes off a finite place set
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_rep_redFst_redSnd_notMem_of_forall_notMem_ssPlaces591 below · cited by 1 · depth 17 - The two level-one prolongations have distinct valuation rings
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.integers_ne_integers124 below · cited by 2 · depth 17 - Value-indexed summed pencil law over the j-line
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.sum_filter_value_eq_ord_add_sum_roots268 below · cited by 1 · depth 17 - Value-filtered pencil divisor law for j+μ j_q, μ̄≠ 0
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.sum_filter_value_eq_sum_roots_add_pencil277 below · cited by 1 · depth 17 - Pinned strict reductions of a Jacobi-inversion divisor at level N
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.mapDomain_fstDiv_eq_and_mapDomain_sndDiv_eq_of_mk_spData_eq_zero_of_pin1,485 below · cited by 1 · depth 17 - Division by a disc parameter at a strict place
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.div_mem_smoothLocalRingFst_of_ord_residue_eq_one668 below · cited by 1 · depth 17 - Value law at a smooth point of the first copy, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_hasValue_of_mem_smoothLocalRingFst_levelOne496 below · cited by 3 · depth 17 - Common Gauss normalisation on both prolongations at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_smul_mem_integers_of_isGoodDiv_of_admissible_levelOne738 below · cited by 1 · depth 17 - A simple pole bound for the primitive part (level one)
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.neg_one_le_ord_residueFst_of_eq_one_add_mul_of_evalAt_ne_levelOne482 below · cited by 2 · depth 17 - At most a simple pole at a second-kind place
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.neg_one_le_ord_residueSnd_of_eq_one_add_mul_of_evalAt_ne_levelOne488 below · cited by 1 · depth 17 - Places over a supersingular node counted by crossing-model branches
ModularCurve.PlaceSpecialization.ProlongationTuple.card_eq_finsum_finrank_quotient_of_forall_iff_evalAt_eq_zero467 below · cited by 3 · depth 17 - Crossing exponent equals place width times e_K
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidth_mul_of_orderLawFixed_levelOne1,037 below · cited by 4 · depth 17 - Places over a node separated by node-ring values
ModularCurve.PlaceSpecialization.ProlongationTuple.eq_of_forall_evalAt_eq_of_reduceFst_eq128 below · cited by 5 · depth 17 - Uniqueness of the place with j of valuation γ^q
ModularCurve.PlaceSpecialization.ProlongationTuple.eq_of_restrictAlong_heckeBetaBar_eq_of_hasValuation_jFun_pow193 below · cited by 1 · depth 17 - Inertia-equivariant one-point mover on X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_ord_eq_one_orderTables_of_realisation_forall_inertia_smul_eq_of_isModel0 below · cited by 1 · depth 17 - Crossing presentation of the node ring at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_levelOne832 below · cited by 4 · depth 17 - Crossing presentation at an arbitrary supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_of_saturated743 below · cited by 1 · depth 17 - Hartogs for R₂-integral functions integral over ℚ̄[j]
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_evalBar_eq_mul_evalBar_of_mem_integersSnd_of_isIntegral184 below · cited by 1 · depth 17 - Local regularity of Gauss-integral functions integral over ℚ̄[j]
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_evalBar_eq_mul_evalBar_of_mem_integers_of_isIntegral178 below · cited by 1 · depth 17 - Inertia-fixed node coordinates at a supersingular place, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_nonempty_nodeCoordinates_levelOne135 below · cited by 1 · depth 17 - Integrality at both prolongations of K-integral functions on X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_isIntegral_adjoin_residue_and_forall_exists_hasValue_of_mem_jIntegralClosure422 below · cited by 3 · depth 17 - Substitution degeneracy and first residues: compatibility with q↦ q^ℓ
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_integersFst_towerSubstBar_and_coe_residueFst_eq0 below · cited by 2 · depth 17 - Prolongation residues are compatible along the degeneracy tower
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mem_integers_towerInclBar_and_coe_residue_eq75 below · cited by 2 · depth 17 - Node ring elements are A∩ K constants modulo non-units
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_not_isUnit_sub_nodeConst_of_evalAt_mem_range_redRestrict_levelOne_of_five_le820 below · cited by 3 · depth 17 - Coefficient rigidity of the crossing presentation
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_ringEquiv_adicCompletion_coeffSubring_forall_apply_nodeConst_eq_const5 below · cited by 5 · depth 17 - Rational depth window for places over a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_xDepth_pow_eq_valuation_pow_of_reduceFst_eq135 below · cited by 1 · depth 17 - Height-one vertical prime is the centre of a Gauss prolongation
ModularCurve.PlaceSpecialization.ProlongationTuple.forall_mem_iff_residueFst_eq_zero_or_forall_mem_iff_residueSnd_eq_zero_of_height_one202 below · cited by 2 · depth 17 - Valuations of j and j_q at a place over the cusps
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValuation_jQFun_eq_pow_or_eq_pow_of_kroneckerCongruence47 below · cited by 1 · depth 17 - Valuation of the q-transform of j on the infinity side
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValuation_jQFun_pow_of_isInftySide0 below · cited by 1 · depth 17 - At a supersingular node, res₂ t = 0 forces res₁ t(w)=0
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_residueFst_zero_of_residueSnd_eq_zero_of_mem_jIntegralClosure544 below · cited by 2 · depth 17 - Both branch residues vanish together at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_residueSnd_zero_iff_residueFst_of_mem_jIntegralClosure1,157 below · cited by 1 · depth 17 - Bijectivity of ι for a level-N prolongation tuple
ModularCurve.PlaceSpecialization.ProlongationTuple.iota_bijective_of_level152 below · cited by 1 · depth 17 - The K-rational node ring is integrally closed
ModularCurve.PlaceSpecialization.ProlongationTuple.isIntegrallyClosed_nodeIntegersOver0 below · cited by 6 · depth 17 - Node ring over K at a supersingular place is noetherian local
ModularCurve.PlaceSpecialization.ProlongationTuple.isLocalRing_and_isNoetherianRing_nodeIntegersOver_levelOne605 below · cited by 3 · depth 17 - Evaluation kernel at V: a non-maximal prime of the node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.ker_evalAt_isPrime_and_ne_maximalIdeal_and_nodeConst_notMem124 below · cited by 6 · depth 17 - Branch lengths in the crossing model equal lengths at a node prime
ModularCurve.PlaceSpecialization.ProlongationTuple.length_localizedModule_quotient_map_eq_of_mem_minimalPrimes468 below · cited by 2 · depth 17 - First-sheet divisor law at ordinary affine φ²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceFst_filter_sheetOne_eq_ord_residueFst_residueField1,889 below · cited by 2 · depth 17 - Second-sheet divisor law at ordinary affine φ²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceSnd_filter_sheetTwo_eq_ord_residueSnd_residueField1,890 below · cited by 1 · depth 17 - Crossing-model constant π forces varpi to generate mathfrak m_{A∩ K}
ModularCurve.PlaceSpecialization.ProlongationTuple.maximalIdeal_coeffSubring_eq_span_of_ringEquiv_apply_nodeConst_eq_const5 below · cited by 5 · depth 17 - Node-ring values reduce to the second residue at Frob· w
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_and_hasValue_nodeResidueSnd_of_hasValue1 below · cited by 3 · depth 17 - Regularity of the first residue at affine φ²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_nonneg_of_forall_reduceFst_eq_ord_nonneg_of_hasValue_atkinLehnerBar_modularUnit125 below · cited by 1 · depth 17 - Regularity of both residues at a φ²-fixed affine place
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residues_nonneg_of_forall_reduceFst_eq_ord_nonneg0 below · cited by 4 · depth 17 - Evaluations along one branch of a node multiply to a norm
ModularCurve.PlaceSpecialization.ProlongationTuple.prod_evalAt_eq_norm_quotient_of_forall_iff_exists_ker_eq179 below · cited by 1 · depth 17 - Section-pair bounds from the regularity law, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.sectionPair_bounds_of_regularityLaw_of_isModel_levelOne732 below · cited by 1 · depth 17 - Finiteness of places with prescribed node-ring vanishing ideal
ModularCurve.PlaceSpecialization.ProlongationTuple.setOf_reduceFst_eq_and_forall_mem_iff_evalAt_eq_zero_finite124 below · cited by 2 · depth 17 - Places over a supersingular node bounded by horizontal primes
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_ord_le_finsum_rank_mul_length_of_ringEquiv_uvCrossingModel195 below · cited by 1 · depth 17 - Order sum at a node place: length times branch rank bound
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_toNat_ord_le_length_mul_finsum_finrank_of_forall_mem_iff_evalAt_eq_zero183 below · cited by 3 · depth 17 - Order at V equals length at the node ring's prime q
ModularCurve.PlaceSpecialization.ProlongationTuple.toNat_ord_eq_length_localizedModule_quotient_of_forall_mem_iff_evalAt_eq_zero126 below · cited by 6 · depth 17 - Principal divisors have trivial depth component class at level one
ModularCurve.PlaceSpecialization.componentGroupProj_depthDual_add_degree_sndDiv_smul_eq_zero_of_div_levelOne_of_five_le1,304 below · cited by 1 · depth 17 - Vanishing of the depth functional on good divisors
ModularCurve.PlaceSpecialization.depthDual_add_degree_sndDiv_smul_eq_branchDegrees_snd_smul_of_isGoodDivisor0 below · cited by 1 · depth 17 - Uniqueness of the ∞-side point over a given j-value
ModularCurve.PlaceSpecialization.eq_of_isInftySide_of_hasValue_jFun79 below · cited by 2 · depth 17 - Inertia-fixed classes represented by admissible divisors, residue-field case
ModularCurve.PlaceSpecialization.exists_degZero_mk_eq_mk_and_forall_inertia_smul_eq_and_isStrictType_or_redFst_mem_of_forall_inertia_smul_coe_eq_residueField1,742 below · cited by 1 · depth 17 - Lifting a Frobenius-fixed regular function on the special fibre
ModularCurve.PlaceSpecialization.exists_lift_of_coeff_pow_eq_of_forall_isAffineGeomPlace_mem753 below · cited by 1 · depth 17 - A place with an A-value of j has one of j_q
ModularCurve.PlaceSpecialization.exists_ord_jqFun_sub_pos_of_ord_jFun_sub_pos51 below · cited by 5 · depth 17 - Pole clearing for functions regular on a strict first-kind disc
ModularCurve.PlaceSpecialization.exists_red_eval_ne_zero_and_isIntegral_mul_evalBar_of_forall_isStrictFst127 below · cited by 1 · depth 17 - Clearing poles on a strict second-kind residue disc
ModularCurve.PlaceSpecialization.exists_red_eval_ne_zero_and_isIntegral_mul_evalBar_of_forall_isStrictSnd127 below · cited by 1 · depth 17 - Completed A∩ K maps to the valuation ring of widehatℚ̄_A
ModularCurve.PlaceSpecialization.exists_ringHom_adicCompletion_coeffSubring_valuationInteger5 below · cited by 2 · depth 17 - Sheet dichotomy for the modular unit at ordinary places
ModularCurve.PlaceSpecialization.hasValue_modularUnit_or_atkinLehnerBar_of_reduceFst_fixed_ordinary589 below · cited by 1 · depth 17 - Non-vanishing of the pencil j + c j_q - a on X₀(q)
ModularCurve.PlaceSpecialization.jFun_add_C_mul_jqFun_sub_algebraMap_ne_zero117 below · cited by 1 · depth 17 - Specialisation commutes with degeneracy at fixed affine places
ModularCurve.PlaceSpecialization.sp_restrictAlong_eq_restrictAlong_sp_of_isModel_of_fixed_of_isAffineGeomPlace940 below · cited by 2 · depth 17 - Node depth along the ℓ-degeneracy leg is a ramified power
ModularCurve.PlaceSpecialization.yDepth_restrictAlong_towerInclBar_eq_yDepth_pow_ramificationIndexAlong_heckeAlphaC_of_prime1,308 below · cited by 1 · depth 17 - Node depth along the substitution degeneracy leg at ℓ≠ q
ModularCurve.PlaceSpecialization.yDepth_restrictAlong_towerSubstBar_eq_yDepth_pow_ramificationIndexAlong_heckeBetaC_of_prime985 below · cited by 1 · depth 17 - A simple pole bound at a strict-type-one reduction
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.neg_one_le_ord_residue_of_eq_one_add_mul478 below · cited by 1 · depth 18 - Inertia-stable representatives of J₀(q)^{I_A} avoiding a finite place set
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_inertiaStable_rep_redFst_redSnd_notMem_of_forall_notMem_ssPlaces702 below · cited by 1 · depth 18 - Residues of j-c along a level-one prolongation pair
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_integers_residue_jFun_sub_algebraMap82 below · cited by 2 · depth 18 - Nonvanishing first residue of j-j₀ on X₀(q)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_integers_residue_jFun_sub_ne_zero86 below · cited by 7 · depth 18 - The parameter j-j₀ has nonzero second residue
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_integers_snd_residue_jFun_sub_ne_zero86 below · cited by 3 · depth 18 - Exact branch orders for a lift keyed on a split datum
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_riemannRochSpace_ord_residue_eq_neg_of_splitDatum534 below · cited by 2 · depth 18 - One-point moving lemma on X₀(q) at level one
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_ord_eq_one_forall_redFst_redSnd_notMem590 below · cited by 1 · depth 18 - Tube equation for the inertial displacement σ V-V
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_tubeEquation_smul_sub_self452 below · cited by 2 · depth 18 - Tube equation for the inertial displacement on an annulus
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_tubeEquation_smul_sub_self_of_annulus4 below · cited by 3 · depth 18 - Two prolongations exhaust the pencil line over X₀(q)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.integers_eq_or_eq_of_forall_mem_iff_pencil242 below · cited by 1 · depth 18 - Exact branch orders make E+divG effective and good
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.isGoodDivisor_add_of_ord_residue_eq_neg182 below · cited by 2 · depth 18 - Node value law at level one over an algebraically closed field
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.nodeValueLaw536 below · cited by 2 · depth 18 - Norm descent to the Gauss order and residue splitting
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.norm_mem_and_residue_norm_eq_core179 below · cited by 2 · depth 18 - Residue of j-j₀ is a uniformiser at `redFst Q`
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.ord_redFst_residue_jFun_sub_eq_one130 below · cited by 4 · depth 18 - Explicit split datum at one supersingular node, level one
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.splitDatum_of_forall_centred_ord_eq632 below · cited by 2 · depth 18 - Fibre-sum law for f over a value c₀ on the base line
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.sum_filter_value_eq_sum_roots_add223 below · cited by 2 · depth 18 - Chord bounds and coupled rigidity along supersingular annuli of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_endOrder_ineq_and_coupledScalings_hasValue_of_isTwistOf_of_mk_spData_eq_zero_of_inertiaStable1,426 below · cited by 1 · depth 18 - Function with uniformising residue is a parameter of the disc
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.ord_eq_one_of_ord_residue_eq_one667 below · cited by 1 · depth 18 - Places on one branch bounded by that branch's W-rank
ModularCurve.PlaceSpecialization.ProlongationTuple.card_le_finrank_quotient_of_forall_ker_eq7 below · cited by 2 · depth 18 - Places over a supersingular node bounded by total branch rank
ModularCurve.PlaceSpecialization.ProlongationTuple.card_le_finsum_finrank_quotient_map_of_forall_mem_iff_evalAt_eq_zero137 below · cited by 1 · depth 18 - One-sided cusp law at infinity, auxiliary level one
ModularCurve.PlaceSpecialization.ProlongationTuple.cuspLawInfty_oneSided_levelOne495 below · cited by 6 · depth 18 - One-sided cusp law at zero, auxiliary level one
ModularCurve.PlaceSpecialization.ProlongationTuple.cuspLawZero_oneSided_levelOne495 below · cited by 6 · depth 18 - One-sided first divisor law at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.divisorLawFst_oneSided_levelOne495 below · cited by 7 · depth 18 - One-sided second divisor law at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.divisorLawSnd_oneSided_levelOne495 below · cited by 7 · depth 18 - Common unit with simple pole over an ordinary Frobenius-fixed place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_of_reduceFst_fixed_ordinary_residueField1,852 below · cited by 1 · depth 18 - Crossing presentation of the K-node ring at a supersingular place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_crossingPresentation_nodeIntegersOver_of_ne_zero_of_ne_1728556 below · cited by 1 · depth 18 - Level-one node ring has fraction field fieldOver(q,K)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_mul_eq_of_mem_fieldOver_nodeIntegersOver_levelOne79 below · cited by 1 · depth 18 - Kronecker pair gives node coordinates at generic supersingular nodes
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_nodeCoordinates_levelOneNodeCoord542 below · cited by 3 · depth 18 - Evaluation at a place extends to the completed node ring
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_ringHom_adicCompletion_nodeIntegersOver_comp_eq_evalAt134 below · cited by 5 · depth 18 - First-sheet divisor law at ordinary φ²-fixed places
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceFst_filter_sheetOne_eq_ord_residueFst_levelOne_univ1,009 below · cited by 3 · depth 18 - Second-sheet divisor law at ordinary φ²-fixed places, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceSnd_filter_sheetTwo_eq_ord_residueSnd_levelOne1,012 below · cited by 1 · depth 18 - Level-one node integers equal the localised plane model
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_modularLocalizedAtPoint_iff_exists_mem_nodeIntegersOver604 below · cited by 6 · depth 18 - Descent of node values of first residues at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.mem_range_redRestrict_of_hasValue_nodeResidueFst_levelOne_of_five_le595 below · cited by 1 · depth 18 - One-sided second-copy regularity at φ²-fixed ordinary places
ModularCurve.PlaceSpecialization.ProlongationTuple.regularityLawSnd_oneSided273 below · cited by 1 · depth 18 - Per-depth zero count at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.sum_toNat_ord_le_length_mul_finsum_finrank_of_forall_mem_iff_evalAt_eq_zero_of_xDepth_pow_eq182 below · cited by 1 · depth 18 - Node maximal ideal generator has valuation below one
ModularCurve.PlaceSpecialization.ProlongationTuple.valuation_coe_lt_one_of_maximalIdeal_eq_span0 below · cited by 9 · depth 18 - Inertia-fixed strict points of both types with distinct reductions
ModularCurve.PlaceSpecialization.exists_families_isStrictTypeOne_isStrictTypeTwo_notMem_forall_inertia_smul_eq442 below · cited by 3 · depth 18 - Inertia-stable divisors modulo fixed strict and glue-trivial divisors
ModularCurve.PlaceSpecialization.exists_fixedStrict_kernelGood_principal1,690 below · cited by 1 · depth 18 - The j-function as an unramified uniformiser lift at a supersingular place
ModularCurve.PlaceSpecialization.exists_ord_sub_pow_sq_eq_one_of_ord_jqModC0 below · cited by 1 · depth 18 - Degree 2q of the pencil j + c j_q on X₀(q)
ModularCurve.PlaceSpecialization.finiteDimensional_and_finrank_adjoin_jFun_add_C_mul_jqFun199 below · cited by 2 · depth 18 - Two-level degeneracy glue for glued specialisations at q'
ModularCurve.PlaceSpecialization.gluedSpecialization_twoLevel_degeneracyGlue_of_isModel_placeWidthChar_restrictAlong1,917 below · cited by 2 · depth 18 - Sheet separation at ordinary places: u or w_q u has unit value
ModularCurve.PlaceSpecialization.hasValue_modularUnit_or_frickeInvolutionBar_of_reduceFst_fixed_ordinary563 below · cited by 2 · depth 18 - Congruent coordinate data force equal strict first reductions
ModularCurve.PlaceSpecialization.isStrictFst_and_reduceFst_eq_of_ord_sub_pos0 below · cited by 1 · depth 18 - Places congruent to a strict second-kind place
ModularCurve.PlaceSpecialization.isStrictSnd_and_reduceSnd_eq_of_ord_sub_pos0 below · cited by 1 · depth 18 - Strict type dichotomy off the Frobenius-square-fixed locus
ModularCurve.PlaceSpecialization.isStrictTypeOne_or_isStrictTypeTwo0 below · cited by 3 · depth 18 - Strict type one or two iff φ² moves `redFst`
ModularCurve.PlaceSpecialization.isStrictTypeOne_or_isStrictTypeTwo_iff_ne0 below · cited by 6 · depth 18 - Level-one gluing datum vanishes at supersingular node places
ModularCurve.PlaceSpecialization.levelOneGlueData_apply_frobNodePair_eq_zero26 below · cited by 2 · depth 18 - Strict type one and strict type two are exclusive
ModularCurve.PlaceSpecialization.not_isStrictTypeOne_and_isStrictTypeTwo0 below · cited by 3 · depth 18 - Value-fibre criterion for the first reduction red₁
ModularCurve.PlaceSpecialization.redFst_eq_charLGeomPlaceOfPoint_iff13 below · cited by 15 · depth 18 - Places above a supersingular j-invariant reduce to supersingular places
ModularCurve.PlaceSpecialization.reduceFst_mem_ssPlaces_of_restrictAlong_towerInclBar_eq147 below · cited by 1 · depth 18 - Ring homomorphism given by t-expansion at a smooth point
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.exists_ringHom_tExpansion473 below · cited by 1 · depth 19 - Simple zero of j-j₀ at a strict-type-one place
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.ord_jFun_sub_eq_one_of_isStrictTypeOne151 below · cited by 4 · depth 19 - Order ≥ m for the first residue at a reduced place
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.residue_eq_zero_or_le_ord_residue_of_tExpansion_red_eq_zero470 below · cited by 1 · depth 19 - Constants from A lie in `smoothLocalRingFst`
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.algebraMap_mem_smoothLocalRingFst0 below · cited by 3 · depth 19 - R₁-integrality of the modular unit with coefficientwise residue
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.coeffEmb_modularUnitSeries_mem_integersFst32 below · cited by 9 · depth 19 - Vanishing of the modular unit's second residue at level one
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.coeffEmb_modularUnitSeries_mem_integersSnd_residue_eq_zero110 below · cited by 4 · depth 19 - One-sided cusp law at ∞ for the first prolongation
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.cuspLawInfty_oneSided214 below · cited by 11 · depth 19 - One-sided cusp law at the zero cusp
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.cuspLawZero_oneSided225 below · cited by 6 · depth 19 - One-sided first-branch divisor law off the φ²-fixed places
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.divisorLawFst_oneSided460 below · cited by 10 · depth 19 - One-sided divisor law for the second level-one prolongation
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.divisorLawSnd_oneSided461 below · cited by 4 · depth 19 - Lifting node-compatible level-one pairs into L(D)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_riemannRochSpace_residue_eq_of_regular_of_nonneg547 below · cited by 2 · depth 19 - Inertia-equivariant one-point moving lemma on X₀(q)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_ord_eq_one_forall_redFst_redSnd_notMem_forall_inertia_smul_eq_residueField693 below · cited by 1 · depth 19 - Branch orders and glued twisted values at a supersingular node
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.le_ord_residue_and_exists_hasValue_of_mul553 below · cited by 1 · depth 19 - R₁-units with no zeros over v are units at v
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.mem_smoothLocalRingFst_and_inv_mem_of_forall_ord_eq_zero0 below · cited by 1 · depth 19 - Constants reduce correctly under the first residue map
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residue_algebraMap_eq_red0 below · cited by 4 · depth 19 - Nonvanishing residue of the modular unit at the first prolongation
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residue_coeffEmb_modularUnitSeries_ne_zero33 below · cited by 6 · depth 19 - Chord bound and rigidity for twisted annulus profiles
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_chord_le_endOrders_and_rigid_of_isTwistOf_of_inertiaStable1,339 below · cited by 1 · depth 19 - A node package over a larger coefficient field
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumLevel.exists_nodePackage_over_of_orderLawFixed1,229 below · cited by 1 · depth 19 - Inertia-stable twisted divisor: fixed strict part plus glue-trivial part
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_fixed_strict_add_kernelGood_of_isTwistOf_of_inertiaStable1,636 below · cited by 1 · depth 19 - Twisting an inertia-stable divisor by a fixed strict pair
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_int_fixed_strict_pair_isTwistOf_sub_of_inertiaStable458 below · cited by 1 · depth 19 - Places over a supersingular node bounded by branch ranks
ModularCurve.PlaceSpecialization.ProlongationTuple.card_le_finsum_finrank_quotient_map_of_xDepth_pow_eq136 below · cited by 1 · depth 19 - Existence of a full annulus datum at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_annulusDatumQ_laws_levelOne928 below · cited by 1 · depth 19 - Common unit with a simple pole at an ordinary fixed place
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_commonUnit_pole_of_reduceFst_fixed_ordinary_levelOne_univ936 below · cited by 2 · depth 19 - Matching node residues at pairs of places from local node rings
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_residueFst_and_residueSnd_of_mem_nodePairsOfPlaces_of_nodePack77 below · cited by 1 · depth 19 - Inertia-stable node telescoping identity at a supersingular crossing
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_residueFst_div_pow_and_residueSnd_div_pow_and_div_eq_angFactor_of_inertiaStable584 below · cited by 2 · depth 19 - Node pack at a supersingular place with no order-one j-difference
ModularCurve.PlaceSpecialization.ProlongationTuple.nodePack_residueField_of_not_ord_sub_pow_sq_eq_one_or1,764 below · cited by 1 · depth 19 - Node pack at supersingular places over κ_A
ModularCurve.PlaceSpecialization.ProlongationTuple.nodePack_residueField_of_ord_sub_pow_sq_eq_one_or1,764 below · cited by 1 · depth 19 - Regularity of the first residue at a second-sheet place
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_nonneg_of_forall_reduceFst_eq_ord_nonneg_of_hasValue_frickeInvolutionBar_modularUnit_levelOne663 below · cited by 1 · depth 19 - Regularity of the second residue at φ v
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueSnd_nonneg_of_forall_reduceFst_eq_ord_nonneg_of_hasValue_modularUnit_levelOne662 below · cited by 1 · depth 19 - Arbitrarily many strict type one and type two places avoiding B
ModularCurve.PlaceSpecialization.exists_families_isStrictTypeOne_isStrictTypeTwo_notMem223 below · cited by 2 · depth 19 - Good representatives whose support avoids a finite set of places
ModularCurve.PlaceSpecialization.exists_isGoodDiv_mem_admissible_mk_eq_reduce_notMem_nodePairsOfPlaces1,719 below · cited by 1 · depth 19 - Sheet separation at ordinary places via the modular unit
ModularCurve.PlaceSpecialization.hasValue_modularUnit_or_frickeInvolutionBar_of_reduceFst_fixed_ordinary_univ563 below · cited by 1 · depth 19 - Degeneracy pushforward of good divisors and glue data
ModularCurve.PlaceSpecialization.isGoodDiv_pushforwardAlong_and_glueData_eq_of_isModel944 below · cited by 1 · depth 19 - Fricke translation exchanges the two reductions at level one
ModularCurve.PlaceSpecialization.redFst_frickeInvolutionBar_smul84 below · cited by 5 · depth 19 - Fricke translation exchanges the two reductions on X₀(q)
ModularCurve.PlaceSpecialization.redSnd_frickeInvolutionBar_smul84 below · cited by 1 · depth 19 - Places over a level-one supersingular node are centred at (a,a^q)
ModularCurve.PlaceSpecialization.reduceFst_eq_iff_centred_levelOne59 below · cited by 5 · depth 19 - Infinitely many strict type one places with distinct first reductions
ModularCurve.PlaceSpecialization.IsStrictTypeOne.exists_family_redFst_injective222 below · cited by 1 · depth 20 - Integral values at strict type-one places of the smooth locus
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.exists_hasValue_of_mem_smoothLocalRingFst462 below · cited by 3 · depth 20 - A-integral t-expansion at a smooth point of the level-q model
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.exists_tExpansion_of_mem_smoothLocalRingFst472 below · cited by 1 · depth 20 - Common unit with a simple zero at a prescribed place
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_commonUnit_ord_eq_one_of_mem_levelOne593 below · cited by 1 · depth 20 - Lifting residues integral over the ̄ u⁻¹-chart
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_isIntegral_and_residue_eq_of_isIntegral_adjoin_residue_modularUnitSeries_inv353 below · cited by 4 · depth 20 - Inertia-equivariant lift of a node-compatible residue pair
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_mem_riemannRochSpace_residue_eq_forall_inertia_smul_eq_of_regular_of_nonneg557 below · cited by 1 · depth 20 - Unit values of the modular unit at strict-type-one places
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_red_ne_zero_and_coeffEmb_modularUnitSeries_inv_sub_algebraMap_mem_nonunits_of_isStrictTypeOne458 below · cited by 1 · depth 20 - Residue of the modular unit has inverse of degree q-1
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.finrank_adjoin_residue_coeffEmb_modularUnitSeries_inv223 below · cited by 4 · depth 20 - Reductions of a bi-integral section of L(D), D good and effective
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residuePair_mem_riemannRochSpace_of_isGoodDivisor153 below · cited by 2 · depth 20 - Inertia-stable divisors have integral circle degrees and depth moments
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.den_circleDeg_eq_one_and_den_depthMoment_eq_one_of_inertiaStable159 below · cited by 3 · depth 20 - Glued classes from inertia-fixed strict divisors with zero twist
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_fixed_strict_mk_glueData_eq482 below · cited by 1 · depth 20 - Strict representative of an inertia-stable, glued-trivial divisor class
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_isGoodDiv_pic0Mk_eq_of_isTwistOf_of_mk_spData_eq_zero_of_inertiaStable1,623 below · cited by 1 · depth 20 - Subtracting a strict twist-zero divisor preserves the twist
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.isTwistOf_sub_and_spData_sub_eq_of_forall_isStrict162 below · cited by 1 · depth 20 - Admissibility of the twisted gluing datum at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.spData_mem_admissible385 below · cited by 1 · depth 20 - Rational node depth: cleared law and inertia invariance
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_depthQ_cleared_law_and_forall_inertia_smul_eq_levelOne467 below · cited by 1 · depth 20 - Inertia-fixed node presentations at all supersingular places, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_inertiaFixed_nodeCoordinates_presentation_levelOne_of_orderLawFixed902 below · cited by 1 · depth 20 - Node unit and normalising constants as values at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_nodeUnit_lam_mu_hasValue_levelOne2 below · cited by 1 · depth 20 - Branch-norm factorisation of the y-product at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_prod_evalAt_y_eq_pow_mul_prod_units_of_forall_iff_evalAt_eq_zero180 below · cited by 1 · depth 20 - Reading first-branch leading coefficients in the crossing model
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidueFst_div_nodeResidueFst_y_pow_of_sub_mul_V_pow_mem_of_hasValue1 below · cited by 1 · depth 20 - Second-branch value of f/x^m at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.hasValue_nodeResidueSnd_div_nodeResidueSnd_x_pow_of_sub_mul_U_pow_mem_of_hasValue1 below · cited by 1 · depth 20 - Node integers at a supersingular place are local and noetherian
ModularCurve.PlaceSpecialization.ProlongationTuple.isLocalRing_and_isNoetherianRing_nodeIntegersOver_of_sp_eq_spPlace870 below · cited by 2 · depth 20 - Saturation of the two node residues at a supersingular place
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidue_saturated_of_sp_eq_spPlace_residueField1,762 below · cited by 2 · depth 20 - Admissible scalings of a function have fixed valuation
ModularCurve.PlaceSpecialization.ProlongationTuple.smul_mem_integers_and_residue_ne_zero_iff_valuation_eq0 below · cited by 2 · depth 20 - Twisted chord bounds and rigidity at a supersingular crossing
ModularCurve.PlaceSpecialization.ProlongationTuple.valuation_pow_le_mul_prod_and_rigid_of_twist1,331 below · cited by 1 · depth 20 - Prime-to-q inertia-invariant classes with equal glued specialisation coincide
ModularCurve.PlaceSpecialization.eq_of_primeToTorsion_of_componentMap_eq_zero_of_gluedSpecialization_eq1,054 below · cited by 1 · depth 20 - Gauss-normalisable basis of L(D) for a good divisor
ModularCurve.PlaceSpecialization.exists_basis_riemannRochSpace_coeffMap_eq_smul_of_isGoodDivisor134 below · cited by 2 · depth 20 - Moving good classes off finite place sets: genus-zero case
ModularCurve.PlaceSpecialization.exists_isGoodDiv_mem_admissible_mk_eq_reduce_notMem_nodePairsOfPlaces_of_not_genusFF_pos1,713 below · cited by 1 · depth 20 - Strict first-kind places reduce onto the Frobenius graph
ModularCurve.PlaceSpecialization.exists_ord_jFun_sub_pos_and_red_eq_pow_of_isStrictFst59 below · cited by 1 · depth 20 - At cuspidal places, finite A-values of u⁻¹ reduce to zero
ModularCurve.PlaceSpecialization.red_eq_zero_of_isCuspidal_of_coeffEmb_modularUnitSeries_inv_sub_algebraMap_mem_nonunits458 below · cited by 1 · depth 20 - Place specialization at level one forces red surjective
ModularCurve.PlaceSpecialization.red_surjective10 below · cited by 7 · depth 20 - Specialization commutes with degeneracy restriction off Frobenius-fixed places
ModularCurve.PlaceSpecialization.sp_restrictAlong_eq_restrictAlong_sp_of_isModel940 below · cited by 1 · depth 20 - Division by j-j₀ in the smooth local ring at a type-one point
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.IsModel.div_jFun_sub_mem_smoothLocalRingFst152 below · cited by 1 · depth 21 - Order 1-q of the reduced modular unit at level one
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.order_residue_coeffEmb_modularUnitSeries33 below · cited by 1 · depth 21 - End-order bounds and coupled scalings for an inertia-stable twist
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_endOrder_ineq_and_coupledScalings_hasValue_of_isTwistOf_of_mk_spData_eq_zero_of_inertiaStable1,622 below · cited by 1 · depth 21 - Gauss prolongations exhaust the valuation rings of L
ModularCurve.PlaceSpecialization.ProlongationTuple.eq_comap_integers_or_of_forall_exists_eq_pow_mul124 below · cited by 2 · depth 21 - Component charts and attached annuli at a supersingular crossing
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_componentCharts_annuli_isAttached_of_crossingPresentation1,325 below · cited by 1 · depth 21 - Node coordinates at wide supersingular crossings, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_nodeCoordinates_nodeEquation_jWidth_of_eq_zero_or_eq_1728_levelOne832 below · cited by 1 · depth 21 - Node ring at a supersingular place is a localisation
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeIntegersOver_isLocalRing_exists_isMaximal_of_regularityLaw1,199 below · cited by 2 · depth 21 - Vanishing of both node residues characterises the ideal (varpi)
ModularCurve.PlaceSpecialization.ProlongationTuple.nodeResidue_eq_zero_and_eq_zero_iff_mem_span_of_orderLawFixed1,250 below · cited by 1 · depth 21 - Node values as a product of branch norms
ModularCurve.PlaceSpecialization.ProlongationTuple.prod_evalAt_eq_prod_norm_quotient_of_forall_iff_evalAt_eq_zero179 below · cited by 1 · depth 21 - One-sided first-side regularity law at level one
ModularCurve.PlaceSpecialization.ProlongationTuple.regularityLawFst_oneSided_levelOne514 below · cited by 3 · depth 21 - Second-side one-sided regularity law, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.regularityLawSnd_oneSided_levelOne517 below · cited by 3 · depth 21 - Hecke transport of node units on the glued fibre at q
ModularCurve.PlaceSpecialization.exists_matrix_gluedSpecialization_nodeUnit_heckeGen_of_ne_of_isModel1,780 below · cited by 1 · depth 21 - Coefficient-field enlargement of a supersingular node package
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatum.exists_nodePackage_over835 below · cited by 3 · depth 22 - Chord bound, rigidity and scaling increment for twisted annulus data
ModularCurve.PlaceSpecialization.ProlongationTuple.AnnulusDatumQ.exists_chord_le_endOrders_and_rigid_of_isTwistOf_of_inertiaStable1,595 below · cited by 1 · depth 22 - First component chart of the special fibre of X₀(Nq)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_componentChart_fst_of_isModel675 below · cited by 1 · depth 22 - Component chart for the second copy of X₀(N)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_componentChart_snd_of_isModel677 below · cited by 1 · depth 22 - Attachment of the node annulus to the first component chart
ModularCurve.PlaceSpecialization.ProlongationTuple.isAttached_fst_of_ringEquiv_uvCrossingModel_of_regularityLaw388 below · cited by 1 · depth 22 - Attachment of the opposite node annulus to the second chart
ModularCurve.PlaceSpecialization.ProlongationTuple.isAttached_snd_of_ringEquiv_uvCrossingModel_of_regularityLaw388 below · cited by 1 · depth 22 - Modular unit residue is a unit at non-supersingular affine places
ModularCurve.PlaceSpecialization.ProlongationTuple.ord_residueFst_eq_zero_of_coe_eq_modularUnitSeries_of_notMem_ssPlaces_levelOne508 below · cited by 2 · depth 22 - Value dictionary forces reduction to w or its Frobenius translate
ModularCurve.PlaceSpecialization.ProlongationTuple.reduceFst_eq_or_eq_arithFrobC_smul_of_forall_hasValue_iff1,185 below · cited by 1 · depth 22 - Genus-zero transport of glued specialisation data along a node-stable automorphism
ModularCurve.PlaceSpecialization.exists_isNodeStable_isGoodClass_iff_isGluedSpecialization_glueMap_of_not_genusFF_pos1,681 below · cited by 1 · depth 22 - Integral node matrix for T_ℓ, ℓ≠ q, on glued specialisations
ModularCurve.PlaceSpecialization.exists_matrix_gluedSpecialization_nodeUnit_heckeGen_of_ne_of_isModel_of_prolongation_of_regularityLaw_nodeValueLaw986 below · cited by 1 · depth 22 - Frobenius equivariance of both level-N reductions of places
ModularCurve.PlaceSpecialization.reduceFst_and_reduceSnd_arithmeticGalois_smul_of_isFrobeniusAt_pow7 below · cited by 2 · depth 22 - Fibre coordinates on the first component chart of X₀(p)
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFst_exists_fibreCoord223 below · cited by 1 · depth 23 - Value law at smooth points of the second prolongation
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_hasValue_of_mem_smoothLocalRingSnd670 below · cited by 3 · depth 23 - Twisted chord bounds and rigidity at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.valuation_pow_le_mul_prod_and_rigid_of_twist_levelOne1,587 below · cited by 1 · depth 23 - Chart laws for the sheet cut out by the modular unit
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFstLaws_sheetOne_of_isModel1,016 below · cited by 1 · depth 24 - Chart supply from the chart laws for model pairs
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFstSupply_of_isModel491 below · cited by 3 · depth 24 - q-expansion unit criterion for the ∞̄-chart
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFst_mem_integers_residue_ne_zero_of_qCoeff1 below · cited by 38 · depth 24 - Residues of j and j(qᵖ) on the first chart
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFst_residue_jFun_jqFun86 below · cited by 14 · depth 24 - Integral q-expansions and coefficientwise residue in the first chart
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.chartFst_residue_of_forall_coeff_mem0 below · cited by 16 · depth 24 - R₁-integrality as a quotient of A-integral Laurent series
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.mem_integersFst_iff_exists_quotient125 below · cited by 14 · depth 24 - q-integrality of coefficients of the Fricke transform
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.padicValRat_coeff_frickeInvolutionFull_nonneg695 below · cited by 2 · depth 24 - Nonvanishing residue at the first prolongation as a quotient of primitive expansions
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residueFst_ne_zero_iff_exists_quotient126 below · cited by 13 · depth 24 - Component charts and attached annuli at a supersingular node
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_componentCharts_annuli_isAttached_of_isModel1,585 below · cited by 1 · depth 24 - Fricke involution exchanges strict types one and two
ModularCurve.PlaceSpecialization.isStrictTypeOne_frickeInvolutionBar_smul_iff84 below · cited by 2 · depth 24 - Fricke involution exchanges strict types one and two
ModularCurve.PlaceSpecialization.isStrictTypeTwo_frickeInvolutionBar_smul_iff84 below · cited by 2 · depth 24 - Fricke separation and covering of the ordinary u-sheet
ModularCurve.PlaceSpecialization.sheetOne_frickeInvolutionBar_separated_and_covering567 below · cited by 1 · depth 24 - A-integrality of q-expansion coefficients of R₁-integral functions
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.coeff_mem_of_mem_integersFst_of_forall_ord_neg119 below · cited by 1 · depth 25 - Integral values of functions on the ∞-side
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.exists_hasValue_of_isInftySide226 below · cited by 1 · depth 25 - Integrality at the second prolongation for functions with poles only at ∞̄
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.mem_integersSnd_of_mem_integersFst_of_forall_ord_nonneg683 below · cited by 1 · depth 25 - Residue pair of a Riemann–Roch section with cuspidal support
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.residuePair_mem_riemannRochSpace_of_isGoodDivisor_or_eq_cuspInftyBar226 below · cited by 1 · depth 25 - Values at strict second-kind places of a level-one model
ModularCurve.PlaceSpecialization.ProlongationTuple.IsModel.exists_hasValue_of_mem_smoothLocalRingSnd_levelOne496 below · cited by 2 · depth 25 - Component charts and a doubly attached annulus at j∈{0,1728}
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_componentCharts_annuli_isAttached_of_isModel_of_eq_zero_or_eq_ofNat17281,584 below · cited by 1 · depth 25 - Values at first-sheet ordinary places reduce (level one)
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_hasValue_residueFst_of_sheetOne_levelOne1,011 below · cited by 1 · depth 25 - The residue function-field map ι is bijective
ModularCurve.PlaceSpecialization.ProlongationTuple.iota_bijective12 below · cited by 2 · depth 25 - First-sheet divisor law at ordinary fixed places, level one
ModularCurve.PlaceSpecialization.ProlongationTuple.mapDomain_reduceFst_filter_sheetOne_eq_ord_residueFst_levelOne1,010 below · cited by 2 · depth 25 - Bijectivity of the residual constant map ̄red
ModularCurve.PlaceSpecialization.ProlongationTuple.redBar_bijective11 below · cited by 2 · depth 25 - Places above a supersingular node are centred at (a,a^q)
ModularCurve.PlaceSpecialization.centred_of_reduceFst_eq_of_mem_ssPlaces4 below · cited by 2 · depth 25 - Regular first residue when poles are confined to ̄ 0
ModularCurve.PlaceSpecialization.LevelOneProlongationPair.ord_residueFst_nonneg_of_forall_ne_cuspZeroBar682 below · cited by 1 · depth 26 - Node crossing exponent equals characteristic-q width times e_K
ModularCurve.PlaceSpecialization.ProlongationTuple.crossingExponent_eq_placeWidthChar_mul_of_orderLawFixed_of_liesOverPrime853 below · cited by 1 · depth 28 - Node data and node equations pass to a larger coefficient field
ModularCurve.PlaceSpecialization.ProlongationTuple.exists_nodeCoordinates_coe_eq_and_mul_eq_nodeConst_pow_mul_of_le0 below · cited by 1 · depth 29
ModularCurve.R2geoDet 1
- Coordinate determinant of a similitude equals its factor
ModularCurve.R2geoDet.coordDet_eq_algebraMap_of_similitude0 below · cited by 1 · depth 13
ModularCurve.RigidWeierstrassData 2
- Representability via a normal-form section, with surjection onto B₀
ModularCurve.RigidWeierstrassData.exists_levelModuliPackageAbs_surjective_of_represents_of_section0 below · cited by 4 · depth 30 - Equal classes in Pt differ by a variable change
ModularCurve.RigidWeierstrassData.exists_eq_act_of_mk_eq_mk0 below · cited by 8 · depth 35
ModularCurve.SSCarrier 1
- Non-vanishing leading coefficient of b at supersingular places
ModularCurve.SSCarrier.lead_qP_mul_thetaL_zpow_ne_zero503 below · cited by 2 · depth 16
ModularCurve.SSHeckeV2 24
- A weight ladder S_k ≃ S_{k+p+1} twisting Hecke by ℓ
ModularCurve.SSHeckeV2.exists_linearEquiv_ssCarrier_forall_ssHeckeFun_eq_smul1,010 below · cited by 1 · depth 15 - Vanishing on the supersingular locus lowers the weight by p-1
ModularCurve.SSHeckeV2.mem_modPMod_sub_of_resQFun_eq_zero971 below · cited by 1 · depth 15 - Additivity of the supersingular restriction map resQFun
ModularCurve.SSHeckeV2.resQFun_add800 below · cited by 1 · depth 15 - Restriction to supersingular points intertwines T_ℓ with T_ℓ^{ss}
ModularCurve.SSHeckeV2.resQFun_heckePS_eq_ssHeckeFun_resQFun906 below · cited by 1 · depth 15 - Homogeneity of the supersingular restriction resQFun
ModularCurve.SSHeckeV2.resQFun_smul800 below · cited by 1 · depth 15 - Additivity of the supersingular Hecke operator T_ℓ
ModularCurve.SSHeckeV2.ssHeckeFun_add887 below · cited by 2 · depth 15 - Homogeneity of the supersingular Hecke operator
ModularCurve.SSHeckeV2.ssHeckeFun_smul887 below · cited by 2 · depth 15 - Window property of supersingular Hecke eigensystems
ModularCurve.SSHeckeV2.ssHeckeFun_window1,279 below · cited by 1 · depth 15 - Cuspidal Hecke eigen-function yields mod-p cusp eigenform of weight 2m'
ModularCurve.SSHeckeV2.exists_isModPEigen_modPCusp_of_eigen_riemannRochSpace997 below · cited by 1 · depth 16 - Hecke eigenfunctions in L(D_m) give mod-p eigenforms
ModularCurve.SSHeckeV2.exists_isModPEigen_of_eigen_riemannRochSpace876 below · cited by 1 · depth 16 - Dual Hecke operators on Ω(D') and the cuspidal exit
ModularCurve.SSHeckeV2.exists_omegaHecke_dualMap_theta_and_exit1,236 below · cited by 1 · depth 16 - Residue pairing Theta: kernel and residue formula
ModularCurve.SSHeckeV2.exists_theta_ker_iff_range_resFnFun_and_apply_weilOfKaehler380 below · cited by 1 · depth 16 - The Hecke multiplier satisfies its defining differential identity
ModularCurve.SSHeckeV2.heckeMultiplier_spec121 below · cited by 11 · depth 16 - Lead coefficients of the weight-2m Hecke image compute T_ℓ^{ss}
ModularCurve.SSHeckeV2.lead_trace_heckeBetaC_mul_pow_eq_ssHeckeFun_of_map893 below · cited by 2 · depth 16 - The chosen lift realises prescribed supersingular leading coefficients
ModularCurve.SSHeckeV2.liftFun_spec373 below · cited by 6 · depth 16 - Hecke multiplier, width identity and supersingularity on α-fibres
ModularCurve.SSHeckeV2.ord_heckeMultiplier_eq_and_width_eq_and_mem_ssPlaces_of_mem_fiber869 below · cited by 5 · depth 16 - Additivity of the supersingular residue map on L(D_m)
ModularCurve.SSHeckeV2.resFnFun_add_of_mem363 below · cited by 1 · depth 16 - Homogeneity of the supersingular leading-coefficient map
ModularCurve.SSHeckeV2.resFnFun_smul_of_mem363 below · cited by 1 · depth 16 - Multiplication by b is ℓ-semilinear for the supersingular Hecke operator
ModularCurve.SSHeckeV2.ssHeckeFun_bMul_eq_smul_bMul_ssHeckeFun1,008 below · cited by 1 · depth 16 - Commutativity of the trace Hecke operators on L(weightDivisor)
ModularCurve.SSHeckeV2.trace_heckeBetaC_mul_pow_comm_of_mem874 below · cited by 2 · depth 16 - Weight-2m Hecke operator preserves the Riemann–Roch space
ModularCurve.SSHeckeV2.trace_heckeBetaC_mul_pow_mem_riemannRochSpace_weightDivisor866 below · cited by 1 · depth 16 - q-expansion of the ℓ-degeneracy Hecke multiplier
ModularCurve.SSHeckeV2.coe_heckeMultiplier_mul_thetaL_eq_smul_qExpand_of_ne_zero124 below · cited by 3 · depth 17 - Uniqueness of the Hecke multiplier on the ℓ-degeneracy roof
ModularCurve.SSHeckeV2.eq_heckeMultiplier_of_D_heckeBetaC_eq_smul_map140 below · cited by 1 · depth 17 - Non-vanishing of the ℓ-degeneracy Hecke multiplier
ModularCurve.SSHeckeV2.heckeMultiplier_ne_zero140 below · cited by 1 · depth 17
ModularCurve.SSLevelDatum 16
- Degeneracy adjoints and Hecke companion relations at the prime s
ModularCurve.SSLevelDatum.degeneracyMatrix_mulVec_padj_and_edgeHecke_companion_laws932 below · cited by 5 · depth 14 - Hecke laws for a supersingular two-level degeneracy datum
ModularCurve.SSLevelDatum.exists_heckeRowSums_and_adjointPair_laws1,035 below · cited by 5 · depth 14 - Hecke laws for the two-level supersingular degeneracy datum
ModularCurve.SSLevelDatum.heckeLaws_of_prime_ne_of_not_dvd435 below · cited by 3 · depth 14 - Atkin–Lehner involution swaps degeneracies and width-adjoints Uₛ
ModularCurve.SSLevelDatum.atkinLehnerPerm_swap_degeneracy_and_width_and_edgeHecke_adjoint847 below · cited by 3 · depth 15 - Second degeneracy map equals first after Atkin–Lehner
ModularCurve.SSLevelDatum.degeneracyData_b_eq_a_atkinLehnerPerm0 below · cited by 4 · depth 15 - Degeneracy degrees s+1 and Tₛ=a_*bᵈagger on supersingular places
ModularCurve.SSLevelDatum.degeneracyMatrix_mulVec_padj_eq_smul_and_eq_vertexHecke_mulVec859 below · cited by 2 · depth 15 - Hecke matrix entries as divisor correspondence coefficients, ℓ ≠ p
ModularCurve.SSLevelDatum.edgeHecke_apply_and_vertexHecke_apply_of_ne0 below · cited by 4 · depth 15 - At ℓ = p both Hecke matrices are Frobenius indicator matrices
ModularCurve.SSLevelDatum.edgeHecke_apply_and_vertexHecke_apply_self0 below · cited by 4 · depth 15 - Hecke-equivariant character duality for the q-adic toric part
ModularCurve.SSLevelDatum.exists_cartierAnchor_toricMonodromyPart_edgeHecke3,377 below · cited by 2 · depth 15 - Surjectivity of the supersingular degeneracy map on character lattices
ModularCurve.SSLevelDatum.exists_mem_characterLattice_degeneracyMatrix_mulVec_eq_pair3,474 below · cited by 4 · depth 15 - Connectedness of the s-isogeny graph of supersingular points of X₀(M)
ModularCurve.SSLevelDatum.eq_empty_or_eq_univ_of_forall_fst_mem_iff_snd_mem3,370 below · cited by 2 · depth 16 - Non-bipartiteness of the s-isogeny graph on supersingular places
ModularCurve.SSLevelDatum.exists_fst_mem_iff_snd_mem_of_nonempty699 below · cited by 1 · depth 16 - Eisenstein cokernel bound for the ribbon component group
ModularCurve.SSLevelDatum.finrank_heckeTorsion_ribbonComponentGroup_le_finrank_quotient_of_addEquiv_prod_characterLattice3,543 below · cited by 1 · depth 16 - Ribet's count: dim X^{old}/𝔪 X^{old} = dim Ψ[𝔪]
ModularCurve.SSLevelDatum.finrank_quotient_eq_finrank_heckeTorsion_ribbonComponentGroup_of_addEquiv_prod_characterLattice3,543 below · cited by 1 · depth 16 - Degeneracy matrices commute with the ℓ-Hecke correspondence, ℓ ∣ M
ModularCurve.SSLevelDatum.degeneracyMatrix_mul_correspondence_heckeAlphaC_heckeBetaC_of_dvd428 below · cited by 2 · depth 17 - Second degeneracy leg equals first composed with Atkin–Lehner
ModularCurve.SSLevelDatum.snd_eq_fst_atkinLehnerPerm0 below · cited by 1 · depth 24
ModularCurve.SerreImage 1
- Irreducible subgroups with a unipotent element contain SL₂(𝔽ₚ)
ModularCurve.SerreImage.contains_SL22 below · cited by 1 · depth 16
ModularCurve.SiegelUnit 17
- Peaked auxiliary form on Γ₁(N) of weight divisible by 12
ModularCurve.SiegelUnit.exists_gamma1_peaked_auxiliary_form_twelve_dvd20 below · cited by 1 · depth 17 - A weight-three form on Γ₁(N) with N-integral expansions at ∞ and 0
ModularCurve.SiegelUnit.exists_modularForm_gamma1_weight_three_isIntegral_qExpansion4 below · cited by 1 · depth 17 - Siegel unit times Δ^t as a weight-12t form on Γ₁(N)
ModularCurve.SiegelUnit.exists_modularForm_gamma1_coe_eq_prod_siegelFun_pow_mul_discriminant_pow9 below · cited by 2 · depth 18 - Existence of a peaked Siegel-unit exponent vector at level N
ModularCurve.SiegelUnit.exists_peaked_exponent3 below · cited by 1 · depth 18 - Integrality up to a power of N of the cusp-0 expansion
ModularCurve.SiegelUnit.isIntegral_qExpansion_slash_S_coeff_of_coe_eq_prod_siegelFun_pow_mul_discriminant_pow10 below · cited by 1 · depth 18 - Exact q-order and integrality of a Siegel-unit form
ModularCurve.SiegelUnit.qExpansion_one_coeff_of_coe_eq_prod_siegelFun_pow_mul_discriminant_pow5 below · cited by 1 · depth 18 - Holomorphy of the Siegel function g_{r,s} on H
ModularCurve.SiegelUnit.differentiableOn_siegelFun1 below · cited by 1 · depth 19 - Integral q-expansion of a product of Siegel function powers
ModularCurve.SiegelUnit.exists_isIntegral_hasSum_prod_siegelFun_pow3 below · cited by 4 · depth 19 - Dilates of the level-N Bernoulli weight span the even functions
ModularCurve.SiegelUnit.mem_span_levelBernoulliWeight_dilate_iff_even0 below · cited by 2 · depth 19 - SL₂(ℤ)-transport and Γ₁(N)-invariance of Siegel-function power products
ModularCurve.SiegelUnit.prod_siegelFun_pow_specialLinearGroup_smul3 below · cited by 4 · depth 19 - Distribution relation for the quadratic Bernoulli weight 6t²-6Nt+N²
ModularCurve.SiegelUnit.sum_levelBernoulliWeight_add_mul0 below · cited by 1 · depth 19 - Vanishing of a totient-weighted Bernoulli sum over ℤ/N
ModularCurve.SiegelUnit.sum_totient_mul_levelBernoulliWeight_dilate_eq_zero0 below · cited by 1 · depth 19 - Integral q^{1/N}-expansion of the Siegel function g_{r,s}
ModularCurve.SiegelUnit.exists_isIntegral_hasSum_siegelFun1 below · cited by 1 · depth 20 - Periodicity of Siegel functions in the index (r,s)
ModularCurve.SiegelUnit.siegelFun_add_level0 below · cited by 1 · depth 20 - Nonnegative exponent vector for level Bernoulli weight sums mod q
ModularCurve.SiegelUnit.exists_exponent_sum_levelBernoulliWeight_mul_eq_indicator_sub1 below · cited by 1 · depth 36 - Integral q-expansion of the Siegel form u_μΔ^t on Γ₁(q)
ModularCurve.SiegelUnit.exists_modularForm_gamma1_isIntegralQExp_coeff_eq_one_and_forall_slash_isIntegral14 below · cited by 1 · depth 36 - Formal Siegel series computes the q-expansion of gₐ^{12q}
ModularCurve.SiegelUnit.hasSum_coeff_siegelSeries_pow_mul_exp_siegelFun_pow_div1 below · cited by 1 · depth 37
ModularCurve.StarBank 11
- In characteristic ℓ, j(qᵖ)notin K(j(q)) for p≠ℓ
ModularCurve.StarBank.starBank65 below · cited by 9 · depth 11 - Divisibility G∘ R ∣ c Gᵖ⁺¹ from q-expansion identities
ModularCurve.StarBank.closure0 below · cited by 1 · depth 12 - Rigidity for G∘ R ∣ c Gᵖ⁺¹ with R monic of degree p
ModularCurve.StarBank.count0 below · cited by 1 · depth 12 - A norm identity for the q-expansion of Δ
ModularCurve.StarBank.deltaNorm0 below · cited by 1 · depth 12 - In characteristic ℓ, Δᵖ is never γ Δ(qᵖ)
ModularCurve.StarBank.delta_pow_ne1 below · cited by 1 · depth 12 - A prime ℓ ≥ 5 does not divide num(B_{ℓ-1})
ModularCurve.StarBank.eisInt_not_dvd_num0 below · cited by 5 · depth 12 - Integral model of num(B_{ℓ-1})E_{ℓ-1}, positive coefficients divisible by ℓ
ModularCurve.StarBank.eisInt_series0 below · cited by 5 · depth 12 - Integral descent: T = G(j) Δ^N with deg G = N
ModularCurve.StarBank.hassePolyDescent12 below · cited by 2 · depth 12 - One-point case: (j-β₀)Δ is a nonzero constant
ModularCurve.StarBank.onePoint0 below · cited by 1 · depth 12 - Monicity and splitting of a polynomial relation R(j(q))=j(qᵖ)
ModularCurve.StarBank.press3 below · cited by 1 · depth 12 - Characteristic ℓ: a unit identity G(j)Δ^M = 1
ModularCurve.StarBank.starK0 below · cited by 1 · depth 12
ModularCurve.TateModule 1
- Agreement of the two Tate module carriers
ModularCurve.TateModule.mem_root_iff0 below · cited by 1 · depth 13
ModularCurve.TatePoint 7
- Monodromy equivariance of the level-N dictionary
ModularCurve.TatePoint.b3Act_dictN_of_monodromy0 below · cited by 1 · depth 14 - Nonvanishing discriminant of the full-kernel quotient at every level
ModularCurve.TatePoint.fullKernelDiscAt14 below · cited by 1 · depth 14 - Injectivity of the full-kernel quotient at level N
ModularCurve.TatePoint.fullKernelInjAt211 below · cited by 2 · depth 14 - Root property of the full-kernel quotient at every level
ModularCurve.TatePoint.fullKernelIsRootAt127 below · cited by 2 · depth 14 - Non-degeneracy of the full-kernel Vélu quotient at odd level
ModularCurve.TatePoint.fullKernelDiscAt_of_odd0 below · cited by 2 · depth 15 - Non-vanishing discriminant of the full-kernel Vélu quotient
ModularCurve.TatePoint.fullKernelDiscAt_univ13 below · cited by 2 · depth 15 - Root property of the full-kernel quotient at odd levels
ModularCurve.TatePoint.fullKernelIsRootAt_of_odd0 below · cited by 1 · depth 15
ModularCurve.TwoChart 7
- The two-chart model admits a two-affine open cover
ModularCurve.TwoChart.nonempty_twoAffineOpenCover0 below · cited by 8 · depth 21 - Morphisms from a local scheme factor through one chart
ModularCurve.TwoChart.exists_eq_specMap_comp_iotaFin_or_exists_eq_specMap_comp_iotaInf0 below · cited by 2 · depth 22 - Finite surjective degeneracy map of two-chart models of X₁(Mp)
ModularCurve.TwoChart.exists_hom_modelTo_comp_eq_and_iotaFin_comp_eq_of_le_laurentBaseChange_x1FunctionField_mul9 below · cited by 1 · depth 22 - The two two-chart models of the j-line agree over Spec A
ModularCurve.TwoChart.exists_iso_twoChartIntegralModel_hom_comp_toBase_eq_modelTo0 below · cited by 23 · depth 22 - Two-chart model over A versus integral model over ℤ₍ₚ₎
ModularCurve.TwoChart.exists_iso_twoChartIntegralModel_ratLocalizedAt_of_isCyclotomicExtension2 below · cited by 1 · depth 22 - Reading b'jⁿ=a in the function field of Z
ModularCurve.TwoChart.germToFunctionField_app_iotaInf_mul_germToFunctionField_app_iotaFin_pow_eq0 below · cited by 1 · depth 22 - Local points hitting the j-finite chart factor through it
ModularCurve.TwoChart.exists_eq_spec_map_comp_iotaFin_of_apply_closedPoint_mem_range0 below · cited by 3 · depth 25
ModularCurve.UVCrossingModel 111
- π is a non-zero-divisor on W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.const_mem_nonZeroDivisors0 below · cited by 20 · depth 16 - Unique normal form a(U)+b(V) in the crossing model
ModularCurve.UVCrossingModel.existsUnique_normalForm0 below · cited by 10 · depth 16 - Horizontal primes at depth p/q counted by the index drop
ModularCurve.UVCrossingModel.finsum_rank_mul_length_eq_circleIndexDrop39 below · cited by 2 · depth 16 - Total zero count equals slope drop of the Gauss polygon
ModularCurve.UVCrossingModel.finsum_rank_mul_length_eq_sInf_sub_sSup35 below · cited by 9 · depth 16 - Additivity of the Gauss order on the crossing model
ModularCurve.UVCrossingModel.gaussOrder_mul0 below · cited by 8 · depth 16 - Scaled Gauss order attained at the normal form
ModularCurve.UVCrossingModel.gaussOrder_scaled_eq_repGaussOrder_normalForm0 below · cited by 6 · depth 16 - Grid reading of Gauss-order differences on the crossing model
ModularCurve.UVCrossingModel.gridSecondDiff_eq_circleIndexDrop_sub_of_forall_offGrid_eq7 below · cited by 1 · depth 16 - Grid reading of the Gauss-order difference at scale e'
ModularCurve.UVCrossingModel.gridSecondDiff_eq_circleIndexDrop_sub_of_forall_offGrid_eq_scaled7 below · cited by 1 · depth 16 - Gauss order of a normal form equals infimum of term orders
ModularCurve.UVCrossingModel.repGaussOrder_normalForm_eq_iInf_termOrder0 below · cited by 6 · depth 16 - Gauss order vanishes and least dominant index equals m
ModularCurve.UVCrossingModel.sInf_dominantIndices_eq_of_sub_mul_U_pow_mem0 below · cited by 11 · depth 16 - Additivity of extremal dominant indices at the annulus ends
ModularCurve.UVCrossingModel.sInf_dominantIndices_zero_mul_and_sSup_dominantIndices_mul33 below · cited by 11 · depth 16 - Largest dominant index equals minus the branch order
ModularCurve.UVCrossingModel.sSup_dominantIndices_eq_neg_of_sub_mul_V_pow_mem0 below · cited by 11 · depth 16 - The chart homomorphism sends constants to constants
ModularCurve.UVCrossingModel.chartHom_C0 below · cited by 14 · depth 17 - Scaling invariance of dominant indices under (v,E,t)↦(qv,qE,qt)
ModularCurve.UVCrossingModel.dominantIndices_scale0 below · cited by 6 · depth 17 - A factorial scale at which dominant indices persist
ModularCurve.UVCrossingModel.exists_forall_factorial_dvd_sInf_dominantIndices_mem_succ2 below · cited by 3 · depth 17 - Every element of the crossing model is a + bU with a,b invariant
ModularCurve.UVCrossingModel.exists_mem_fixedSubring_eq_add_mul_U8 below · cited by 8 · depth 17 - Rational depth of horizontal primes in the crossing model
ModularCurve.UVCrossingModel.exists_mul_length_eq_mul_finrank_of_ne_bot_of_const_notMem35 below · cited by 6 · depth 17 - A crossing-model prime with norm polynomial vanishing at c₀
ModularCurve.UVCrossingModel.exists_prime_const_notMem_and_norm_sub_eq_eval_of_pow_eq_mul43 below · cited by 1 · depth 17 - Branch quotient of the uv=π crossing model
ModularCurve.UVCrossingModel.exists_ringEquiv_quotient_span_U_powerSeries0 below · cited by 15 · depth 17 - The V-branch of the crossing model is (W/π)[[T]]
ModularCurve.UVCrossingModel.exists_ringEquiv_quotient_span_V_powerSeries0 below · cited by 11 · depth 17 - Dominant indices of χ_U shift those of x
ModularCurve.UVCrossingModel.exists_sInf_sSup_dominantIndices_charpoly_eq_add33 below · cited by 1 · depth 17 - Free finite quotient of rank the Gauss polygon drop
ModularCurve.UVCrossingModel.free_finite_finrank_quotient_span_of_isUnit_coeff26 below · cited by 2 · depth 17 - Multiplicativity of the scaled Gauss order at depth p/q
ModularCurve.UVCrossingModel.gaussOrder_mul_scale0 below · cited by 3 · depth 17 - Left secant of the Gauss order equals the top dominant index
ModularCurve.UVCrossingModel.gaussOrder_sub_pred_eq_sSup_dominantIndices0 below · cited by 4 · depth 17 - Right slope of the Gauss order at depth p
ModularCurve.UVCrossingModel.gaussOrder_succ_sub_eq_sInf_dominantIndices0 below · cited by 4 · depth 17 - The crossing model W[[u,v]]/(uv-varpi^e) is a normal local domain
ModularCurve.UVCrossingModel.isIntegrallyClosed_of_uniformizer_pow24 below · cited by 6 · depth 17 - Normality of the crossing model over a complete discrete valuation ring
ModularCurve.UVCrossingModel.isIntegrallyClosed_of_uniformizer_pow_of_isAdicComplete25 below · cited by 43 · depth 17 - The crossing model W[[u,v]]/(uv-π) is Noetherian
ModularCurve.UVCrossingModel.isNoetherianRing1 below · cited by 32 · depth 17 - Maximal ideal of the crossing model W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.maximalIdeal_eq_map_maximalIdeal_sup_span_pair0 below · cited by 23 · depth 17 - Chart image equals swap-invariant subring of the crossing model
ModularCurve.UVCrossingModel.range_chartHom_eq_fixedSubring3 below · cited by 8 · depth 17 - Krull dimension at most two for the crossing model
ModularCurve.UVCrossingModel.ringKrullDim_le_two26 below · cited by 21 · depth 17 - Additivity of extreme dominant indices under multiplication
ModularCurve.UVCrossingModel.sInf_dominantIndices_mul_and_sSup_dominantIndices_mul32 below · cited by 4 · depth 17 - Finitely many horizontal primes contain a nonzero element
ModularCurve.UVCrossingModel.setOf_horizontal_mem_finite26 below · cited by 11 · depth 17 - Two-element slope law on the crossing-model annulus
ModularCurve.UVCrossingModel.slopeDrop_sub_eq_circleIndexDrop_sub_of_forall_circleIndexDrop_eq6 below · cited by 2 · depth 17 - Branch ideals of the crossing model UV=π^E
ModularCurve.UVCrossingModel.span_inf_span_eq_and_pow_notMem_and_iInf_sup_span_pow_eq0 below · cited by 3 · depth 17 - The relation U· V=π in the crossing model
ModularCurve.UVCrossingModel.U_mul_V0 below · cited by 16 · depth 18 - U is not fixed by the branch-exchange involution
ModularCurve.UVCrossingModel.U_notMem_fixedSubring4 below · cited by 2 · depth 18 - Dominant indices at interior depth: finite and nonempty
ModularCurve.UVCrossingModel.dominantIndices_finite_nonempty0 below · cited by 1 · depth 18 - Crossing model as a quadratic algebra over D[[s]]
ModularCurve.UVCrossingModel.exists_ringEquiv_adjoinRoot18 below · cited by 3 · depth 18 - Crossing model as a quadratic algebra over W[[s]]
ModularCurve.UVCrossingModel.exists_ringEquiv_adjoinRoot_of_isPrecomplete22 below · cited by 3 · depth 18 - Horizontal primes of the crossing model give finite free W-quotients
ModularCurve.UVCrossingModel.free_and_finite_quotient_of_ne_bot_of_const_notMem36 below · cited by 3 · depth 18 - W[[u,v]]/(uv-π) is local for π a non-unit
ModularCurve.UVCrossingModel.isLocalRing_of_not_isUnit1 below · cited by 22 · depth 18 - Module-finiteness of crossing-model quotients by ideals avoiding both branches
ModularCurve.UVCrossingModel.moduleFinite_quotient_of_not_le_span_pair35 below · cited by 6 · depth 18 - Chart image lies in the swap-fixed subring of the crossing model
ModularCurve.UVCrossingModel.range_chartHom_le_fixedSubring1 below · cited by 4 · depth 18 - Distinctness of the two branch coordinates u ≠ v
ModularCurve.UVCrossingModel.U_ne_V2 below · cited by 1 · depth 19 - Vieta relation U²-S U+π=0 in the crossing model
ModularCurve.UVCrossingModel.U_quadratic1 below · cited by 2 · depth 19 - The chart homomorphism sends X to S
ModularCurve.UVCrossingModel.chartHom_X0 below · cited by 2 · depth 19 - Injectivity of the crossing chart into W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.chartHom_injective3 below · cited by 2 · depth 19 - The crossing involution sends U to V
ModularCurve.UVCrossingModel.crossingSwap_U0 below · cited by 2 · depth 19 - Chart values are fixed by the branch swap
ModularCurve.UVCrossingModel.crossingSwap_chartHom0 below · cited by 1 · depth 19 - Invariants of W[[u,v]]/(uv-π) lift to symmetric series
ModularCurve.UVCrossingModel.exists_uvSwapEquiv_eq_and_mk_eq_of_mem_fixedSubring3 below · cited by 2 · depth 19 - Nontriviality of W[[u,v]]/(uv-π) for non-unit π
ModularCurve.UVCrossingModel.nontrivial_of_not_isUnit0 below · cited by 4 · depth 19 - In the crossing model, U does not lie in (V)
ModularCurve.UVCrossingModel.U_notMem_span_V1 below · cited by 7 · depth 20 - Vanishing on the crossing chart forces divisibility by π
ModularCurve.UVCrossingModel.exists_C_mul_of_chartHom_eq_zero1 below · cited by 1 · depth 20 - Adic completeness of the uv=π crossing model
ModularCurve.UVCrossingModel.isAdicComplete_maximalIdeal3 below · cited by 17 · depth 20 - Leading residue and dominance at the index m
ModularCurve.UVCrossingModel.leadingResidue_nfCoeff_eq_residue_constantCoeff_and_mem_dominantIndices_of_sub_mul_U_pow_mem0 below · cited by 2 · depth 20 - Leading residue at index -n equals ̄ G(0,0), -n dominant
ModularCurve.UVCrossingModel.leadingResidue_nfCoeff_neg_eq_residue_constantCoeff_and_mem_dominantIndices_of_sub_mul_V_pow_mem0 below · cited by 2 · depth 20 - Multiplicativity of extreme dominant leading residues
ModularCurve.UVCrossingModel.leadingResidue_nfCoeff_sInf_dominantIndices_zero_mul_and_sSup_mul34 below · cited by 1 · depth 20 - End-ratio reciprocity on the crossing model W[[U,V]]/(UV-varpi^e)
ModularCurve.UVCrossingModel.leadingResidue_nfCoeff_sSup_mul_finprod_residue_unitPart_norm_pow_eq54 below · cited by 1 · depth 20 - Residue of norms on finite free quotients of the crossing model
ModularCurve.UVCrossingModel.residue_norm_quotient_mk_eq_residue_constantCoeff_pow_finrank3 below · cited by 1 · depth 20 - Fixed ring of a tangentially diagonal automorphism of uv=π
ModularCurve.UVCrossingModel.exists_algHom_range_eq_fixedPoints_apply_U_apply_V_of_tangent13 below · cited by 4 · depth 21 - Horizontal prime of the crossing model through V-varpiᵈ
ModularCurve.UVCrossingModel.exists_prime_V_sub_const_pow_mem26 below · cited by 1 · depth 21 - Reciprocity between end leading residues and branch norms of V
ModularCurve.UVCrossingModel.leadingResidue_nfCoeff_sSup_mul_finprod_residue_unitPart_norm_pow_eq_of_isUnit_coeff51 below · cited by 1 · depth 21 - Rescaling crossing coordinates by a unit and its inverse
ModularCurve.UVCrossingModel.exists_algEquiv_apply_U_eq_mul1 below · cited by 1 · depth 22 - μₑ-invariants of a crossing model form the e-fold model
ModularCurve.UVCrossingModel.exists_powMap_range_eq_fixedPoints3 below · cited by 1 · depth 22 - Linearising a tame finite-order automorphism of a crossing model
ModularCurve.UVCrossingModel.exists_unit_eigen_of_tangent5 below · cited by 2 · depth 22 - End leading residues and χ_U(0) on a crossing model
ModularCurve.UVCrossingModel.leadingResidue_charpoly_coeff_zero_mul_leadingResidue_nfCoeff_sInf47 below · cited by 1 · depth 22 - Branch decomposition of the norm on R/xR for W[[U,V]]/(UV-varpi^e)
ModularCurve.UVCrossingModel.norm_quotient_span_eq_finprod_norm_quotient_pow_length1 below · cited by 1 · depth 22 - u is a non-zero-divisor on W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.U_mem_nonZeroDivisors2 below · cited by 6 · depth 23 - Fixed points of the diagonal automorphism of W[[x,y]]/(xy-π)
ModularCurve.UVCrossingModel.exists_diagAut_fixed_iff0 below · cited by 1 · depth 23 - Zero total drop forces monomial form in the crossing model
ModularCurve.UVCrossingModel.exists_eq_unit_mul_U_pow_mul_V_pow_mul_const_pow_of_sInf_eq_sSup41 below · cited by 1 · depth 23 - Weight-zero series are e-th expansions modulo X₀X₁-π
ModularCurve.UVCrossingModel.exists_mk_eq_mk_expand_of_dvd_sub0 below · cited by 1 · depth 23 - Descent of the nodal normal form uv=t, 𝔪=(u,v)
ModularCurve.UVCrossingModel.exists_mul_eq_and_maximalIdeal_eq_span_pair_of_ringEquiv_adicCompletion26 below · cited by 1 · depth 23 - Injectivity of the e-th power map between crossing models
ModularCurve.UVCrossingModel.exists_powMap_injective0 below · cited by 1 · depth 23 - The two branches of the crossing model are distinct
ModularCurve.UVCrossingModel.V_notMem_span_U1 below · cited by 6 · depth 24 - Divisors of varpi^m in the crossing model are monomials
ModularCurve.UVCrossingModel.exists_eq_unit_mul_U_pow_mul_V_pow_mul_const_pow_of_dvd20 below · cited by 1 · depth 24 - Descent of node coordinates from the crossing-model completion
ModularCurve.UVCrossingModel.exists_mul_eq_pow_mul_and_maximalIdeal_eq_span_of_ringEquiv_adicCompletion_pow5 below · cited by 4 · depth 24 - The branch ideal (U) in the crossing model is prime
ModularCurve.UVCrossingModel.isPrime_span_U2 below · cited by 2 · depth 24 - Primality of the branch ideal (V) in the crossing model
ModularCurve.UVCrossingModel.isPrime_span_V2 below · cited by 2 · depth 24 - Regularity of W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.isRegularLocalRing10 below · cited by 4 · depth 24 - Maximal ideal of W[[u,v]]/(uv-π) is (u,v)
ModularCurve.UVCrossingModel.maximalIdeal_eq_span_pair1 below · cited by 5 · depth 24 - The two non-maximal primes through t are the formal branches
ModularCurve.UVCrossingModel.mem_iff_ringEquiv_adicCompletion_mem_span_or_swap_of_ne0 below · cited by 17 · depth 24 - The two branches are the minimal primes of (UV)
ModularCurve.UVCrossingModel.minimalPrimes_span_U_mul_V8 below · cited by 2 · depth 24 - dim W[[u,v]]/(uv-π)=2 over a discrete valuation ring
ModularCurve.UVCrossingModel.ringKrullDim_eq_two9 below · cited by 5 · depth 24 - Completion of a cyclic cover of a thickness-n crossing
ModularCurve.UVCrossingModel.exists_ringEquiv_adicCompletion_uvCrossingModel_of_moduleFinite_of_isUnramifiedAt_of_isGalois73 below · cited by 1 · depth 25 - R_π/(U) is a domain when W/(π) is
ModularCurve.UVCrossingModel.isDomain_quotient_span_U1 below · cited by 1 · depth 25 - The crossing model modulo V is a domain
ModularCurve.UVCrossingModel.isDomain_quotient_span_V1 below · cited by 1 · depth 25 - Regularity modulo (U) of elements outside (varpi,U)
ModularCurve.UVCrossingModel.mem_span_U_of_mul_mem_of_notMem1 below · cited by 2 · depth 25 - Crossing model over a DVR has Krull dimension at least 2
ModularCurve.UVCrossingModel.two_le_ringKrullDim2 below · cited by 8 · depth 25 - Tame chart rung: crossing thickness mw descends to m
ModularCurve.UVCrossingModel.exists_ringEquiv_adicCompletion_uvCrossingModel_pow_of_isInvariant_of_isLocalization_atPrime113 below · cited by 1 · depth 26 - Tame cyclic covers of the node xy=πⁿ are nodes
ModularCurve.UVCrossingModel.exists_ringEquiv_uvCrossingModel_of_isGalois_of_isCyclic_of_isUnramifiedAt_of_residue51 below · cited by 1 · depth 26 - Every element of the crossing model is a constant mod 𝔪
ModularCurve.UVCrossingModel.exists_sub_const_mem_maximalIdeal1 below · cited by 7 · depth 26 - Cyclic covers of xy=πⁿ unramified in codimension one are Kummer
ModularCurve.UVCrossingModel.exists_pow_eq_unit_mul_U_of_isCyclic_of_isUnramifiedAt_of_residue_thicknessOne49 below · cited by 1 · depth 27 - Tame descent of crossing thickness: mw → m after completion
ModularCurve.UVCrossingModel.exists_ringEquiv_adicCompletion_uvCrossingModel_pow_of_isInvariant_of_card_inertia_eq_of_isUnit112 below · cited by 1 · depth 27 - Extracting an n-th root of a branch coordinate
ModularCurve.UVCrossingModel.exists_ringEquiv_integralClosure_uvCrossingModel_pow_of_pow_eq_unit_mul_U31 below · cited by 2 · depth 27 - Units away from varpi in the crossing model are monomials
ModularCurve.UVCrossingModel.exists_mul_const_pow_eq_of_isUnit_of_isLocalization_away20 below · cited by 1 · depth 28 - Finite étale local algebras over a crossing model are crossing models
ModularCurve.UVCrossingModel.exists_ringEquiv_uvCrossingModel_of_etale_of_isLocalRing_of_isAdicComplete19 below · cited by 1 · depth 28 - Tame cyclic degree-n covers of the node of thickness mn
ModularCurve.UVCrossingModel.exists_ringEquiv_uvCrossingModel_pow_of_isGalois_of_isCyclic_of_isUnramifiedAt_of_residue_of_isUnit81 below · cited by 1 · depth 28 - Crossing model base change along a finite free coefficient extension
ModularCurve.UVCrossingModel.exists_algEquiv_tensorProduct_map_of_finite_of_free0 below · cited by 2 · depth 29 - Kummer descent: cyclic unramified cover of a node of thickness mn
ModularCurve.UVCrossingModel.exists_ringEquiv_uvCrossingModel_pow_of_isGalois_of_isCyclic_of_isUnramifiedAt_of_residue_of_isPrimitiveRoot_mul52 below · cited by 1 · depth 29 - Kummer form of cyclic covers of the crossing model
ModularCurve.UVCrossingModel.exists_pow_eq_unit_mul_U_of_isCyclic_of_isUnramifiedAt_of_residue_of_isPrimitiveRoot_mul50 below · cited by 1 · depth 30 - Monomial germs descend exactly through the tame cyclic quotient
ModularCurve.UVCrossingModel.dvd_and_exists_eq_mul_pow_of_apply_eq_mul_pow_of_range_eq_fixedPoints32 below · cited by 3 · depth 31 - Local n-th powers on the crossing model are u U^jhⁿ
ModularCurve.UVCrossingModel.exists_eq_unit_mul_U_pow_mul_pow_of_forall_height_eq_one_of_isPrimitiveRoot_mul49 below · cited by 1 · depth 31 - Automorphisms of the crossing model rescale V by a unit
ModularCurve.UVCrossingModel.exists_isUnit_apply_V_eq_mul_V_of_ringEquiv_apply_const_eq_of_apply_U_not_mem_span30 below · cited by 3 · depth 31 - The two branch quotients of W[[u,v]]/(uv-π) are isomorphic
ModularCurve.UVCrossingModel.exists_ringEquiv_quotient_span_U_quotient_span_V2 below · cited by 1 · depth 31 - A discrete valuation ring along the U-branch of the crossing model
ModularCurve.UVCrossingModel.exists_ringHom_isDiscreteValuationRing_U_mem_isUnit_V_of_uniformizer_pow27 below · cited by 15 · depth 31 - Tangent scalars of a branch-preserving automorphism of the crossing model
ModularCurve.UVCrossingModel.exists_pow_eq_one_tangent_of_ringEquiv_apply_const_eq_of_iterate_apply_V_mul_eq34 below · cited by 2 · depth 32 - Branch exchange carries the ideal (U) onto (V)
ModularCurve.UVCrossingModel.map_crossingSwap_span_U1 below · cited by 1 · depth 32 - The element V is a non-zero-divisor on the crossing model
ModularCurve.UVCrossingModel.V_mem_nonZeroDivisors2 below · cited by 2 · depth 33 - Functoriality of the crossing model in the coefficient ring
ModularCurve.UVCrossingModel.exists_ringEquiv_apply_U_apply_V_apply_const_of_ringEquiv0 below · cited by 2 · depth 33 - Universal property of the crossing ring W[[u,v]]/(uv-π)
ModularCurve.UVCrossingModel.existsUnique_algHom_apply_U_eq_apply_V_eq_of_isAdicComplete2 below · cited by 2 · depth 34
ModularCurve.UniformizedHeckeCurve 4
- Transport of a uniformised Hecke curve along a ℂ-algebra isomorphism
ModularCurve.UniformizedHeckeCurve.exists_transport_of_algEquiv0 below · cited by 1 · depth 26 - Existence of a uniformized Hecke curve for cocompact Γ
ModularCurve.UniformizedHeckeCurve.exists_of_isCompact_of_discrete70 below · cited by 1 · depth 27 - Comparison of two germwise meromorphic realisations of function fields
ModularCurve.UniformizedHeckeCurve.exists_algEquiv_realize_eventuallyEq_of_meromorphicAt_of_separatesOrbits119 below · cited by 1 · depth 28 - Hecke cycles of places produce fixed points in Γ
ModularCurve.UniformizedHeckeCurve.exists_mul_prod_smul_eq_of_forall_mem_support_corr0 below · cited by 1 · depth 28
ModularCurve.UnramifiedOutside 1
- Inertia acts trivially on p-power torsion outside Np
ModularCurve.UnramifiedOutside.of_specializationExists0 below · cited by 1 · depth 11
ModularCurve.XH 1
- Correspondence α_*β^* on J_H induced by an endomorphism
ModularCurve.XH.pic0Correspondence_pts_eq_comp_of_poincare_pullbackAlong_iso_laurentBaseChange180 below · cited by 3 · depth 12
ModularCurve.XHDRLevel 54
- Frobenius place clauses for π and w at p ∥ M
ModularCurve.XHDRLevel.comp1_pi_place_and_pi_w_comp0_place_of_chart_atkinLehner1,003 below · cited by 1 · depth 13 - Frobenius component of the fibre admits no section
ModularCurve.XHDRLevel.comp_one_comp_fibreMap_ne_id_of_theta_iota0_eq_qExpand_of_liesOverPrime994 below · cited by 1 · depth 13 - Uniqueness of closed-immersion sections of the fibre map
ModularCurve.XHDRLevel.eq_comp_zero_of_isClosedImmersion_of_comp_fibreMap_eq_id2 below · cited by 1 · depth 13 - Crossings of the Deligne–Rapoport fibre enumerated by supersingular places
ModularCurve.XHDRLevel.exists_nodeEquiv_placeOfPoint_eq_and_eq_qExpFrobeniusPlaceModL645 below · cited by 1 · depth 13 - Two q-expansion readings and a retraction on the j-chart ring
ModularCurve.XHDRLevel.exists_ringHom_laurentSeries_pair_and_retraction_pair_chartAlgFin_gammaH990 below · cited by 6 · depth 13 - Crossings of the fibre at p ‖ M lie over the j-finite chart
ModularCurve.XHDRLevel.fst_fst_pullback_comp_mem_range_iotaFin_and_fst_snd_pullback_comp_mem_range_iotaFin_of_chart_atkinLehner1,005 below · cited by 1 · depth 13 - Reducedness mod p of the two chart rings of X_H(M)
ModularCurve.XHDRLevel.isReduced_chartAlgFin_quotient_and_chartAlgInf_quotient_span_natCast_gammaH407 below · cited by 7 · depth 13 - Cusp ∞ reduces onto comp₀, off comp₁
ModularCurve.XHDRLevel.range_sectionFibre_epsInf_subset_compl_range_and_subset_range_of_comp_fibreMap_eq_id1,000 below · cited by 1 · depth 13 - Section avoiding the second component lies in the smooth open
ModularCurve.XHDRLevel.range_section_subset_of_forall_range_sectionFibre_subset_compl_range_comp_one14 below · cited by 1 · depth 13 - Degree p+1 of X_H(M) over X_{H'}(M/p)
ModularCurve.XHDRLevel.relfinrank_qExpFunctionFieldC_gammaH_infSubgroup_gammaH_eq_add_one239 below · cited by 6 · depth 13 - Retraction reads the forgetful chart map as p-th power
ModularCurve.XHDRLevel.retraction_one_tmul_iota0_eq_pow_of_theta_iota0_eq_qExpand_of_liesOverPrime991 below · cited by 3 · depth 13 - Chart-pinned morphisms base-change to the κ-fibre charts
ModularCurve.XHDRLevel.chart_comp_fibreMap_eq_specMap_tensor_comp_chart0 below · cited by 3 · depth 14 - Atkin–Lehner pullback of the Gauss ring is a distinct branch ring
ModularCurve.XHDRLevel.comap_atkinLehner_valuationSubring_gauss_gammaH10 below · cited by 7 · depth 14 - Chart-level Frobenius for comp₁ followed by π
ModularCurve.XHDRLevel.exists_chart_frobenius_comp_one_fibreMap_pi_of_liesOverPrime994 below · cited by 1 · depth 14 - Fibre charts compatible with an Atkin–Lehner chart square
ModularCurve.XHDRLevel.exists_chart_pair_fibreMap_atkinLehner_eq0 below · cited by 1 · depth 14 - Distinct minimal primes over p on the pole chart
ModularCurve.XHDRLevel.exists_minimalPrimes_chartAlgInf_map_le_of_mem_range_comp_gammaH992 below · cited by 1 · depth 14 - Ogg's unit pair Δ(q)/Δ(qᵖ) in the j-finite chart ring
ModularCurve.XHDRLevel.exists_ogg_unit_pair_chartAlgFin_gammaH338 below · cited by 10 · depth 14 - Place at a crossing is the Frobenius translate
ModularCurve.XHDRLevel.exists_placeOfPoint_fst_pullback_comp_eq_qExpFrobeniusPlaceModL0 below · cited by 3 · depth 14 - Crossings of the fibre lie at supersingular places
ModularCurve.XHDRLevel.exists_placeOfPoint_snd_pullback_comp_mem_ssPlacesQExp640 below · cited by 1 · depth 14 - Generic Atkin–Lehner automorphism descends to ℚ and to the j-chart
ModularCurve.XHDRLevel.exists_ratAlgEquiv_chartAlgFin_algEquiv_of_atkinLehner_generic325 below · cited by 4 · depth 14 - Chart retraction from a section of the degeneracy map mod p
ModularCurve.XHDRLevel.exists_retraction_chart_comp_zero_eq0 below · cited by 8 · depth 14 - Retraction onto the lower-level chart ring from a q-expansion reading
ModularCurve.XHDRLevel.exists_retraction_of_ringHom_laurentSeries_chartAlgFin_gammaH893 below · cited by 1 · depth 14 - Two mod-p readings of the Γ_H(M) chart ring
ModularCurve.XHDRLevel.exists_ringHom_laurentSeries_pair_chartAlgFin_gammaH426 below · cited by 2 · depth 14 - Supersingular places come from crossings of the two components
ModularCurve.XHDRLevel.exists_snd_pullback_comp_eq_of_mem_ssPlacesQExp642 below · cited by 1 · depth 14 - A Gauss valuation subring of the q-expansion function field at p
ModularCurve.XHDRLevel.exists_valuationSubring_gauss_qExpFunctionFieldC6 below · cited by 19 · depth 14 - Exactly two branch valuation rings of F(Γ_H(M)) above j mod p
ModularCurve.XHDRLevel.exists_valuationSubring_pair_gammaH396 below · cited by 10 · depth 14 - Two minimal primes in the geometric special fibre at p ∥ M
ModularCurve.XHDRLevel.finite_minimalPrimes_tensor_chartAlgFin_gammaH_and_ncard_eq_two461 below · cited by 2 · depth 14 - Integrality of the two-chart integral model X p Γ
ModularCurve.XHDRLevel.isIntegral_X1 below · cited by 1 · depth 14 - p is a uniformiser at the minimal primes above p
ModularCurve.XHDRLevel.map_span_natCast_eq_maximalIdeal_of_mem_minimalPrimes_chartAlg_gammaH401 below · cited by 1 · depth 14 - Chartwise p-th power endomorphism acts on places by q-expansion Frobenius
ModularCurve.XHDRLevel.placeOfPoint_inv_efib_comp_eq_qExpFrobeniusPlaceModL_of_chart_pow9 below · cited by 2 · depth 14 - Second projection of a fibre product of closed immersions is injective
ModularCurve.XHDRLevel.snd_pullback_comp_injective0 below · cited by 1 · depth 14 - Distinct minimal primes over p together with 1/j generate everything
ModularCurve.XHDRLevel.sup_sup_span_jInvChartInf_eq_top_of_mem_minimalPrimes_gammaH407 below · cited by 1 · depth 14 - No third branch above the Gauss point for Γ_H(M)
ModularCurve.XHDRLevel.valuationSubring_eq_gauss_or_eq_comap_atkinLehner_gammaH360 below · cited by 8 · depth 14 - Rigidity over the level-M/p q-expansion field
ModularCurve.XHDRLevel.algEquiv_eq_refl_of_forall_coe_eq_gammaH_infSubgroup228 below · cited by 2 · depth 15 - Pole-chart retraction for the mod p fibre of X_H(M)
ModularCurve.XHDRLevel.exists_retraction_chartInf_comp_zero_eq_of_dvd0 below · cited by 2 · depth 15 - Ogg's unit on the j-chart at p ∥ M
ModularCurve.XHDRLevel.exists_retraction_tmul_theta_eq_zero_and_mem_iff_exists_mem_ssJSet631 below · cited by 2 · depth 15 - Two mod-p readings of the j-finite chart ring of X_H(M)
ModularCurve.XHDRLevel.exists_ringHom_laurentSeries_zmod_pair_chartAlgFin_gammaH431 below · cited by 3 · depth 15 - Generic Frobenius of a p-th-power endomorphism of the fibre
ModularCurve.XHDRLevel.fromSpecStalk_comp_inv_efib_comp_eq_specMap_qExpFrobeniusModL_of_chart_pow3 below · cited by 1 · depth 15 - Gauss prime of the pole chart is a minimal prime
ModularCurve.XHDRLevel.map_ker_mem_minimalPrimes_and_le_ker_of_chartAlgInf7 below · cited by 1 · depth 15 - Only the Gauss branch and its Atkin–Lehner transform
ModularCurve.XHDRLevel.valuationSubring_eq_gauss_or_eq_comap_atkinLehner_of_unique_of_relfinrank_gammaH352 below · cited by 1 · depth 15 - Uniqueness of the branch ring at p for Γ_{H'}(M/p)
ModularCurve.XHDRLevel.valuationSubring_unique_gammaH_infSubgroup_of_not_sq_dvd326 below · cited by 3 · depth 15 - Chart automorphism agrees with the field automorphism σ
ModularCurve.XHDRLevel.coe_theta_eq_of_forall_coe_iota0_of_qExpand430 below · cited by 2 · depth 16 - Two primes over the Gauss ring with residue degrees 1 and p
ModularCurve.XHDRLevel.exists_primesOver_pair_integralClosure_comap_gauss_gammaH351 below · cited by 1 · depth 16 - Any mod-p retraction of ι₀ reads q-expansions
ModularCurve.XHDRLevel.exists_ringHom_laurentSeries_and_embedding_comp_retraction_gammaH443 below · cited by 2 · depth 16 - Supersingular points of the first copy lie on the second
ModularCurve.XHDRLevel.retraction_map_theta_eq_zero_mem_of_mem_ssJSet_gammaH630 below · cited by 1 · depth 16 - Generic Atkin–Lehner θ as base change of rational σ
ModularCurve.XHDRLevel.algEquiv_coeffEmb_eq_coeffEmb_ratAlgEquiv_of_atkinLehner_generic324 below · cited by 2 · depth 27 - Integrality of j(qᵖ)j(q)⁻ᵖ and its congruence to 1 mod p
ModularCurve.XHDRLevel.exists_chartAlgInf_coe_eq_qExpand_jqModC_mul_inv_pow_and_coeffMap_eq_one_add97 below · cited by 5 · depth 27 - Non-Gauss minimal prime over p as Atkin–Lehner branch
ModularCurve.XHDRLevel.mem_iff_coe_mem_nonunits_comap_atkinLehner_of_mem_minimalPrimes_chartAlgInf363 below · cited by 1 · depth 27 - Flatness of the two chart algebras over ℤ₍ₚ₎
ModularCurve.XHDRLevel.flat_chartAlgFin_and_flat_chartAlgInf1 below · cited by 5 · depth 28 - Fraction field of the j-finite chart ring at level Γ_H(M)
ModularCurve.XHDRLevel.isFractionRing_chartAlgFin_qExpFunctionFieldC121 below · cited by 2 · depth 29 - Reduced q-expansion function field generated by integral chart functions
ModularCurve.XHDRLevel.qExpFunctionFieldC_residueField_le_adjoin_coeffMap_residue_of_mem_chartAlgFin892 below · cited by 1 · depth 29 - Elements of W as chart fractions with denominator off r₀
ModularCurve.XHDRLevel.exists_fraction_not_mem_comap_maximalIdeal_of_mem_valuationSubring_of_map_maximalIdeal_localization_eq5 below · cited by 2 · depth 30 - Chart functions lie in the Gauss prolongation, with explicit residues
ModularCurve.XHDRLevel.mem_integers_and_residue_tmul_eq_smul_coeffMap_of_regularProlongation_gauss3 below · cited by 3 · depth 30 - Integral q-expansion differentials are g dj with g in W₀
ModularCurve.XHDRLevel.exists_mem_gauss_and_eq_smul_D_jAt_of_diffQExp_eq_ofPowerSeries122 below · cited by 1 · depth 31
ModularCurve.XHDRModelAtP 313
- Degeneracy pull-backs between relative Pic⁰ representing objects
ModularCurve.XHDRModelAtP.exists_degPull_classifies_pullback_and_mul4 below · cited by 2 · depth 12 - Degeneracy embeddings and pinned level-M/p generic fibre
ModularCurve.XHDRModelAtP.exists_degeneracyEmb_curveModel_iso_genericFibre_restrictAlong_chartPin_of_atkinLehner_generic132 below · cited by 2 · depth 12 - Level-M/p generic fibre and the two degeneracy embeddings
ModularCurve.XHDRModelAtP.exists_degeneracyEmb_curveModel_iso_genericFibre_restrictAlong_of_atkinLehner_generic132 below · cited by 1 · depth 12 - Degeneracy morphisms D → D₀ and Ribet's special-fibre formula
ModularCurve.XHDRModelAtP.exists_degeneracyHom_mul_pts_special1,698 below · cited by 1 · depth 12 - Hecke degeneracy pair for the Γ_H model over ℤ₍ₚ₎
ModularCurve.XHDRModelAtP.exists_heckeDegeneracyPair_chartPin_flat283 below · cited by 3 · depth 12 - Diamond operators induced by endomorphisms of the Pic⁰ scheme
ModularCurve.XHDRModelAtP.exists_hom_mul_and_pts_diamondHBar_eq_comp163 below · cited by 1 · depth 12 - Hecke operators at ℓ≠ p on a relative Pic⁰ of X_H(M)
ModularCurve.XHDRModelAtP.exists_hom_mul_and_pts_heckeOperatorHAlong_eq_comp_of_ne719 below · cited by 1 · depth 12 - Uₚ at p ∥ M as a group endomorphism of Pic⁰
ModularCurve.XHDRModelAtP.exists_hom_mul_and_pts_heckeOperatorHAlong_self_eq_comp719 below · cited by 1 · depth 12 - Diamond ⟨ e⟩ on places of the mod-p dictionary model
ModularCurve.XHDRModelAtP.exists_placeOfPoint_fibreMap_dia0_eq_diamondActionModL_smul_of_ker_le347 below · cited by 10 · depth 12 - Glued special-fibre dictionary for relative Pic⁰ at p ‖ M
ModularCurve.XHDRModelAtP.exists_ptsSp_gluedPic0_dictionary_specialFibre1,294 below · cited by 1 · depth 12 - Special-fibre Pic⁰ dictionary for the level-Γ_N model
ModularCurve.XHDRModelAtP.exists_ptsSp_levelN_pic0_equiv_of_representsRelSubPic1,183 below · cited by 2 · depth 12 - Generic fibre and points dictionary for the level-M/p Pic⁰ object
ModularCurve.XHDRModelAtP.exists_representsRelSubPic_abelJacobi_pts_levelN_of_representsRelSubPic299 below · cited by 2 · depth 12 - Abel–Jacobi dictionary for the relative Pic⁰ of X_H(M)
ModularCurve.XHDRModelAtP.exists_representsRelSubPic_abelJacobi_pts_of_representsRelSubPic299 below · cited by 1 · depth 12 - Relative Pic⁰ of the X_H(M) model at p
ModularCurve.XHDRModelAtP.exists_representsRelSubPic_algEquivZeroCut_epsInf_of_atkinLehner_generic_of_ker_le1,758 below · cited by 1 · depth 12 - Representability of relative Pic⁰ for the level-M/p model
ModularCurve.XHDRModelAtP.exists_representsRelSubPic_levelN_comp_epsInf_pi1,564 below · cited by 4 · depth 12 - Atkin–Lehner translate of a configured point exchanges special-fibre components
ModularCurve.XHDRModelAtP.exists_schemeHomOver_comp_w_inv_pointEquivPlace_eq_smul_placeOfPoint_eq0 below · cited by 5 · depth 12 - Matching the generic and special Pic⁰ dictionaries by an A-section
ModularCurve.XHDRModelAtP.exists_schemeHomOver_pts_eq_and_ptsSp_symm_eq_mk_of_sameComponent30 below · cited by 2 · depth 12 - Push-down and reduction agree on level-(M/p) dictionaries
ModularCurve.XHDRModelAtP.exists_schemeHomOver_pts_levelN_degPts_eq_and_ptsSp_levelN_symm_eq_mk166 below · cited by 2 · depth 12 - Hensel lifting of a smooth special point to an A-section
ModularCurve.XHDRModelAtP.exists_schemeHomOver_range_subset_smoothLocus_of_mem_smoothLocus4 below · cited by 9 · depth 12 - Two-sided pools of étale blocks in the smooth locus
ModularCurve.XHDRModelAtP.exists_twoSided_pools_smoothLocus_of_atkinLehner_generic_of_ker_le1,160 below · cited by 2 · depth 12 - Inertia differences σ x - x extend over the place
ModularCurve.XHDRModelAtP.extendsToPlace_pts_smul_sub_of_mem_inertiaSubgroupIn1,446 below · cited by 1 · depth 12 - Properness and geometric connectedness of the generic fibre of Pic⁰
ModularCurve.XHDRModelAtP.isProper_and_geometricallyConnected_pullback_snd_rat_of_representsRelSubPic392 below · cited by 1 · depth 12 - Flatness, surjectivity and quasi-finiteness of [n] on D
ModularCurve.XHDRModelAtP.nsmul_flat_surjective_locallyQuasiFinite_of_representsRelSubPic1,960 below · cited by 1 · depth 12 - Degeneracy pull-backs match the point dictionaries
ModularCurve.XHDRModelAtP.pts_alphaPull_eq_pts_levelN_comp_degPull364 below · cited by 1 · depth 12 - Ramification index one along α_H at formally unramified points
ModularCurve.XHDRModelAtP.ramificationIndexAlong_degeneracyEmb_pointEquivPlace_eq_one_of_formallyUnramified2 below · cited by 3 · depth 12 - Frobenius squared is the inverse diamond at supersingular places
ModularCurve.XHDRModelAtP.smul_frob_mem_ssPlacesQExp_and_frob_smul_frob_eq_of_mem_ssPlacesQExp0 below · cited by 7 · depth 12 - Special fibre of the degeneracy pull-backs on Pic⁰ coordinates
ModularCurve.XHDRModelAtP.toPic0Pair_ptsSp_symm_schemeHomOverComp_degPull_eq1,100 below · cited by 1 · depth 12 - Component immersions commute with twists of the geometric point
ModularCurve.XHDRModelAtP.baseChangeSnd_comp_comp943 below · cited by 1 · depth 13 - H⁰ of every base change of the model at p is A
ModularCurve.XHDRModelAtP.bijective_algebraMap_sections_baseChange211 below · cited by 2 · depth 13 - Component maps commute with a residual base twist at pole-chart points
ModularCurve.XHDRModelAtP.comp_base_baseTwist_eq_baseTwist_comp_base_of_mem_range_iotaInf53 below · cited by 1 · depth 13 - Classifying morphisms D₀ → D respect group law and zero
ModularCurve.XHDRModelAtP.degPull_mul_and_zeroSection_comp_of_classifies_pullback5 below · cited by 1 · depth 13 - Splitting along π of a section divisor on the Γ_H(M) model
ModularCurve.XHDRModelAtP.exists_comap_curveChange_pi_ofPoint_eq_mul_prod_pow_of_ker_le410 below · cited by 1 · depth 13 - Degeneracy morphisms of relative Pic⁰ as norm maps
ModularCurve.XHDRModelAtP.exists_degeneracyHom_classifies_normModule78 below · cited by 1 · depth 13 - Constant arithmetic genus of the geometric fibres at p
ModularCurve.XHDRModelAtP.exists_forall_finrank_H1_fibre_eq245 below · cited by 2 · depth 13 - Formal unramifiedness of π near a section off the crossings
ModularCurve.XHDRModelAtP.exists_opens_formallyUnramified_pi_of_comp_zero_of_forall_ne_placeOn00 below · cited by 2 · depth 13 - Relative Frobenius twists places of closed fibre points
ModularCurve.XHDRModelAtP.exists_placeOfPoint_frobeniusTwist_eq_smul60 below · cited by 1 · depth 13 - Torus and abelian quotient on the special fibre of Pic⁰
ModularCurve.XHDRModelAtP.exists_representsRelSubPic_torus_abq_specialFibre1,035 below · cited by 3 · depth 13 - Unramifiedness of π and Frobenius on a second preimage
ModularCurve.XHDRModelAtP.exists_schemeHomOver_comp_one_frob_placeOfPoint_eq_of_comp_pi_eq_of_ne5 below · cited by 3 · depth 13 - Atkin–Lehner translate of a section: other component, diamond-twisted place
ModularCurve.XHDRModelAtP.exists_schemeHomOver_comp_w_inv_placeOfPoint_eq0 below · cited by 1 · depth 13 - An A-point of relative Pic⁰ carrying 𝒪(u₁)⊗𝒪(u₂)⁻¹
ModularCurve.XHDRModelAtP.exists_schemeHomOver_poincare_iso_ofPoint_tensor_idealModule_of_sameComponent1,204 below · cited by 3 · depth 13 - Frobenius conjugate of a smooth A-point of X
ModularCurve.XHDRModelAtP.exists_schemeHomOver_specMap_decomposition_comp_pointEquivPlace_eq_smul_placeOfPoint_eq_smul0 below · cited by 1 · depth 13 - Two glued smooth curves in non-smooth fibres of X_H(M)
ModularCurve.XHDRModelAtP.exists_twoGluedSmoothCurveDegeneration_of_not_smooth140 below · cited by 2 · depth 13 - Geometric closed fibres as two transversally glued smooth curves
ModularCurve.XHDRModelAtP.exists_twoGluedSmoothCurves_isReduced_pullback_of_ker_ne_bot194 below · cited by 2 · depth 13 - Two-sided pools of étale blocks at the closed prime
ModularCurve.XHDRModelAtP.exists_twoSidedPool_smoothLocus_closedPrime_of_five_le_of_atkinLehner_generic1,156 below · cited by 2 · depth 13 - Two-sided pools of étale blocks at the closed prime, p=3
ModularCurve.XHDRModelAtP.exists_twoSidedPool_smoothLocus_closedPrime_three_of_atkinLehner_generic1,156 below · cited by 2 · depth 13 - Two-sided pools of étale blocks in the smooth locus, p=2
ModularCurve.XHDRModelAtP.exists_twoSidedPool_smoothLocus_closedPrime_two_of_atkinLehner_generic1,156 below · cited by 2 · depth 13 - Generic-prime two-sided pools in the Γ_H smooth locus
ModularCurve.XHDRModelAtP.exists_twoSidedPool_smoothLocus_genericPrime_of_atkinLehner_generic1,159 below · cited by 1 · depth 13 - Inertia displacement at a non-crossing place extends over A
ModularCurve.XHDRModelAtP.extendsToPlace_pts_mk_smul_single_sub_single_of_not_mem_range_comp_inter1,214 below · cited by 1 · depth 13 - Inertial displacement of a place extends: crossing case
ModularCurve.XHDRModelAtP.extendsToPlace_pts_mk_smul_single_sub_single_of_range_subset_range_comp_inter1,429 below · cited by 1 · depth 13 - Degeneracy map on geometric generic fibres restricts to Specα
ModularCurve.XHDRModelAtP.fromSpecStalk_genericPoint_comp_eq_specMap_ffEquiv_degeneracyEmb_of_chartPin0 below · cited by 2 · depth 13 - Diamond ⟨ e⟩₀ on the j-finite chart
ModularCurve.XHDRModelAtP.iotaFin_comp_dia0_hom_eq_spec_map_comp_iotaFin158 below · cited by 1 · depth 13 - Generic fibre of the two Hecke degeneracy legs stays finite flat
ModularCurve.XHDRModelAtP.isFinite_flat_finrank_curveChange_heckeDegeneracy_rat5 below · cited by 2 · depth 13 - Forgetful map of the model at p is finite flat of rank p+1
ModularCurve.XHDRModelAtP.isFinite_flat_finrank_pi290 below · cited by 4 · depth 13 - Geometric fibres of the X_H(M) model at p are reduced
ModularCurve.XHDRModelAtP.isReduced_pullback_toBase_of_isAlgClosed11 below · cited by 5 · depth 13 - Local quasi-finiteness of [n] on fibres of relative Pic⁰
ModularCurve.XHDRModelAtP.locallyQuasiFinite_fibre_schemeNsmul_of_not_isUnit1,950 below · cited by 1 · depth 13 - Smooth locus criterion on the special fibre of X_H(M)
ModularCurve.XHDRModelAtP.mem_preimage_smoothLocus_iff_not_mem_range_comp_inter940 below · cited by 20 · depth 13 - Algebraically trivial invertible sheaves with a section on geometric fibres
ModularCurve.XHDRModelAtP.nonempty_iso_unit_of_isAlgEquivZero_of_ne_zero_fibre463 below · cited by 1 · depth 13 - Poincaré bundle at ℚ̄-points of the integral model
ModularCurve.XHDRModelAtP.nonempty_poincare_pullbackAlong_iso_ofPoint_tensor_ofPoint_idealModule_of_eq_comp_ajbar15 below · cited by 8 · depth 13 - Special-fibre formula for the degeneracy push-forwards at p ∥ M
ModularCurve.XHDRModelAtP.ptsSp_levelN_symm_schemeHomOverComp_degeneracyHom_eq_of_pts_levelN_degPts_eq_comp1,637 below · cited by 1 · depth 13 - Degeneracy push-forwards as norm homomorphisms on ℚ̄-points
ModularCurve.XHDRModelAtP.pts_levelN_degPts_eq_comp_degeneracyHom_of_classifies_normModule230 below · cited by 1 · depth 13 - Smooth locus of the Γ_H(M) model: smooth and maximal
ModularCurve.XHDRModelAtP.smoothOfRelativeDimension_one_smoothLocus_and_maximal0 below · cited by 1 · depth 13 - Geometric fibres of the Γ_H(M) model at p∣ M are connected
ModularCurve.XHDRModelAtP.connectedSpace_pullback_toBase_specMap_of_isAlgClosed131 below · cited by 3 · depth 14 - Atkin–Lehner map read on the j-finite chart
ModularCurve.XHDRModelAtP.exists_chartAlgFin_algEquiv_iotaFin_comp_w_eq_of_atkinLehner_generic_of_unitsMap330 below · cited by 3 · depth 14 - Ogg's unit and the components of the fibre at p
ModularCurve.XHDRModelAtP.exists_chartAlgFin_forall_mem_range_comp_zero_and_not_mem_range_comp_one1,029 below · cited by 4 · depth 14 - Inertia line bundle 𝒪(σ V-V) at a crossing
ModularCurve.XHDRModelAtP.exists_isInvertible_iso_ofPoint_tensor_idealModule_iso_tensorUnit_of_range_subset_range_comp_inter1,176 below · cited by 2 · depth 14 - Pools of level polynomials on the j-finite chart ring
ModularCurve.XHDRModelAtP.exists_levelPolynomials_of_chartAlgFin572 below · cited by 3 · depth 14 - A one-sided pool of étale blocks in the smooth locus
ModularCurve.XHDRModelAtP.exists_oneSidedPool_baseChange_of_levelPolynomials957 below · cited by 3 · depth 14 - Function-field map of π has degree p+1
ModularCurve.XHDRModelAtP.exists_ringHom_functionField_fromSpecStalk_comp_pi_eq_and_finrank_eq_add_one248 below · cited by 1 · depth 14 - Splitting of π⁻¹[u] over the geometric generic fibre
ModularCurve.XHDRModelAtP.exists_sections_comap_genericFibre_ofPoint_pi_eq_mul_prod_pow400 below · cited by 1 · depth 14 - Trivialised line bundle on mathfrak X_A makes pts([y₁]-[y₂]) extend
ModularCurve.XHDRModelAtP.extendsToPlace_pts_pic0Mk_single_sub_single_of_isInvertible_of_iso_ofPoint_tensor_idealModule_of_iso_tensorUnit1,205 below · cited by 1 · depth 14 - At non-smooth fibres, w moves the ε_∞-component off itself
ModularCurve.XHDRModelAtP.fibre_w_mem_diff_connectedComponentIn_and_cuspZero_mem_baseChange160 below · cited by 3 · depth 14 - Level-M/p diamond acts on the pole chart as Specσ
ModularCurve.XHDRModelAtP.iotaInf_comp_dia0_hom_eq_spec_map_comp_iotaInf155 below · cited by 1 · depth 14 - Finiteness and finite presentation of π for X_H(M) at p‖M
ModularCurve.XHDRModelAtP.isFinite_and_locallyOfFinitePresentation_pi122 below · cited by 1 · depth 14 - Membership in the component comp₀ via vanishing q-expansions
ModularCurve.XHDRModelAtP.mem_range_comp_zero_iff_map_ker_le52 below · cited by 5 · depth 14 - Geometric generic points lie in the smooth locus
ModularCurve.XHDRModelAtP.mem_smoothLocus_of_mem_range_fst_geomGeneric0 below · cited by 4 · depth 14 - Non-supersingular points avoid crossings and lie over the smooth locus
ModularCurve.XHDRModelAtP.not_mem_range_comp_one_and_mem_smoothLocus_of_placeOfPoint_not_mem_ssPlacesQExp944 below · cited by 4 · depth 14 - Geometric characteristic-p fibres of the model are not smooth
ModularCurve.XHDRModelAtP.not_smooth_pullback_snd_toBase_of_charP147 below · cited by 2 · depth 14 - A-point with Poincaré bundle 𝒪(u₁-u₂) computes pts([y₁]-[y₂])
ModularCurve.XHDRModelAtP.pts_pic0Mk_eq_barPt_comp_of_poincare_pullbackAlong_iso_ofPoint_tensor_idealModule30 below · cited by 2 · depth 14 - Inertia fixes the reduction of the A-section at a place
ModularCurve.XHDRModelAtP.residue_comp_section_smul_eq_of_mem_inertia1 below · cited by 1 · depth 14 - w preserves the smooth locus; the model is separated
ModularCurve.XHDRModelAtP.w_preimage_smoothLocus_eq_and_isSeparated_toBase0 below · cited by 3 · depth 14 - Pole-chart sections read as coefficient embeddings of q-expansions
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_app_iotaInf_eq_coeffEmb1 below · cited by 10 · depth 15 - Factorisation of the pulled-back point ideal on the generic fibre
ModularCurve.XHDRModelAtP.comap_curveChange_pi_ofPoint_genericFibre_eq_mul_prod_pow_of_restrictAlong_pointEquivPlace_eq3 below · cited by 1 · depth 15 - Étale level sets of the modular unit on the j-finite chart
ModularCurve.XHDRModelAtP.exists_finite_etale_quotient_span_aeval_chartAlgFin569 below · cited by 1 · depth 15 - Section through a crossing factors through the crossing chart
ModularCurve.XHDRModelAtP.exists_lift_comp_crossingChart_eq_specMap_lift_of_base_closedPoint_eq0 below · cited by 1 · depth 15 - Two branch primes over p on the j-finite chart
ModularCurve.XHDRModelAtP.exists_minimalPrimes_chartAlgFin_le_of_mem_range_comp1,000 below · cited by 3 · depth 15 - Crossing chart and section complement cover X_A
ModularCurve.XHDRModelAtP.exists_opens_sup_eq_top_and_forall_mem_basicOpen_of_crossingChart_of_sections0 below · cited by 1 · depth 15 - Oriented crossing charts over the valuation ring A
ModularCurve.XHDRModelAtP.forall_exists_orientedCrossingChart_valuationSubring1,137 below · cited by 1 · depth 15 - Primes over I avoiding v lie in the smooth locus
ModularCurve.XHDRModelAtP.iotaFin_mem_smoothLocus_of_le_of_sup_span_singleton_eq_top943 below · cited by 1 · depth 15 - Chart points are not w-translates under comaximality
ModularCurve.XHDRModelAtP.iotaFin_ne_w_iotaFin_of_span_singleton_sup_span_singleton_theta_eq_top0 below · cited by 1 · depth 15 - Diamond operator on the j=∞ chart as Specσ
ModularCurve.XHDRModelAtP.iotaInf_comp_dia_hom_eq_spec_map_comp_iotaInf4 below · cited by 2 · depth 15 - Smooth chart points lie on the ε_∞-component of geometric fibres
ModularCurve.XHDRModelAtP.mem_connectedComponentIn_baseChange_of_fst_eq_iotaFin82 below · cited by 1 · depth 15 - Crossing-chart glued module restricts to 𝒪(̄ y₁)⊗𝒪(̄ y₂)⁻¹ generically
ModularCurve.XHDRModelAtP.nonempty_pullback_baseChangeSnd_iso_ofPoint_tensor_idealModule_of_isFrameOn_of_map_eq_smul22 below · cited by 1 · depth 15 - Triviality of the glued module on both special-fibre components
ModularCurve.XHDRModelAtP.nonempty_pullback_comp_iso_unit_of_isFrameOn_of_map_eq_smul12 below · cited by 1 · depth 15 - Cusp sections miss the j-finite chart of the Γ_H model
ModularCurve.XHDRModelAtP.range_epsInf_inter_range_iotaFin_eq_empty_and_range_epsZero_inter_range_iotaFin_eq_empty0 below · cited by 1 · depth 15 - Uniqueness of the fibre component through the cusp at infinity
ModularCurve.XHDRModelAtP.subsingleton_minimalPrimes_le_ker_cusp7 below · cited by 1 · depth 15 - Pole-chart sections on the special fibre reduce q-expansions
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_app_iotaInf_eq_coeffMap_of_mfib3 below · cited by 5 · depth 16 - Formally unramified level sets of Ogg's unit on the Γ_H chart
ModularCurve.XHDRModelAtP.exists_avoid_forall_formallyUnramified_quotient_quotient_span_aeval_chartAlgFin489 below · cited by 1 · depth 16 - Finiteness of chart ring modulo a monic polynomial in the modular unit
ModularCurve.XHDRModelAtP.exists_forall_finite_quotient_span_aeval_and_finrank_le_chartAlgFin290 below · cited by 1 · depth 16 - Generic unramifiedness of the Γ_H chart ring over Rₚ[X]
ModularCurve.XHDRModelAtP.exists_forall_isUnramifiedAt_polynomial_of_aeval_notMem_chartAlgFin127 below · cited by 1 · depth 16 - Ogg's unit distinguishes the two branches above the Gauss point
ModularCurve.XHDRModelAtP.exists_modularUnit_mem_and_inv_mem_and_div_mem_of_valuationSubring_pair_chartAlgFin444 below · cited by 2 · depth 16 - Inertia-equivariant crossing chart over A from a chart over O
ModularCurve.XHDRModelAtP.exists_orientedCrossingChart_valuationSubring_of_chart3 below · cited by 1 · depth 16 - Oriented étale crossing charts uv=p^e on the Deligne–Rapoport model
ModularCurve.XHDRModelAtP.forall_exists_orientedEtaleCrossingChart1,124 below · cited by 3 · depth 16 - Crossings of the special fibre are κ_A-rational points
ModularCurve.XHDRModelAtP.forall_exists_spec_hom_fibre_comp_snd_eq_id_and_base_closedPoint_eq_comp_fst0 below · cited by 3 · depth 16 - At a crossing: p a non-zero-divisor, stalk dimension ≥ 2
ModularCurve.XHDRModelAtP.baseGerm_mem_nonZeroDivisors_and_two_le_ringKrullDim_stalk1,017 below · cited by 1 · depth 17 - Both branch generic points specialise to each crossing point
ModularCurve.XHDRModelAtP.bcMap_genericPoint_specializes_crossingPt0 below · cited by 2 · depth 17 - Special fibre points specialise from ξ_∞ or ξ₀
ModularCurve.XHDRModelAtP.bcMap_genericPoint_specializes_or0 below · cited by 1 · depth 17 - Branch ideals at a crossing meet in (p)
ModularCurve.XHDRModelAtP.branchIdeal_xiInf_inf_branchIdeal_xiZero_eq_span_baseGerm19 below · cited by 2 · depth 17 - Transversal crossing: branch ideals sum to the maximal ideal
ModularCurve.XHDRModelAtP.branchIdeal_xiInf_sup_branchIdeal_xiZero_eq_maximalIdeal1,052 below · cited by 2 · depth 17 - Each branch at a crossing is generated by two elements (a,p)
ModularCurve.XHDRModelAtP.exists_span_pair_baseGerm_eq_branchIdeal1,072 below · cited by 1 · depth 17 - Ogg's unit at a crossing: tt' = p¹² in the stalk
ModularCurve.XHDRModelAtP.exists_stalk_mul_eq_baseGerm_pow_and_isUnit_stalkSpecializes_of_crossing1,035 below · cited by 1 · depth 17 - Integrality of the base change X_O of the Γ_H(M) model
ModularCurve.XHDRModelAtP.isIntegral_xO3 below · cited by 4 · depth 17 - Incomparable branch ideals at a crossing point
ModularCurve.XHDRModelAtP.not_branchIdeal_le_branchIdeal_crossingPt1,012 below · cited by 2 · depth 17 - Special-fibre components map onto closures of their generic points
ModularCurve.XHDRModelAtP.range_comp_bcMap_eq_closure_and_isClosed1,011 below · cited by 2 · depth 17 - Rationality and closedness of a crossing point
ModularCurve.XHDRModelAtP.residue_baseGerm_surjective_and_isClosed_crossingPt0 below · cited by 1 · depth 17 - Crossing points lie in the finite-j chart
ModularCurve.XHDRModelAtP.crossingPt_mem_preimage_iotaFin3 below · cited by 3 · depth 18 - Minimal primes over p in a stalk are the two branch ideals
ModularCurve.XHDRModelAtP.eq_comap_or_eq_comap_of_mem_minimalPrimes_natCast_of_specializes0 below · cited by 1 · depth 18 - Maximality of ξ_∞ and ξ₀ in the special fibre
ModularCurve.XHDRModelAtP.eq_xi_of_specializes64 below · cited by 2 · depth 18 - Each branch at a crossing is regular: 𝔪 = P+(t)
ModularCurve.XHDRModelAtP.exists_maximalIdeal_eq_branchIdeal_sup_span_singleton1,021 below · cited by 1 · depth 18 - Finiteness of the crossings of the special fibre at A
ModularCurve.XHDRModelAtP.finite_crossings60 below · cited by 7 · depth 18 - Distinctness of the two branch points ξ_∞ ≠ ξ₀
ModularCurve.XHDRModelAtP.xiInf_ne_xiZero1,001 below · cited by 4 · depth 18 - Both crossing generic points lie over the smooth locus
ModularCurve.XHDRModelAtP.xi_mem_preimage_smoothLocus950 below · cited by 1 · depth 18 - Special-fibre components are saturated for the comparison map
ModularCurve.XHDRModelAtP.preimage_closure_image_range_comp_eq_of_comp_fst_eq1,001 below · cited by 1 · depth 19 - Place-specialization kit for X_H(M) at p ∥ M
ModularCurve.XHDRModelAtP.exists_jHPlaceSpecialization_prolongationDatum_gluedSpecialization_componentGroup_offDiag_of_wgen2,517 below · cited by 2 · depth 24 - Diamond translate of a configured point on the Deligne–Rapoport model
ModularCurve.XHDRModelAtP.exists_schemeHomOver_comp_dia_pointEquivPlace_eq_smul_placeOfPoint_eq_smul0 below · cited by 2 · depth 24 - Place action determines the Atkin–Lehner field automorphism
ModularCurve.XHDRModelAtP.algEquiv_eq_of_forall_pointEquivPlace_eq_ofAlgAut_smul_of_comp_w_eq16 below · cited by 6 · depth 25 - ∞-side cusp law for the prolongation datum at p ∥ M
ModularCurve.XHDRModelAtP.cuspLawInfty_prolongationDatum_offDiag_of_residue1,248 below · cited by 1 · depth 25 - Zero-side cusp law for the Deligne–Rapoport prolongation datum
ModularCurve.XHDRModelAtP.cuspLawZero_prolongationDatum_offDiag1,249 below · cited by 1 · depth 25 - Orientation of cuspidal reductions: ∞-side and 0-side places
ModularCurve.XHDRModelAtP.cuspOrientationInf_and_cuspOrientationZero_of_jHPlaceSpecialization_of_offDiag493 below · cited by 3 · depth 25 - Off-diagonal reading: δ-twisted specialisation equals δ-twisted Frobenius
ModularCurve.XHDRModelAtP.delta_sp_restrictAlong_comp_eq_delta_qExpFrobeniusPlaceModL_placeOfPoint_of_comp_zero1,020 below · cited by 1 · depth 25 - Disc laws at affine readings on the Deligne–Rapoport model
ModularCurve.XHDRModelAtP.discLawFst_and_discLawSnd_of_jHPlaceSpecialization_of_offDiag1,052 below · cited by 2 · depth 25 - First divisor law for the prolongation datum at p ∥ M
ModularCurve.XHDRModelAtP.divisorLawFst_prolongationDatum_of_norm_of_typeDichotomy_of_localSemicontinuity_of_poleCancellation271 below · cited by 1 · depth 25 - Second divisor law from the first at p ∥ M
ModularCurve.XHDRModelAtP.divisorLawSnd_prolongationDatum_of_divisorLawFst_of_norm_of_typeDichotomy58 below · cited by 1 · depth 25 - Atkin–Lehner automorphism on the geometric function field of X_H(M)
ModularCurve.XHDRModelAtP.exists_algEquiv_pointEquivPlace_eq_ofAlgAut_smul_and_arithmeticGalois_comm_of_comp_w_eq54 below · cited by 5 · depth 25 - Both cuspidal sides lie above a non-affine place
ModularCurve.XHDRModelAtP.exists_isInftySide_reduceFst_eq_and_isZeroSide_reduceSnd_eq_of_not_isAffinePlace_prolongationDatum1,102 below · cited by 8 · depth 25 - Reduction of the norm along α of a doubly integral function
ModularCurve.XHDRModelAtP.exists_mapDomain_sp_eq_ord_and_ord_frob_eq_add_of_norm_of_prolongationDatum433 below · cited by 3 · depth 25 - Valuative extension of a geometric point to an A-section
ModularCurve.XHDRModelAtP.exists_schemeHomOver_barPt_eq_and_fibre_lift_and_comp_base_closedPoint_eq1 below · cited by 12 · depth 25 - Unit pair and one-sided divisor laws at the X_H model
ModularCurve.XHDRModelAtP.exists_unit_pair_divisor_oneSidedLaws_jump_prolongationDatum_of_isModel_of_nodeValueLaw1,394 below · cited by 4 · depth 25 - Node annuli at supersingular crossings, attached at both ends
ModularCurve.XHDRModelAtP.exists_width_annulus_attachedBothEnds_of_jHPlaceSpecialization_of_offDiag1,448 below · cited by 2 · depth 25 - Bidegree (0,0) divisor classes extend to A-points
ModularCurve.XHDRModelAtP.extendsToPlace_pts_pic0Mk_of_forall_sum_filter_eq_zero1,218 below · cited by 2 · depth 25 - Vertical-slope functions on the node annuli at p ∥ M
ModularCurve.XHDRModelAtP.forall_annulus_exists_smul_mem_integers_isGoodDiv_ord_eq_zero_verticalSlope_of_dvd_of_offDiag1,368 below · cited by 1 · depth 25 - Off-diagonal semicontinuity for the prolongation datum at p ‖ M
ModularCurve.XHDRModelAtP.localSemicontinuity_prolongationDatum_offDiag1,284 below · cited by 1 · depth 25 - Strict places land on their own component, off the crossings
ModularCurve.XHDRModelAtP.mem_range_comp_and_not_crossing_of_isStrict_of_placeSpecializationKit_offDiag261 below · cited by 2 · depth 25 - Order zero of one-sided Gauss residues at fixed non-node places
ModularCurve.XHDRModelAtP.ord_residue_eq_zero_of_fixed_of_forall_ord_eq_zero_fst_and_snd_of_offDiag1,410 below · cited by 1 · depth 25 - One-sided regularity of residues at fixed affine non-node places
ModularCurve.XHDRModelAtP.ord_residue_nonneg_of_fixed_of_isAffinePlace_of_forall_ord_nonneg_fst_and_snd_of_offDiag1,410 below · cited by 1 · depth 25 - Fixed-place order law from the reduced norm datum
ModularCurve.XHDRModelAtP.orderLawFixed_prolongationDatum_of_norm56 below · cited by 1 · depth 25 - Reduction on the component 1 as a diamond-twisted specialised place
ModularCurve.XHDRModelAtP.placeOfPoint_eq_delta_sp_restrictAlong_of_comp_one_of_gauss1,045 below · cited by 1 · depth 25 - Reduction along a fibral component as Gauss specialisation of places
ModularCurve.XHDRModelAtP.placeOfPoint_eq_sp_restrictAlong_of_comp_zero_of_gauss1,017 below · cited by 2 · depth 25 - Pole cancellation for common units of a prolongation datum
ModularCurve.XHDRModelAtP.poleCancellation_prolongationDatum64 below · cited by 1 · depth 25 - Regularity law from order law and type dichotomy
ModularCurve.XHDRModelAtP.regularityLaw_of_orderLawFixed_of_typeDichotomy_of_prolongationDatum1,055 below · cited by 1 · depth 25 - Frobenius reading of the π-specialisation on the 0-component
ModularCurve.XHDRModelAtP.sp_restrictAlong_eq_qExpFrobeniusPlaceModL_placeOfPoint_of_comp_one1,019 below · cited by 2 · depth 25 - q-expansions on the pole chart of the Deligne–Rapoport model
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_inf_eq_coeffEmb0 below · cited by 2 · depth 26 - Pole-chart functions read on the special fibre reduce coefficientwise
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_inf_eq_coeffMap_residue0 below · cited by 3 · depth 26 - Cusp local semicontinuity for both prolongation residues
ModularCurve.XHDRModelAtP.cuspLocalSemicontinuity_prolongationDatum_of_residue1,218 below · cited by 2 · depth 26 - Annulus at a node from an oriented étale crossing chart
ModularCurve.XHDRModelAtP.exists_annulus_mem_dom_iff_and_param_eq_read_chart_and_modulus_eq_pow_of_chart_of_residue_surjective1,128 below · cited by 1 · depth 26 - Integral Taylor expansions on residue discs of strict places
ModularCurve.XHDRModelAtP.exists_discParameter_ringHom_powerSeries_taylor_and_ord_residue_eq_of_isStrict1,050 below · cited by 1 · depth 26 - Common node value at supersingular gluing pairs
ModularCurve.XHDRModelAtP.exists_hasValue_residue_pair_of_mem_ssNodePairs_of_orderLawFixed_of_prolongationDatum468 below · cited by 1 · depth 26 - Invertible module framing nodes, fixed places and a base point
ModularCurve.XHDRModelAtP.exists_isInvertible_presentation_frames_slopeLaw_fixed_base_strict_of_dvd_width296 below · cited by 1 · depth 26 - Component-wise trivial invertible module is a node-unit module
ModularCurve.XHDRModelAtP.exists_isNodeUnitModule_pullback_of_forall_nonempty_pullback_comp_iso_unit11 below · cited by 1 · depth 26 - Unit germ at a crossing from unit values on the tube
ModularCurve.XHDRModelAtP.exists_isUnit_and_read_eq_and_ord_placeOn_eq_zero_of_forall_ord_eq_zero_of_forall_isUnit_evalAt_of_chart_of_residue_surjective1,123 below · cited by 1 · depth 26 - Reading an oriented crossing chart in the geometric function field
ModularCurve.XHDRModelAtP.exists_read_chart_mul_eq_and_isUnit_germ_and_smul_eq_and_evalAt_eq_of_chart1,020 below · cited by 4 · depth 26 - An A-point of Pic⁰ classifying a given line bundle
ModularCurve.XHDRModelAtP.exists_schemeHomOver_pts_pic0Mk_eq_barPt_comp_and_poincare_pullbackAlong_iso_of_isInvertible_of_iso_ofPoint_tensor_idealModule_of_iso_tensorUnit1,205 below · cited by 1 · depth 26 - A-section through a non-crossing point of the special fibre
ModularCurve.XHDRModelAtP.exists_section_of_not_mem_range_comp946 below · cited by 1 · depth 26 - Local section cutting k times a branch at a crossing
ModularCurve.XHDRModelAtP.exists_section_slopeLaw_isUnit_ord_eq_zero_at_crossing_of_dvd_width1,340 below · cited by 1 · depth 26 - Sections through a crossing: first reduction and non-strictness
ModularCurve.XHDRModelAtP.exists_section_through_crossing_iff_reduceFst_eq_and_not_isStrict_of_offDiag_of_surjective1,247 below · cited by 2 · depth 26 - Vertical-slope function from a framed invertible module
ModularCurve.XHDRModelAtP.exists_smul_mem_integers_isGoodDiv_ord_eq_zero_verticalSlope_of_isInvertible_frames943 below · cited by 1 · depth 26 - Pairs of places on one component extend to A-points
ModularCurve.XHDRModelAtP.extendsToPlace_pts_pic0Mk_single_sub_single_of_mem_range_comp1,211 below · cited by 1 · depth 26 - Crossing points of the special fibre: finite and point-determined
ModularCurve.XHDRModelAtP.finite_and_injective_and_forall_exists_schemeHomOver_crossing_baseChange62 below · cited by 1 · depth 26 - Residue-field rationality of crossing points on the O-model
ModularCurve.XHDRModelAtP.forall_exists_spec_residueField_hom_comp_snd_eq_and_base_closedPoint_eq_crossingPt_of_surjective1 below · cited by 2 · depth 26 - Cuspidal generic place iff non-affine special point
ModularCurve.XHDRModelAtP.isCuspidal_iff_not_isAffinePlace_placeOfPoint_of_section_comp905 below · cited by 4 · depth 26 - Non-affine first reduction forces a cuspidal place
ModularCurve.XHDRModelAtP.isCuspidal_of_not_isAffinePlace_reduceFst_prolongationDatum597 below · cited by 7 · depth 26 - Cuspidal place specialising into component 0 is ∞-side
ModularCurve.XHDRModelAtP.isInftySide_of_isCuspidal_of_section_comp_zero430 below · cited by 6 · depth 26 - Integrality of mathfrak X_{P_ℓ} and density of its geometric generic fibre
ModularCurve.XHDRModelAtP.isIntegral_pullback_specMap_and_nonempty_preimage_of_nonempty_and_isOpenImmersion5 below · cited by 15 · depth 26 - Geometric generic fibre: open immersion and function field
ModularCurve.XHDRModelAtP.isOpenImmersion_and_exists_functionField_ringEquiv_of_genericFibre2 below · cited by 3 · depth 26 - Cuspidal section closing on component 1 lies on the zero side
ModularCurve.XHDRModelAtP.isZeroSide_of_isCuspidal_of_section_comp_one512 below · cited by 4 · depth 26 - Norm identity for the first reduction map at level Γ_H
ModularCurve.XHDRModelAtP.mapDomain_reduceFst_eq_ord_add_ord_of_norm_prolongationDatum55 below · cited by 2 · depth 26 - Norm identity for a Γ_H prolongation datum
ModularCurve.XHDRModelAtP.mapDomain_reduceFst_eq_ord_add_ord_of_norm_prolongationDatum_min55 below · cited by 1 · depth 26 - Zero-side places: Frobenius of the second reading is non-affine
ModularCurve.XHDRModelAtP.not_isAffinePlace_frob_reduceSnd_of_isZeroSide_prolongationDatum987 below · cited by 4 · depth 26 - ∞-side places reduce to non-affine places under the first reading
ModularCurve.XHDRModelAtP.not_isAffinePlace_reduceFst_of_isInftySide_prolongationDatum595 below · cited by 8 · depth 26 - Sections closing on component 1 are not ∞-side
ModularCurve.XHDRModelAtP.not_isInftySide_of_section_comp_one452 below · cited by 1 · depth 26 - Cusps of the special fibre lie on one component only
ModularCurve.XHDRModelAtP.not_mem_range_comp_of_not_isAffinePlace_placeOfPoint54 below · cited by 2 · depth 26 - One-sided first laws for the modular unit Δ(q)/Δ(qᵖ)
ModularCurve.XHDRModelAtP.oneSidedFst_laws_of_coe_eq_coeffEmb_modularUnitSeries_prolongationDatum_of_isModel1,368 below · cited by 2 · depth 26 - One-sided second laws for p¹²u⁻¹ on a Deligne–Rapoport model
ModularCurve.XHDRModelAtP.oneSidedSnd_laws_pow_twelve_mul_inv_of_coe_eq_coeffEmb_modularUnitSeries_prolongationDatum_of_isModel1,384 below · cited by 1 · depth 26 - Order at non-supersingular points of the Atkin–Lehner fibre component
ModularCurve.XHDRModelAtP.ord_placeOfPoint_eq_sum_ite_comp_one_of_not_mem_ssPlacesQExp_of_mul_coeffMap_eq_coeffMap1,012 below · cited by 2 · depth 26 - Order of the reduced function at non-supersingular places
ModularCurve.XHDRModelAtP.ord_placeOfPoint_eq_sum_ite_of_not_mem_ssPlacesQExp_of_mul_coeffMap_eq_coeffMap1,011 below · cited by 5 · depth 26 - Regularity of residues at affine places on both components
ModularCurve.XHDRModelAtP.ord_residue_nonneg_of_mem_integers_of_isAffinePlace_of_forall_reduce_eq_ord_nonneg_of_prolongationDatum1,052 below · cited by 1 · depth 26 - Place of special point as Gauss specialisation of restricted place
ModularCurve.XHDRModelAtP.placeOfPoint_eq_sp_restrictAlong_of_specializes_levelN_of_gauss1,018 below · cited by 1 · depth 26 - Integrality and residues of a section at both special-fibre components
ModularCurve.XHDRModelAtP.read_mem_integers_and_residue_eq_restrict_comp_of_mem916 below · cited by 3 · depth 26 - Zero-side places: first reading equals Frobenius of second reading
ModularCurve.XHDRModelAtP.reduceFst_eq_frob_reduceSnd_of_isZeroSide_prolongationDatum449 below · cited by 3 · depth 26 - Strong pole cancellation for a prolongation datum of X_H(M)
ModularCurve.XHDRModelAtP.strongPoleCancellation_prolongationDatum97 below · cited by 2 · depth 26 - Galois invariance of sections read over a coefficient ring
ModularCurve.XHDRModelAtP.arithmeticGalois_smul_read_eq_of_forall_apply_eq50 below · cited by 1 · depth 27 - Finite-chart functions on the component Σ^∞ reduce to q-expansions
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_app_comp_zero_iotaFin_eq_coeffMap_of_mfib411 below · cited by 3 · depth 27 - Pole-chart sections on the zeroth fibre component reduce q-expansions
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_app_comp_zero_iotaInf_eq_coeffMap_of_mfib_of_not_sq_dvd410 below · cited by 7 · depth 27 - Local semicontinuity at the ∞-side cusps, first prolongation
ModularCurve.XHDRModelAtP.cuspLocalSemicontinuityInfty_prolongationDatum_of_residue905 below · cited by 2 · depth 27 - Semicontinuity of 0-side cusp orders under the second reduction
ModularCurve.XHDRModelAtP.cuspLocalSemicontinuityZero_prolongationDatum_of_residue1,217 below · cited by 1 · depth 27 - Unique extension of ℚ̄-points over valuation rings
ModularCurve.XHDRModelAtP.existsUnique_hom_comp_toBase_eq_and_specMap_comp_eq_of_point0 below · cited by 1 · depth 27 - Crossing coordinates of an A-section through the node
ModularCurve.XHDRModelAtP.exists_comp_eq_specMap_and_mem_maximalIdeal_and_mul_eq_of_section_of_chart0 below · cited by 2 · depth 27 - Residue-disc expansion of stalk germs at a strict place of the first kind
ModularCurve.XHDRModelAtP.exists_discParameter_ringHom_powerSeries_range_stalk_read_of_isStrictFst939 below · cited by 1 · depth 27 - Residue-disc expansion of germs at strict places of the second kind
ModularCurve.XHDRModelAtP.exists_discParameter_ringHom_powerSeries_range_stalk_read_of_isStrictSnd942 below · cited by 1 · depth 27 - Cuspidal sections factor through the pole chart
ModularCurve.XHDRModelAtP.exists_eq_specMap_comp_iotaInf_of_isCuspidal_of_section2 below · cited by 5 · depth 27 - Common value at supersingular nodes of the reduced fibre
ModularCurve.XHDRModelAtP.exists_hasValue_residue_pair_of_mem_ssNodePairs_of_forall_reduceFst_eq_reduceSnd_eq_ord_nonneg_of_prolongationDatum467 below · cited by 2 · depth 27 - Sections over A and rational places of X_H
ModularCurve.XHDRModelAtP.exists_isRational_comp_eq_pointEquivPlace_and_eq_of_comp_eq2 below · cited by 2 · depth 27 - Horizontal primes of a crossing stalk as kernels of sections
ModularCurve.XHDRModelAtP.exists_isRational_section_forall_mem_iff_stalkClosedPointTo_eq_zero_of_isPrime_of_not_mem17 below · cited by 2 · depth 27 - Unit germs at the crossing point arise from unit sections
ModularCurve.XHDRModelAtP.exists_isUnit_section_read_eq_stalkRead_of_isUnit0 below · cited by 1 · depth 27 - Minimal primes of (p) on the pole chart of X_H(M)
ModularCurve.XHDRModelAtP.exists_minimalPrimes_chartAlgInf_eq_pair_and_mem_iff_gauss_and_mem_range_comp_iff_le426 below · cited by 3 · depth 27 - Hensel lifting of A-points on the étale crossing chart
ModularCurve.XHDRModelAtP.exists_section_base_closedPoint_eq_and_comp_eq_specMap_of_chart3 below · cited by 2 · depth 27 - Strict second-kind places as A-sections closing on the second component
ModularCurve.XHDRModelAtP.exists_section_comp_one_placeOfPoint_eq_reduceSnd_of_isStrictSnd261 below · cited by 2 · depth 27 - Places give A-valued sections of the base-changed model
ModularCurve.XHDRModelAtP.exists_section_comp_snd_eq_barPt_comp_eq_pointEquivPlace_symm0 below · cited by 1 · depth 27 - Sections realising strict places of the first kind
ModularCurve.XHDRModelAtP.exists_section_comp_zero_placeOfPoint_eq_reduceFst_of_isStrictFst0 below · cited by 2 · depth 27 - Unit principle on the node annulus of X_H(M)
ModularCurve.XHDRModelAtP.exists_zpow_unit_principle_evalAt_read_chart_of_section_of_chart1,113 below · cited by 1 · depth 27 - Function field elements as fractions from a crossing stalk
ModularCurve.XHDRModelAtP.forall_exists_isDiscreteValuationRing_specializes_and_mul_stalkRead_eq_stalkRead129 below · cited by 2 · depth 27 - Primes of Pl⊗ B as points of a chart
ModularCurve.XHDRModelAtP.injective_and_exists_pointEquivPlace_mem_iff_of_tmul_eq_smul_coeffEmb3 below · cited by 6 · depth 27 - Node stalk over a second DVR: injectivity, coordinates, evaluation
ModularCurve.XHDRModelAtP.injective_stalkRead_and_stalkRead_germ_eq_read_chart_and_forall_section_evalAt_stalkRead_eq_of_chart7 below · cited by 2 · depth 27 - Nodal completion of the stalk after coefficient change
ModularCurve.XHDRModelAtP.isNoetherianRing_stalk_and_exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_of_chart55 below · cited by 2 · depth 27 - Atkin–Lehner twist exchanges zero-side and infinity-side places
ModularCurve.XHDRModelAtP.isZeroSide_iff_isInftySide_smul_prolongationDatum1,055 below · cited by 2 · depth 27 - Hartogs criterion for germs at a smooth special-fibre point
ModularCurve.XHDRModelAtP.mem_range_stalk_read_of_mem_integers_of_forall_isStrictFst_mem1,015 below · cited by 1 · depth 27 - Hartogs regularity at a strict place of the second kind
ModularCurve.XHDRModelAtP.mem_range_stalk_read_of_mem_integers_of_forall_isStrictSnd_mem1,022 below · cited by 1 · depth 27 - Crossing chart coordinates have order one on both components
ModularCurve.XHDRModelAtP.ord_placeOn_germ_chart_eq_one_of_chart_of_residue_surjective2 below · cited by 2 · depth 27 - Unit sections have order zero at both places of a node
ModularCurve.XHDRModelAtP.ord_placeOn_germ_eq_zero_of_isUnit_section2 below · cited by 1 · depth 27 - Crossing chart coordinate minus its value is a uniformiser
ModularCurve.XHDRModelAtP.ord_read_chart_sub_algebraMap_eq_one_of_section_of_chart5 below · cited by 1 · depth 27 - Readings of sections at the two components: integrality and residue
ModularCurve.XHDRModelAtP.readA_mem_integers_and_residue_eq_restrict_comp_of_mem915 below · cited by 7 · depth 27 - Section through a crossing: red₁ and non-strictness
ModularCurve.XHDRModelAtP.reduceFst_eq_and_not_isStrict_of_section_closedPoint_eq_crossing_of_offDiag1,242 below · cited by 1 · depth 27 - Strict-second places as θ⁻¹-translates of strict-first places
ModularCurve.XHDRModelAtP.reduceSnd_ofAlgAut_symm_smul_eq_reduceFst_and_isStrictSnd_iff_isStrictFst_ofAlgAut_symm_smul_prolongationDatum379 below · cited by 1 · depth 27 - A non-strict section reduces to the prescribed crossing
ModularCurve.XHDRModelAtP.section_closedPoint_eq_crossing_of_reduceFst_eq_of_not_isStrict_of_offDiag3 below · cited by 1 · depth 27 - Uniqueness of A-points through a crossing on a crossing chart
ModularCurve.XHDRModelAtP.section_eq_of_comp_chart_eq_of_base_closedPoint_eq_of_chart2 below · cited by 1 · depth 27 - A point dominated by R₁ equals ξ_∞
ModularCurve.XHDRModelAtP.eq_xiInf_of_base_eq_closedPoint_of_forall_isUnit_germ_iff_residue_ne_zero14 below · cited by 1 · depth 28 - A centre for the prolongation R₁ on the model over A
ModularCurve.XHDRModelAtP.exists_base_eq_closedPoint_and_forall_readA_mem_integers_and_isUnit_germ_iff0 below · cited by 1 · depth 28 - Base change of a crossing chart along a second coefficient DVR
ModularCurve.XHDRModelAtP.exists_chart_baseChange_mem_and_flat_and_map_maximalIdeal_eq_and_isIso_residueFieldMap_and_germ_eq_of_chart8 below · cited by 1 · depth 28 - Rational functions of X_H(M) as fractions in a crossing stalk
ModularCurve.XHDRModelAtP.exists_coeffEmb_mul_stalkRead_eq_stalkRead126 below · cited by 1 · depth 28 - Coefficient descent to a DVR with rational special point
ModularCurve.XHDRModelAtP.exists_coeffRing_isIso_residueFieldMap_and_mul_stalkRead_eq138 below · cited by 2 · depth 28 - Étale coordinate at a smooth point of the special fibre
ModularCurve.XHDRModelAtP.exists_etale_chart_affineLine_of_isStrictFst5 below · cited by 1 · depth 28 - Étale coordinate to A¹_A at a strict second-kind point
ModularCurve.XHDRModelAtP.exists_etale_chart_affineLine_of_isStrictSnd5 below · cited by 1 · depth 28 - Existence of an ∞-side cusp chart for the first prolongation
ModularCurve.XHDRModelAtP.exists_isCuspChartFstAt_of_isInftySide_prolongationDatum884 below · cited by 1 · depth 28 - Denominators outside varpi' for Gauss-integral functions
ModularCurve.XHDRModelAtP.exists_notMem_span_and_mul_stalkRead_eq_of_mem_integers_of_isIso_residueFieldMap_of_not_mem_range_comp_one982 below · cited by 1 · depth 28 - Clearing denominators outside the vertical prime at a non-crossing point
ModularCurve.XHDRModelAtP.exists_notMem_span_and_mul_stalkRead_eq_of_mem_integers_of_isIso_residueFieldMap_of_not_mem_range_comp_zero999 below · cited by 1 · depth 28 - Uniformiser at one ∞-side cusp over v
ModularCurve.XHDRModelAtP.exists_ord_eq_one_section_of_isInftySide_prolongationDatum834 below · cited by 1 · depth 28 - Branch primes, Gauss localisations and values at a crossing
ModularCurve.XHDRModelAtP.exists_primes_tensorProduct_chartAlgFin_crossing_gauss_iff_and_section_and_hasValue460 below · cited by 1 · depth 28 - Algebraisation of a crossing chart: nodal completion and horizontal primes
ModularCurve.XHDRModelAtP.exists_ringEquiv_adicCompletion_uvCrossingModel_and_forall_section_exists_ringHom_evalAt_eq_of_chart216 below · cited by 1 · depth 28 - Horizontal primes at a special-fibre point as kernels of sections
ModularCurve.XHDRModelAtP.exists_section_forall_mem_iff_stalkClosedPointTo_eq_zero_of_point17 below · cited by 2 · depth 28 - Transporting the residue dictionary from ξ_∞ to ξ₀
ModularCurve.XHDRModelAtP.forall_readA_mem_integers_snd_of_forall_readA_mem_integers_fst55 below · cited by 1 · depth 28 - Integrality of the X_H(M) model over a discrete valuation ring
ModularCurve.XHDRModelAtP.isIntegral_xO_of_mem_maximalIdeal3 below · cited by 7 · depth 28 - Normal two-dimensional stalk at a non-crossing rational point
ModularCurve.XHDRModelAtP.isIntegrallyClosed_stalk_and_ringKrullDim_eq_two_of_isIso_residueFieldMap_of_not_mem_range_comp955 below · cited by 4 · depth 28 - Reading germs at a point in the geometric function field
ModularCurve.XHDRModelAtP.isLocalHom_and_injective_stalkRead_and_forall_section_evalAt_eq_of_point5 below · cited by 3 · depth 28 - Maximal ideal at a crossing: p and the chart coordinates
ModularCurve.XHDRModelAtP.maximalIdeal_stalk_crossing_eq_span_germ_chart_of_residue_surjective0 below · cited by 1 · depth 28 - Vanishing on the zero component forces Gauss nonunit
ModularCurve.XHDRModelAtP.mem_nonunits_gauss_of_ffEquiv_symm_germToFunctionField_app_comp_zero_eq_zero53 below · cited by 1 · depth 28 - θ∘θ acts as an inverse diamond on places
ModularCurve.XHDRModelAtP.ofAlgAut_smul_ofAlgAut_smul_eq_ofAlgAut_diamondAutHBar_inv_smul_of_unitsMap_mul_eq_one_prolongationDatum0 below · cited by 2 · depth 28 - Étale coordinate is a uniformiser at a rational place
ModularCurve.XHDRModelAtP.ord_read_chart_sub_algebraMap_eq_one_of_section_of_etale_chart_of_isStrictFst8 below · cited by 1 · depth 28 - Étale chart coordinate minus its value is a uniformiser
ModularCurve.XHDRModelAtP.ord_read_chart_sub_algebraMap_eq_one_of_section_of_etale_chart_of_isStrictSnd8 below · cited by 1 · depth 28 - Frobenius squared equals the diamond ⟨ p⟩⁻¹ at supersingular places
ModularCurve.XHDRModelAtP.qExpFrobeniusPlaceModL_qExpFrobeniusPlaceModL_eq_diamondActionModL_smul_of_mem_ssPlacesQExp0 below · cited by 1 · depth 28 - Diamond equivariance of the two place reductions
ModularCurve.XHDRModelAtP.reduceFst_smul_diamondAutHBar_eq_and_reduceSnd_smul_eq_of_section_comp_prolongationDatum972 below · cited by 1 · depth 28 - Diamond equivariance of both readings of a configured place
ModularCurve.XHDRModelAtP.reduceFst_smul_diamondAutHBar_eq_and_reduceSnd_smul_eq_of_section_comp_prolongationDatum_all348 below · cited by 1 · depth 28 - First unit coefficient computes the residue order at a strict place
ModularCurve.XHDRModelAtP.residue_ne_zero_and_ord_residue_eq_of_forall_coeff_mem_of_isStrictFst918 below · cited by 1 · depth 28 - First unit coefficient computes the order of the reduced germ
ModularCurve.XHDRModelAtP.residue_ne_zero_and_ord_residue_eq_of_forall_coeff_mem_of_isStrictSnd918 below · cited by 1 · depth 28 - R₁-residue of a germ equals restriction along the zero component
ModularCurve.XHDRModelAtP.residue_readA_eq_restrict_comp_zero_of_forall_isUnit_germ_iff_residue_ne_zero894 below · cited by 1 · depth 28 - Uniqueness of A-sections with a common étale coordinate
ModularCurve.XHDRModelAtP.section_eq_of_specMap_residue_comp_eq_of_comp_etale_chart_eq_of_isStrictFst0 below · cited by 1 · depth 28 - Uniqueness of A-sections in an étale chart
ModularCurve.XHDRModelAtP.section_eq_of_specMap_residue_comp_eq_of_comp_etale_chart_eq_of_isStrictSnd0 below · cited by 1 · depth 28 - Étaleness of the ∞-cusp chart at first-reduction places
ModularCurve.XHDRModelAtP.chartEtaleAt_cuspChartSetInf_of_isInftySide_prolongationDatum740 below · cited by 1 · depth 29 - Branch points ξ_∞,ξ₀ are maximal in the special fibre
ModularCurve.XHDRModelAtP.eq_xi_of_specializes_of_maximalIdeal_eq_span64 below · cited by 1 · depth 29 - Branch primes at a crossing: Gauss criterion and evaluation
ModularCurve.XHDRModelAtP.exists_branch_primes_gauss_iff_and_hasValue_of_crossing_prime453 below · cited by 1 · depth 29 - Base change of the crossing chart along O → O'
ModularCurve.XHDRModelAtP.exists_chart_baseChange_comp_spec_eq_morphismRestrict_comp_and_etale_of_chart2 below · cited by 1 · depth 29 - Injective stalk reading of rational functions at a point
ModularCurve.XHDRModelAtP.exists_coeffEmb_mul_stalkRead_eq_stalkRead_of_point126 below · cited by 1 · depth 29 - A level-M/p lift uniformising a fibre place along cuspidal sections
ModularCurve.XHDRModelAtP.exists_lift_regular_section_comp_zero_ord_placeOfPoint_eq_one_prolongationDatum418 below · cited by 1 · depth 29 - Clearing denominators inside the ∞-side cusp chart
ModularCurve.XHDRModelAtP.exists_mul_eq_of_mem_integers_of_forall_sp_eq_cuspChartSetInf_prolongationDatum856 below · cited by 1 · depth 29 - A crossing of the special fibre as a prime of the j-chart
ModularCurve.XHDRModelAtP.exists_prime_tensorProduct_chartAlgFin_crossing_and_section_closes9 below · cited by 1 · depth 29 - A separator t_∞-a at zero-side places over an ∞-side reading
ModularCurve.XHDRModelAtP.exists_sub_algebraMap_residue_ne_and_ord_pos_of_isZeroSide_of_isInftySide145 below · cited by 1 · depth 29 - Gauss-dominated closed-fibre point is not a closed point
ModularCurve.XHDRModelAtP.ne_comp_efib_of_mem_closedPoints_of_forall_isUnit_germ_iff_residue_ne_zero0 below · cited by 1 · depth 29 - A point dominated by R₁ is not ξ₀
ModularCurve.XHDRModelAtP.ne_xiZero_of_forall_isUnit_germ_iff_residue_ne_zero10 below · cited by 1 · depth 29 - Chart function readings agree with the reduced q-expansion
ModularCurve.XHDRModelAtP.residue_readA_chart_eq_and_restrict_comp_zero_chart_eq_coeffMap_of_coeffMap_eq_coeffEmb0 below · cited by 2 · depth 29 - Values of the level-M/p pole chart at a cuspidal place
ModularCurve.XHDRModelAtP.exists_hasValue_and_hasValue_sp_residue_of_mem_closure_chartAlgInf_of_cusp578 below · cited by 1 · depth 30 - Separating two non-affine places by pole-chart functions
ModularCurve.XHDRModelAtP.exists_mem_closure_chartAlgInf_hasValue_residue_zero_and_not_hasValue_of_not_isAffinePlace_of_ne6 below · cited by 1 · depth 30 - Second residue of the cusp coordinate j(qᵖ)j⁻ᵖ
ModularCurve.XHDRModelAtP.exists_mem_integers_residue_eq_jqModC_mul_inv_qExpand_pow_of_residue_eq_qExpFrobeniusModL666 below · cited by 1 · depth 30 - Uniformiser at a pole-chart point of the special fibre
ModularCurve.XHDRModelAtP.exists_ord_placeOfPoint_sum_smul_ffEquiv_symm_germToFunctionField_app_iotaInf_eq_one_of_mem_preimage_iotaInf1 below · cited by 1 · depth 30 - Second branch prime at a supersingular crossing
ModularCurve.XHDRModelAtP.exists_snd_branch_prime_of_crossing_prime_of_regularProlongation452 below · cited by 1 · depth 30 - Chart fraction at a crossing: common value on both branches
ModularCurve.XHDRModelAtP.hasValue_placeOn0_and_placeOn1_of_mul_eq_of_not_mem_crossing_prime447 below · cited by 2 · depth 30 - Chart function restricted to the Σ⁰ component reads ̄ y(qᵖ)
ModularCurve.XHDRModelAtP.restrict_comp_one_chart_eq_qExpand_coeffMap_of_coeffMap_eq_coeffEmb7 below · cited by 1 · depth 30 - θ acts on functions as pull-back along w
ModularCurve.XHDRModelAtP.algEquiv_ffEquiv_symm_germToFunctionField_eq_of_pointEquivPlace_eq_ofAlgAut_smul55 below · cited by 2 · depth 31 - Chart function on the branch `comp 1` reads as ̄ y(qᵖ)
ModularCurve.XHDRModelAtP.coe_ffEquiv_symm_germToFunctionField_app_comp_one_iotaFin_iota0_eq_qExpand_coeffMap_of_mfib5 below · cited by 1 · depth 31 - Reductions of q-expansions as chart values on the fibre
ModularCurve.XHDRModelAtP.coeffMap_residue_mem_and_sub_algebraMap_mem_nonunits_pointEquivPlace_mfib_of_comp_efib_eq5 below · cited by 1 · depth 31 - Gauss witness for a section and its reduction at ∞
ModularCurve.XHDRModelAtP.exists_gaussWitness_and_ffEquiv_symm_germToFunctionField_mfib_eq_of_mem_opens420 below · cited by 7 · depth 31 - Mod p reduction of g dj along the Gauss branch
ModularCurve.XHDRModelAtP.exists_kaehlerDifferential_diffQExp_eq_intSeriesC_of_eq_smul_D_jAt_of_mem_gauss138 below · cited by 1 · depth 31 - Regularity of g dj at smooth special-fibre chart points
ModularCurve.XHDRModelAtP.exists_pow_smul_eq_sum_smul_D_chartAlgFin_of_mem_gauss_of_mem_smoothLocus143 below · cited by 1 · depth 31 - Regularity of g dj at smooth special-fibre points of the j⁻¹-chart
ModularCurve.XHDRModelAtP.exists_pow_smul_eq_sum_smul_D_chartAlgInf_of_mem_gauss_of_mem_smoothLocus143 below · cited by 1 · depth 31 - Distinctness and exhaustiveness of the two prolongations at p ∥ M
ModularCurve.XHDRModelAtP.integers_ne_and_forall_valuationSubring_eq_or_eq_of_residue_eq_qExpFrobeniusModL617 below · cited by 1 · depth 31 - Chart-local presentation of η gives regularity after reduction
ModularCurve.XHDRModelAtP.isRegularAt_placeOfPoint_of_smul_eq_sum_smul_D_chartAlgFin_of_diffQExp_eq_intSeriesC424 below · cited by 1 · depth 31 - Chart-local presentation gives regularity of the reduced differential
ModularCurve.XHDRModelAtP.isRegularAt_placeOfPoint_of_smul_eq_sum_smul_D_chartAlgInf_of_diffQExp_eq_intSeriesC423 below · cited by 1 · depth 31 - Gauss valuation and prime p at a finite-chart point
ModularCurve.XHDRModelAtP.mem_gauss_and_isPrime_map_span_of_mem_range_comp_zero_of_not_mem_range_comp_one_chartAlgFin1,001 below · cited by 1 · depth 31 - Pole-chart point off the second component: Gauss ring, p prime
ModularCurve.XHDRModelAtP.mem_gauss_and_isPrime_map_span_of_mem_range_comp_zero_of_not_mem_range_comp_one_chartAlgInf436 below · cited by 1 · depth 31 - Pole-chart morphism to the special fibre intertwining a diamond
ModularCurve.XHDRModelAtP.exists_chartInf_fibre_spec_map_tensor_comp_eq_comp_fibreMap_dia0 below · cited by 2 · depth 32 - Diamond at e lifts to germs and reduces mod l
ModularCurve.XHDRModelAtP.exists_etaEmb_eq_diamondAutHBar_and_etaRes_eq_diamondActionModL349 below · cited by 1 · depth 32 - Gauss residue of the Uₚ-pushed function is c·Frobenius
ModularCurve.XHDRModelAtP.exists_mem_integers_algebraMap_mul_smul_norm_heckeBetaHBar_and_coe_residue_eq_C_mul_coeffMap_frobenius_coe_residue_of_mem_integers_of_algEquiv332 below · cited by 1 · depth 32 - Two minimal primes of (p) on the finite chart, oriented
ModularCurve.XHDRModelAtP.exists_minimalPrimes_chartAlgFin_eq_pair_and_mem_iff_gauss_and_mem_range_comp_iff_le1,000 below · cited by 1 · depth 32 - Generic-point stalk of the model: a maximal proper valuation subring
ModularCurve.XHDRModelAtP.injective_etaEmb_and_exists_valuationSubring_range_eq_rankOne_and_const428 below · cited by 2 · depth 32 - Genericity of the second prolongation and p-th powers along α
ModularCurve.XHDRModelAtP.integers_snd_isGeneric_and_forall_exists_valuation_alpha_sub_pow_lt_one_of_residue_eq_qExpFrobeniusModL3 below · cited by 1 · depth 32 - Diamond invariance of the Σ^∞ condition on the pole chart
ModularCurve.XHDRModelAtP.map_ker_le_asIdeal_iff_map_ker_le_spec_map_tensor_asIdeal55 below · cited by 1 · depth 32 - Germs at the Σ^∞ generic point form the Gauss ring
ModularCurve.XHDRModelAtP.mem_range_etaEmb_iff_mem_integers_and_coe_etaRes_eq_coe_residue429 below · cited by 1 · depth 32 - Diamond translate of a p-division datum on J_H(M)
ModularCurve.XHDRModelAtP.mk_degZeroSMulHom_diamondAutHBar_eq_genOpH_dia_and_forall_mul_smul_eq_ord0 below · cited by 1 · depth 32 - Reduced diamond automorphisms stabilise the range of comp₀
ModularCurve.XHDRModelAtP.range_comp_zero_fibreMap_dia0 below · cited by 2 · depth 32 - Function-field automorphism attached to X.w: existence, uniqueness
ModularCurve.XHDRModelAtP.existsUnique_algEquiv_pointEquivPlace_eq_ofAlgAut_smul_of_comp_w_eq56 below · cited by 2 · depth 33 - Atkin–Lehner translation pulls back the Poincaré bundle
ModularCurve.XHDRModelAtP.nonempty_poincare_pullbackAlong_pts_smul_iso_pullback_w_of_abelJacobiPin59 below · cited by 3 · depth 33 - Model automorphism w realised on the function field
ModularCurve.XHDRModelAtP.exists_algEquiv_pointEquivPlace_eq_ofAlgAut_smul_of_comp_w_eq0 below · cited by 1 · depth 34 - Atkin–Lehner automorphism of the X_H(M) model over A
ModularCurve.XHDRModelAtP.exists_iso_pullback_toBase_specMap_atkinLehner_complement_placePin180 below · cited by 1 · depth 34 - Divisible supersingular root orders give a good class
ModularCurve.XHDRModelAtP.isGoodClass_of_forall_dvd_ord_residue_of_annulus_offDiag_of_wgen1,754 below · cited by 1 · depth 34 - Unit sections have reduction of order zero at ̄ P
ModularCurve.XHDRModelAtP.ord_placeOfPoint_eq_zero_of_isUnit_of_ffEquiv_symm_germToFunctionField_eq421 below · cited by 1 · depth 34 - Balanced configurations on a supersingular node annulus are principal
ModularCurve.XHDRModelAtP.exists_ord_eq_and_smul_mem_integers_of_isUnit_mul_modulus_zpow_eq_prod_neg_evalAt_zpow_of_annulus_offDiag_of_wgen1,561 below · cited by 1 · depth 35 - Moving a class of J_H(M) off δ-fixed places
ModularCurve.XHDRModelAtP.exists_pic0Mk_eq_forall_isStrict_or_reduceFst_mem_of_prolongationDatum_offDiag_of_wgen1,734 below · cited by 1 · depth 35 - Existence of a residue carrier for a prescribed jump varpi
ModularCurve.XHDRModelAtP.exists_residueCarrier_of_ne_zero_of_prolongationDatum_offDiag_of_wgen1,334 below · cited by 1 · depth 35 - Hensel lifting of a special point off one component
ModularCurve.XHDRModelAtP.exists_section_smoothLocus_specialPoint_eq_of_notMem_range_comp9 below · cited by 2 · depth 35 - Placement of an effective configuration on a node annulus
ModularCurve.XHDRModelAtP.exists_ord_residue_eq_and_ord_eq_of_nonneg_of_isUnit_mul_modulus_zpow_eq_prod_neg_evalAt_zpow_of_annulus_offDiag_of_wgen1,560 below · cited by 1 · depth 36 - Band of residue carriers from a vertical unit
ModularCurve.XHDRModelAtP.exists_residueCarrier_pow_of_verticalUnit_of_prolongationDatum_offDiag_of_wgen1,320 below · cited by 1 · depth 36 - Existence of Ogg's vertical unit with Atkin–Lehner relation
ModularCurve.XHDRModelAtP.exists_verticalUnit_atkinLehner_eq_mul_inv_residue_eq_prod_ssJSet_of_prolongationDatum_offDiag_of_wgen493 below · cited by 1 · depth 36 - Atkin–Lehner swap of j, j(qᵖ) and cuspidal sides
ModularCurve.XHDRModelAtP.atkinLehner_swap_and_exists_coe_eq_modularUnitSeries_and_isInftySide_smul_iff_isZeroSide_of_wgen191 below · cited by 1 · depth 37 - Lifting node-compatible fibre pairs to L(D₀-E) across a supersingular annulus
ModularCurve.XHDRModelAtP.exists_forall_exists_mem_riemannRochSpace_sub_and_residue_eq_of_hasValue_leading_of_nonneg_of_annulus_offDiag_of_wgen1,540 below · cited by 1 · depth 37 - Non-strict zeros of a pole-free factor lie over supersingular nodes
ModularCurve.XHDRModelAtP.exists_mem_reduceFst_eq_of_ord_ne_zero_of_mul_commonUnit_of_ord_nonneg_of_ord_residue_eq_zero_of_prolongationDatum_offDiag_of_wgen1,310 below · cited by 1 · depth 37 - Residues at a node of bi-integral sections of L(D₀-E)
ModularCurve.XHDRModelAtP.residue_mem_riemannRochSpace_sub_and_hasValue_of_mem_riemannRochSpace_sub_of_annulus_offDiag_of_wgen1,332 below · cited by 1 · depth 38
ModularCurve.XOne 18
- Places of X₁(M) in characteristic p give chart points
ModularCurve.XOne.exists_injective_algHom_tensorProduct_chartAlgFin_apply_eq_evalAt_x1FunctionFieldC952 below · cited by 1 · depth 24 - Correspondence α_*β^* on J₁(N) realised by an endomorphism
ModularCurve.XOne.pic0Correspondence_pts_eq_comp_of_poincare_pullbackAlong_iso_laurentBaseChange180 below · cited by 1 · depth 24 - Mod p charts of X₁(M) lie in Gauss reductions
ModularCurve.XOne.chartRing_le_adjoin_gaussReductions_chartAlg_x1947 below · cited by 1 · depth 25 - Good reduction of X₁(M) at p∤ M, model form
ModularCurve.XOne.smooth_toBase_and_isIntegral_pullback_twoChartIntegralModel_x1936 below · cited by 5 · depth 25 - Prime-to-p level raising is finite étale on the j-finite chart
ModularCurve.XOne.finite_and_etale_chartAlgFin_levelRaise_x11,222 below · cited by 2 · depth 26 - Gauss reductions of the integral charts of X₁(M) have DVR localisations
ModularCurve.XOne.isDiscreteValuationRing_localization_atPrime_adjoin_gaussReductions_chartAlg_x1944 below · cited by 1 · depth 26 - Level raising by ℓ is finite and flat on j-charts
ModularCurve.XOne.finite_and_flat_chartAlgFin_levelRaise_x1949 below · cited by 1 · depth 27 - Special fibre of the j-finite chart of X₁(M) is prime
ModularCurve.XOne.isPrime_map_maximalIdeal_chartAlgFin_twoChartIntegralModel_x1945 below · cited by 1 · depth 27 - Separability of the residue extension at a vertical height-one prime
ModularCurve.XOne.isSeparable_residueField_of_height_eq_one_of_map_maximalIdeal_le_chartAlgFin_levelRaise_x11,186 below · cited by 1 · depth 27 - Mod-varpi local rings of the j-finite chart are DVRs
ModularCurve.XOne.exists_isDomain_isDiscreteValuationRing_localization_quotient_span_chartAlgFin_twoChartIntegralModel_x1939 below · cited by 2 · depth 28 - At good level (varpi) is prime, not maximal, in the j-finite chart
ModularCurve.XOne.isDomain_quotient_span_and_not_isMaximal_chartAlgFin_twoChartIntegralModel_x1939 below · cited by 2 · depth 28 - Local rings of the j-finite chart ring of X₁(M) off the special fibre are principal
ModularCurve.XOne.isPrincipalIdealRing_localization_atPrime_chartAlgFin_of_not_mem_twoChartIntegralModel_x17 below · cited by 1 · depth 28 - No three-step prime chain above varpi in the finite chart
ModularCurve.XOne.not_lt_of_lt_of_mem_of_isPrime_chartAlgFin_twoChartIntegralModel_x17 below · cited by 3 · depth 28 - Local ring of the Gauss sheet is a discrete valuation ring
ModularCurve.XOne.isDiscreteValuationRing_and_isFractionRing_of_mem_iff_exists_mul_residue_eq_of_gamma1_inf_gamma0_le983 below · cited by 1 · depth 29 - Primes above the Gauss prime in the j-chart are maximal
ModularCurve.XOne.isMaximal_asIdeal_of_forall_mem_nonunits_imp_mem_of_exists_not_mem_nonunits1 below · cited by 1 · depth 29 - Discrete valuation ring at a closed point of X₁(M)
ModularCurve.XOne.isDiscreteValuationRing_and_isFractionRing_of_mem_iff_exists_mul_residue_eq_x1948 below · cited by 1 · depth 30 - Gauss valuation ring as a localisation of the chart ring
ModularCurve.XOne.forall_exists_mul_residue_eq_and_mem_iff_exists_mul_eq_gaussValuationSubring_chartAlgFin_x112 below · cited by 1 · depth 31 - Gauss centre of the j-finite chart of X₁(M) is (varpi)
ModularCurve.XOne.mem_span_of_coe_mem_nonunits_gaussValuationSubring_chartAlgFin_twoChartIntegralModel_x1941 below · cited by 1 · depth 31
ModularCurve.XOneGammaZeroP 24
- Tame supersingular node: widehat𝒪≅ W[[U,V]]/(UV-varpiᵖ⁻¹)
ModularCurve.XOneGammaZeroP.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_pow_of_mem_ssJSet_twoChartIntegralModel_of_tame2,530 below · cited by 1 · depth 25 - Chart-level package at a supersingular point over the X₀(Mp) node
ModularCurve.XOneGammaZeroP.chartPackage_floorHom_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_tame2,438 below · cited by 1 · depth 26 - Tame inertia at supersingular points: cyclic, order prime to p
ModularCurve.XOneGammaZeroP.isCyclic_inertia_and_not_dvd_card_inertia_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_tame1,598 below · cited by 1 · depth 26 - Inertia of order at most 3 at supersingular chart primes
ModularCurve.XOneGammaZeroP.card_inertia_le_three_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_five_le1,560 below · cited by 1 · depth 27 - Cyclic inertia of order prime to 3 at supersingular points
ModularCurve.XOneGammaZeroP.isCyclic_inertia_and_not_dvd_card_inertia_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_eq_three1,586 below · cited by 1 · depth 27 - Cyclic inertia of odd order at a supersingular point, p=2
ModularCurve.XOneGammaZeroP.isCyclic_inertia_and_not_dvd_card_inertia_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_eq_two1,586 below · cited by 1 · depth 27 - Unramified off the closed point at supersingular points
ModularCurve.XOneGammaZeroP.isUnramifiedAt_stalk_of_not_isMaximal_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma0_of_tame1,082 below · cited by 1 · depth 27 - Automorphism with σ(j)=j(qᵖ) moving the Gauss valuation ring
ModularCurve.XOneGammaZeroP.exists_algEquiv_map_j_eq_qExpand_and_chartAlgFin_iff_and_comap_ne_x1x0_gamma0170 below · cited by 3 · depth 28 - Ramification index one for valuation rings over A in K₁
ModularCurve.XOneGammaZeroP.exists_eq_mul_of_mem_nonunits_valuationSubring_x1x01,027 below · cited by 1 · depth 28 - Gauss residue fields at the floor: X₁(M) and j(q),j(q^M)
ModularCurve.XOneGammaZeroP.exists_ringEquiv_residueField_gauss_x1FunctionFieldC_and_modularFunctionFieldC_x1x0_gamma01,041 below · cited by 1 · depth 28 - Gauss residue field of the Γ₀(p)-floor equals κ(X₁(M))
ModularCurve.XOneGammaZeroP.exists_ringEquiv_residueField_x1FunctionFieldC_of_gaussPresentation_x1x01,029 below · cited by 4 · depth 28 - Inertia at a supersingular point is H/⟨-1⟩
ModularCurve.XOneGammaZeroP.exists_subgroup_units_zmod_sq_sub_mul_add_one_eq_zero_and_nonempty_quotient_mulEquiv_inertia_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma01,582 below · cited by 2 · depth 28 - Unique discrete place of the Gauss sheet at a closed point
ModularCurve.XOneGammaZeroP.exists_valuationSubring_residueField_unique_and_forall_exists_sub_residue_mem_nonunits_twoChartIntegralModel_x1x0986 below · cited by 3 · depth 28 - Gauss valuations of X₀(Mp) are inert in X(Γ₁(M)∩Γ₀(p))
ModularCurve.XOneGammaZeroP.finrank_residueField_valuationSubring_eq_finrank_and_isSeparable_of_gauss_x1x0_gamma01,040 below · cited by 5 · depth 28 - Supersingular special-fibre points lie on the Gauss sheet
ModularCurve.XOneGammaZeroP.mem_of_coe_mem_nonunits_of_mem_ssJSet_twoChartIntegralModel_x1x0_gamma01,479 below · cited by 3 · depth 28 - Group actions fixing K₂ preserve the Gauss valuation ring
ModularCurve.XOneGammaZeroP.smul_mem_gaussValuationSubring_of_forall_smul_eq_x1x0_gamma01,047 below · cited by 4 · depth 28 - Branches above the j-line for Γ₁(M)∩Γ₀(p)
ModularCurve.XOneGammaZeroP.valuationSubring_eq_or_eq_comap_and_uniformizer_and_gaussReduction_eq_x1x01,004 below · cited by 4 · depth 28 - Trivial inertia at the Gauss valuation of K₁/K₂
ModularCurve.XOneGammaZeroP.eq_one_of_forall_smul_sub_mem_nonunits_gauss_x1x0_gamma01,049 below · cited by 1 · depth 29 - An Atkin–Lehner automorphism sending j to j(qᵖ)
ModularCurve.XOneGammaZeroP.exists_algEquiv_map_j_eq_qExpand_x1x0_gamma0129 below · cited by 1 · depth 29 - Compatible Atkin–Lehner automorphisms of K₁ supseteq K₂
ModularCurve.XOneGammaZeroP.exists_algEquiv_pair_map_j_eq_qExpand_and_coe_comp_eq_x1x0_gamma0312 below · cited by 4 · depth 29 - Gauss reduction of the Γ₁(M)∩Γ₀(p) q-expansion field at level M
ModularCurve.XOneGammaZeroP.gaussReduction_mem_x1FunctionFieldC_of_x1x0137 below · cited by 2 · depth 29 - Diamond group over X₀(Mp) has order [(ℤ/M)^×:± 1]
ModularCurve.XOneGammaZeroP.natCard_eq_index_zpowers_neg_one_of_faithful_fixed_x1x0_gamma0240 below · cited by 1 · depth 29 - Atkin–Lehner automorphism of the Γ₁(M)∩Γ₀(p) q-expansion field
ModularCurve.XOneGammaZeroP.exists_algEquiv_map_j_eq_qExpand_and_coe_eq_atkinLehnerInvolutionFull_x1x0_gamma0202 below · cited by 1 · depth 30 - Gauss reduction preserves the level Γ₁(M)∩Γ₀(p) q-expansion field
ModularCurve.XOneGammaZeroP.gaussReduction_mem_x1x0FunctionFieldC_of_x1x07 below · cited by 1 · depth 30
ModularCurve.XOneP 376
- Frobenius twist on the Igusa component is coefficientwise
ModularCurve.XOneP.addEquiv_proj_fst_eq_frob_smul_of_pts_eq_frobenius_comp_of_gaussReading_twoChartModel_x1_mul2,278 below · cited by 4 · depth 21 - Uₚ acts as p Frob⁻¹ on the Igusa component
ModularCurve.XOneP.addEquiv_proj_fst_eq_natCast_smul_frob_inv_smul_of_pts_reduction_heckeGenOne_of_normFreePart_of_surjective_residue_of_gaussReading_twoChartModel_x1_mul3,358 below · cited by 4 · depth 21 - q-expansion pin for the Igusa component of J₁(Mp) at p
ModularCurve.XOneP.addEquiv_proj_fst_eq_pic0Mk_conorm_laurentPlaceReduction_of_points_of_gaussReading_twoChartModel_x1_mul1,336 below · cited by 4 · depth 21 - Sum of p-diamond operators kills the norm-free subscheme's special fibre
ModularCurve.XOneP.comp_heckeHom_sum_diamondGen_eq_one_of_factors_normFreePart_specialFibre_twoChartModel_x1_mul6 below · cited by 1 · depth 21 - q-divisible norm-free systems reducing into the torus vanish
ModularCurve.XOneP.eq_zero_of_proj_eq_zero_of_qDivisible_normFreePart_points_twoChartModel_x1_mul1,250 below · cited by 4 · depth 21 - Two smooth components of the bad fibre, ordered by a section
ModularCurve.XOneP.exists_components_specialFibre_card_pos_and_section_comp_eq_twoChartModel_x1_mul2,880 below · cited by 1 · depth 21 - Cusp ∞ lies on the Gauss component, read by q-expansions
ModularCurve.XOneP.exists_curveModel_igusaFunctionFieldX1C_iso_specialFibre_components_gaussReading_fst_of_section_eq_comp_iotaInf_twoChartModel_x1_mul1,780 below · cited by 1 · depth 21 - Geometric generic fibre model of X₁(Mp) with chart pin
ModularCurve.XOneP.exists_curveModel_x1FunctionFieldBar_iso_pullback_chartPin_galoisCompat_twoChartModel_x1_mul126 below · cited by 1 · depth 21 - Special fibre of J₁(Mp) as glued Pic⁰ of Igusa curves
ModularCurve.XOneP.exists_gluedPic0_addEquiv_neronSpecialFibreGeom_toPic0Pair_eq_proj_of_curveModel_igusa_twoChartModel_x1_mul1,721 below · cited by 5 · depth 21 - Abel–Jacobi-normalised Hecke and Galois action on Pic⁰ of X₁(Mp)
ModularCurve.XOneP.exists_heckeHom_galoisHom_pts_smul_eq_comp_abelJacobi_of_representsRelSubPic_twoChartModel_x1_mul3,203 below · cited by 1 · depth 21 - Abelian subscheme of relative Pic⁰ cutting out the norm-free part
ModularCurve.XOneP.exists_isClosedImmersion_isProper_smooth_normFreePart_of_representsRelSubPic_twoChartModel_x1_mul3,371 below · cited by 1 · depth 21 - Special-fibre geometry of Pic⁰ for the X₁(Mp) model
ModularCurve.XOneP.exists_neronSpecialFibreGeom_of_representsRelSubPic_baseChange_twoChartModel_x1_mul1,246 below · cited by 2 · depth 21 - Hecke, diamond and inertia operators on the Néron special fibre of J₁(Mp)
ModularCurve.XOneP.exists_neronSpecialFibreOpsV3_of_heckeHom_galoisHom_of_representsRelSubPic_of_isAlgebraic_twoChartModel_x1_mul_of_baseChangeIso_of_abelJacobi_of_gaussReading3,339 below · cited by 1 · depth 21 - A p-divisible group over A for the norm-free part of J₁(Mp)
ModularCurve.XOneP.exists_pDivisibleGroup_normFreePart_points_tateModule_valuation_lt_one_of_reduction_eq_zeroSection_twoChartModel_x1_mul742 below · cited by 4 · depth 21 - Inertia-fixed norm-free classes extend over the invariant subring
ModularCurve.XOneP.exists_points_fixedValuationSubring_of_smul_eq_self_of_mem_normFreePart_twoChartModel_x1_mul1 below · cited by 4 · depth 21 - Galois transport of O-points of the Pic⁰ model
ModularCurve.XOneP.exists_points_smul_eq_and_reduction_eq_comp_galoisHom_of_points_twoChartModel_x1_mul0 below · cited by 8 · depth 21 - Reduction bijective on prime-to-p torsion of O_I-points
ModularCurve.XOneP.exists_reduction_torsion_bijective_points_fixedValuationSubring_of_representsRelSubPic_twoChartModel_x1_mul16 below · cited by 1 · depth 21 - Representability of Pic⁰ for the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_representsRelSubPic_algEquivZeroCut_twoChartModel_x1_mul3,368 below · cited by 1 · depth 21 - Proper flat regular two-chart model of X₁(Mp) with semistable fibres
ModularCurve.XOneP.isProper_and_flat_and_isRegularLocalRing_and_twoGluedSmoothCurveDegeneration_twoChartModel_x1_mul2,883 below · cited by 29 · depth 21 - Reducedness of all geometric fibres of the X₁(Mp) two-chart model
ModularCurve.XOneP.isReduced_pullback_modelTo_of_isAlgClosed_twoChartModel_x1_mul1,189 below · cited by 34 · depth 21 - Crossings in the special fibre count supersingular places
ModularCurve.XOneP.natCard_pullback_specialFibre_eq_natCard_evalAt_mem_ssJSet_twoChartModel_x1_mul1,454 below · cited by 4 · depth 21 - Level monotonicity and component-wise compatibility of the specialisation family
ModularCurve.XOneP.normFreePartFamily_dom_mono_and_toPic0Pair_sp_eq_of_le_twoChartModel_x1_mul_opsV30 below · cited by 1 · depth 21 - Uₚ acts through an automorphism on the étale Igusa component
ModularCurve.XOneP.normFreePartFamily_exists_addEquiv_toPic0Pair_sp_heckeOperatorOneBar_snd_eq_twoChartModel_x1_mul4,670 below · cited by 1 · depth 21 - Diamond action on first components of specialised norm-free classes
ModularCurve.XOneP.normFreePartFamily_exists_addMonoidHom_toPic0Pair_sp_diamondOneBar_fst_eq_twoChartModel_x1_mul2,964 below · cited by 1 · depth 21 - Decomposition group acts on second Igusa projection of norm-free points
ModularCurve.XOneP.normFreePartFamily_exists_addMonoidHom_toPic0Pair_sp_smul_snd_eq_of_mem_decompositionSubgroup_twoChartModel_x1_mul3,011 below · cited by 1 · depth 21 - Specialisation datum for the norm-free part of J₁(Mp)
ModularCurve.XOneP.normFreePartFamily_exists_dom_sp_interface_twoChartModel_x1_mul_opsV32 below · cited by 1 · depth 21 - Inertia-invariant functionals annihilate Tate vectors with vanishing Igusa specialisation
ModularCurve.XOneP.normFreePartFamily_forall_apply_eq_zero_of_tateModule_toPic0Pair_sp_eq_zero_twoChartModel_x1_mul2,215 below · cited by 1 · depth 21 - Level independence of the specialisation family on J₁(Mp)
ModularCurve.XOneP.normFreePartFamily_level_pushout_and_sp_eq_twoChartModel_x1_mul_opsV30 below · cited by 1 · depth 21 - Inertia-fixed norm-free classes lie in the specialisation domain
ModularCurve.XOneP.normFreePartFamily_mem_dom_of_forall_smul_eq_self_twoChartModel_x1_mul_opsV32 below · cited by 1 · depth 21 - Trivial Weil pairing for vanishing glued specialisations
ModularCurve.XOneP.normFreePartFamily_pairing_eq_one_of_toPic0Pair_sp_eq_zero_twoChartModel_x1_mul4,591 below · cited by 1 · depth 21 - Inertia twisted by a diamond fixes the second Igusa component
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_diamondOneBar_smul_snd_eq_of_mem_inertiaSubgroupIn_twoChartModel_x1_mul_opsV31,268 below · cited by 1 · depth 21 - q-expansion pin of the specialisation on the Gauss component
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_fst_eq_pic0Mk_conorm_laurentPlaceReduction_twoChartModel_x1_mul1,337 below · cited by 1 · depth 21 - Frobenius acts coefficientwise on the first Igusa component
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_fst_smul_of_isFrobeniusAt_twoChartModel_x1_mul2,332 below · cited by 1 · depth 21 - Uₚ acts as p Fr⁻¹ on norm-free specialisations
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_heckeOperatorOneBar_fst_eq_natCast_smul_frob_inv_smul_twoChartModel_x1_mul3,359 below · cited by 1 · depth 21 - Triangularity of Uₚ on specialisations of the norm-free part
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_heckeOperatorOneBar_snd_eq_zero_twoChartModel_x1_mul3,359 below · cited by 1 · depth 21 - Frobenius acts coefficientwise on the first Igusa-component specialisation
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_smul_fst_eq_frob_smul_of_isFrobeniusAt_twoChartModel_x1_mul0 below · cited by 1 · depth 21 - Inertia fixes the cuspidal component of reductions of norm-free points
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_smul_fst_eq_of_mem_inertiaSubgroupIn_twoChartModel_x1_mul1,268 below · cited by 1 · depth 21 - Inertia fixing μₚ preserves the second Igusa component of specialisations
ModularCurve.XOneP.normFreePartFamily_toPic0Pair_sp_smul_snd_eq_of_mem_inertiaSubgroupIn_of_forall_pow_eq_one_twoChartModel_x1_mul_opsV31 below · cited by 1 · depth 21 - Inertia and diamond act trivially on special-fibre components
ModularCurve.XOneP.proj_fst_eq_and_proj_snd_eq_of_opoints_pts_eq_comp_galoisHom_diamondGen_of_mem_inertiaSubgroupIn_gaussPin_cuspPin_abelJacobi_twoChartModel_x1_mul1,266 below · cited by 7 · depth 21 - Hecke generator at p preserves vanishing étale component
ModularCurve.XOneP.proj_snd_eq_zero_of_proj_snd_eq_zero_of_pts_reduction_heckeGenOne_of_normFreePart_of_surjective_residue_of_gaussReading_twoChartModel_x1_mul3,358 below · cited by 4 · depth 21 - Group-law form of the special-fibre points dictionary
ModularCurve.XOneP.pts_add_eq_relativeGroupLaw_mul_and_pts_zero_eq_one_specialFibre_twoChartModel_x1_mul1 below · cited by 10 · depth 21 - Section through one special-fibre component misses the other
ModularCurve.XOneP.sectionBaseChange_not_mem_range_of_comp_eq_sectionBaseChange_twoChartModel_x1_mul2,902 below · cited by 1 · depth 21 - Galois invariance of the cusp section of the two-chart model
ModularCurve.XOneP.section_comp_eq_spec_comp_section_of_iotaFin_comp_eq_of_coeff_zero_twoChartModel_x1_mul5 below · cited by 1 · depth 21 - Generic fibre of the two-chart model of X₁(Mp): smooth, geometrically integral
ModularCurve.XOneP.smoothOfRelativeDimension_one_and_geometricallyIntegral_baseChange_twoChartModel_x1_mul11 below · cited by 18 · depth 21 - Trivial Weil pairing for classes reducing into the torus
ModularCurve.XOneP.weilDatum_pairing_eq_one_of_proj_eq_zero_of_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul3,869 below · cited by 3 · depth 21 - Frobenius pull-back acts as coefficientwise Frobenius on the Igusa component
ModularCurve.XOneP.addEquiv_eq_frob_smul_of_nonempty_poincare_pullbackAlong_iso_pullback_frobeniusTwist_fst_twoChartModel_x1_mul1,412 below · cited by 1 · depth 22 - Eichler–Shimura on the cusp component: Uₚ reduces to p frob⁻¹
ModularCurve.XOneP.addEquiv_proj_fst_eq_natCast_smul_frob_inv_smul_of_pts_reduction_heckeGenOne_of_points_pic0Mk_valuationSubring_of_forall_mem_support_gaussReduces_twoChartModel_x1_mul1,520 below · cited by 1 · depth 22 - Galois action on A is trivial modulo the maximal ideal
ModularCurve.XOneP.algebraMap_smul_eq_of_isCyclotomicExtension_of_charP1 below · cited by 1 · depth 22 - Universal H⁰ equals the base for the X₁(Mp) two-chart model
ModularCurve.XOneP.bijective_algebraMap_sections_baseChange_twoChartModel_x1_mul2,901 below · cited by 1 · depth 22 - Diamonds ⟨ d⟩, d≡ 1 (M), fix the gluing torus
ModularCurve.XOneP.comp_heckeHom_diamondGen_eq_of_comp_torus_specialFibre_of_representsRelSubPic_abelJacobi_twoChartModel_x1_mul2,995 below · cited by 2 · depth 22 - Geometric fibres of the two-chart model of X₁(Mp) are connected
ModularCurve.XOneP.connectedSpace_pullback_modelTo_of_isAlgClosed_twoChartModel_x1_mul2,886 below · cited by 5 · depth 22 - Inertia annihilates the Weil pairing on σ z - z
ModularCurve.XOneP.divisorialWeilPairingData_pair_eq_one_of_coe_eq_smul_sub_of_smul_eq_of_mem_inertia_of_not_dvd_x12 below · cited by 1 · depth 22 - Triviality of the Gal(L/ℚ)-action on the toric part
ModularCurve.XOneP.eq_of_galois_of_postComp_eq_one_points_specialFibre_of_gaussReading_twoChartModel_x1_mul_of_abelJacobi1,268 below · cited by 1 · depth 22 - Unique homomorphic factorisation through D₁×_k D₂
ModularCurve.XOneP.existsUnique_schemeHomOver_prodStr_comp_eq_of_comp_splitTorus_eq_one_specialFibre_baseChange_x1_mul2 below · cited by 1 · depth 22 - Uₚ on the étale component J_E of the special fibre
ModularCurve.XOneP.exists_addEquiv_proj_snd_eq_of_pts_reduction_heckeGenOne_of_normFreePart_of_eichlerShimura_twoChartModel_x1_mul2 below · cited by 1 · depth 22 - Residue-field twists act on J_E through a single additive map
ModularCurve.XOneP.exists_addMonoidHom_proj_snd_eq_of_pts_eq_spec_map_comp_specialFibre_twoChartModel_x1_mul1,207 below · cited by 1 · depth 22 - Hecke endomorphisms act additively on the geometric special fibre
ModularCurve.XOneP.exists_addMonoidHom_pts_comp_eq_comp_and_eq_of_pts_reduction_specialFibre_twoChartModel_x1_mul5 below · cited by 4 · depth 22 - Level-p automorphism with σ(j)=j(qᵖ) moving the Gauss ring
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_chartAlgFin_iff_and_comap_ne_x1_mul123 below · cited by 14 · depth 22 - Geometric base change of the two chart algebras of X₁(Mp)
ModularCurve.XOneP.exists_algEquiv_tensor_chartAlgFin_chartRing_and_chartAlgInf_x1FunctionFieldBar_twoChartModel_x1_mul5 below · cited by 3 · depth 22 - First special-fibre component of X₁(Mp) is an Igusa model
ModularCurve.XOneP.exists_curveModel_igusaFunctionFieldX1C_iso_fst_twoChartModel_x1_mul1,217 below · cited by 2 · depth 22 - Second special fibre component of X₁(Mp) is Igusa
ModularCurve.XOneP.exists_curveModel_igusaFunctionFieldX1C_iso_snd_twoChartModel_x1_mul1,217 below · cited by 3 · depth 22 - Both special-fibre components are Igusa curves over k
ModularCurve.XOneP.exists_curveModel_igusaFunctionFieldX1C_iso_specialFibre_components_twoChartModel_x1_mul1,223 below · cited by 1 · depth 22 - Diamond operators descend to both special-fibre components
ModularCurve.XOneP.exists_descent_diamondGen_of_coprime_specialFibre_components_of_abelJacobi_twoChartModel_x1_mul2,961 below · cited by 3 · depth 22 - Diagonal descent of T_ℓ (ℓ≠ p) to the special fibre
ModularCurve.XOneP.exists_descent_heckeGenOne_of_ne_specialFibre_components_of_abelJacobi_twoChartModel_x1_mul3,243 below · cited by 1 · depth 22 - Toric prime-to-p torsion classes of J₁(Mp) are γ· w-w
ModularCurve.XOneP.exists_forall_exists_eq_smul_sub_of_proj_eq_zero_of_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul3,837 below · cited by 1 · depth 22 - Semilinear Galois action on the relative Picard model of X₁(Mp)
ModularCurve.XOneP.exists_galoisHom_pts_smul_eq_specMap_comp_comp_abelJacobi_of_representsRelSubPic_twoChartModel_x1_mul169 below · cited by 1 · depth 22 - Galois twists of the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_galoisModelHom_comp_modelTo_eq_and_iotaFin_comp_eq_twoChartModel_x1_mul2 below · cited by 3 · depth 22 - Common affine neighbourhoods in the smooth locus over an affine base
ModularCurve.XOneP.exists_isAffineOpen_of_finset_smoothLocus_twoChartModel_x1_mul5 below · cited by 1 · depth 22 - Geometric special fibre of the two-chart model: two components
ModularCurve.XOneP.exists_isClosedImmersion_pair_specialFibre_twoChartIntegralModel_x1_mul2,872 below · cited by 2 · depth 22 - Diamond automorphism of the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_iso_modelTo_eq_and_iotaFin_comp_eq_of_diamondAut_twoChartModel_x1_mul48 below · cited by 4 · depth 22 - Crossing equation uv=varpi at singular points of the special fibre
ModularCurve.XOneP.exists_mul_eq_and_maximalIdeal_eq_span_pair_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul2,856 below · cited by 4 · depth 22 - Prime-to-p divisibility of finite torsion classes in J₁(Mp)
ModularCurve.XOneP.exists_nsmul_eq_of_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul1,774 below · cited by 1 · depth 22 - Smoothness over the base at one-branch points of the special fibre
ModularCurve.XOneP.exists_opens_smooth_comp_toBase_of_subsingleton_minimalPrimes_fibre_twoChartIntegralModel_x1_mul1,734 below · cited by 3 · depth 22 - Good generators of the special fibre from cusp-component points
ModularCurve.XOneP.exists_place_schemeHomOver_valuationSubring_pts_reduction_proj_fst_eq_pic0Mk_proj_snd_eq_zero_of_notMem_range_crossings_of_mem_range_iotaFin_twoChartModel_x1_mul3,008 below · cited by 2 · depth 22 - Points dictionary of the p-divisible group into J₁(Mp)
ModularCurve.XOneP.exists_points_injective_iff_normFreePart_galois_read_of_pDivisibleGroup_abelianSubscheme_twoChartModel_x1_mul0 below · cited by 1 · depth 22 - Uₚ on the second Picard factor of the special fibre
ModularCurve.XOneP.exists_postComp_heckeGenOne_eq_apply_postComp_and_map_mul_and_bijective_points_snd_specialFibre_of_factors_normFreePart_of_gaussReading_twoChartModel_x1_mul3,528 below · cited by 1 · depth 22 - Hensel lifting of k-points of D to Pl-points
ModularCurve.XOneP.exists_pts_reduction_and_exists_schemeHomOver_valuationSubring_of_pts_specialFibre_twoChartModel_x1_mul5 below · cited by 4 · depth 22 - Affine split torus kernel in Pic⁰ of the special fibre
ModularCurve.XOneP.exists_relativeGroupLaw_isAffine_isClosedImmersion_iff_postComp_pullbackHom_eq_one_splitTorus_specialFibre_baseChange_x1_mul54 below · cited by 1 · depth 22 - Kernel of special-fibre Picard projections: a split torus of rank n-1
ModularCurve.XOneP.exists_relativeGroupLaw_isClosedImmersion_iff_postComp_pullbackHom_eq_one_splitTorus_specialFibre_baseChange_x1_mul54 below · cited by 4 · depth 22 - Points dictionary and Abel–Jacobi map for X₁(Mp)'s relative Pic⁰
ModularCurve.XOneP.exists_representsRelSubPic_abelJacobi_pts_of_representsRelSubPic_twoChartModel_x1_mul301 below · cited by 1 · depth 22 - Residue field of a Gauss valuation subring is the Igusa field
ModularCurve.XOneP.exists_ringEquiv_residueField_igusaFunctionFieldX1C_of_gaussPresentation1 below · cited by 5 · depth 22 - An A-algebra field of characteristic p receives A through 𝔽ₚ
ModularCurve.XOneP.exists_ringHom_zmod_castHom_comp_eq_algebraMap_of_isCyclotomicExtension1 below · cited by 2 · depth 22 - Reduction of an A-point meets one of two fibre components
ModularCurve.XOneP.exists_schemeHomOver_comp_eq_sectionBaseChange_or_of_isClosedImmersion_pair_specialFibre_twoChartModel_x1_mul0 below · cited by 2 · depth 22 - Maximal smooth locus of the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_smoothLocus_maximal_twoChartModel_x1_mul0 below · cited by 13 · depth 22 - Two-sided pools of étale multisections on the X₁(Mp) two-chart model
ModularCurve.XOneP.exists_twoSidedPool_smoothLocus_twoChartModel_x1_mul2,950 below · cited by 1 · depth 22 - Chart functions read in a component of the special fibre
ModularCurve.XOneP.exists_valuationSubring_algEquiv_fractionRing_tensorProduct_apply_germ_eq_of_curveModel_component_twoChartModel_x1_mul23 below · cited by 3 · depth 22 - Two valuation subrings of L(X₁(Mp)) above p, with Igusa residue field
ModularCurve.XOneP.exists_valuationSubring_pair_x1_mul1,179 below · cited by 38 · depth 22 - 𝔽ₚ-models of special-fibre components and their relative Pic⁰
ModularCurve.XOneP.exists_zmodp_models_components_and_pic0_specialFibre_twoChartModel_x1_mul_of_poincare_iso2,244 below · cited by 1 · depth 22 - Euler characteristic of the special fibre equals 1-g(X₁(Mp))
ModularCurve.XOneP.finrank_H0_sub_finrank_H1_sectionsOf_specialFibre_eq_one_sub_genusFF_twoChartIntegralModel_x1_mul337 below · cited by 2 · depth 22 - Hecke generators as endomorphisms of the relative Pic⁰ model
ModularCurve.XOneP.forall_prime_exists_hom_mul_and_pts_heckeGenOne_smul_eq_comp_abelJacobi_of_representsRelSubPic_twoChartModel_x1_mul3,122 below · cited by 1 · depth 22 - Gauss reductions on X₁(Mp) give exactly the Igusa function field
ModularCurve.XOneP.gaussReduction_mem_igusaFunctionFieldX1C_and_surjective_x1_mul1,178 below · cited by 4 · depth 22 - Uniformiser germ in mathfrak m_z² at singular points of the special fibre
ModularCurve.XOneP.germ_mem_maximalIdeal_sq_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul2,857 below · cited by 1 · depth 22 - Rigidity of Hecke–diamond endomorphisms on the Jacobian model
ModularCurve.XOneP.heckeHom_eq_of_forall_smul_eq_and_diamondGen_congr_of_representsRelSubPic_twoChartModel_x1_mul4 below · cited by 2 · depth 22 - Components of the special fibre meet in a finite reduced scheme
ModularCurve.XOneP.isReduced_and_card_pos_pullback_of_isClosedImmersion_pair_specialFibre_twoChartIntegralModel_x1_mul2,861 below · cited by 2 · depth 22 - Reducedness of the mod p fibre of the two-chart model of X₁(Mp)
ModularCurve.XOneP.isReduced_pullback_toBase_twoChartIntegralModel_x1_mul1,186 below · cited by 9 · depth 22 - Generating Pic⁰ of the Igusa curve by chart point differences
ModularCurve.XOneP.mem_closure_pic0Mk_single_pointEquivPlace_sub_single_of_notMem_range_crossings_of_mem_range_iotaFin_igusaModel_twoChartModel_x1_mul49 below · cited by 3 · depth 22 - Galois twists respect the relative group law on D
ModularCurve.XOneP.mul_comp_galoisHom_eq_mul_comp_of_pts_smul_eq_comp_abelJacobi_of_representsRelSubPic_twoChartModel_x1_mul3 below · cited by 1 · depth 22 - Triviality of sectioned algebraically trivial bundles on fibres of X₁(Mp)
ModularCurve.XOneP.nonempty_iso_unit_fibre_of_isAlgEquivZero_of_ne_zero_twoChartModel_x1_mul2,948 below · cited by 1 · depth 22 - Frobenius twist commutes with restricting the Poincaré bundle
ModularCurve.XOneP.nonempty_poincare_pullbackAlong_postComp_pullbackHom_iso_pullback_obj_of_comp_fst_eq_frobenius_comp_twoChartModel_x1_mul10 below · cited by 1 · depth 22 - Section at ∞ of the two-chart model of X₁(Mp)
ModularCurve.XOneP.nonempty_schemeHomOver_id_modelTo_twoChartModel_x1_mul2 below · cited by 1 · depth 22 - Non-smooth closed fibre of the two-chart model of X₁(Mp)
ModularCurve.XOneP.not_smooth_pullback_snd_modelTo_of_not_injective_twoChartModel_x1_mul1,498 below · cited by 10 · depth 22 - Galois action on points versus places for the twisted X₁(Mp) model
ModularCurve.XOneP.pointEquivPlace_eq_arithmeticGalois_smul_of_chartPin_of_galoisTwist_twoChartModel_x1_mul125 below · cited by 2 · depth 22 - Frobenius twist of a point twists its Igusa place by `frobIg`
ModularCurve.XOneP.pointEquivPlace_eq_frob_smul_pointEquivPlace_of_comp_eq_frobenius_comp_of_gaussReading_twoChartModel_x1_mul1,197 below · cited by 3 · depth 22 - Galois acts trivially on C₁, through a diamond on C₂
ModularCurve.XOneP.postComp_pullbackHom_galois_eq_and_postComp_diamond_comp_galoisInv_eq_of_gaussReading_specialFibre_twoChartModel_x1_mul_of_abelJacobi1,265 below · cited by 2 · depth 22 - Reduction of Uₚ preserves the Néron special fibre torus
ModularCurve.XOneP.proj_eq_zero_of_proj_eq_zero_of_pts_reduction_heckeGenOne_of_surjective_residue_of_gaussReading_twoChartModel_x1_mul1,687 below · cited by 2 · depth 22 - Triangularity of Uₚ on the Néron special fibre of J₁(Mp)
ModularCurve.XOneP.proj_snd_eq_zero_of_proj_snd_eq_zero_of_pts_reduction_heckeGenOne_of_surjective_residue_of_gaussReading_twoChartModel_x1_mul3,357 below · cited by 2 · depth 22 - Diamond operator realised as Picard transport on the Jacobian model
ModularCurve.XOneP.pts_diamondGen_smul_eq_comp_transport_of_abelJacobi_of_diamondModelAut_twoChartModel_x1_mul170 below · cited by 4 · depth 22 - A-sections of the X₁(Mp) model land in the smooth locus
ModularCurve.XOneP.range_section_subset_smoothLocus_twoChartModel_x1_mul2,891 below · cited by 2 · depth 22 - Prime-to-p torsion with a Pl-integral point is inertia-fixed
ModularCurve.XOneP.smul_eq_self_of_mem_inertiaSubgroupIn_of_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul148 below · cited by 2 · depth 22 - Bad geometric fibres of the two-chart model of X₁(Mp)
ModularCurve.XOneP.twoGluedSmoothCurveDegenerations_twoChartModel_x1_mul2,894 below · cited by 3 · depth 22 - Component meeting the cusp ∞ has Gauss valuation ring
ModularCurve.XOneP.valuationSubring_eq_gauss_of_ringEquiv_stalk_germ_eq_of_section_eq_comp_iotaInf_twoChartModel_x1_mul1,745 below · cited by 1 · depth 22 - The two branches of X₁(Mp) above A: count and degrees
ModularCurve.XOneP.valuationSubring_eq_or_eq_comap_and_uniformizer_and_relfinrank_gaussReduction_x1_mul1,162 below · cited by 15 · depth 22 - Abel–Jacobi commutes with reduction onto the Igusa component
ModularCurve.XOneP.addEquiv_proj_fst_eq_pic0Mk_mapDomain_of_points_eq_reduction_of_surjective_residue_of_forall_mem_support_exists_section_twoChartModel_x1_mul1,437 below · cited by 1 · depth 23 - Igusa-component class of the reduction of 𝒪(ξ₁)⊗𝒪(ξ₂)⁻¹
ModularCurve.XOneP.addEquiv_proj_fst_eq_pic0Mk_single_sub_single_of_points_eq_reduction_of_poincare_iso_ofPoint_valuationSubring_twoChartModel_x1_mul291 below · cited by 2 · depth 23 - Abel–Jacobi commutes with reduction onto the étale component
ModularCurve.XOneP.addEquiv_proj_snd_eq_pic0Mk_mapDomain_of_points_eq_reduction_of_surjective_residue_of_forall_mem_support_exists_section_twoChartModel_x1_mul1,437 below · cited by 3 · depth 23 - Igusa function field inside the Gauss reductions of both charts
ModularCurve.XOneP.coe_mem_adjoin_gaussReductions_chartAlg_igusaFunctionFieldX1C_x1_mul1,181 below · cited by 4 · depth 23 - Galois model automorphism acts trivially on the Gauss component
ModularCurve.XOneP.comp_fibreAut_eq_of_galoisModelAut_of_gaussPin_twoChartModel_x1_mul1,184 below · cited by 2 · depth 23 - Inertia-twisted diamond is trivial on the Igusa branch
ModularCurve.XOneP.comp_fibreIso_eq_of_diamondModelAut_galoisModelHom_of_gaussPin_twoChartModel_x1_mul1,197 below · cited by 1 · depth 23 - Uniqueness of the minimal special-fibre point over a valuation point
ModularCurve.XOneP.eq_of_forall_specializes_imp_eq_of_ringEquiv_stalk_of_fst_eq_twoChartModel_x1_mul1,184 below · cited by 7 · depth 23 - Partial Atkin–Lehner automorphism at p of L·ℚ(X₁(Mp))
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_chartAlgFin_iff_and_comap_ne_and_coe_eq_atkinLehnerInvolutionFull_and_diamondConj_and_galoisConj_x1_mul172 below · cited by 2 · depth 23 - An L-automorphism of the X₁(Mp) function field sending j to j(qᵖ)
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_x1_mul80 below · cited by 1 · depth 23 - Hecke degeneracies preserve special-fibre components for ℓ ≠ p
ModularCurve.XOneP.exists_comp_eq_fst_comp_heckeDegeneracy_baseChange_of_ne_specialFibre_components_twoChartModel_x1_mul3,025 below · cited by 1 · depth 23 - Étale entry of Uₚ on good generators of J_E
ModularCurve.XOneP.exists_coprime_algEquiv_finset_addMonoidHom_proj_snd_heckeGenOne_eq_symm_frob_smul_and_proj_snd_diamondGen_eq_smul_of_pic0Mk_single_sub_single_snd_specialFibre_twoChartModel_x1_mul_of_atkinLehner_of_diamondConj3,174 below · cited by 1 · depth 23 - Igusa reading of the second special-fibre component via σ
ModularCurve.XOneP.exists_curveModel_iso_snd_gaussReading_algEquiv_of_gaussReading_fst_twoChartModel_x1_mul1,225 below · cited by 1 · depth 23 - Branch valuation rings of X₁(Mp) are unramified over A
ModularCurve.XOneP.exists_eq_mul_of_mem_nonunits_valuationSubring_x1_mul1,178 below · cited by 3 · depth 23 - Base change of a semilinear automorphism to the geometric fibre
ModularCurve.XOneP.exists_fibreIso_comp_fst_eq_of_modelHom_comp_modelTo_eq_of_algebraMap_smul_eq_twoChartModel_x1_mul0 below · cited by 2 · depth 23 - Inertia-invariant prime-to-p torsion of J₁(Mp) bounded by its finite part
ModularCurve.XOneP.exists_forall_natCard_torsion_inertiaInvariants_le_mul_natCard_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul3,235 below · cited by 1 · depth 23 - Gauss reduction and Igusa degree p-1 for p<5
ModularCurve.XOneP.exists_gaussReduction_eq_hasseRootFn_and_relfinrank_igusaFunctionFieldX1C_of_lt_five968 below · cited by 2 · depth 23 - A Gauss valuation subring of the function field of X₁(Mp)
ModularCurve.XOneP.exists_gaussValuationSubring_x1_mul0 below · cited by 8 · depth 23 - Hecke degeneracy pair on the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_heckeDegeneracyPair_chartPin_flat_twoChartModel_x1_mul253 below · cited by 3 · depth 23 - Level-p Hecke divisor of a Gauss-reducing place of X₁(Mp)
ModularCurve.XOneP.exists_heckeDivOneBar_single_eq_sum_and_red_eq_frob_inv_smul_of_gaussReduces_of_surjective_residue_twoChartModel_x1_mul1,242 below · cited by 2 · depth 23 - Hecke endomorphism T_ℓ of the relative Pic⁰ of X₁(Mp)
ModularCurve.XOneP.exists_hom_classifies_norm_pullback_poincare_heckeDegeneracyPair_twoChartModel_x1_mul403 below · cited by 2 · depth 23 - Non-regular points above supersingular j-invariants on X₁(Mp)
ModularCurve.XOneP.exists_injective_not_isRegularLocalRing_stalk_specialFibre_of_mem_ssJSet_twoChartIntegralModel_x1_mul1,488 below · cited by 2 · depth 23 - Residue field 𝔽ₚ for a valuation ring of ℚ(ζₚ)
ModularCurve.XOneP.exists_intCast_sub_mem_maximalIdeal_of_isCyclotomicExtension0 below · cited by 2 · depth 23 - Gauss centre on the j-finite chart specialises to the cusp
ModularCurve.XOneP.exists_iotaFin_specializes_section_closedPoint_and_iff_mem_nonunits_gauss_twoChartModel_x1_mul4 below · cited by 1 · depth 23 - Diamond ⟨ d⟩ preserves components and crossings mod p
ModularCurve.XOneP.exists_iso_comp_eq_and_comp_eq_of_crossing_specialFibre_of_apply_eq_diamondAut_twoChartModel_x1_mul2,924 below · cited by 1 · depth 23 - Diamond model automorphism preserves both special-fibre components
ModularCurve.XOneP.exists_iso_comp_eq_specialFibre_components_of_apply_eq_diamondAut_of_coprime_twoChartModel_x1_mul2,912 below · cited by 1 · depth 23 - A level-p involution swapping bad-fibre components of X₁(Mp)
ModularCurve.XOneP.exists_iso_modelTo_swap_components_twoChartModel_x1_mul2,910 below · cited by 1 · depth 23 - Component Jacobians of the special fibre descend to 𝔽ₚ
ModularCurve.XOneP.exists_iso_pic0_baseChange_and_descent_projections_specialFibre_twoChartModel_x1_mul10 below · cited by 1 · depth 23 - Components of the geometric special fibre over chart primes
ModularCurve.XOneP.exists_mem_minimalPrimes_iotaFin_eq_and_eq_of_isDomain_tensorProduct_quotient_specialFibre_twoChartModel_x1_mul127 below · cited by 4 · depth 23 - Crossing point in the j-finite chart of the integral model
ModularCurve.XOneP.exists_not_subsingleton_minimalPrimes_span_germ_iotaFin_twoChartIntegralModel_x1_mul1,482 below · cited by 1 · depth 23 - One-sided pools of étale multisections on the ε-component
ModularCurve.XOneP.exists_oneSidedPool_smoothLocus_twoChartModel_x1_mul2,935 below · cited by 1 · depth 23 - Inertia displacements on J₁(Mp) reduce into the toric part
ModularCurve.XOneP.exists_points_valuationSubring_and_proj_eq_zero_smul_sub_self_of_mem_inertia_of_curveModel_igusa_twoChartModel_x1_mul3,106 below · cited by 1 · depth 23 - Galois transport of Pic⁰ on the special fibre
ModularCurve.XOneP.exists_postComp_eq_of_comp_fst_eq_comp_galoisTransport_of_classifies_fibre_twoChartModel_x1_mul9 below · cited by 1 · depth 23 - Completed local ring at a singular point of the X₁(Mp) model
ModularCurve.XOneP.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_unramified_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul2,852 below · cited by 1 · depth 23 - Pl-point of relative Pic⁰ representing 𝒪(ξ₁)⊗𝒪(ξ₂)⁻¹
ModularCurve.XOneP.exists_schemeHomOver_poincare_iso_ofPoint_tensor_idealModule_of_reduction_fst_valuationSubring_twoChartModel_x1_mul1,411 below · cited by 3 · depth 23 - Hensel lifting of off-crossing k-points of the second component
ModularCurve.XOneP.exists_schemeHomOver_valuationSubring_reduction_eq_and_generic_eq_pointEquivPlace_of_notMem_range_crossings_snd_twoChartModel_x1_mul2,894 below · cited by 4 · depth 23 - Henselian lift of a k-point off the crossings
ModularCurve.XOneP.exists_schemeHomOver_valuationSubring_reduction_eq_and_generic_eq_pointEquivPlace_of_notMem_range_crossings_twoChartModel_x1_mul2,894 below · cited by 1 · depth 23 - Gauss reduction maps the j-charts onto the Igusa charts
ModularCurve.XOneP.exists_surjective_tensorProduct_chartAlg_to_chartRing_igusaFunctionFieldX1C_of_algEquiv_x1_mul2,869 below · cited by 1 · depth 23 - Integral j-charts surject onto the Igusa curve charts
ModularCurve.XOneP.exists_surjective_tensorProduct_chartAlg_to_chartRing_igusaFunctionFieldX1C_x1_mul2,867 below · cited by 1 · depth 23 - Divisibility of toric torsion classes on J₁(Mp)
ModularCurve.XOneP.exists_toric_nsmul_eq_of_toric_of_points_valuationSubring_of_curveModel_igusa_twoChartModel_x1_mul1,736 below · cited by 1 · depth 23 - Geometric closed fibres of the two-chart model of X₁(Mp)
ModularCurve.XOneP.exists_twoGluedSmoothCurves_isReduced_pullback_twoChartModel_x1_mul_of_ker_ne_bot2,901 below · cited by 2 · depth 23 - Special-fibre stalks of the X₁(Mp) two-chart model are valuation rings
ModularCurve.XOneP.exists_valuationSubring_ringEquiv_stalk_apply_germ_eq_of_ringKrullDim_le_one_twoChartIntegralModel_x1_mul14 below · cited by 6 · depth 23 - Components of the special fibre of X₁(Mp) descend to 𝔽ₚ
ModularCurve.XOneP.exists_zmodp_curves_isPullback_components_specialFibre_twoChartModel_x1_mul1,750 below · cited by 1 · depth 23 - Finite and toric parts of J₁(Mp) form subgroups
ModularCurve.XOneP.finitePart_toricPart_zero_mem_add_mem_neg_mem_sub_mem_points_valuationSubring_twoChartModel_x1_mul2 below · cited by 2 · depth 23 - Geometric generic fibre of the X₁(Mp) two-chart model: h⁰=1, h¹=g
ModularCurve.XOneP.finrank_H0_sectionsOf_eq_one_and_finrank_H1_eq_genusFF_pullback_toBase_of_isAlgClosed_twoChartIntegralModel_x1_mul245 below · cited by 1 · depth 23 - Degree over the j-line unchanged on reduction at p
ModularCurve.XOneP.finrank_adjoin_j_eq_relfinrank_adjoin_jqModC_x1FunctionFieldC_of_x1949 below · cited by 2 · depth 23 - Finite surjective maps onto a regular two-chart model are flat
ModularCurve.XOneP.flat_and_locallyOfFinitePresentation_of_isRegularLocalRing_of_isFinite_heckeRoof_twoChartModel_x1_mul31 below · cited by 2 · depth 23 - Crossings in a geometric fibre lie in the j-finite chart
ModularCurve.XOneP.fst_mem_chartFinOpen_of_mem_irreducibleComponents_pair_specialFibre_twoChartModel_x1_mul1,738 below · cited by 3 · depth 23 - Pinned Galois transport on relative Pic⁰ equals τ(s)
ModularCurve.XOneP.galoisHom_eq_of_classifies_rigidify_pullback_of_modelHom_inv_twoChartModel_x1_mul_of_abelJacobi173 below · cited by 2 · depth 23 - Germ of a uniformiser and stalk dimension at a component's generic point
ModularCurve.XOneP.germ_mem_maximalIdeal_and_ringKrullDim_stalk_le_one_of_isGenericPoint_component_twoChartModel_x1_mul12 below · cited by 6 · depth 23 - Pic⁰ of the Igusa field generated by differences of chart points
ModularCurve.XOneP.mem_closure_pic0Mk_single_pointEquivPlace_sub_single_of_notMem_range_crossings_of_mem_range_iotaFin_of_notMem_finset_igusaModel_snd_twoChartModel_x1_mul49 below · cited by 1 · depth 23 - Toric and finite m-torsion counts for J₁(Mp) at p
ModularCurve.XOneP.natCard_toricTorsion_mul_natCard_finiteTorsion_eq_natCard_torsion_jOne_of_curveModel_igusa_twoChartModel_x1_mul_of_not_dvd2,063 below · cited by 1 · depth 23 - Monodromy bound for ℓ-power torsion on J₁(Mp)
ModularCurve.XOneP.natCard_torsion_le_natCard_image_smul_sub_mul_natCard_inertiaInvariants_of_forall_smul_sub_toric_of_curveModel_igusa_twoChartModel_x1_mul0 below · cited by 1 · depth 23 - Function field of the first special-fibre component is Igusa's
ModularCurve.XOneP.nonempty_algEquiv_igusaFunctionFieldX1C_of_curveModel_fst_twoChartModel_x1_mul1,197 below · cited by 1 · depth 23 - Second component of the special fibre has Igusa function field
ModularCurve.XOneP.nonempty_algEquiv_igusaFunctionFieldX1C_of_curveModel_snd_twoChartModel_x1_mul1,197 below · cited by 1 · depth 23 - Poincaré bundle along the Abel–Jacobi image of a ℚ̄-point
ModularCurve.XOneP.nonempty_poincare_pullbackAlong_iso_ofPoint_tensor_ofPoint_idealModule_of_eq_comp_ajbar_twoChartModel_x1_mul15 below · cited by 2 · depth 23 - Poincaré bundle along gpts([P]-[Q]) is 𝒪(x_P)⊗ I_{x_Q}
ModularCurve.XOneP.nonempty_poincare_pullbackAlong_points_pic0Mk_single_sub_single_iso_ofPoint_tensor_idealModule_twoChartModel_x1_mul31 below · cited by 7 · depth 23 - At least two branches through singular special-fibre points
ModularCurve.XOneP.not_subsingleton_minimalPrimes_span_germ_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul1,727 below · cited by 2 · depth 23 - Diamond model automorphism moves places by the inverse diamond automorphism
ModularCurve.XOneP.pointEquivPlace_eq_diamondAutBar_inv_smul_of_chartPin_of_diamondModelAut_twoChartModel_x1_mul126 below · cited by 2 · depth 23 - Galois transport fixes Pic⁰ of pointwise-fixed components
ModularCurve.XOneP.postComp_pullbackHom_eq_and_postComp_eq_of_comp_fst_eq_comp_galoisTransport_of_comp_fibreIso_eq_twoChartModel_x1_mul0 below · cited by 1 · depth 23 - Vanishing étale coordinate for classes reducing off the crossings
ModularCurve.XOneP.proj_snd_eq_zero_of_points_eq_reduction_of_surjective_residue_of_forall_mem_support_exists_section_twoChartModel_x1_mul1,438 below · cited by 5 · depth 23 - Vanishing étale coordinate for Hecke reductions of Gauss-reducing divisors
ModularCurve.XOneP.proj_snd_eq_zero_of_pts_reduction_heckeGenOne_of_points_pic0Mk_valuationSubring_of_forall_mem_support_gaussReduces_twoChartModel_x1_mul1,521 below · cited by 1 · depth 23 - Endomorphism classifying the norm bundle realises T_ℓ on points
ModularCurve.XOneP.pts_heckeGenOne_smul_eq_comp_abelJacobi_of_classifies_norm_pullback_poincare_heckeDegeneracyPair_twoChartModel_x1_mul312 below · cited by 2 · depth 23 - Galois transport of J₁(Mp)-points by the Picard automorphism N
ModularCurve.XOneP.pts_smul_eq_specMap_comp_comp_of_galoisModelAut_of_classifies_abelJacobi_twoChartModel_x1_mul168 below · cited by 2 · depth 23 - Pl-points reducing off the crossings lie in the smooth locus
ModularCurve.XOneP.range_subset_smoothLocus_of_reduction_eq_of_not_mem_range_valuationSubring_twoChartModel_x1_mul1,194 below · cited by 3 · depth 23 - Relative degree of X₁(Mp) over X₁(M) at most p²-1
ModularCurve.XOneP.relfinrank_laurentBaseChange_x1FunctionField_le_x1_mul53 below · cited by 2 · depth 23 - Reduction of 𝒪(ξ₁)⊗𝒪(ξ₂)⁻¹ read on the Igusa component
ModularCurve.XOneP.addEquiv_proj_snd_eq_pic0Mk_single_sub_single_of_points_eq_reduction_of_poincare_iso_ofPoint_valuationSubring_twoChartModel_x1_mul291 below · cited by 3 · depth 24 - Vanishing at k-points of a chart function with zero reduction
ModularCurve.XOneP.apply_eq_zero_of_mul_eq_of_map_eq_zero_of_comp_eq_specMap_comp_iotaFin_of_gaussReading_twoChartModel_x1_mul0 below · cited by 2 · depth 24 - Integral j-charts generate the Igusa curve's affine rings
ModularCurve.XOneP.chartRing_le_adjoin_gaussReductions_chartAlg_x1_mul2,866 below · cited by 2 · depth 24 - Igusa charts inside Gauss reductions of σ-twisted j-charts
ModularCurve.XOneP.chartRing_le_adjoin_gaussReductions_map_chartAlg_of_algEquiv_x1_mul2,868 below · cited by 1 · depth 24 - Diamonds with d≡ 1 mod M fix supersingular points
ModularCurve.XOneP.comap_eq_self_and_sub_mem_of_apply_eq_diamondAut_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,555 below · cited by 1 · depth 24 - Diamonds prime to Mp fix each minimal prime over varpi
ModularCurve.XOneP.comap_eq_self_of_mem_minimalPrimes_span_of_apply_eq_diamondAut_of_coprime_twoChartIntegralModel_x1_mul1,191 below · cited by 2 · depth 24 - Diamond operators with d ≡ 1 (mod M) fix each finite-chart component
ModularCurve.XOneP.comap_eq_self_of_mem_minimalPrimes_span_of_apply_eq_diamondAut_twoChartIntegralModel_x1_mul1,229 below · cited by 1 · depth 24 - Identity on the second component of the X₁(Mp) special fibre
ModularCurve.XOneP.comp_fibreIso_eq_of_forall_sub_mem_of_mem_minimalPrimes_of_gaussPin_twoChartModel_x1_mul1 below · cited by 1 · depth 24 - Components of the geometric fibre are determined by their chart prime
ModularCurve.XOneP.eq_of_iotaFin_eq_fst_genericPoint_of_mem_irreducibleComponents_specialFibre_twoChartModel_x1_mul1,197 below · cited by 4 · depth 24 - Geometric fibre in characteristic 0 has one component
ModularCurve.XOneP.eq_of_mem_irreducibleComponents_pullback_modelTo_of_map_uniformizer_ne_zero_twoChartModel_x1_mul12 below · cited by 1 · depth 24 - Partial Atkin–Lehner automorphism on L·ℚ(X₁(Mp)) with conjugation laws
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_coe_eq_atkinLehnerInvolutionFull_and_diamondConj_and_galoisConj_x1_mul166 below · cited by 1 · depth 24 - j-integral ℚ̄-points factor through the finite chart
ModularCurve.XOneP.exists_comp_eq_specMap_comp_iotaFin_of_jChartFin_mem_pointEquivPlace_twoChartModel_x1_mul1 below · cited by 4 · depth 24 - Place-level Eichler–Shimura relation on the non-Gauss component
ModularCurve.XOneP.exists_coprime_algEquiv_finset_red_diamondAutBar_smul_eq_smul_and_red_eq_smul_frob_smul_red_of_reducesSnd_of_gaussReading_snd_algEquiv_twoChartModel_x1_mul_of_atkinLehner_of_diamondConj3,028 below · cited by 1 · depth 24 - Geometric generic fibre of the two-chart Hecke roof model
ModularCurve.XOneP.exists_curveModel_x1x0FunctionFieldC_iso_pullback_chartPin_galoisCompat_twoChartModel_x1_mul125 below · cited by 1 · depth 24 - Sorting Uₚ at places reducing into the twisted component
ModularCurve.XOneP.exists_finset_heckeDivOneBar_single_eq_single_add_sum_of_red_notMem_of_reducesSnd_of_gaussReading_snd_algEquiv_twoChartModel_x1_mul2,966 below · cited by 1 · depth 24 - Fibre-singular points of the two-chart model are supersingular
ModularCurve.XOneP.exists_iotaFin_eq_and_map_jChartFin_mem_ssJSet_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul1,725 below · cited by 6 · depth 24 - Crossing points of the special fibre are supersingular
ModularCurve.XOneP.exists_iotaFin_eq_and_mem_and_map_jChartFin_mem_ssJSet_of_mem_irreducibleComponents_pair_specialFibre_twoChartModel_x1_mul1,483 below · cited by 3 · depth 24 - Level polynomials for the modular unit on X₁(Mp)
ModularCurve.XOneP.exists_levelPolynomials_chartAlgFin_twoChartModel_x1_mul1,274 below · cited by 1 · depth 24 - Two minimal primes over varpi match the two special-fibre components
ModularCurve.XOneP.exists_minimalPrimes_pair_mem_range_iff_le_of_fst_eq_iotaFin_specialFibre_twoChartModel_x1_mul1,238 below · cited by 2 · depth 24 - Modular units Δ(q)/Δ(qᵖ) in the finite-j chart of X₁(Mp)
ModularCurve.XOneP.exists_modularUnit_mem_chartAlgFin_twoChartModel_x1_mul104 below · cited by 2 · depth 24 - Kronecker congruence in Gauss form for X₁(Mp) chart functions
ModularCurve.XOneP.exists_monic_map_eq_prod_X_sub_C_qTwist_and_gaussPresentation_chartAlgFin_x1_mul86 below · cited by 3 · depth 24 - One-sided pool from level polynomials on the X₁(Mp) model
ModularCurve.XOneP.exists_oneSidedPool_of_levelPolynomials_twoChartModel_x1_mul2,891 below · cited by 1 · depth 24 - Two irreducible components in the mod p fibre of X₁(Mp)
ModularCurve.XOneP.exists_pair_irreducibleComponents_pullback_modelTo_zmod_eq_or_eq_x1_mul1,185 below · cited by 3 · depth 24 - Lifting two good C₂-points of X₁(Mp) to Pic⁰
ModularCurve.XOneP.exists_place_schemeHomOver_valuationSubring_pts_reduction_proj_snd_eq_pic0Mk_proj_fst_eq_zero_and_reduction_eq_of_generic_eq_of_notMem_range_crossings_snd_of_mem_range_iotaFin_twoChartModel_x1_mul3,003 below · cited by 1 · depth 24 - Abel–Jacobi payload on the non-Gauss component of X₁(Mp)
ModularCurve.XOneP.exists_place_schemeHomOver_valuationSubring_pts_reduction_proj_snd_eq_pic0Mk_proj_fst_eq_zero_of_notMem_range_crossings_snd_of_mem_range_iotaFin_twoChartModel_x1_mul3,003 below · cited by 1 · depth 24 - Twisting a k-point of the first component off the crossings
ModularCurve.XOneP.exists_point_fst_comp_eq_and_forall_notMem_range_of_comp_eq_specMap_ringEquiv_comp_specialFibre_twoChartModel_x1_mul1,195 below · cited by 2 · depth 24 - Inertia displacement σ P-P is toric above p
ModularCurve.XOneP.exists_points_valuationSubring_and_proj_eq_zero_pic0Mk_single_sub_single_of_mem_inertia_of_curveModel_igusa_twoChartModel_x1_mul3,103 below · cited by 1 · depth 24 - Inertia-invariant classes of J₁(Mp) extend over Pl after multiplication by n
ModularCurve.XOneP.exists_points_valuationSubring_nsmul_of_forall_smul_eq_self_of_curveModel_igusa_twoChartModel_x1_mul3,105 below · cited by 1 · depth 24 - Existence of place-level reductions red₁,red₂ for X₁(Mp)
ModularCurve.XOneP.exists_red_place_eq_pointEquivPlace_of_generic_eq_of_reduction_eq_components_twoChartModel_x1_mul1 below · cited by 2 · depth 24 - Tame crossing-model form of the completed stalk at a fibre-singular point
ModularCurve.XOneP.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_unramified_of_not_isRegularLocalRing_fibre_twoChartIntegralModel_x1_mul_of_tameLevel2,812 below · cited by 1 · depth 24 - Crossing normal form descends along prime-to-p level raising
ModularCurve.XOneP.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_unramified_of_primeToP_levelRaise_twoChartIntegralModel_x1_mul1,591 below · cited by 1 · depth 24 - Residue field of the σ-pull-back of a Gauss branch
ModularCurve.XOneP.exists_ringEquiv_residueField_comap_igusaFunctionFieldX1C_of_gaussPresentation2 below · cited by 6 · depth 24 - Pic⁰-point of D classifying 𝒪(ξ₁-ξ₂)
ModularCurve.XOneP.exists_schemeHomOver_poincare_iso_ofPoint_tensor_idealModule_of_reduction_snd_valuationSubring_twoChartModel_x1_mul1,411 below · cited by 4 · depth 24 - An 𝔽ₚ-section on every component of the mod p fibre
ModularCurve.XOneP.exists_section_range_subset_of_mem_irreducibleComponents_pullback_modelTo_zmod_x1_mul1,737 below · cited by 1 · depth 24 - Integral j-charts map to the Igusa curve through σ
ModularCurve.XOneP.exists_tensorProduct_chartAlg_to_chartRing_igusaFunctionFieldX1C_of_algEquiv_x1_mul1,180 below · cited by 2 · depth 24 - Integral j-charts of X₁(Mp) reduce into the Igusa function field
ModularCurve.XOneP.exists_tensorProduct_chartAlg_to_chartRing_igusaFunctionFieldX1C_x1_mul1,180 below · cited by 2 · depth 24 - Component function fields of X₁(Mp)_k from valuation subrings
ModularCurve.XOneP.exists_valuationSubring_algEquiv_fractionRing_tensorProduct_of_curveModel_fst_twoChartModel_x1_mul24 below · cited by 1 · depth 24 - Second special-fibre component's function field from a branch valuation ring
ModularCurve.XOneP.exists_valuationSubring_algEquiv_fractionRing_tensorProduct_of_curveModel_snd_twoChartModel_x1_mul24 below · cited by 1 · depth 24 - Degeneracy legs at ℓ≠ p preserve the Gauss centre
ModularCurve.XOneP.forall_valuationSubring_heckeRoof_gaussCentre_alpha_iff_beta_x1_mul1,232 below · cited by 1 · depth 24 - Hecke legs track chart primes under ι_α, ι_β
ModularCurve.XOneP.fst_heckeDegeneracy_baseChange_eq_iotaFin_comap_and_ker_comp_eq_of_fst_eq_iotaFin_twoChartModel_x1_mul0 below · cited by 2 · depth 24 - Gauss reductions land in the residual q-expansion field of X₁(M)
ModularCurve.XOneP.gaussReduction_mem_x1FunctionFieldC_of_x14 below · cited by 5 · depth 24 - Component generic points of geometric fibres lie in the finite-j chart
ModularCurve.XOneP.genericPoint_mem_preimage_chartFinOpen_of_mem_irreducibleComponents_twoChartModel_x1_mul2,890 below · cited by 6 · depth 24 - Non-regular fibre ring at a crossing point of the special fibre
ModularCurve.XOneP.germ_mem_maximalIdeal_and_not_isRegularLocalRing_fibre_fst_of_mem_irreducibleComponents_pair_twoChartModel_x1_mul24 below · cited by 1 · depth 24 - Diamond action on ℚ̄-points of the Pic⁰ model
ModularCurve.XOneP.gpts_diamondAutBar_smul_eq_comp_heckeHom_diamondGen_twoChartModel_x1_mul273 below · cited by 1 · depth 24 - ℓ≠ p: the second Hecke leg preserves crossing points
ModularCurve.XOneP.heckeDegeneracy_baseChange_mem_range_inter_range_of_mem_specialFibre_components_twoChartModel_x1_mul2,954 below · cited by 1 · depth 24 - Base-changed automorphism preserves each component of the geometric fibre
ModularCurve.XOneP.image_eq_self_of_comap_eq_of_mem_irreducibleComponents_specialFibre_twoChartModel_x1_mul0 below · cited by 3 · depth 24 - Reducedness of a special-fibre component's preimage under π_α, ℓ≠ p
ModularCurve.XOneP.isReduced_pullback_heckeDegeneracy_baseChange_specialFibre_component_of_ne_of_specializes_twoChartModel_x1_mul2,893 below · cited by 1 · depth 24 - Chart functions of X₁(Mp) at ℚ̄-points of the finite chart
ModularCurve.XOneP.mem_and_evalAt_pointEquivPlace_eq_of_comp_eq_specMap_comp_iotaFin_and_exists_point_and_algebraMap_twoChartModel_x1_mul1 below · cited by 4 · depth 24 - Chart prime below a component's generic point is minimal over varpi
ModularCurve.XOneP.mem_minimalPrimes_span_of_iotaFin_eq_fst_genericPoint_specialFibre_twoChartModel_x1_mul0 below · cited by 5 · depth 24 - Gauss valuation as the only such valuation subring of K
ModularCurve.XOneP.mem_valuationSubring_iff_exists_powerSeries_of_x1940 below · cited by 3 · depth 24 - Dictionary for the modular unit on the two-chart model of X₁(Mp)
ModularCurve.XOneP.modularUnit_dictionary_or_twoChartModel_x1_mul2,899 below · cited by 1 · depth 24 - Special-fibre Poincaré bundle at reductions of point divisors
ModularCurve.XOneP.nonempty_poincare_pullbackAlong_iso_ofPoint_lineBundle_tensor_idealModule_and_isInvertible_of_points_eq_reduction_twoChartModel_x1_mul26 below · cited by 4 · depth 24 - Two branches at supersingular points of the X₁(Mp) model
ModularCurve.XOneP.not_subsingleton_minimalPrimes_span_germ_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,480 below · cited by 1 · depth 24 - Two branches at supersingular points of the X₁(Mp) model
ModularCurve.XOneP.not_subsingleton_minimalPrimes_span_germ_of_mem_ssJSet_twoChartIntegralModel_x1_mul1,479 below · cited by 6 · depth 24 - Étale entry of Uₚ on the special fibre of J₁(Mp)
ModularCurve.XOneP.proj_snd_addMonoidHom_eq_symm_frob_mul_ofAlgAut_smul_proj_snd_of_pts_reduction_of_diamondRead_of_frobRead_of_sort_specialFibre_twoChartModel_x1_mul_of_atkinLehner_of_diamondConj1,480 below · cited by 1 · depth 24 - Vanishing of the J^E-component off the second component
ModularCurve.XOneP.proj_snd_eq_zero_of_points_eq_reduction_of_poincare_iso_ofPoint_valuationSubring_twoChartModel_x1_mul292 below · cited by 1 · depth 24 - Pl-points reducing into C₂ off C₁ meet the smooth locus
ModularCurve.XOneP.range_subset_smoothLocus_of_reduction_snd_eq_of_not_mem_range_valuationSubring_twoChartModel_x1_mul1,194 below · cited by 4 · depth 24 - The twist swaps the two sides of every non-smooth geometric fibre
ModularCurve.XOneP.sectionBaseChange_swap_connectedComponentIn_of_valuationSubring_comap_ne_twoChartModel_x1_mul2,903 below · cited by 1 · depth 24 - Generator law conjugates ⟨ d⟩ into ⟨ d'⟩ on X₁(Mp)
ModularCurve.XOneP.algEquiv_diamond_symm_eq_diamond_of_generatorLaw_x1_mul32 below · cited by 1 · depth 25 - Diamond automorphisms preserve the Gauss valuation subring
ModularCurve.XOneP.apply_mem_gaussValuationSubring_iff_of_apply_eq_diamondAut_x1_mul1,184 below · cited by 1 · depth 25 - Distinct restrictions of the two Gauss branches to K₁
ModularCurve.XOneP.comap_inclusion_ne_comap_comap_inclusion_of_gaussPresentation_of_algEquiv_x1_mul_x1x01,221 below · cited by 1 · depth 25 - Pull-back along a surjective chart endomorphism preserves minimality over varpi
ModularCurve.XOneP.comap_mem_minimalPrimes_span_of_surjective_of_apply_eq_diamondAut_chartAlgFin_x1_mul0 below · cited by 1 · depth 25 - Unique supersingular prime with trivial residue extension
ModularCurve.XOneP.eq_of_comap_eq_and_forall_exists_sub_mem_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,553 below · cited by 2 · depth 25 - At most two minimal primes over varpi in the j-finite chart
ModularCurve.XOneP.eq_of_mem_minimalPrimes_span_of_ne_of_ne_chartAlgFin_twoChartIntegralModel_x1_mul1,183 below · cited by 1 · depth 25 - Modular equation between the two Hecke legs on j
ModularCurve.XOneP.eval_fibrePoly_apply_heckePin_jChartFin_eq_zero_of_modularPolynomialData_twoChartModel_x1_mul0 below · cited by 1 · depth 25 - Function field of a special-fibre component as a fraction field of k⊗_Amathcal O_{X,z}
ModularCurve.XOneP.exists_algEquiv_fractionRing_tensorProduct_stalk_of_curveModel_fst_twoChartModel_x1_mul6 below · cited by 1 · depth 25 - Function field of the second component as a base-changed stalk
ModularCurve.XOneP.exists_algEquiv_fractionRing_tensorProduct_stalk_of_curveModel_snd_twoChartModel_x1_mul6 below · cited by 1 · depth 25 - A level-p twist of the X₁(Mp) function field
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_chartAlgFin_iff_and_comap_ne_and_coe_eq_atkinLehnerInvolutionFull_x1_mul161 below · cited by 7 · depth 25 - Partial Atkin–Lehner automorphism of L·ℚ(X₁(Mp)), with generator law
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_coe_eq_atkinLehnerInvolutionFull_and_generatorLaw_x1_mul155 below · cited by 1 · depth 25 - Kronecker branch test on the mod p fibre of X₁(Mp)
ModularCurve.XOneP.exists_comp_fst_iff_and_exists_comp_snd_iff_apply_jChartFin_of_pow_pow_ne_of_gaussReading_algEquiv_specialFibre_twoChartModel_x1_mul2,926 below · cited by 1 · depth 25 - Eichler–Shimura relation read through σ on the Igusa curve
ModularCurve.XOneP.exists_coprime_algEquiv_finset_red_smul_diamondAutBar_smul_eq_and_red_smul_eq_smul_frob_smul_of_gaussReduces_smul_twoChartModel_x1_mul_of_atkinLehner_of_diamondConj2,993 below · cited by 1 · depth 25 - Prime-to-p level raising is étale at supersingular points
ModularCurve.XOneP.exists_etale_away_comp_chartAlgFin_primeToP_levelRaise_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,233 below · cited by 1 · depth 25 - Finite étale level quotients of the modular unit on X₁(Mp)
ModularCurve.XOneP.exists_finite_etale_quotient_span_aeval_chartAlgFin_twoChartModel_x1_mul1,271 below · cited by 1 · depth 25 - Readings of j outside 𝔽_{p²} off finitely many places
ModularCurve.XOneP.exists_finset_red_notMem_imp_apply_jChartFin_pow_ne_of_gaussReading_snd_algEquiv_twoChartModel_x1_mul263 below · cited by 1 · depth 25 - Finite surjective morphism of two-chart models over the Γ₀(p) floor
ModularCurve.XOneP.exists_floorHom_isFinite_twoChartIntegralModel_x1_mul_x1x09 below · cited by 2 · depth 25 - Canonical-subgroup sorting of Uₚ at a place of X₁(Mp)
ModularCurve.XOneP.exists_heckeDivOneBar_single_eq_single_add_sum_and_apply_jChartFin_eq_pow_of_apply_eq_pow_of_spec_comp_iotaFin_twoChartModel_x1_mul311 below · cited by 1 · depth 25 - Cusp section and its translate meet all components mod p
ModularCurve.XOneP.exists_hom_pullback_snd_forall_mem_or_mem_of_irreducibleComponents_of_cuspSection_twoChartModel_x1_mul1,735 below · cited by 1 · depth 25 - Fibre-singular points are supersingular j-points, under a genus inequality
ModularCurve.XOneP.exists_iotaFin_eq_and_map_jChartFin_mem_ssJSet_of_not_isRegularLocalRing_fibre_of_genusFF_add_one_le_twoChartIntegralModel_x1_mul1,693 below · cited by 1 · depth 25 - Crossing points of the special fibre are supersingular
ModularCurve.XOneP.exists_iotaFin_eq_and_natCast_mem_and_map_jChartFin_mem_ssJSet_of_mem_range_inter_range_specialFibre_twoChartModel_x1_mul2,917 below · cited by 2 · depth 25 - Picard–Lefschetz bundle for inertia displacement at a crossing
ModularCurve.XOneP.exists_isInvertible_pullback_iso_ofPoint_tensor_and_pullback_iso_unit_of_reduction_crossing_of_mem_inertia_of_curveModel_igusa_twoChartModel_x1_mul2,993 below · cited by 1 · depth 25 - Centre of the Gauss valuation is minimal over varpi
ModularCurve.XOneP.exists_mem_minimalPrimes_span_forall_mem_iff_coe_mem_nonunits_gaussValuationSubring_chartAlgFin_x1_mul1,183 below · cited by 1 · depth 25 - Monic lift through q↦ qᵖ of prodᵢ (X-g(ζⁱ q))
ModularCurve.XOneP.exists_monic_map_eq_prod_X_sub_C_qTwist_chartAlgFin_x1_mul81 below · cited by 1 · depth 25 - Toric divisor class from two Pl-points with common reduction
ModularCurve.XOneP.exists_points_valuationSubring_and_proj_eq_zero_pic0Mk_of_poincare_iso_ofPoint_tensor_idealModule_of_reduction_eq_of_sameComponent_of_curveModel_igusa_twoChartModel_x1_mul2,995 below · cited by 1 · depth 25 - Line bundle trivial on both components extends [P']-[P] torically
ModularCurve.XOneP.exists_points_valuationSubring_and_proj_eq_zero_pic0Mk_single_sub_single_of_isInvertible_of_pullback_iso_ofPoint_tensor_idealModule_of_pullback_iso_unit_twoChartModel_x1_mul1,422 below · cited by 1 · depth 25 - Extending the n-th power of a Pic⁰ point over O
ModularCurve.XOneP.exists_rigidifiedLineBundle_fibrewiseAlgEquivZero_and_pullbackAlong_iso_tensorPow_poincare_of_map_maximalIdeal_eq_twoChartModel_x1_mul3,095 below · cited by 1 · depth 25 - The Pic⁰-point of 𝒪(u₁-u₂) for same-component sections
ModularCurve.XOneP.exists_schemeHomOver_poincare_iso_ofPoint_tensor_idealModule_of_sameComponent_of_curveModel_igusa_twoChartModel_x1_mul2,987 below · cited by 1 · depth 25 - Inertia translates extend to Pl-sections with equal reduction
ModularCurve.XOneP.exists_sections_valuationSubring_extending_and_reduction_eq_of_mem_inertia_of_curveModel_twoChartModel_x1_mul1 below · cited by 1 · depth 25 - Spanning set with residue-independent Gauss units on a Hecke leg
ModularCurve.XOneP.exists_span_and_residueIndependent_gauss_heckeRoof_x1_mul1,226 below · cited by 1 · depth 25 - Stalk package at a supersingular point of X₁(Mp)
ModularCurve.XOneP.exists_stalkPackage_floorHom_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,578 below · cited by 1 · depth 25 - A Gauss valuation subring of the Hecke roof function field
ModularCurve.XOneP.exists_valuationSubring_gauss_heckeRoof_x1_mul0 below · cited by 1 · depth 25 - Stalks of the two-chart model of X₁(Mp) are valuation rings
ModularCurve.XOneP.exists_valuationSubring_ringEquiv_stalk_of_ringKrullDim_le_one_twoChartIntegralModel_x1_mul14 below · cited by 2 · depth 25 - Pull-back of V equals the Gauss ring iff non-units match
ModularCurve.XOneP.forall_gaussPresentation_map_mem_nonunits_iff_iff_comap_eq0 below · cited by 1 · depth 25 - Irreducible fibre pieces: u or u' vanishes identically
ModularCurve.XOneP.forall_mem_or_forall_mem_range_of_irreducibleSpace_fibre_twoChartModel_x1_mul0 below · cited by 1 · depth 25 - Uniformiser germ and stalk dimension at the generic point of C₁
ModularCurve.XOneP.germ_mem_maximalIdeal_and_ringKrullDim_stalk_le_one_of_isGenericPoint_fst_twoChartModel_x1_mul11 below · cited by 1 · depth 25 - Generic points of the second special-fibre component have codimension one
ModularCurve.XOneP.germ_mem_maximalIdeal_and_ringKrullDim_stalk_le_one_of_isGenericPoint_snd_twoChartModel_x1_mul11 below · cited by 1 · depth 25 - No component of a bad geometric fibre is fixed
ModularCurve.XOneP.image_ne_of_mem_irreducibleComponents_pullback_of_not_smooth_twoChartModel_x1_mul2,893 below · cited by 1 · depth 25 - Far component of the geometric special fibre descends to ̄ k₀
ModularCurve.XOneP.image_range_eq_range_of_geometric_specialFibre_twoChartModel_x1_mul0 below · cited by 1 · depth 25 - Localisations of the Gauss reduction of the X₁(Mp) charts are DVRs
ModularCurve.XOneP.isDiscreteValuationRing_localization_atPrime_adjoin_gaussReductions_chartAlg_x1_mul2,865 below · cited by 1 · depth 25 - Regularity of the σ-twisted reduced j⁻¹-chart of X₁(Mp)
ModularCurve.XOneP.isDiscreteValuationRing_localization_atPrime_adjoin_gaussReductions_map_chartAlgInf_of_algEquiv_x1_mul2,865 below · cited by 1 · depth 25 - Geometric integrality of the Gauss branches of the X₁(Mp) two-chart model
ModularCurve.XOneP.isDomain_tensorProduct_quotient_of_mem_minimalPrimes_span_of_map_eq_zero_twoChartModel_x1_mul1,192 below · cited by 3 · depth 25 - Both branches meet every supersingular point of X₁(Mp)
ModularCurve.XOneP.le_of_mem_minimalPrimes_span_of_mem_ssJSet_chartAlgFin_x1_mul1,477 below · cited by 5 · depth 25 - Crossing points of the special fibre have supersingular j
ModularCurve.XOneP.map_jChartFin_mem_ssJSet_of_not_subsingleton_minimalPrimes_span_germ_twoChartIntegralModel_x1_mul1,482 below · cited by 1 · depth 25 - Saturation of the geometric special fibre components of X₁(Mp)
ModularCurve.XOneP.mem_range_iff_mem_range_of_fst_eq_specialFibre_components_twoChartModel_x1_mul1,204 below · cited by 2 · depth 25 - Supersingular chart points of the special fibre are crossing points
ModularCurve.XOneP.mem_range_inter_range_of_iotaFin_eq_of_map_jChartFin_mem_ssJSet_specialFibre_twoChartModel_x1_mul1,480 below · cited by 1 · depth 25 - Unramifiedness of valuation rings of the Hecke roof field
ModularCurve.XOneP.mul_inv_mem_of_mem_nonunits_valuationSubring_heckeRoof_of_forall_aeval_mem_x1_mul1,179 below · cited by 1 · depth 25 - Two minimal primes over varpi in the j-chart of X₁(Mp)
ModularCurve.XOneP.ncard_minimalPrimes_span_chartAlgFin_eq_two_and_jInvChartInf_notMem_x1_mul1,182 below · cited by 2 · depth 25 - Hecke degeneracy legs pull the j-finite chart back exactly
ModularCurve.XOneP.preimage_opensRange_iotaFin_eq_heckeDegeneracy_twoChartModel_x1_mul255 below · cited by 1 · depth 25 - Vanishing J_I-coordinate when both reductions avoid C₁
ModularCurve.XOneP.proj_fst_eq_zero_of_points_eq_reduction_snd_of_poincare_iso_ofPoint_valuationSubring_twoChartModel_x1_mul292 below · cited by 2 · depth 25 - Reduction into the second component equals Gauss reduction of σ̄ P
ModularCurve.XOneP.red_eq_red_smul_of_reducesSnd_of_gaussReading_snd_algEquiv_twoChartModel_x1_mul2,937 below · cited by 1 · depth 25 - Reduction into C₂ versus Gauss reduction of the σ̄-translate
ModularCurve.XOneP.reducesSnd_iff_gaussReduces_smul_of_gaussReading_algEquiv_twoChartModel_x1_mul2,934 below · cited by 1 · depth 25 - Galois conjugate of σ equals σ twisted by a diamond
ModularCurve.XOneP.ringEquiv_algEquiv_symm_eq_algEquiv_diamond_of_generatorLaw_x1_mul44 below · cited by 1 · depth 25 - Surjectivity of a diamond endomorphism of the finite chart algebra
ModularCurve.XOneP.surjective_of_apply_eq_diamondAut_chartAlgFin_twoChartIntegralModel_x1_mul32 below · cited by 1 · depth 25 - An L-automorphism exchanging the two minimal primes over varpi
ModularCurve.XOneP.comap_eq_and_comap_eq_of_mem_minimalPrimes_of_gauss_algEquiv_chartAlgFin_x1_mul1,182 below · cited by 2 · depth 26 - Supersingular special-fibre points fixed by relative chart automorphisms
ModularCurve.XOneP.comap_eq_of_forall_apply_eq_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,544 below · cited by 1 · depth 26 - Automorphisms fixing j and j(qᵖ) stabilise both branch valuation rings
ModularCurve.XOneP.comap_eq_of_map_j_eq_of_map_jp_eq_valuationSubring_pair_x1_mul124 below · cited by 2 · depth 26 - No minimal prime of the special fibre is fixed
ModularCurve.XOneP.comap_ne_of_mem_minimalPrimes_map_maximalIdeal_chartAlgFin_twoChartModel_x1_mul1,182 below · cited by 1 · depth 26 - Gauss valuation of X₁(Mp) is the unique extension from the floor
ModularCurve.XOneP.eq_of_comap_inclusion_eq_comap_inclusion_of_gaussPresentation_x1_mul_x1x01,220 below · cited by 3 · depth 26 - Primes of the j-chart of X₁(Mp) are conjugate over the floor
ModularCurve.XOneP.exists_algEquiv_chartAlgFin_comap_eq_of_comap_eq_twoChartIntegralModel_x1_mul238 below · cited by 1 · depth 26 - Diamond action on Gauss-reduced places of the Igusa field
ModularCurve.XOneP.exists_algEquiv_forall_gaussReduces_diamondAutBar_smul_and_red_eq_smul_red_of_coprime_twoChartModel_x1_mul2,923 below · cited by 1 · depth 26 - Atkin–Lehner involution on X₁(Mp) over ℚ(ζₚ)
ModularCurve.XOneP.exists_algEquiv_map_j_eq_qExpand_and_coe_eq_atkinLehnerInvolutionFull_x1_mul155 below · cited by 1 · depth 26 - Unramified level sets of p¹²/u on the far branch
ModularCurve.XOneP.exists_avoid_forall_formallyUnramified_quotient_farPrime_sup_span_aeval_twoChartModel_x1_mul164 below · cited by 1 · depth 26 - Unramified level sets of the modular unit on the Gauss component
ModularCurve.XOneP.exists_avoid_forall_formallyUnramified_quotient_gaussPrime_sup_span_aeval_twoChartModel_x1_mul163 below · cited by 2 · depth 26 - Oriented branch dictionary for the special fibre of X₁(Mp)
ModularCurve.XOneP.exists_comp_fst_iff_and_exists_comp_snd_iff_of_mem_minimalPrimes_of_gaussReading_specialFibre_twoChartModel_x1_mul2,924 below · cited by 2 · depth 26 - σ-transport of non-nodal k-points between the two special-fibre components
ModularCurve.XOneP.exists_comp_snd_iff_exists_comp_fst_specMap_comp_ringEquiv_symm_of_gaussReading_algEquiv_specialFibre_twoChartModel_x1_mul2,926 below · cited by 2 · depth 26 - Atkin–Lehner transport of Uₚ-supports, up to one diamond
ModularCurve.XOneP.exists_coprime_forall_smul_mem_support_heckeDivOneBar_single_diamondAutBar_smul_smul_of_mem_support_of_atkinLehner394 below · cited by 1 · depth 26 - Two Igusa branches in the geometric special fibre of X₁(Mp)
ModularCurve.XOneP.exists_curveModel_pair_hom_specialFibre_birational_twoChartIntegralModel_x1_mul1,209 below · cited by 1 · depth 26 - Crossings of the two-chart model enumerated by k-points
ModularCurve.XOneP.exists_fin_hom_pullback_comp_eq_id_and_injective_base_closedPoint_twoChartModel_x1_mul0 below · cited by 1 · depth 26 - Hecke legs span the roof field by d elements
ModularCurve.XOneP.exists_fin_span_heckeRoof_x1_mul220 below · cited by 1 · depth 26 - Finiteness of modular-unit level quotients on the j-chart
ModularCurve.XOneP.exists_forall_finite_quotient_span_aeval_and_finrank_le_chartAlgFin_twoChartModel_x1_mul281 below · cited by 1 · depth 26 - Unramified level sets of the modular unit on the j-chart
ModularCurve.XOneP.exists_forall_isUnramifiedAt_quotient_span_aeval_of_comap_eq_bot_chartAlgFin_twoChartModel_x1_mul9 below · cited by 1 · depth 26 - Surjectivity of Gauss reduction onto κ(X₁(M))
ModularCurve.XOneP.exists_gaussPresentation_reduction_eq_of_mem_x1FunctionFieldC_of_x10 below · cited by 1 · depth 26 - Uₚ at a place reducing into the Igusa component
ModularCurve.XOneP.exists_heckeDivOneBar_single_eq_sum_and_red_eq_frob_inv_smul_of_gaussReduces_of_surjective_residue_igusaModel_twoChartModel_x1_mul1,242 below · cited by 1 · depth 26 - Component-swapping endomorphism of the 𝔽ₚ-fibre of X₁(Mp)
ModularCurve.XOneP.exists_hom_pullback_snd_forall_exists_ne_mapsTo_irreducibleComponents_twoChartModel_x1_mul1,733 below · cited by 1 · depth 26 - Sections through a crossing factor through the crossing chart
ModularCurve.XOneP.exists_lift_comp_crossingChart_eq_specMap_lift_of_base_closedPoint_eq_twoChartModel_x1_mul0 below · cited by 1 · depth 26 - Fixed component of a bad fibre yields a fixed minimal prime
ModularCurve.XOneP.exists_mem_minimalPrimes_comap_eq_of_image_eq_of_mem_irreducibleComponents_twoChartModel_x1_mul2,891 below · cited by 1 · depth 26 - Finite diamond action on L·ℚ(X₁(Mp)) fixing j, j(qᵖ)
ModularCurve.XOneP.exists_mulSemiringAction_isInvariant_laurentBaseChange_gamma0_smul_j_eq_x1_mul219 below · cited by 2 · depth 26 - Nodal geometric special fibre of X₁(Mp), chartwise
ModularCurve.XOneP.exists_mul_eq_zero_and_maximalIdeal_eq_span_pair_localization_tensor_chartAlg_x1_mul2,864 below · cited by 2 · depth 26 - Two-open cover at a crossing with invertible tube functions
ModularCurve.XOneP.exists_opens_sup_eq_top_and_forall_mem_basicOpen_of_crossingChart_of_sections_twoChartModel_x1_mul9 below · cited by 1 · depth 26 - A Pl-point of relative Pic⁰ classifying an invertible module
ModularCurve.XOneP.exists_points_valuationSubring_and_poincare_pullbackAlong_iso_pic0Mk_single_sub_single_of_isInvertible_of_pullback_iso_ofPoint_tensor_idealModule_of_pullback_iso_unit_twoChartModel_x1_mul1,420 below · cited by 1 · depth 26 - Component-trivial Pl-point of Pic⁰ reduces with proj=0
ModularCurve.XOneP.exists_pts_comp_fst_eq_and_proj_eq_zero_of_pullback_poincare_pullbackAlong_iso_unit_twoChartModel_x1_mul0 below · cited by 1 · depth 26 - Poincaré bundle at a T'-point as a divisor difference
ModularCurve.XOneP.exists_relEffCartierDiv_pair_isInvertible_and_pullback_lineBundle_tensor_idealModule_iso_poincare_of_map_maximalIdeal_eq_twoChartModel_x1_mul3,031 below · cited by 1 · depth 26 - Residue fields of the Gauss rings of X₁(Mp) and X₁(M)
ModularCurve.XOneP.exists_ringHom_residueField_eq_of_gaussPresentation_x1_mul1,180 below · cited by 2 · depth 26 - Bidegree-zero twist of L̄^{⊗ n} on the special fibre
ModularCurve.XOneP.exists_tensorPow_tensor_tensorPow_eulerChar_sectionsOf_pullback_eq_of_relEffCartierDiv_twoChartModel_x1_mul2,987 below · cited by 1 · depth 26 - Inertia-equivariant oriented étale crossing chart for X₁(Mp) over Pl
ModularCurve.XOneP.forall_exists_orientedEtaleCrossingChart_valuationSubring_twoChartModel_x1_mul2,963 below · cited by 1 · depth 26 - Residue degree one at supersingular points over the Γ₀(p)-floor
ModularCurve.XOneP.forall_exists_sub_mem_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,550 below · cited by 1 · depth 26 - Generic Abel–Jacobi reading of a Pl-point of relative Pic⁰
ModularCurve.XOneP.gpts_pic0Mk_single_sub_single_eq_comp_of_pullback_poincare_pullbackAlong_iso_ofPoint_tensor_idealModule_twoChartModel_x1_mul1,419 below · cited by 2 · depth 26 - Components of the special fibre of X₁(Mp): I₁ I₂ = Iₛ
ModularCurve.XOneP.isInvertible_ker_and_ker_mul_ker_eq_ker_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,925 below · cited by 4 · depth 26 - Chart rings of X₁(Mℓ p) as a pushout
ModularCurve.XOneP.isPushout_chartAlgFin_levelRaise_twoChartIntegralModel_x1_mul1,232 below · cited by 1 · depth 26 - Reduced special fibre of the j-chart ring of X₁(Mp)
ModularCurve.XOneP.isReduced_chartAlgFin_quotient_map_maximalIdeal_twoChartModel_x1_mul1,187 below · cited by 1 · depth 26 - Unramifiedness at non-maximal primes of the X₁(Mp) chart ring
ModularCurve.XOneP.isUnramifiedAt_chartAlgFin_of_not_isMaximal_twoChartIntegralModel_x1_mul_x1x01,200 below · cited by 1 · depth 26 - Crossing points on the finite chart of X₁(Mp) are supersingular
ModularCurve.XOneP.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_le_chartAlgFin_x1_mul1,480 below · cited by 1 · depth 26 - Modular unit at the far branch of the j-finite chart
ModularCurve.XOneP.modularUnit_mem_and_notMem_farPrime_chartAlgFin_twoChartModel_x1_mul262 below · cited by 1 · depth 26 - Modular unit Δ(q)/Δ(qᵖ) at the Gauss prime
ModularCurve.XOneP.modularUnit_notMem_and_mem_gaussPrime_chartAlgFin_twoChartModel_x1_mul262 below · cited by 1 · depth 26 - Triviality on both special-fibre components of a crossing-glued module
ModularCurve.XOneP.nonempty_pullback_comp_iso_unit_of_isFrameOn_of_map_eq_smul_twoChartModel_x1_mul12 below · cited by 1 · depth 26 - Generic fibre of a crossing-glued module on X₁(Mp)
ModularCurve.XOneP.nonempty_pullback_iso_ofPoint_tensor_idealModule_of_isFrameOn_of_map_eq_smul_twoChartModel_x1_mul34 below · cited by 1 · depth 26 - Conjugation of diamond operators on places of X₁(Mp)
ModularCurve.XOneP.ofAlgAut_smul_diamondAutBar_smul_eq_diamondAutBar_smul_ofAlgAut_smul_place_of_diamondConj_x1_mul0 below · cited by 1 · depth 26 - Gauss and σ-twisted readings give the same Igusa place
ModularCurve.XOneP.pointEquivPlace_snd_eq_pointEquivPlace_fst_of_comp_eq_spec_map_comp_iotaFin_of_gaussReading_algEquiv_twoChartModel_x1_mul1,182 below · cited by 1 · depth 26 - Transport of j-finite chart points under σ̄
ModularCurve.XOneP.pointEquivPlace_symm_comp_eq_iff_smul_ofAlgAut_of_coe_eq_algEquiv_twoChartModel_x1_mul49 below · cited by 2 · depth 26 - Stalks of the mod p fibre of the X₁(Mp) model have dimension ≤ 1
ModularCurve.XOneP.ringKrullDim_stalk_pullback_toBase_le_one_twoChartIntegralModel_x1_mul10 below · cited by 3 · depth 26 - Euler characteristics of twists by the component C₂
ModularCurve.XOneP.eulerChar_sectionsOf_pullback_tensorPow_module_ker_and_invModule_ker_twoChartModel_x1_mul2,983 below · cited by 1 · depth 27 - Atkin–Lehner automorphism at p exchanging the two degeneracy legs
ModularCurve.XOneP.exists_coprime_algEquiv_algEquiv_apply_heckeAlphaOneBar_eq_heckeBetaOneBar_diamondAutBar_and_apply_heckeBetaOneBar_eq_of_atkinLehnerInvolutionFull211 below · cited by 1 · depth 27 - A component-moving self-map on the mod p fibre of X₁(Mp)
ModularCurve.XOneP.exists_opens_hom_mapsTo_irreducibleComponents_pullback_modelTo_zmod_x1_mul1,730 below · cited by 1 · depth 27 - Oriented crossing chart of X₁(Mp) over a valuation subring
ModularCurve.XOneP.exists_orientedCrossingChart_valuationSubring_of_chart_twoChartModel_x1_mul3 below · cited by 1 · depth 27 - Relative Cartier extension of a generic divisor on X₁(Mp)
ModularCurve.XOneP.exists_relEffCartierDiv_pullbackAlong_eq_and_isInvertible_comap_ker_of_isInvertible_ker_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,950 below · cited by 1 · depth 27 - Residue degree p-1 over the floor for Gauss valuation rings
ModularCurve.XOneP.finrank_residueField_valuationSubring_eq_sub_one_of_gauss_x1_mul_x1x01,215 below · cited by 1 · depth 27 - Oriented étale crossing chart uv=varpi^e at the X₁(Mp) crossings
ModularCurve.XOneP.forall_exists_orientedEtaleCrossingChart_baseChange_of_injective_twoChartModel_x1_mul2,954 below · cited by 1 · depth 27 - Crossings of the two-chart model as residue-field points
ModularCurve.XOneP.forall_exists_spec_hom_pullback_comp_snd_eq_and_base_closedPoint_eq_of_surjective_twoChartModel_x1_mul0 below · cited by 1 · depth 27 - Trivial residue action at supersingular points of X₁(Mp)
ModularCurve.XOneP.forall_smul_sub_mem_of_smul_eq_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,544 below · cited by 1 · depth 27 - Valuation rings at a supersingular point pinned by floor residues
ModularCurve.XOneP.forall_valuationSubring_residueField_eq_and_forall_exists_sub_residue_mem_nonunits_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul_x1x01,243 below · cited by 2 · depth 27 - Two special-fibre components of X₁(Mp) as distinct prime divisors
ModularCurve.XOneP.isIntegral_subscheme_ker_and_ker_eq_vanishingIdeal_closure_and_ker_inf_ker_eq_ker_of_map_maximalIdeal_eq_twoChartModel_x1_mul1,234 below · cited by 6 · depth 27 - Components of the special fibre of X₁(Mp) as Cartier divisors
ModularCurve.XOneP.isInvertible_ker_and_tensor_iso_unit_and_pullback_invModule_iso_foldr_ofPoint_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,973 below · cited by 2 · depth 27 - Invertibility of the component ideal sheaves over O
ModularCurve.XOneP.isInvertible_ker_comp_baseChange_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,923 below · cited by 2 · depth 27 - Supersingular points of the j-finite chart of X₁(Mp) are closed
ModularCurve.XOneP.isMaximal_and_finite_quotient_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_x1_mul1,480 below · cited by 3 · depth 27 - Chart functions in the Gauss maximal ideal vanish on C₁
ModularCurve.XOneP.mem_asIdeal_of_mem_nonunits_of_iotaFin_eq_fst_of_gaussReading_specialFibre_twoChartModel_x1_mul1,180 below · cited by 1 · depth 27 - Poincaré bundle along the geometric Abel–Jacobi map of X₁(Mp)
ModularCurve.XOneP.nonempty_poincare_pullbackAlong_iso_ofPoint_tensor_ofPoint_idealModule_of_eq_comp_ajbar_of_curveModel_twoChartModel_x1_mul15 below · cited by 2 · depth 27 - Frobenius twist of k-points acts by coefficientwise Frobenius on Igusa places
ModularCurve.XOneP.pointEquivPlace_eq_frob_smul_pointEquivPlace_of_comp_eq_frobenius_comp_of_gaussReading_igusaModel_twoChartModel_x1_mul1,197 below · cited by 1 · depth 27 - Branch ideals at a crossing meet in (varpi_A)
ModularCurve.XOneP.branchIdeal_inf_branchIdeal_eq_span_germ_stalk_of_forall_mem_range_twoChartModel_x1_mul1,191 below · cited by 1 · depth 28 - Branch ideals proper, stalk dimension ≤ 2, uniformiser nonzero
ModularCurve.XOneP.branchIdeal_ne_maximalIdeal_and_ringKrullDim_stalk_le_two_and_germ_ne_zero_twoChartModel_x1_mul1,196 below · cited by 3 · depth 28 - Transversality of the two branches in the stalk over A
ModularCurve.XOneP.branchIdeal_sup_branchIdeal_eq_maximalIdeal_stalk_of_isReduced_pullback_twoChartModel_x1_mul1,208 below · cited by 2 · depth 28 - Exchanging j(q^e) and j(q^{ep}) determines wₚ
ModularCurve.XOneP.coe_apply_coeffEmb_eq_coeffEmb_atkinLehnerInvolutionFull_of_atkinLehnerSlash_p78 below · cited by 1 · depth 28 - Slash transport along Wₚ swaps j(q^e) and j(q^{ep})
ModularCurve.XOneP.coe_apply_eq_coeffEmb_qExpand_mul_jq_of_atkinLehnerSlash_p7 below · cited by 1 · depth 28 - Atkin–Lehner pull-back at p swaps the degeneracy embeddings
ModularCurve.XOneP.comp_alpha_eq_beta_and_comp_beta_eq_alpha_comp_diamondAutBar_of_atkinLehnerSlash_p1 below · cited by 1 · depth 28 - Atkin–Lehner automorphism W_{p²} of the Γ₁(Mp)∩Γ₀(Mp²) function field
ModularCurve.XOneP.exists_algEquiv_laurentBaseChange_x1x0FunctionFieldC_coeffMap_apply_eq_atkinLehnerSlash_sq112 below · cited by 1 · depth 28 - Atkin–Lehner automorphism wₚ of ℚ̄(X₁(Mp))
ModularCurve.XOneP.exists_algEquiv_x1FunctionFieldBar_coeffMap_apply_eq_atkinLehnerSlash_p81 below · cited by 2 · depth 28 - Atkin–Lehner leg laws: Wα=β⟨ d⟩τ, Wβ=α⟨ d'⟩τ
ModularCurve.XOneP.exists_coprime_apply_heckeAlphaOneBar_eq_heckeBetaOneBar_diamondAutBar_of_atkinLehnerSlash_p_of_atkinLehnerSlash_sq33 below · cited by 1 · depth 28 - Transport of a crossing presentation to the base change over O
ModularCurve.XOneP.exists_crossingPresentation_stalk_baseChange_of_branchIdeal_eq_span_twoChartModel_x1_mul2,919 below · cited by 1 · depth 28 - Local principality of the closure divisor along the special fibre
ModularCurve.XOneP.exists_mem_ideal_and_map_ideal_eq_span_singleton_and_mem_nonZeroDivisors_of_I_eq_ker_twoChartModel_x1_mul2,928 below · cited by 1 · depth 28 - Branch generic points specialise to each crossing point
ModularCurve.XOneP.genericPoint_specializes_crossing_and_baseChange_twoChartModel_x1_mul0 below · cited by 3 · depth 28 - Points over the closed point specialise from a branch
ModularCurve.XOneP.genericPoint_specializes_or_of_not_mem_basicOpen_baseChange_twoChartModel_x1_mul0 below · cited by 2 · depth 28 - Component ideal sheaves cut out the reduced crossing divisor
ModularCurve.XOneP.isInvertible_comap_ker_and_comap_ker_eq_prod_ofPoint_of_map_maximalIdeal_eq_twoChartModel_x1_mul1,250 below · cited by 1 · depth 28 - Regularity of the X₁(Mp) two-chart model after unramified base change
ModularCurve.XOneP.isRegularLocalRing_stalk_pullback_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,887 below · cited by 1 · depth 28 - Kernel of the geometric fibre comparison is the invertible ideal (varpi)
ModularCurve.XOneP.ker_baseChange_eq_comap_ker_residue_and_isInvertible_and_nonempty_invModule_iso_twoChartModel_x1_mul2,903 below · cited by 1 · depth 28 - Closure of a generic-fibre divisor avoids special-fibre generic points
ModularCurve.XOneP.notMem_support_of_closure_mem_irreducibleComponents_of_I_eq_ker_twoChartModel_x1_mul6 below · cited by 2 · depth 28 - Images of the two components in X_O are closed
ModularCurve.XOneP.range_comp_baseChange_eq_closure_genericPoint_and_isClosed_twoChartModel_x1_mul1,205 below · cited by 2 · depth 28 - Rationality and closedness at a crossing over O
ModularCurve.XOneP.residue_germ_surjective_and_isClosed_crossing_baseChange_twoChartModel_x1_mul0 below · cited by 1 · depth 28 - Stalk dimensions on X_O: ≤ 2 on the closed fibre, 1 at component generic points
ModularCurve.XOneP.ringKrullDim_stalk_le_two_of_snd_eq_closedPoint_and_ringKrullDim_stalk_genericPoint_eq_one_of_map_maximalIdeal_eq_twoChartModel_x1_mul2,918 below · cited by 1 · depth 28 - Stalk at a crossing of the two-chart model has dimension ≥ 2
ModularCurve.XOneP.two_le_ringKrullDim_stalk_crossing_baseChange_of_injective_twoChartModel_x1_mul2,892 below · cited by 1 · depth 28 - Branch ideals extend to the geometric special fibre at a crossing
ModularCurve.XOneP.map_stalkMap_branchIdeal_eq_branchIdeal_specialFibre_twoChartModel_x1_mul1,208 below · cited by 1 · depth 29 - Specialisation onto the other vertical component forces a crossing
ModularCurve.XOneP.mem_range_of_comp_baseChange_mem_closure_genericPoint_of_map_maximalIdeal_eq_twoChartModel_x1_mul1,235 below · cited by 1 · depth 29
ModularCurve.XZeroP 10
- Supersingular primes contain every minimal prime of (varpi) on X₀(Mp)
ModularCurve.XZeroP.le_of_mem_minimalPrimes_span_of_mem_ssJSet_chartAlgFin_gamma0_mul976 below · cited by 2 · depth 26 - Node thickness (p-1)·width at supersingular points of X₀(pM)
ModularCurve.XZeroP.exists_place_ringEquiv_adicCompletion_stalk_uvCrossingModel_pow_placeWidthChar_of_mem_ssJSet_twoChartIntegralModel_gamma0_mul_of_forall_exists_pow_eq_self1,833 below · cited by 1 · depth 27 - Crossing point of the special fibre of X₀(Mp) is supersingular
ModularCurve.XZeroP.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_le_chartAlgFin_gamma0_mul979 below · cited by 1 · depth 27 - Coefficient ring for the completed stalk at a supersingular point
ModularCurve.XZeroP.exists_completeDVR_algebraMap_ringHom_adicCompletion_stalk_twoChartIntegralModel_gamma0_mul24 below · cited by 1 · depth 28 - Supersingular chart point yields supersingular place of level-M fibre
ModularCurve.XZeroP.exists_mem_ssPlaces_ringHom_eq_residueFst_of_mem_ssJSet_chartAlgFin_twoChartIntegralModel_gamma0_mul979 below · cited by 1 · depth 28 - Base change of the X₀(Mp) chart algebra to ℚ̄
ModularCurve.XZeroP.exists_ringHom_valuationSubring_algebra_ringEquiv_chartAlgFin_coeffSubring_fieldOver_twoChartIntegralModel_gamma0_mul190 below · cited by 1 · depth 28 - Supersingular point of the finite chart of X₀(Mp) is closed with finite residue field
ModularCurve.XZeroP.isMaximal_and_finite_quotient_of_map_jChartFin_mem_ssJSet_twoChartIntegralModel_gamma0_mul15 below · cited by 2 · depth 28 - Gauss-reduction lifting of the Γ₀(M) q-expansion field
ModularCurve.XZeroP.exists_gaussPresentation_of_mem_qExpFunctionFieldC_gamma0_of_gamma0_mul332 below · cited by 1 · depth 29 - Two Gauss valuation rings on X₀(Mp) above p
ModularCurve.XZeroP.valuationSubring_eq_or_eq_comap_and_uniformizer_and_gaussReduction_eq_gamma0_mul202 below · cited by 4 · depth 29 - Crossings in the special fibre of X₀(Mp) are supersingular
ModularCurve.XZeroP.map_jChartFin_mem_ssJSet_of_exists_two_minimalPrimes_span_le_chartAlgFin_gamma0_mul_of_embedding1,027 below · cited by 3 · depth 32
ModularCurve.XZeroPM 7
- Thickness of the X₀(Mp) node above a supersingular point
ModularCurve.XZeroPM.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_pow_card_inertia_of_mem_ssJSet_twoChartIntegralModel_gamma02,397 below · cited by 2 · depth 26 - Inertia order equals the supersingular place width
ModularCurve.XZeroPM.card_inertia_eq_placeWidthChar_of_forall_exists_pow_eq_self_of_mem_ssJSet_twoChartIntegralModel_gamma01,599 below · cited by 1 · depth 27 - Factorisation of a supersingular attachment through the Gauss residue field
ModularCurve.XZeroPM.exists_ringHom_residueField_gauss_comp_residue_eq_of_ringEquiv_modularFunctionFieldC_gamma0_mul345 below · cited by 1 · depth 28 - Inertia at a supersingular chart point via the Gauss place
ModularCurve.XZeroPM.mem_inertia_iff_smul_eq_and_forall_sub_mem_nonunits_gaussPlace_twoChartIntegralModel_gamma01 below · cited by 1 · depth 28 - Kernel of the supersingular attachment map is the Gauss centre
ModularCurve.XZeroPM.mem_ker_iff_coe_mem_nonunits_gaussValuationSubring_of_map_jChartFin_eq_jGeomGen_gamma0_mul344 below · cited by 1 · depth 29 - Kernel of ρ is a minimal prime over (varpi)
ModularCurve.XZeroPM.ker_mem_minimalPrimes_span_of_map_jChartFin_eq_jGeomGen_gamma0_mul0 below · cited by 1 · depth 30 - The chart kernel is not the σ-twisted Gauss centre
ModularCurve.XZeroPM.not_forall_mem_ker_iff_coe_mem_nonunits_comap_of_map_jChartFin_eq_jGeomGen_gamma0_mul135 below · cited by 1 · depth 30