Namespace AlgebraicCurve 1,577 theorems
— 533 · AlgEquiv 1 · Annulus 37 · CartierB 2 · CellDissection 11 · ComponentChart 17 · ConstantReduction 13 · CurveModel 68 · Differential 9 · Divisor 80 · DivisorialWeilPairingData 9 · FunctionField 2 · GluedPic0 10 · GluingData 1 · IsConfluentPattern 1 · IsCurveOver 3 · IsFrobeniusEndo 3 · KummerCover 1 · KwPke 1 · NodalPic0 1 · NodeAnnulusEngine 25 · NodeRingLayers 2 · Pic0 107 · Place 289 · RROpens 9 · RadialRegion 9 · RationalFunctionField 47 · RegularProlongation 91 · RiemannGenusReachedAt 1 · SemilinearAut 18 · SemistableCovering 21 · SemistableModel 20 · TranscendenceTower 2 · TwoChartIntegralModel 127 · WeilDatum 6
directly in AlgebraicCurve 533
- Finiteness along a composite of K-algebra maps
AlgebraicCurve.finiteAlong_comp0 below · cited by 15 · depth 8 - Pushforward norm formula along a finite separable K-morphism
AlgebraicCurve.normFormulaAlong6 below · cited by 54 · depth 8 - Integral extensions in characteristic zero are separable along φ
AlgebraicCurve.separableAlong_of_charZero0 below · cited by 85 · depth 8 - Over an algebraically closed base, the constants are K
AlgebraicCurve.constantsAreBase_of_isAlgClosed45 below · cited by 20 · depth 9 - Differential of a uniformiser generates Ω_{F/K}
AlgebraicCurve.dCoordGenerates_of_isCurveOver1 below · cited by 136 · depth 9 - Surjective K-algebra maps of fields are finite
AlgebraicCurve.finiteAlong_of_surjective0 below · cited by 7 · depth 9 - Riemann–Roch over an algebraically closed base field
AlgebraicCurve.functionFieldRiemannRoch_of_isAlgClosed8 below · cited by 9 · depth 9 - A principal divisor P-Q with deg Q=1 forces genus zero
AlgebraicCurve.genus_eq_zero_of_isPrincipal_single_sub_single28 below · cited by 4 · depth 9 - Principal divisors of degree zero on finite extensions of K(X)
AlgebraicCurve.hasPrincipalDivisors_of_finiteDimensional_ratFunc24 below · cited by 2 · depth 9 - The rational function field K(X) is a curve over K
AlgebraicCurve.instIsCurveOverRatFunc26 below · cited by 12 · depth 9 - Constants are the base field, given a rational place
AlgebraicCurve.constantsAreBase_of_exists_isRational4 below · cited by 4 · depth 10 - Degree of a canonical divisor is 2g-2
AlgebraicCurve.degree_canonicalDivisor_eq_of_riemannRoch0 below · cited by 14 · depth 10 - Vanishing of ℓ(D) when deg D<0
AlgebraicCurve.ell_eq_zero_of_degree_neg0 below · cited by 23 · depth 10 - Finite extensions of K(x) are essentially of finite type
AlgebraicCurve.essFiniteType_of_transcendental_of_finiteDimensional0 below · cited by 117 · depth 10 - Constant reduction of a constant-field extension of function fields
AlgebraicCurve.exists_constantReduction_of_constantFieldExtension9 below · cited by 1 · depth 10 - Riemann's index theorem for curves over a perfect field
AlgebraicCurve.exists_genus_riemannIndex_of_isCurveOver26 below · cited by 13 · depth 10 - Deuring reduction of divisors at good constant reduction
AlgebraicCurve.exists_placeMap_mapDomain_eq_ord_of_good_constantReduction74 below · cited by 5 · depth 10 - Finite-dimensionality of all L(D) from that of L(0)
AlgebraicCurve.finiteDimensional_lSpace0 below · cited by 77 · depth 10 - Riemann–Roch from the residue theorem, K algebraically closed
AlgebraicCurve.functionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed0 below · cited by 1 · depth 10 - Fundamental identity along a finite separable embedding
AlgebraicCurve.fundamentalIdentityAlong6 below · cited by 63 · depth 10 - Genus invariance under algebraically closed constant field extension
AlgebraicCurve.genusFF_eq_of_constantFieldExtension_of_isAlgClosed59 below · cited by 14 · depth 10 - Existence of canonical divisors on a curve over a perfect field
AlgebraicCurve.hasCanonicalDivisor_of_isCurveOver3 below · cited by 89 · depth 10 - Principal divisors on K(x,T) with x transcendental, T integral
AlgebraicCurve.hasPrincipalDivisors_adjoin_of_transcendental25 below · cited by 1 · depth 10 - Finite separable extensions of K(x) are curves over K
AlgebraicCurve.isCurveOver_of_transcendental_of_isSeparable38 below · cited by 35 · depth 10 - Integrality over L[j] descends to an intermediate field
AlgebraicCurve.isIntegral_adjoin_intermediateField_mk0 below · cited by 22 · depth 10 - Integrality over K[j] transports along K-algebra maps
AlgebraicCurve.isIntegral_adjoin_map_algHom0 below · cited by 13 · depth 10 - Integrality over K[j] passes to L[j] in a tower
AlgebraicCurve.isIntegral_adjoin_of_isScalarTower0 below · cited by 11 · depth 10 - Function fields are linearly disjoint from constant field extensions
AlgebraicCurve.linearIndependent_of_constantFieldExtension0 below · cited by 19 · depth 10 - Powers of a transcendental element are linearly independent
AlgebraicCurve.linearIndependent_pow_of_transcendental23 below · cited by 4 · depth 10 - Residue theorem over an algebraically closed base field
AlgebraicCurve.residueTheoremK_of_isAlgClosed6 below · cited by 5 · depth 10 - Riemann–Hurwitz bookkeeping for a j-type map
AlgebraicCurve.twelve_mul_eq_of_sum_ordDiff_eq0 below · cited by 2 · depth 10 - Constants are the base field given a degree-one place
AlgebraicCurve.constantsAreBase_of_deg_eq_one6 below · cited by 53 · depth 11 - Degree of a canonical divisor is 2g-2 over ̄ K
AlgebraicCurve.degree_canonicalDivisor_eq_of_isAlgClosed56 below · cited by 3 · depth 11 - Existence of the constant field extension F₀K'
AlgebraicCurve.exists_constantFieldExtension43 below · cited by 9 · depth 11 - Adelic Riemann–Roch from existence of the Stichtenoth genus
AlgebraicCurve.exists_genus_riemannIndex_of_stichtenothGenusExists0 below · cited by 2 · depth 11 - Prolongation of a K-rational place of K' to F'
AlgebraicCurve.exists_valuationSubring_section_of_constantField_valuationSubring7 below · cited by 1 · depth 11 - Finiteness over K(t) for every transcendental t
AlgebraicCurve.finiteDimensional_adjoin_of_transcendental0 below · cited by 66 · depth 11 - Regular differentials form a space of dimension the genus
AlgebraicCurve.finite_and_finrank_regularDifferentials_eq_genus61 below · cited by 25 · depth 11 - Riemann–Roch over an algebraically closed constant field
AlgebraicCurve.functionFieldRiemannRoch_of_isAlgClosed_of_isCurveOver9 below · cited by 19 · depth 11 - Genus does not increase under algebraically closed constant field extension
AlgebraicCurve.genusFF_le_of_constantFieldExtension_of_isAlgClosed57 below · cited by 1 · depth 11 - Genus of an unramified Kummer cover of prime degree
AlgebraicCurve.genusFF_sub_one_eq_of_isSplittingField_of_forall_dvd_ord97 below · cited by 1 · depth 11 - Degree-zero principal divisors over K(x), characteristic 0
AlgebraicCurve.hasPrincipalDivisors_of_transcendental24 below · cited by 24 · depth 11 - Principal divisors have degree zero for finite separable extensions of K(x)
AlgebraicCurve.hasPrincipalDivisors_of_transcendental_of_isSeparable29 below · cited by 11 · depth 11 - Transcendence degree one: F algebraic over K(t) for any transcendental t
AlgebraicCurve.isAlgebraic_adjoin_of_transcendental0 below · cited by 16 · depth 11 - Curves over a perfect field: separating transcendence element
AlgebraicCurve.isCurveOver_iff_exists_transcendental_finiteDimensional39 below · cited by 33 · depth 11 - Finite separable extensions of K(x) are curves over K
AlgebraicCurve.isCurveOver_of_transcendental37 below · cited by 42 · depth 11 - The rational function field K(t) is a curve over K
AlgebraicCurve.isCurveOver_ratFunc0 below · cited by 25 · depth 11 - Genus does not drop under algebraically closed constant field extension
AlgebraicCurve.le_genusFF_of_constantFieldExtension_of_isAlgClosed51 below · cited by 1 · depth 11 - Residue theorem for K(x), K algebraically closed
AlgebraicCurve.residueTheoremK_ratFunc_of_isAlgClosed0 below · cited by 1 · depth 11 - Residue–trace commutation through the completion, F/E separable
AlgebraicCurve.residueTraceCompletionCommute4 below · cited by 1 · depth 11 - Degree of the pole divisor of x equals [F:K(x)]
AlgebraicCurve.degree_poleDivisor_eq_finrank_adjoin_of_isAlgClosed_of_transcendental61 below · cited by 25 · depth 12 - Invariance of ℓ(D) under constant field extension
AlgebraicCurve.ell_mapDomain_eq_of_constantFieldExtension_of_isAlgClosed26 below · cited by 2 · depth 12 - Base change of a curve correspondence to a constant field extension
AlgebraicCurve.exists_baseChange_correspondence_of_constantFieldExtension74 below · cited by 4 · depth 12 - Descent to a countable algebraically closed field of constants
AlgebraicCurve.exists_constantFieldDescent43 below · cited by 2 · depth 12 - Horizontal lift of a constant derivation to a constant field extension
AlgebraicCurve.exists_derivation_constantFieldExtension_map_mem61 below · cited by 2 · depth 12 - Eventual exactness of ℓ(N· D) for the pole divisor of x
AlgebraicCurve.exists_ell_nsmul_eq_of_isAlgClosed_of_transcendental66 below · cited by 10 · depth 12 - Vanishing index of specialty for a lifted divisor
AlgebraicCurve.exists_indexOfSpecialty_mapDomain_eq_zero_of_constantFieldExtension_of_isAlgClosed40 below · cited by 1 · depth 12 - Existence of the pole divisor of a transcendental element
AlgebraicCurve.exists_poleDivisor_of_transcendental43 below · cited by 12 · depth 12 - Deuring's Gauss prolongation of a K-rational place
AlgebraicCurve.exists_regularProlongation_retraction_of_constantField_valuationSubring6 below · cited by 4 · depth 12 - Existence of a separating transcendental over a perfect field
AlgebraicCurve.exists_separating_transcendental_of_perfectField2 below · cited by 27 · depth 12 - Nonempty opens on a smooth curve have finite complement
AlgebraicCurve.finite_compl_of_isOpen2 below · cited by 19 · depth 12 - Canonical-degree genus equals the adelic genus dim_K H¹(0)
AlgebraicCurve.genus_eq_genusFF2 below · cited by 26 · depth 12 - Genus invariance under algebraically closed constant field extension
AlgebraicCurve.genus_eq_of_constantFieldExtension_of_isAlgClosed9 below · cited by 3 · depth 12 - Canonical local residue data exist at every place
AlgebraicCurve.hasCanonicalLocalResidueK0 below · cited by 3 · depth 12 - Degree-zero principal divisors for separable extensions of K(t)
AlgebraicCurve.hasPrincipalDivisors_of_finiteDimensional_of_isSeparable28 below · cited by 1 · depth 12 - Index of specialty equals dim_K H¹(D)
AlgebraicCurve.indexOfSpecialty_eq_finrank_H10 below · cited by 17 · depth 12 - Index of specialty at an attained Riemann genus
AlgebraicCurve.indexOfSpecialty_eq_of_genusReached0 below · cited by 15 · depth 12 - Index of specialty vanishes at a genus-realising divisor
AlgebraicCurve.indexOfSpecialty_eq_zero_of_genusReached0 below · cited by 2 · depth 12 - Curve criterion over an algebraically closed base
AlgebraicCurve.isCurveOver_of_isAlgClosed_of_transcendental42 below · cited by 62 · depth 12 - Function fields of smooth integral curves over K
AlgebraicCurve.isCurveOver_of_ringEquiv_functionField_of_isIntegral_of_smoothOfRelativeDimension_one39 below · cited by 38 · depth 12 - Ω_{F/K} is free of rank one over a separable extension of K(x)
AlgebraicCurve.kaehlerRankOne_of_transcendental0 below · cited by 2 · depth 12 - Unit derivatives are regular: du/dπ_w∈mathcal O_w
AlgebraicCurve.localUnitDerivativeRegular_of_isCurveOver2 below · cited by 20 · depth 12 - Nonvanishing of pulled-back differentials in tame extensions
AlgebraicCurve.map_ne_zero_of_tame6 below · cited by 4 · depth 12 - L(D)· L(E)⊆ L(D+E)
AlgebraicCurve.mul_mem_lSpace_add0 below · cited by 5 · depth 12 - Two descriptions of the regular differentials agree
AlgebraicCurve.regularDiffs_eq_regularDifferentials56 below · cited by 6 · depth 12 - Residue theorem for curves over an algebraically closed field
AlgebraicCurve.residueTheorem_of_isAlgClosed8 below · cited by 2 · depth 12 - Existence of the Stichtenoth genus for a curve over a perfect field
AlgebraicCurve.stichtenothGenusExists_of_isCurveOver25 below · cited by 48 · depth 12 - Tate's residue agrees with the local residue trace
AlgebraicCurve.tateAgreement0 below · cited by 1 · depth 12 - Chain rule for Tate's residue along F/E
AlgebraicCurve.tateChainRule0 below · cited by 2 · depth 12 - Tate's commutator has finite K-rank at every place
AlgebraicCurve.tateCommFinite0 below · cited by 2 · depth 12 - Trace compatibility of Tate's local residue for separable F/E
AlgebraicCurve.tateTraceCompat_of_isSeparable0 below · cited by 2 · depth 12 - Hurwitz genus formula for tame separable extensions
AlgebraicCurve.two_mul_genus_sub_two_eq_of_degree_canonical6 below · cited by 5 · depth 12 - Weil duality from Riemann–Roch and existence of the genus
AlgebraicCurve.weilDualityAdelic_of_functionFieldRiemannRoch_of_stichtenothGenusExists2 below · cited by 25 · depth 12 - Degree invariance of the constant-field pullback of divisors
AlgebraicCurve.constantFieldDegreeFormula_of_isConstantFieldExtension_of_isCurveOver47 below · cited by 4 · depth 13 - Constants are the base field when K is algebraically closed
AlgebraicCurve.constantsAreBase_of_isAlgClosed_of_transcendental49 below · cited by 27 · depth 13 - ℓ of a canonical divisor equals the genus
AlgebraicCurve.ell_canonicalDivisor_eq_genus_of_riemannRoch0 below · cited by 1 · depth 13 - Riemann–Roch over an algebraically closed field
AlgebraicCurve.exists_canonicalDivisor_genus_riemannRoch43 below · cited by 21 · depth 13 - Descent to a countable algebraically closed constant field
AlgebraicCurve.exists_constantFieldDescent_finset39 below · cited by 1 · depth 13 - Constant field extension of a separably generated function field
AlgebraicCurve.exists_finiteDimensional_isSeparable_adjoin_of_constantFieldExtension_of_isAlgClosed3 below · cited by 1 · depth 13 - Integral closure of K[x] spanned by x^jL(M₁D)
AlgebraicCurve.exists_forall_mem_span_pow_mul_of_forall_ord_nonneg69 below · cited by 2 · depth 13 - Degree-zero divisors are equivalent to sumᵢ [vᵢ] - r[v₀]
AlgebraicCurve.exists_list_isPrincipal_sub_sum_single_sub_smul_single45 below · cited by 7 · depth 13 - Differentials are integral at a place: dx = c dπ
AlgebraicCurve.exists_mem_D_eq_smul_D_of_isCurveOver1 below · cited by 2 · depth 13 - Genus bound attained at a multiple of any single place
AlgebraicCurve.exists_riemannGenusReachedAt_nsmul_single_of_stichtenothGenusExists4 below · cited by 2 · depth 13 - Separating transcendental element when dim_F Ω_{F/K} = 1
AlgebraicCurve.exists_transcendental_isSeparable_of_finrank_kaehlerDifferential_eq_one0 below · cited by 13 · depth 13 - Riemann–Roch with a Weil canonical divisor
AlgebraicCurve.exists_weilCanonical_riemannRoch35 below · cited by 18 · depth 13 - Finiteness of zeros and poles in a finite separable extension of K(X)
AlgebraicCurve.finite_setOf_ord_ne_zero_of_finiteDimensional3 below · cited by 1 · depth 13 - Multiplicativity of degree along a composite of function-field maps
AlgebraicCurve.finrankAlong_comp0 below · cited by 31 · depth 13 - Degree along a map equals relative degree over its image
AlgebraicCurve.finrankAlong_eq_relfinrank_fieldRange147 below · cited by 30 · depth 13 - [M:Mᵖ]=p for a one-variable function field over a perfect field
AlgebraicCurve.finrank_frobeniusSubfield_eq_of_transcendental0 below · cited by 3 · depth 13 - Riemann–Roch for one-variable function fields over algebraically closed fields
AlgebraicCurve.functionFieldRiemannRoch_of_isAlgClosed_of_transcendental58 below · cited by 16 · depth 13 - Invariance of the repartition genus under K-algebra isomorphism
AlgebraicCurve.genusFF_eq_of_algEquiv0 below · cited by 25 · depth 13 - Points other than the generic point of a smooth curve are closed
AlgebraicCurve.isClosed_singleton_of_ne_genericPoint1 below · cited by 40 · depth 13 - One-variable function fields over perfect fields are curves
AlgebraicCurve.isCurveOver_of_transcendental_of_perfectField41 below · cited by 97 · depth 13 - Descent of Riemann–Roch spaces along a constant field extension
AlgebraicCurve.lSpace_mapDomain_subset_span_image_lSpace_of_constantFieldExtension_of_isAlgClosed25 below · cited by 3 · depth 13 - Linear disjointness of a function field from constant field extensions
AlgebraicCurve.linearIndependent_of_constantFieldExtension_of_isAlgClosed3 below · cited by 9 · depth 13 - Functions integral at all new places lie in K'· F
AlgebraicCurve.mem_span_range_algebraMap_of_constantFieldExtension16 below · cited by 2 · depth 13 - Regularity at all new places forces membership in K'· F
AlgebraicCurve.mem_span_range_algebraMap_of_constantFieldExtension_of_isAlgClosed20 below · cited by 2 · depth 13 - Pushforward norm formula along a finite separable embedding
AlgebraicCurve.normFormulaAlong_of_separableAlong7 below · cited by 36 · depth 13 - Residue theorem from its family-universal form
AlgebraicCurve.residueTheorem_of_residueTheoremK0 below · cited by 1 · depth 13 - Hurwitz genus inequality for F/k(x) in differential form
AlgebraicCurve.sum_ordDiff_D_le_two_mul_genusFF_of_isSeparable106 below · cited by 4 · depth 13 - Hurwitz ramification bound for a separable function field over the line
AlgebraicCurve.sum_ord_sub_one_le_two_mul_genusFF_of_isSeparable107 below · cited by 5 · depth 13 - Adelic Weil duality over an algebraically closed constant field
AlgebraicCurve.weilDualityAdelic_of_isAlgClosed69 below · cited by 2 · depth 13 - Riemann's inequality: ℓ(D)≤deg D+ℓ(0)
AlgebraicCurve.ell_le_degree_add_ellZero0 below · cited by 4 · depth 14 - Single-place step: ℓ(D)≤ℓ(D-P)+deg P
AlgebraicCurve.ell_le_ell_sub_single_add_deg0 below · cited by 8 · depth 14 - Function field of an integral curve model is essentially of finite type
AlgebraicCurve.essFiniteType_functionField0 below · cited by 30 · depth 14 - Class number of a curve as P(1)
AlgebraicCurve.eval_one_eq_natCard_pic0_of_natCard_fixedPoints_restrictAlong_eq80 below · cited by 1 · depth 14 - Existence of a degree-r constant field extension
AlgebraicCurve.exists_constantFieldExtension_of_finite47 below · cited by 3 · depth 14 - Jacobi inversion for complex algebraic function fields
AlgebraicCurve.exists_degree_eq_zero_and_abelJacobiDiv_sub_mem_pathPeriodLattice8 below · cited by 3 · depth 14 - Strong approximation: i(nQ)=0 for some n
AlgebraicCurve.exists_indexOfSpecialty_nsmul_single_eq_zero_of_genusReached1 below · cited by 2 · depth 14 - Fixed points of Frobenius on places and the L-polynomial
AlgebraicCurve.exists_monic_natCard_fixedPoints_restrictAlong_eq_of_constantFieldExtension121 below · cited by 5 · depth 14 - Existence of a differential of the third kind
AlgebraicCurve.exists_ordDifferential_eq_neg_one_of_ne0 below · cited by 3 · depth 14 - Simple-root place criterion along a finite morphism
AlgebraicCurve.exists_place_over_of_simple_root_along9 below · cited by 8 · depth 14 - L(0) is finite-dimensional and ℓ(0)=1 over algebraically closed k
AlgebraicCurve.finiteDimensional_lSpace_zero_and_ell_zero_eq_one_of_isAlgClosed_of_transcendental65 below · cited by 7 · depth 14 - Degree along the identity is 1
AlgebraicCurve.finrankAlong_id147 below · cited by 11 · depth 14 - Generic germ commutes with pull-back of sections
AlgebraicCurve.germToFunctionField_app_eq_of_fromSpecStalk_comp_eq0 below · cited by 5 · depth 14 - Index of specialty equals ℓ(W-D) at a Weil differential
AlgebraicCurve.indexOfSpecialty_eq_ell_sub_of_rankOne_max0 below · cited by 1 · depth 14 - Function field of a smooth integral curve is a curve over K
AlgebraicCurve.isCurveOver_of_isIntegral_of_smoothOfRelativeDimension_one38 below · cited by 20 · depth 14 - L(D)=0 for divisors of negative degree
AlgebraicCurve.lSpace_eq_bot_of_degree_neg0 below · cited by 38 · depth 14 - Stabilisation of the pole filtration past the Riemann–Roch threshold
AlgebraicCurve.lSpace_nsmul_succ_poleDivisor_le_sup_map_mulLeft_of_ell_eq1 below · cited by 2 · depth 14 - Linear independence of the products x^j uᵢ
AlgebraicCurve.linearIndependent_pow_mul6 below · cited by 1 · depth 14 - Coefficients of a constant-field-extension Riemann–Roch element descend
AlgebraicCurve.mem_riemannRochSpace_of_sum_basis_smul_algebraMap_mem_mapDomain3 below · cited by 1 · depth 14 - Every place of a curve has degree at least one
AlgebraicCurve.one_le_deg0 below · cited by 6 · depth 14 - Relative norm of a maximal ideal equals p^f
AlgebraicCurve.relNorm_eq_pow_of_isMaximal_of_isSeparable0 below · cited by 3 · depth 14 - Weil differentials form a rank-one F-module for curves
AlgebraicCurve.weilDifferentialRankOne_of_isCurveOver26 below · cited by 5 · depth 14 - Weil reciprocity for function fields in characteristic zero
AlgebraicCurve.weilReciprocity68 below · cited by 11 · depth 14 - Class number formula: L(1)=#Pic⁰ for function fields
AlgebraicCurve.LPolynomial_eval_one_eq_natCard_pic065 below · cited by 1 · depth 15 - Abel's theorem, necessity: principal divisors have lattice periods
AlgebraicCurve.abelJacobiDiv_mem_pathPeriodLattice_of_isPrincipal26 below · cited by 2 · depth 15 - Recursion nAₙ=sum_{r≤ n} Nᵣ Aₙ₋ᵣ for effective divisor counts
AlgebraicCurve.card_effectiveDivisors_mul_eq_sum3 below · cited by 3 · depth 15 - Path periods form a lattice in ℂⁿ
AlgebraicCurve.discreteTopology_pathPeriodLattice_and_span_eq_top191 below · cited by 2 · depth 15 - Logarithmic differentials with p-divisible orders are regular
AlgebraicCurve.dlog_mem_regularDifferentials_of_forall_dvd_ord3 below · cited by 4 · depth 15 - Existence of the L-polynomial of a function field
AlgebraicCurve.exists_LPolynomial_of_finite73 below · cited by 2 · depth 15 - Existence of a good constant reduction, after Deuring
AlgebraicCurve.exists_constantReduction_isGood_and_forall_residueField_pow_pow_eq_self174 below · cited by 2 · depth 15 - Sum of pole orders of x equals [F:k(x)]
AlgebraicCurve.exists_finset_sum_neg_ord_eq_finrank_of_isAlgClosed63 below · cited by 7 · depth 15 - A smooth proper curve is covered by two affine opens
AlgebraicCurve.exists_isAffineOpen_sup_eq_top3 below · cited by 9 · depth 15 - Primitives of regular differentials along paths exist, unique up to constants
AlgebraicCurve.exists_isPrimitiveAlong_of_mem_regularDifferentials4 below · cited by 6 · depth 15 - A single integral characteristic polynomial for a correspondence on Tₚ(Pic⁰)
AlgebraicCurve.exists_monic_charpoly_tateModule_rep_correspondence_eq_map293 below · cited by 1 · depth 15 - Degree-zero divisors as residue divisors of differentials
AlgebraicCurve.exists_ordDifferential_ge_neg_one_and_evalAt_eq_of_degree_eq_zero62 below · cited by 1 · depth 15 - Period normalisation of a third-kind differential
AlgebraicCurve.exists_regular_pathIntegral_sub_eq_of_abelJacobiDiv_mem_pathPeriodLattice165 below · cited by 1 · depth 15 - Finite-dimensionality of L(0) on a curve
AlgebraicCurve.finiteDimensional_lSpace_zero0 below · cited by 16 · depth 15 - dim_K Ω_{reg} = g over an algebraically closed base
AlgebraicCurve.finite_and_finrank_regularDiffs_eq_genusFF_of_isAlgClosed107 below · cited by 4 · depth 15 - Fixed places of iterated Frobenius count places of F₀
AlgebraicCurve.finite_fixedPoints_restrictAlong_iterate_and_natCard_eq_sum_divisors7 below · cited by 7 · depth 15 - Genus invariance under constant-field extension in Frobenius form
AlgebraicCurve.genusFF_eq_of_constantFieldExtension_of_finite_of_isAlgClosed94 below · cited by 1 · depth 15 - A smooth curve over a field has infinitely many closed points
AlgebraicCurve.infinite_setOf_isClosed_singleton2 below · cited by 50 · depth 15 - Integrality over K[t] from membership in all places containing t
AlgebraicCurve.isIntegral_adjoin_of_forall_mem_toValuationSubring1 below · cited by 11 · depth 15 - No poles where t is regular implies integrality over K[t]
AlgebraicCurve.isIntegral_adjoin_of_forall_ord_nonneg3 below · cited by 15 · depth 15 - Regular differentials already form a K-subspace
AlgebraicCurve.mem_regularDiffs_iff11 below · cited by 2 · depth 15 - Multiplication by x on the pole filtration is graded-injective
AlgebraicCurve.mul_mem_lSpace_nsmul_succ_and_reflects_of_poleDivisor0 below · cited by 3 · depth 15 - Finite-dimensionality of the space Ω(D)
AlgebraicCurve.omegaSpace_finite_of_genusReached0 below · cited by 1 · depth 15 - Heights of roots bounded by height of coefficients
AlgebraicCurve.sum_absLogHeight_roots_le_coeff1 below · cited by 2 · depth 15 - Weil differentials have rank one, given the Stichtenoth genus
AlgebraicCurve.weilDifferentialRankOne_of_stichtenothGenusExists0 below · cited by 1 · depth 15 - Weil reciprocity for f against a function from the base field
AlgebraicCurve.weilReciprocity_algebraMap29 below · cited by 1 · depth 15 - Height of coefficients bounded by heights of roots
AlgebraicCurve.absLogHeight_coeff_le_sum_roots2 below · cited by 1 · depth 16 - Height of a root bounded by height of coefficients
AlgebraicCurve.absLogHeight_root_le_coeff2 below · cited by 1 · depth 16 - Counting effective divisors in a divisor class over a finite field
AlgebraicCurve.card_effective_sub_isPrincipal_of_finite1 below · cited by 1 · depth 16 - Counting effective divisors of large degree over a finite field
AlgebraicCurve.card_sub_one_mul_card_effectiveDivisors_eq33 below · cited by 1 · depth 16 - Čech Riemann–Roch on a two-chart cover of a curve
AlgebraicCurve.cechRiemannRoch_of_genusReached12 below · cited by 6 · depth 16 - Full constant field from a relative q-Frobenius
AlgebraicCurve.constantsAreBase_of_apply_algebraMap_eq_pow_card51 below · cited by 6 · depth 16 - Adding a principal divisor does not change ℓ(D)
AlgebraicCurve.ell_add_of_forall_eq_ord0 below · cited by 2 · depth 16 - ℓ(D₂)-ℓ(D₁)≤deg D₂-deg D₁ for D₁≤ D₂
AlgebraicCurve.ell_sub_ell_le_degree_sub_degree0 below · cited by 2 · depth 16 - Uniqueness of the genus in Riemann–Roch
AlgebraicCurve.eq_genusFF_of_forall_ell_sub_ell_eq32 below · cited by 5 · depth 16 - A point is determined by its local ring in K(C)
AlgebraicCurve.eq_of_range_stalk_eq0 below · cited by 14 · depth 16 - Local constancy of AJ(f^*t) modulo periods
AlgebraicCurve.eventually_abelJacobiDiv_fibre_sub_mem_pathPeriodLattice25 below · cited by 1 · depth 16 - Constant-field Frobenius extends to a K-endomorphism of F
AlgebraicCurve.exists_algHom_apply_algebraMap_eq_pow_card_of_constantsAreBase1 below · cited by 1 · depth 16 - Places of a proper smooth curve come from closed points
AlgebraicCurve.exists_closedPoint_range_stalk_eq3 below · cited by 9 · depth 16 - F. K. Schmidt: existence of a degree-one divisor
AlgebraicCurve.exists_divisor_degree_eq_one_of_finite63 below · cited by 2 · depth 16 - Meromorphic functions on the space of places come from F
AlgebraicCurve.exists_eventuallyEq_evalAt_of_meromorphicAt70 below · cited by 2 · depth 16 - Period group of a differential basis has ≤ 2n generators
AlgebraicCurve.exists_finset_card_le_span_eq_pathPeriodLattice190 below · cited by 1 · depth 16 - Uniform lower bound for the base height
AlgebraicCurve.exists_forall_neg_le_baseHt3 below · cited by 4 · depth 16 - One integral matrix for a correspondence on all Tate modules
AlgebraicCurve.exists_int_matrix_forall_toMatrix_tateModule_rep_correspondence_eq_map292 below · cited by 2 · depth 16 - Finite sets of points on a smooth proper curve lie in an affine open
AlgebraicCurve.exists_isAffineOpen_forall_mem_of_finset69 below · cited by 6 · depth 16 - Canonical loops and Riemann's bilinear period relations
AlgebraicCurve.exists_loops_pathIntegral_reciprocity164 below · cited by 1 · depth 16 - Interpolation of twisted values in a Riemann–Roch space
AlgebraicCurve.exists_mem_riemannRochSpace_forall_hasValue_mul_of_exists_not_mem0 below · cited by 3 · depth 16 - Meromorphic function realising the residue divisor of θ
AlgebraicCurve.exists_meromorphicOrderAt_eq_of_forall_pathIntegral_eq_two_pi_I_mul6 below · cited by 1 · depth 16 - Transport of places and divisors along F ≃_K F'
AlgebraicCurve.exists_placeEquiv_ord_eq_and_ell_mapDomain_eq0 below · cited by 1 · depth 16 - Closed points of a smooth curve give places of its function field
AlgebraicCurve.exists_place_range_stalk_eq2 below · cited by 12 · depth 16 - Riemann–Roch in two-chart Čech form for 𝒪(D)
AlgebraicCurve.finrank_H0_H1_sectionsOf_of_range_eq_lSpaceOn76 below · cited by 5 · depth 16 - Codimension of twisted node conditions on L(E₁)× L(E₂)
AlgebraicCurve.finrank_add_card_le_of_forall_exists_mem_riemannRochSpace_hasValue_mul0 below · cited by 3 · depth 16 - Riemann–Roch space of a constant field conorm
AlgebraicCurve.lSpace_pullbackConstants_eq_span_of_isConstantFieldExtension3 below · cited by 1 · depth 16 - Function fields are K-algebra isomorphic under isomorphism over K
AlgebraicCurve.nonempty_algEquiv_functionField_of_iso0 below · cited by 13 · depth 16 - Čech cohomology of mathcal O_C computed by places
AlgebraicCurve.nonempty_linearEquiv_cechH0_and_cechH18 below · cited by 3 · depth 16 - Two-chart Čech cohomology of an invertible sheaf as L(D) Čech cohomology
AlgebraicCurve.nonempty_linearEquiv_cechH0_and_cechH1_sectionsOf11 below · cited by 2 · depth 16 - Two proper opens of a smooth proper curve exhaust the places
AlgebraicCurve.placesOf_union_eq_univ_of_sup_eq_top7 below · cited by 9 · depth 16 - Additivity of point heights under products of coordinate families
AlgebraicCurve.pointHt_mul_eq_add1 below · cited by 13 · depth 16 - Real span of the period vectors is all of ℂⁿ
AlgebraicCurve.span_real_pathPeriodLattice_eq_top5 below · cited by 3 · depth 16 - Abel–Jacobi intertwines a correspondence with its differential matrix
AlgebraicCurve.abelJacobiDiv_correspondence_sub_vecMul_mem_pathPeriodLattice99 below · cited by 2 · depth 17 - Rational places in a constant field extension
AlgebraicCurve.card_places_deg_one_eq_sum_divisors_of_constantFieldExtension0 below · cited by 2 · depth 17 - Two-chart Čech H¹ computes the répartition H¹(D)
AlgebraicCurve.cechH1ToH1_bijective7 below · cited by 2 · depth 17 - Rescaling by regular multipliers multiplies the jet determinant
AlgebraicCurve.det_taylorCoeff_mul_eq_prod_evalAt_mul_det_jetMatrix9 below · cited by 1 · depth 17 - Weil differentials bounded by a divisor are λ_{fω_0}
AlgebraicCurve.eq_zero_or_exists_eq_weilOfKaehler_smul_of_mem_omegaSpace24 below · cited by 1 · depth 17 - Local Jacobi inversion for the lifted Abel–Jacobi map
AlgebraicCurve.exists_abelJacobiDiv_sub_mem_pathPeriodLattice_and_image_mem_nhds5 below · cited by 1 · depth 17 - Local holomorphic lift of the Abel–Jacobi vector
AlgebraicCurve.exists_ball_abelJacobiVec_sub_sub_mem_pathPeriodLattice8 below · cited by 3 · depth 17 - Lefschetz principle: descent of a curve with finitely many correspondences
AlgebraicCurve.exists_constantFieldDescent_correspondence40 below · cited by 2 · depth 17 - Uniform lower bound for the chordal pair height
AlgebraicCurve.exists_forall_neg_le_pairHt2 below · cited by 3 · depth 17 - Existence and uniqueness of primitives along a path
AlgebraicCurve.exists_isPrimitiveAlong_of_forall_ordDifferential_nonneg4 below · cited by 5 · depth 17 - Places in general position for regular differentials
AlgebraicCurve.exists_isUnit_det_evalAt_differentialCoeff81 below · cited by 1 · depth 17 - Raw form of Riemann's bilinear relations
AlgebraicCurve.exists_loops_pathIntegral_reciprocity_raw161 below · cited by 1 · depth 17 - Functions in L(D) from orthogonality to Ω(D-E)
AlgebraicCurve.exists_mem_riemannRochSpace_forall_adicValuation_sub_le_of_forall_omegaSpace0 below · cited by 1 · depth 17 - One loop realising an integer combination of periods
AlgebraicCurve.exists_path_forall_pathIntegral_eq_sum_mul5 below · cited by 1 · depth 17 - A function regular exactly away from one closed point
AlgebraicCurve.exists_transcendental_mem_range_stalk_iff_ne60 below · cited by 1 · depth 17 - Weil differentials bounded by W are F-proportional
AlgebraicCurve.exists_weilSmul_eq_of_riemannIndexFormula23 below · cited by 2 · depth 17 - Čech h⁰=1 and h¹= genus for smooth proper curves
AlgebraicCurve.finite_H0_H1_structureSheaf_of_isAlgClosed79 below · cited by 3 · depth 17 - Finiteness of H¹(D) and duality with Ω_F(D)
AlgebraicCurve.finite_H1_and_exists_linearEquiv_dual_H1_omegaSpace31 below · cited by 2 · depth 17 - Finiteness of the répartition space H¹(D)
AlgebraicCurve.finite_H1_of_genusReached3 below · cited by 1 · depth 17 - Degree and trace under constant field extension
AlgebraicCurve.finrankAlong_eq_and_trace_eq_of_constantFieldExtension1 below · cited by 2 · depth 17 - Riemann–Hurwitz for tame separable covers of P¹
AlgebraicCurve.finsum_ramificationIndex_ratFunc_sub_one_eq_of_tame100 below · cited by 5 · depth 17 - Riemann–Hurwitz for an unramified Kummer cover of prime degree
AlgebraicCurve.genusFF_sub_one_eq_of_isSplittingField_of_forall_dvd_ord_of_natCast_ne_zero101 below · cited by 1 · depth 17 - Existence of canonical divisors on P¹_K, K perfect
AlgebraicCurve.instHasCanonicalDivisorRatFuncPerfectField21 below · cited by 4 · depth 17 - Maximal domain of a transcendental function is affine
AlgebraicCurve.isAffineOpen_of_maximal_domain6 below · cited by 1 · depth 17 - Chart primitives give primitives along paths inside a chart
AlgebraicCurve.isPrimitiveAlong_comp_extChartAt_of_hasDerivAt_readDifferential4 below · cited by 1 · depth 17 - Jet matrix invertible iff span meets L(A-sum Pᵢ) trivially
AlgebraicCurve.isUnit_det_jetMatrix_iff_span_inf_riemannRochSpace_eq_bot7 below · cited by 1 · depth 17 - Two-chart Čech cohomology of a module realised as L(D)
AlgebraicCurve.nonempty_linearEquiv_cechH0_and_cechH1_sectionsOf_of_range_eq_lSpaceOn1 below · cited by 2 · depth 17 - Symmetry of the pair height pairHt
AlgebraicCurve.pairHt_comm1 below · cited by 2 · depth 17 - Affine sections as intersection of places centred in U
AlgebraicCurve.range_algebraMap_functionField_eq_iInf_of_isAffineOpen5 below · cited by 10 · depth 17 - Residue of a norm equals norm of the residue
AlgebraicCurve.residue_norm_eq_norm_residue_of_retraction1 below · cited by 1 · depth 17 - Attained Riemann genus equals the canonical genus
AlgebraicCurve.riemannGenusReached_of_stichtenothGenusExists23 below · cited by 1 · depth 17 - Adelic index formula from an attained Riemann genus
AlgebraicCurve.riemannIndexFormula_of_genusReached23 below · cited by 1 · depth 17 - Vanishing of the fibre sum of a regular differential's coefficient
AlgebraicCurve.sum_fibre_evalAt_eq_zero_of_smul_D_mem_regularDifferentials17 below · cited by 1 · depth 17 - Trace integrality along a finite separable extension of fields
AlgebraicCurve.traceIntegralAlong_of_separableAlong0 below · cited by 3 · depth 17 - Weil–Kähler agreement from the residue theorem
AlgebraicCurve.weilKaehlerAgree_of_residueTheorem1 below · cited by 1 · depth 17 - Canonical divisor bounds any D with λ_ω ∈ Ω(D)
AlgebraicCurve.weilOfKaehler_omegaSpace_le_canonical0 below · cited by 1 · depth 17 - Local calculus for a differential's coefficient in an analytic chart
AlgebraicCurve.coeffIn_local_calculus8 below · cited by 8 · depth 18 - Dimension of L(mD) via an adapted monomial family
AlgebraicCurve.ell_nsmul_eq_card_of_flagAdaptedBasisAt0 below · cited by 1 · depth 18 - Local holomorphic lift of AJ∘ T modulo periods
AlgebraicCurve.exists_ball_abelJacobiDiv_correspondence_sub_sub_mem_pathPeriodLattice98 below · cited by 1 · depth 18 - Cell dissection of a compact complex curve with marked places
AlgebraicCurve.exists_cellDissection137 below · cited by 1 · depth 18 - Flag-adapted basis for the pole filtration of x
AlgebraicCurve.exists_flagAdaptedBasis_lSpace_nsmul_poleDivisor5 below · cited by 1 · depth 18 - Strong approximation away from one place on a curve
AlgebraicCurve.exists_forall_adicValuation_sub_le_of_riemannGenusReachedAt5 below · cited by 3 · depth 18 - Degree-one places killing a Riemann–Roch space, avoiding a finite set
AlgebraicCurve.exists_injective_ell_sub_sum_single_eq_zero_of_card_le1 below · cited by 1 · depth 18 - Adelic index quotient is K-linearly isomorphic to H¹(D)
AlgebraicCurve.exists_linearEquiv_adeleSpaceQuot_H13 below · cited by 1 · depth 18 - Attained valuation in partial Riemann–Roch spaces L_S(D)
AlgebraicCurve.exists_mem_lSpaceOn_adicValuation_eq_of_riemannGenusReachedAt15 below · cited by 1 · depth 18 - Base-point-free pencils inside a base-point-free linear system
AlgebraicCurve.exists_pair_forall_ord_add_eq_zero_of_subset_riemannRochSpace0 below · cited by 2 · depth 18 - Riemann–Roch in genus 0: dim_k L(E)=max(deg E+1,0)
AlgebraicCurve.finiteDimensional_and_finrank_riemannRochSpace_of_ringEquiv_ratFunc45 below · cited by 2 · depth 18 - Čech H⁰,H¹ of mathcal O_C on a two-affine cover
AlgebraicCurve.finite_H0_H1_structureSheaf_of_smoothProperCurve26 below · cited by 1 · depth 18 - Hasse–Witt bound for the p-torsion of Pic⁰
AlgebraicCurve.finite_and_card_torsion_le_pow_finrank18 below · cited by 1 · depth 18 - Base-point-free pencil trick for Riemann–Roch spaces
AlgebraicCurve.finrank_span_pair_mul_riemannRochSpace_add_finrank0 below · cited by 2 · depth 18 - Vanishing of the adelic genus of K(X) over ̄ K
AlgebraicCurve.genusFF_ratFunc_eq_zero_of_isAlgClosed80 below · cited by 2 · depth 18 - Adèle space equals the repartition algebra
AlgebraicCurve.mem_adeleSpace_iff_mem_repartitions0 below · cited by 3 · depth 18 - Bounded repartitions are bounded adèles
AlgebraicCurve.mem_repartitionsOf_iff_coe_mem_adeleBdd0 below · cited by 2 · depth 18 - Vanishing of Ω(D) when deg D>deg(ω)
AlgebraicCurve.omegaSpace_eq_bot_of_degree_canonicalDivisorOf_lt25 below · cited by 1 · depth 18 - Linearity of the path integral in the differential
AlgebraicCurve.pathIntegral_finset_sum_smul3 below · cited by 2 · depth 18 - Chordal proximity in a normalised chart
AlgebraicCurve.prox_eq_neg_log_iSup_sub_of_chart0 below · cited by 17 · depth 18 - Vanishing of chordal proximity for far points in a chart
AlgebraicCurve.prox_eq_zero_of_far_of_chart0 below · cited by 4 · depth 18 - Riemann–Hurwitz formula for tame separable extensions
AlgebraicCurve.two_mul_genusFF_sub_two_eq_of_isSeparable_of_tame96 below · cited by 3 · depth 18 - Weil functional of ω lies in Ω(div ω)
AlgebraicCurve.weilOfKaehler_mem_omegaSpace_of_residueTheorem0 below · cited by 3 · depth 18 - Adèles split as A(D₀)+F at a genus-attaining divisor
AlgebraicCurve.adeleSpace_eq_of_genusReached0 below · cited by 1 · depth 19 - Boundedness of normalised Chow form values along a pencil
AlgebraicCurve.bddAbove_chowLogAt_range0 below · cited by 3 · depth 19 - Existence and uniqueness of the Cartier operator on Ω¹_{F/K}
AlgebraicCurve.cartierOperator_existsUnique6 below · cited by 11 · depth 19 - Existence of a separating element on a curve
AlgebraicCurve.exists_D_ne_zero0 below · cited by 11 · depth 19 - Local primitive for the Abel–Jacobi vector pulled back along ψ
AlgebraicCurve.exists_ball_abelJacobiVec_restrictAlong_sub_sub_mem_pathPeriodLattice14 below · cited by 1 · depth 19 - Flag-adapted basis of L(MD) up to level M₁
AlgebraicCurve.exists_flagAdaptedBasisAt_lSpace_nsmul_poleDivisor2 below · cited by 1 · depth 19 - Twisted Lagrange interpolation in L(E) on a rational function field
AlgebraicCurve.exists_mem_riemannRochSpace_forall_hasValue_zpow_mul_of_ringEquiv_ratFunc45 below · cited by 1 · depth 19 - Genus-zero Riemann–Roch: L(E-w) is properly contained in L(E)
AlgebraicCurve.exists_mem_riemannRochSpace_not_mem_sub_single_of_ringEquiv_ratFunc45 below · cited by 1 · depth 19 - Interpolation on a rational function field with prescribed simple poles
AlgebraicCurve.exists_mem_riemannRochSpace_ord_sub_eq_one_hasValue_of_ringEquiv_ratFunc45 below · cited by 3 · depth 19 - Existence of a paired cell family on a compact complex curve
AlgebraicCurve.exists_pairedCellFamily136 below · cited by 1 · depth 19 - Degree and trace unchanged by algebraically closed constant field extension
AlgebraicCurve.finrankAlong_eq_and_trace_eq_of_constantFieldExtension_of_isAlgClosed4 below · cited by 2 · depth 19 - dim_K Ω_F(D) = ℓ((ω) - D)
AlgebraicCurve.finrank_omegaSpace_eq_ell_canonical_sub_of_genusReached23 below · cited by 1 · depth 19 - Dimension bound for twisted node-compatible pairs of sections
AlgebraicCurve.finrank_twistedNodeCompatible_add_min_card_le_of_ringEquiv_ratFunc45 below · cited by 1 · depth 19 - The rational function field has genus zero over a perfect field
AlgebraicCurve.genus_ratFunc_eq_zero_of_perfectField21 below · cited by 2 · depth 19 - Relative q-Frobenius: image F^q, bijective on places, inertia degree 1
AlgebraicCurve.isFrobeniusEndo_and_bijective_restrictAlong_of_apply_algebraMap_eq_pow_card0 below · cited by 2 · depth 19 - Weil: P(Fr_*) annihilates ℓ^m-torsion divisor classes
AlgebraicCurve.isPrincipal_aeval_pushforwardAlong_torsion_of_natCard_fixedPoints_restrictAlong_eq1,206 below · cited by 2 · depth 19 - Regularity of dlog g when all orders are p-divisible
AlgebraicCurve.isRegularDiff_dlog_of_dvd_ord10 below · cited by 2 · depth 19 - Global independence of a pole-filtration-adapted family x^j y_σ
AlgebraicCurve.linearIndependent_pow_mul_of_flagAdaptedBasisAt_of_ell_eq1 below · cited by 1 · depth 19 - Principal repartitions are the constant families
AlgebraicCurve.mem_principalRepartitions_iff_coe_mem_globalSub0 below · cited by 1 · depth 19 - Weil's Riemann hypothesis for curves over finite fields
AlgebraicCurve.norm_eq_sqrt_of_mem_roots_of_natCard_fixedPoints_restrictAlong_eq113 below · cited by 4 · depth 19 - Unique p-digit expansion in a separating element
AlgebraicCurve.pDigits_existsUnique2 below · cited by 5 · depth 19 - Scaling invariance of chordal proximity
AlgebraicCurve.prox_smul_smul0 below · cited by 16 · depth 19 - Defining property of the trace map on differentials
AlgebraicCurve.traceDiff_apply0 below · cited by 6 · depth 19 - Unique p-digit expansion in a function field
AlgebraicCurve.existsUnique_pDigits_of_transcendental0 below · cited by 1 · depth 20 - Scale data for dissecting a compact complex curve
AlgebraicCurve.exists_dissectionScaleData57 below · cited by 1 · depth 20 - Inductive step for a reduced basis adapted to the pole filtration
AlgebraicCurve.exists_flagAdaptedBasisAt_lSpace_nsmul_poleDivisor_succ1 below · cited by 1 · depth 20 - Twisting a curve over a finite field by a commuting automorphism
AlgebraicCurve.exists_isCurveOver_adjoin_range_eq_top_apply_algEquiv_eq_pow_of_pow_eq_one44 below · cited by 1 · depth 20 - Bombieri's lower bound for Nᵣ along multiples of some m
AlgebraicCurve.exists_sub_le_sum_divisors_mul_card_places95 below · cited by 1 · depth 20 - Fibre residue identity along a finite separable map
AlgebraicCurve.fibreResidueIdentityAlong_of_separableAlong_of_dCoordGenerates12 below · cited by 3 · depth 20 - Constant field extension: degree and zero/pole counts
AlgebraicCurve.finrank_le_and_natCard_places_le_of_constantFieldExtension_adjoin4 below · cited by 3 · depth 20 - Glued sections on two rational curves: a dimension bound
AlgebraicCurve.finrank_nodeCompatible_add_min_card_le_of_ringEquiv_ratFunc45 below · cited by 2 · depth 20 - Genus-zero Riemann–Roch for effective divisors
AlgebraicCurve.finrank_riemannRochSpace_eq_degree_add_one_of_ringEquiv_ratFunc45 below · cited by 2 · depth 20 - Degree-zero principal divisors over finite separable extensions of K(X)
AlgebraicCurve.hasPrincipalDivisors_of_finiteDimensional_ratFunc_of_isSeparable24 below · cited by 1 · depth 20 - The totalised trace of differentials satisfies its defining identity
AlgebraicCurve.isTraceDiff_traceDiff0 below · cited by 4 · depth 20 - Cartier operator fixes g^{ℓ-1} dg
AlgebraicCurve.kw_cart_C_pow_pred_smul_D0 below · cited by 2 · depth 20 - Cartier operator annihilates gⁱ dg for i+1<ℓ
AlgebraicCurve.kw_cart_C_pow_smul_D_eq_zero0 below · cited by 2 · depth 20 - Automorphisms preserve regular differentials on a curve
AlgebraicCurve.pullbackAlong_mem_regularDifferentials_of_mem_of_isCurveOver3 below · cited by 1 · depth 20 - Bombieri's bound on places of a function field
AlgebraicCurve.sum_divisors_mul_card_places_lt_of_even64 below · cited by 1 · depth 20 - Trace of dlog h equals dlog of the norm
AlgebraicCurve.traceDiff_inv_smul_D_eq_inv_norm_smul_D_norm2 below · cited by 2 · depth 20 - Completion trace sum for separable extensions of function fields
AlgebraicCurve.completionTraceSum_of_isSeparable6 below · cited by 2 · depth 21 - Chart-supported degree-zero representatives of inertia-invariant Tate vectors
AlgebraicCurve.exists_chartSupported_repr_of_mem_invariants_rationalTateModule_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel185 below · cited by 6 · depth 21 - Existence of chartwise reduction on inertia invariants
AlgebraicCurve.exists_linearMap_rationalTateModule_reduction_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel187 below · cited by 4 · depth 21 - Local normal form g=ζ^{ordᵥ g} at a place
AlgebraicCurve.exists_localCoordinate_evalAt_eq_pow0 below · cited by 1 · depth 21 - Bombieri's Galois-closure counting identity for places
AlgebraicCurve.finrank_mul_natCard_fixedPoints_restrictAlong_eq_sum_natCard6 below · cited by 1 · depth 21 - Riemann–Hurwitz formula for the cover F/K(f)
AlgebraicCurve.finsum_ramificationIndexAlong_sub_one_eq109 below · cited by 1 · depth 21 - Genus is invariant under a finite separable constant extension
AlgebraicCurve.genusFF_eq_of_constantFieldExtension_of_finiteDimensional1 below · cited by 1 · depth 21 - Evaluation at a transcendental element is a covering off finitely many points
AlgebraicCurve.isCoveringMapOn_evalAt48 below · cited by 1 · depth 21 - Bombieri's bound on fixed places of a twisted Frobenius
AlgebraicCurve.natCard_fixedPoints_restrictAlong_lt_of_isFrobeniusEndo_sq81 below · cited by 1 · depth 21 - Semilinearity of the pullback on Kähler differentials
AlgebraicCurve.pullbackDiff_smul0 below · cited by 1 · depth 21 - Vanishing of chart reduction on S-invariants equals augmentation span
AlgebraicCurve.red_eq_zero_iff_mem_span_smul_sub_of_forall_smul_eq_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel497 below · cited by 3 · depth 21 - Residue commutes with trace through the completion
AlgebraicCurve.residueTraceCompletionCommute_v24 below · cited by 3 · depth 21 - Relative ramification–inertia identity from the degree identity
AlgebraicCurve.sumRamificationInertia_of_fundamentalIdentity0 below · cited by 2 · depth 21 - Residue-pairing adjunction for a correspondence on a stable place set
AlgebraicCurve.sum_kaehlerResidueTerm_traceFunAlong_mul_eq_sum_kaehlerResidueTerm_traceAlong_smul_pullbackAlong0 below · cited by 1 · depth 21 - Tropically principal divisors are chart-representable in chart-degrees zero
AlgebraicCurve.exists_add_sum_sub_sum_mem_principal_of_degree_add_sum_eq_zero_of_valuation_mul_prod_eq_of_lattice_of_semistableCovering_of_discFibres_of_rankOne166 below · cited by 1 · depth 22 - Kummer-normalised representatives of ℓ^k-torsion classes on a semistable covering
AlgebraicCurve.exists_mk_eq_forall_mem_support_pow_evalAt_param_eq_of_zsmul_eq_zero_of_semistableCovering_of_discFibres_of_rankOne_of_charZero_of_semistableModel165 below · cited by 3 · depth 22 - Function with a single pole avoiding T and simple zeros on T
AlgebraicCurve.exists_place_notMem_ord_neg_and_forall_ord_eq_one40 below · cited by 1 · depth 22 - Triviality of annulus Kummer values along dual-graph cycles
AlgebraicCurve.exists_residue_prod_zpow_eq_one_of_forall_mapDomain_placeMap_eq_zero_of_forall_annulus_sum_eq_zero_of_prod_valuation_evalAt_zpow_eq_one_of_semistableCovering_of_discFibres_of_rankOne31 below · cited by 1 · depth 22 - Slope formula for a function on a semistable covering
AlgebraicCurve.exists_slopes_degree_add_sum_eq_zero_and_valuation_mul_prod_eq_of_ord_of_semistableCovering_of_discFibres_of_rankOne2 below · cited by 1 · depth 22 - Genus of K(X) is zero in characteristic zero
AlgebraicCurve.genus_ratFunc_eq_zero22 below · cited by 1 · depth 22 - Intermediate fields of a one-variable function field are curves
AlgebraicCurve.isCurveOver_and_essFiniteType_intermediateField_of_transcendental_mem42 below · cited by 2 · depth 22 - Coordinate Cartier operator agrees with any operator satisfying the Cartier laws
AlgebraicCurve.kw_cart_C_eq_of_cartierLaws7 below · cited by 2 · depth 22 - Chart components of a principal divisor push forward to principal divisors
AlgebraicCurve.mapDomain_placeMap_mem_principal_of_forall_support_subset_dom_of_degree_eq_zero2 below · cited by 1 · depth 22 - Reduction-killed invariant Tate vectors lie in the monodromy span
AlgebraicCurve.mem_span_smul_sub_of_red_eq_zero_of_forall_smul_eq_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel496 below · cited by 1 · depth 22 - Monodromy differences lie in the kernel of chartwise reduction
AlgebraicCurve.red_eq_zero_of_mem_span_smul_sub_of_forall_smul_eq_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel188 below · cited by 3 · depth 22 - Tate's residue equals the trace of the local residue
AlgebraicCurve.tateAgreement_v20 below · cited by 1 · depth 22 - Recognition of one-variable function fields via valuations
AlgebraicCurve.transcendental_and_finiteDimensional_adjoin_of_valuations0 below · cited by 1 · depth 22 - Chordal proximity under a bounded change of coordinates
AlgebraicCurve.abs_prox_evalVec_sub_prox_le_of_coordinate_change0 below · cited by 1 · depth 23 - Bounded change of basis shifts chordal proximity by at most 4logβ
AlgebraicCurve.abs_prox_mulVec_mulVec_sub_prox_le_of_abv_le0 below · cited by 12 · depth 23 - Uniqueness of the centre of a place on a special fibre component
AlgebraicCurve.eq_of_specializes_of_forall_residue_mem_valuationSubring_of_isCurveOver_residue4 below · cited by 3 · depth 23 - Places and points of a normal proper model
AlgebraicCurve.existsUnique_point_localRing_eq_and_specializes_closedPoint_and_forall_eq_of_isProper_of_isIntegrallyClosed5 below · cited by 3 · depth 23 - Reciprocal annulus pair at a crossing-model node
AlgebraicCurve.exists_annulusPair_isAttached_of_ringEquiv_uvCrossingModel_of_nodeCoordinates3 below · cited by 6 · depth 23 - Existence and uniqueness of the centre of a place on a proper model
AlgebraicCurve.exists_closedPoint_specializes_reads_and_unique_of_isProper0 below · cited by 6 · depth 23 - Evaluation vector of a family equals a scalar times M applied to a normalised family
AlgebraicCurve.exists_evalVec_eq_smul_mulVec_of_eq_sum_smul1 below · cited by 13 · depth 23 - Zero sum of x-a equals [F:k(x)] over ̄ k
AlgebraicCurve.exists_finset_sum_ord_sub_algebraMap_eq_finrank_of_isAlgClosed63 below · cited by 13 · depth 23 - Riemann's inequality with a uniform constant γ
AlgebraicCurve.exists_int_forall_degree_add_one_sub_le_ell27 below · cited by 1 · depth 23 - Functions in L((2g+1)c) separating two places
AlgebraicCurve.exists_mem_riemannRochSpace_smul_single_ord_pos_and_ord_eq_zero91 below · cited by 4 · depth 23 - Multidegree map modulo dual-graph Laplacians for semistable coverings
AlgebraicCurve.exists_multidegree_of_semistableCovering1 below · cited by 3 · depth 23 - Lifting prescribed annulus divisors and Laplacian degrees on semistable coverings
AlgebraicCurve.exists_ne_zero_ord_eq_of_sum_eq_zero_of_semistableCovering_of_discFibres_of_rankOne161 below · cited by 5 · depth 23 - Branch places and node coordinates at an ordinary double point
AlgebraicCurve.exists_nodeRing_coords_and_branch_unique_and_residue_surjective_of_ringEquiv_adicCompletion_stalk_of_isUnit_of_isIntegrallyClosed45 below · cited by 3 · depth 23 - The bound deg D-ℓ(D)≤γ-1 is attained
AlgebraicCurve.exists_riemannGenusReachedAt_of_bounded0 below · cited by 2 · depth 23 - Residue disc has a smooth centre and is its formal fibre
AlgebraicCurve.exists_smoothCentre_of_isResidueDisc_of_reads_smooth53 below · cited by 3 · depth 23 - Residue-disc package at a smooth closed point of a model
AlgebraicCurve.exists_smoothPointPackage_localRing_of_mem_smoothLocus_of_isProper11 below · cited by 5 · depth 23 - Reading place, locality and residue surjectivity at a smooth point
AlgebraicCurve.exists_smoothPointRing_mem_iff_and_locality_and_residue_surjective_of_mem_smoothLocus_of_isProper32 below · cited by 3 · depth 23 - Vanishing cycles span the monodromy differences, naturally
AlgebraicCurve.exists_vanishingCycles_smul_sub_mem_span_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_src_ne_tgt_of_charZero_of_semistableModel_of_forall_pow_eq_self_of_algEquiv1,137 below · cited by 3 · depth 23 - Degree bound from a fibre with bounded multiplicities
AlgebraicCurve.finrank_adjoin_le_mul_natCard_place_ord_sub_algebraMap_pos_of_forall_ord_le64 below · cited by 1 · depth 23 - Toric bound for the kernel of chartwise reduction
AlgebraicCurve.finrank_ker_reduction_add_le_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel298 below · cited by 2 · depth 23 - One S-element moving an ℓ-th root of π cuts out all invariants
AlgebraicCurve.ker_sub_one_eq_iInf_ker_of_pow_eq_of_baseAut_ne_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel285 below · cited by 1 · depth 23 - Slope formula: chart parts of div f are principal
AlgebraicCurve.mapDomain_placeMap_mem_principal_of_degree_eq_zero_of_forall_annulus_sum_eq_zero_of_prod_valuation_evalAt_zpow_eq_one2 below · cited by 2 · depth 23 - Level-two monodromy law on the rational Tate module
AlgebraicCurve.rationalGaloisRep_apply_sub_eq_sub_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel131 below · cited by 2 · depth 23 - Chartwise reduction vanishes on averages of chart-trivial automorphisms
AlgebraicCurve.red_apply_eq_zero_of_sum_rationalGaloisRep_eq_zero_of_forall_inducesOnChart_refl_of_mem_invariants1 below · cited by 3 · depth 23 - Naturality of chartwise ℓ-adic reduction under a chart-stabilising automorphism
AlgebraicCurve.red_rationalGaloisRep_apply_eq_rationalGaloisRep_red_of_inducesOnChart_of_placeMap_smul_of_isRational_of_mem_invariants0 below · cited by 7 · depth 23 - Finite index of speciality bounds deg D-ℓ(D)
AlgebraicCurve.riemannGenusBounded_of_indexFinite0 below · cited by 1 · depth 23 - Riemann–Hurwitz inequality for a Kummer extension
AlgebraicCurve.two_mul_genusFF_sub_one_ge_of_isSplittingField_X_pow_sub_C100 below · cited by 1 · depth 23 - Riemann–Roch equality for deg D ≥ 2g-1
AlgebraicCurve.ell_eq_degree_add_one_sub_genusFF_of_isAlgClosed_of_isSeparable87 below · cited by 20 · depth 24 - Specialisations of a non-closed special point are closed
AlgebraicCurve.eq_of_specializes_of_specializes_of_ne_of_regularProlongation_of_isCurveOver2 below · cited by 1 · depth 24 - Smooth special point reduces to a unique rational place
AlgebraicCurve.existsUnique_place_residue_localRing_surjective_of_mem_smoothLocus7 below · cited by 2 · depth 24 - Residue disc of a point: sections, valuations, locality
AlgebraicCurve.exists_disc_sections_locality_of_henselSections_of_weierstrassPreparation0 below · cited by 1 · depth 24 - Floor divisor of m(div f-min(0,div x))/d and its degree
AlgebraicCurve.exists_divisor_eq_floor_and_mul_degree_le_of_min_ord_le64 below · cited by 3 · depth 24 - Existence of the divisor of zeros of x-a
AlgebraicCurve.exists_divisor_eq_max_ord_sub_algebraMap0 below · cited by 1 · depth 24 - Finiteness of the ramification locus of a tame separable cover
AlgebraicCurve.exists_finset_forall_not_mem_ramificationIndex_eq_one6 below · cited by 2 · depth 24 - Residue discs lie in the formal fibre of a smooth centre
AlgebraicCurve.exists_forall_specializes_of_isResidueDisc_of_reads_smooth50 below · cited by 1 · depth 24 - One monodromy operator N with ρ(s)-1=t N
AlgebraicCurve.exists_linearMap_forall_sub_one_eq_smul_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel284 below · cited by 1 · depth 24 - Every Laplacian multidegree is realised by a nonzero function
AlgebraicCurve.exists_ne_zero_apply_ord_eq_sum_lap_of_semistableCovering_of_discFibres_of_rankOne150 below · cited by 1 · depth 24 - Lifting annulus divisors of vanishing interior multidegree
AlgebraicCurve.exists_ne_zero_ord_eq_of_forall_eq_zero_of_semistableCovering_of_discFibres_of_rankOne127 below · cited by 1 · depth 24 - A rational place from a local subring with principal-plus-q maximal ideal
AlgebraicCurve.exists_place_residue_eq_algebraMap_of_maximalIdeal_eq_span_sup0 below · cited by 4 · depth 24 - Normal noetherian local subrings of a curve are valuation rings of places
AlgebraicCurve.exists_place_toSubring_eq_of_isIntegrallyClosedIn_of_isNoetherianRing0 below · cited by 2 · depth 24 - Correspondences with equal fibre supports have isomorphic composita
AlgebraicCurve.exists_ringEquiv_closure_of_support_correspondence_single_eq_of_essFiniteType50 below · cited by 2 · depth 24 - Dedekind: x dt is regular where t is finite
AlgebraicCurve.exists_smul_D_eq_smul_dCoord_of_forall_isIntegral_trace_mul_eq_aeval26 below · cited by 2 · depth 24 - Constant residues and node generators for the local ring at x
AlgebraicCurve.exists_sub_algebraMap_not_isUnit_and_exists_eq_mul_add_of_iso_pullback_of_maximalIdeal_eq_span21 below · cited by 1 · depth 24 - Stalk of a model descends to a local subring of F
AlgebraicCurve.exists_subring_ringEquiv_stalk_of_iso_pullback0 below · cited by 1 · depth 24 - Dedekind: regular x dt gives Tr(xb) polynomial in t
AlgebraicCurve.exists_trace_mul_eq_aeval_of_forall_exists_smul_D_eq_smul_dCoord31 below · cited by 2 · depth 24 - Vanishing cycles span the kernel of chartwise reduction, naturally
AlgebraicCurve.exists_vanishingCycles_ker_reduction_le_span_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_src_ne_tgt_of_charZero_of_semistableModel_of_forall_pow_eq_self_of_algEquiv1,130 below · cited by 1 · depth 24 - Finiteness of Pic⁰[n] and finite level of its Galois action
AlgebraicCurve.finite_pic0Torsion_and_exists_intermediateField_smul_eq_of_descent246 below · cited by 1 · depth 24 - Distinct non-maximal branch kernels on a descended node ring
AlgebraicCurve.ker_residue_ne_and_ne_maximalIdeal_of_iso_pullback_of_specializes_of_ne2 below · cited by 1 · depth 24 - Semistable covering: n≤ m+1 and toric k-torsion bound
AlgebraicCurve.le_add_one_and_exists_finset_card_le_pow_of_chartSupported_principal_of_semistableCovering_of_discFibres_of_rankOne_of_charZero_of_semistableModel240 below · cited by 1 · depth 24 - Node local ring lies in both branches and all S-places
AlgebraicCurve.localRing_le_integers_and_forall_mem_toValuationSubring_and_algebraMap_mem_localRing0 below · cited by 1 · depth 24 - Residue discs are the formal fibres at smooth special points
AlgebraicCurve.mem_iff_specializes_of_isResidueDisc_of_mem_smoothLocus_of_isCurveOver49 below · cited by 1 · depth 24 - Regularity at a node from integrality on both branches
AlgebraicCurve.mem_localRing_of_mem_integers_of_forall_mem_toValuationSubring_of_ringEquiv_adicCompletion_stalk_of_isIntegrallyClosed31 below · cited by 1 · depth 24 - Locality of the local ring at a smooth special point
AlgebraicCurve.mem_localRing_of_mem_integers_of_forall_mem_valuationSubring_of_mem_smoothLocus31 below · cited by 2 · depth 24 - Exactness of floors when the degree bound is attained
AlgebraicCurve.mul_apply_eq_mul_sub_min_ord_of_eq_floor_of_mul_degree_eq_finrank64 below · cited by 3 · depth 24 - Places over a point bounded by double cosets
AlgebraicCurve.natCard_place_ord_sub_pos_le_natCard_doubleCoset29 below · cited by 2 · depth 24 - Existence of a place for a finite separable extension of k(x)
AlgebraicCurve.nonempty_place_of_transcendental_of_finiteDimensional5 below · cited by 9 · depth 24 - Level-two unipotence of inertia on prime-to-p torsion of Pic⁰
AlgebraicCurve.smul_smul_sub_eq_smul_sub_of_isUnit_natCast_of_nsmul_eq_zero_of_genusFF_of_semistableCovering_of_charZero_of_semistableModel129 below · cited by 2 · depth 24 - Rigidity of curve embeddings agreeing on all places
AlgebraicCurve.algHom_eq_of_forall_restrictAlong_eq_of_charZero9 below · cited by 1 · depth 25 - Constants and denominators in the local ring at x
AlgebraicCurve.algebraMap_mem_localRing_and_exists_mul_eq_of_mem_integers_of_specializes0 below · cited by 1 · depth 25 - Degree-zero chart divisors force unit reductions at the nodes
AlgebraicCurve.exists_forall_ord_residue_smul_eq_zero_of_forall_degree_eq_zero_of_semistableCovering8 below · cited by 1 · depth 25 - Rational place centred at an L-point of an affine model
AlgebraicCurve.exists_isRational_forall_mem_and_evalAt_eq_of_algHom45 below · cited by 1 · depth 25 - Descent of a function to a discrete-valuation level
AlgebraicCurve.exists_level_mem_functionField_of_iso_pullback_of_isAlgebraic2 below · cited by 2 · depth 25 - Weierstrass preparation at a smooth special point
AlgebraicCurve.exists_monic_eval2_eq_mul_of_inv_mem_integers_of_ord_eq_one_of_mem_smoothLocus13 below · cited by 1 · depth 25 - Potential lifting on a semistable covering by width-one annuli
AlgebraicCurve.exists_ne_zero_apply_ord_eq_sum_lap_of_semistableCovering_of_discFibres_of_rankOne_of_width_one125 below · cited by 1 · depth 25 - Exact annulus lifting for depth-balanced divisors, rank-one base
AlgebraicCurve.exists_ne_zero_ord_eq_of_depthMass_eq_zero_of_semistableCovering_of_rankOne126 below · cited by 1 · depth 25 - Places centred at a smooth special point with prescribed value
AlgebraicCurve.exists_place_evalAt_eq_and_forall_evalAt_eq_zero_of_ord_eq_one_of_mem_smoothLocus23 below · cited by 1 · depth 25 - Equivariant family of vanishing cycles of full rank
AlgebraicCurve.exists_vanishingCycles_red_eq_zero_and_add_le_finrank_span_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_src_ne_tgt_of_charZero_of_semistableModel_of_forall_pow_eq_self_of_algEquiv1,117 below · cited by 1 · depth 25 - Local ring at a node: intersection over proper generisations
AlgebraicCurve.mem_localRing_of_forall_specializes_mem_localRing_of_ringEquiv_adicCompletion_stalk_of_isIntegrallyClosed27 below · cited by 1 · depth 25 - Functions integral on both branches are regular above the node
AlgebraicCurve.mem_localRing_of_specializes_of_mem_integers_of_forall_mem_toValuationSubring_of_isIntegrallyClosed2 below · cited by 1 · depth 25 - Cancelling Weierstrass factors T-c one at a time
AlgebraicCurve.mem_of_mul_eval2_mem_of_forall_coeff_mem_maximalIdeal0 below · cited by 1 · depth 25 - Divisibility descent for chart-supported divisors on a semistable model
AlgebraicCurve.mem_principal_of_zsmul_mem_principal_of_forall_mapDomain_placeMap_eq_zero_of_genusFF_of_semistableModel_of_descent128 below · cited by 6 · depth 25 - Monodromy on ℓ^k-torsion of Pic⁰ factors through roots of π
AlgebraicCurve.nsmul_smul_sub_eq_nsmul_smul_sub_of_forall_pow_eq_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel282 below · cited by 1 · depth 25 - Chart independence of the Serre residue pairing
AlgebraicCurve.serrePairing_eq_of_cechH1ToH1_eq1 below · cited by 2 · depth 25 - Correspondence adjunction for the Serre residue pairing
AlgebraicCurve.serrePairing_traceAlong_eq_serrePairing_traceAlong_pullbackAlong_of_cechH1ToH1_eq4 below · cited by 1 · depth 25 - Stalk at a node over an intermediate discrete-valuation level
AlgebraicCurve.stalk_level_of_isPullback_of_ringEquiv_adicCompletion_stalk16 below · cited by 2 · depth 25 - Annulus Tate classes: rank bound m+1≤dimspan+n
AlgebraicCurve.add_le_finrank_span_tmul_of_forall_proj_eq_mk_single_sub_single_quadruples_annulus_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel39 below · cited by 1 · depth 26 - Correspondence identity in H¹ descends along trivial base change
AlgebraicCurve.cechH1ToH1_corrH1_of_pullback_specMap_self15 below · cited by 1 · depth 26 - Same kernel, generation and equal degree force an isomorphism
AlgebraicCurve.exists_algEquiv_comp_eq_of_ker_eq_of_closure_range_eq_top_of_finrankAlong_eq0 below · cited by 1 · depth 26 - Cartier data for a balanced divisor on a semistable model
AlgebraicCurve.exists_cartierData_eq_ord_and_pt_mem_iff_of_forall_mapDomain_placeMap_eq_zero_of_balanced_of_semistableModel2 below · cited by 1 · depth 26 - Descent of Cartier–Kummer data to a finite henselian level
AlgebraicCurve.exists_cartierData_kummer_finiteLevel_of_cartierData_of_balanced_of_semistableModel_of_descent93 below · cited by 1 · depth 26 - Constant field extension from ℚ̄ to ℂ with base change of places
AlgebraicCurve.exists_constantFieldExtension_place_of_isAlgClosed48 below · cited by 1 · depth 26 - Constant reduction from a smooth proper model over a Galois-fixed base
AlgebraicCurve.exists_constantReduction_of_smoothOfRelativeDimension_one_liesOverPrime_fixedBase809 below · cited by 1 · depth 26 - Places off the chart U₁ are the section points
AlgebraicCurve.exists_embedding_place_range_eq_compl_placesOf_of_isSectional7 below · cited by 3 · depth 26 - Local ring at a smooth special point: localisation of a finitely presented flat algebra
AlgebraicCurve.exists_finitePresentation_isLocalizationAtPrime_localRing_of_mem_smoothLocus6 below · cited by 2 · depth 26 - Component constants making g/hₐ^k a chart unit
AlgebraicCurve.exists_forall_smul_div_pow_mem_integers_of_cartierData_of_balanced_of_semistableModel8 below · cited by 1 · depth 26 - Regularity of generic germs at places centred in U
AlgebraicCurve.exists_kaehlerToFunctionField_eq_smul_dCoord_of_mem_placesOf0 below · cited by 2 · depth 26 - Finite families of rational functions descend to a DVR level
AlgebraicCurve.exists_level_mem_functionField_of_iso_pullback_of_isAlgebraic_min2 below · cited by 1 · depth 26 - Descent of finitely many functions to a discrete valuation level
AlgebraicCurve.exists_level_mem_functionField_of_iso_pullback_of_isAlgebraic_min22 below · cited by 1 · depth 26 - Symmetry of the two-chart Čech H¹ in its charts
AlgebraicCurve.exists_linearEquiv_cechH1_swap0 below · cited by 2 · depth 26 - Germ comparison of Čech H¹ with function-field H¹(0)
AlgebraicCurve.exists_linearEquiv_structureSheafH1_cechH18 below · cited by 2 · depth 26 - Annulus classes are S-invariant and killed by chartwise reduction
AlgebraicCurve.exists_mem_iInf_ker_red_eq_zero_of_forall_proj_eq_mk_single_sub_single_quadruples_annulus_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel242 below · cited by 1 · depth 26 - Change of uniformiser: dπ' = u dπ with u a v-unit
AlgebraicCurve.exists_ord_eq_zero_D_eq_smul_D_of_isCurveOver1 below · cited by 5 · depth 26 - A place of F/L from a section with kernel (T-c)
AlgebraicCurve.exists_place_evalAt_eq_of_section_of_ker_eq_span1 below · cited by 1 · depth 26 - A-sections through a smooth special point with prescribed value
AlgebraicCurve.exists_section_localRing_apply_eq_of_ord_eq_one_of_mem_smoothLocus20 below · cited by 1 · depth 26 - Annulus Tate classes for a semistable covering exist
AlgebraicCurve.exists_tateModule_forall_proj_eq_mk_single_sub_single_quadruples_annulus_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel_of_forall_pow_eq_self1,107 below · cited by 1 · depth 26 - Integral maps into essentially finite type fields are finite
AlgebraicCurve.finiteAlong_of_isIntegral0 below · cited by 2 · depth 26 - Flatness of the level projection and primes of the level stalk
AlgebraicCurve.flat_fst_and_forall_isPrime_eq_comap_maximalIdeal_of_isPullback_of_ringEquiv_adicCompletion_stalk1 below · cited by 1 · depth 26 - Flatness of X→ X₁ and primes of the level stalk
AlgebraicCurve.flat_fst_and_forall_isPrime_eq_comap_maximalIdeal_of_isPullback_of_ringEquiv_adicCompletion_stalk_min1 below · cited by 1 · depth 26 - Stalk at a node over an intermediate discrete-valuation level
AlgebraicCurve.forall_exists_not_isUnit_sub_germ_and_maximalIdeal_le_map_sup_span_of_isPullback_of_ringEquiv_adicCompletion_stalk4 below · cited by 1 · depth 26 - Level stalk at a crossing point: constants modulo non-units
AlgebraicCurve.forall_exists_not_isUnit_sub_germ_and_maximalIdeal_le_map_sup_span_of_isPullback_of_ringEquiv_adicCompletion_stalk_min4 below · cited by 1 · depth 26 - Genus of the Kummer cover cⁿ = X - X^q
AlgebraicCurve.genusFF_eq_of_finrankAlong_eq_of_pow_eq_X_sub_X_pow113 below · cited by 1 · depth 26 - Sections on U∩ V are regular at places centred in U and in V
AlgebraicCurve.germToFunctionField_inf_mem_lSpaceOn_inter_placesOf2 below · cited by 4 · depth 26 - Noetherian stalk of dimension ≥ 2 at a node over a level
AlgebraicCurve.isNoetherianRing_stalk_and_two_le_ringKrullDim_and_exists_eq_mul_pow_of_isPullback_of_ringEquiv_adicCompletion_stalk9 below · cited by 1 · depth 26 - Node stalk over a discrete-valuation level: noetherian, dimension ≥ 2
AlgebraicCurve.isNoetherianRing_stalk_and_two_le_ringKrullDim_and_exists_eq_mul_pow_of_isPullback_of_ringEquiv_adicCompletion_stalk_min9 below · cited by 1 · depth 26 - Residue term vanishes for a differential regular at v
AlgebraicCurve.kaehlerResidueTerm_diagonalHom_eq_zero_of_eq_smul_dCoord0 below · cited by 1 · depth 26 - Tensor kernel determined by the induced bijection of places
AlgebraicCurve.ker_productMap_eq_of_placeEquiv_restrictAlong48 below · cited by 1 · depth 26 - Regular differentials are preserved by a K-isomorphism of function fields
AlgebraicCurve.pullbackAlong_mem_regularDifferentials_of_mem_of_algEquiv4 below · cited by 2 · depth 26 - Naturality of the annulus Tate classes under L-algebra automorphisms
AlgebraicCurve.rationalGaloisRep_tmul_eq_tmul_perm_of_forall_proj_eq_mk_single_sub_single_quadruples_annulus_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_src_ne_tgt_of_charZero_of_semistableModel_of_algEquiv378 below · cited by 1 · depth 26 - Regular differentials: span of ordDiff≥ 0 equals the valuation description
AlgebraicCurve.regularDiffs_eq_regularDifferentials_of_perfectField17 below · cited by 2 · depth 26 - Residue theorem for the function field of a smooth curve
AlgebraicCurve.residueTheorem_functionField_of_smoothOfRelativeDimension_one71 below · cited by 3 · depth 26 - Perfectness of the Čech residue pairing on a curve
AlgebraicCurve.serrePairing_bijective_and_flip_bijective45 below · cited by 1 · depth 26 - Serre pairing is adjoint for pull-back and trace along φ
AlgebraicCurve.serrePairing_pullbackAlong_eq_serrePairing_traceAlong0 below · cited by 1 · depth 26 - Trace of differentials is adjoint to pull-back of Čech classes
AlgebraicCurve.serrePairing_traceAlong_eq_serrePairing_pullbackAlong0 below · cited by 1 · depth 26 - Automorphisms fixing ℓ^k-th roots of π fix ℓ^k-torsion
AlgebraicCurve.smul_eq_of_forall_pow_eq_baseAut_eq_of_zsmul_eq_zero_of_semistableCovering_of_discFibres_of_rankOne_of_charZero_of_semistableModel280 below · cited by 1 · depth 26 - Cover independence of the Čech-to-répartition map in H¹(0)
AlgebraicCurve.cechH1ToH1_mk_eq_cechH1ToH1_mk_of_crossSections0 below · cited by 2 · depth 27 - Reflection of H¹(0)-class equality along an isomorphism
AlgebraicCurve.cechH1ToH1_mk_eq_of_cechH1ToH1_pullbackAlong_mk_eq1 below · cited by 1 · depth 27 - Regular dlog forces orders divisible by p
AlgebraicCurve.dvd_ord_of_isRegularDiff_dlog15 below · cited by 2 · depth 27 - Reduction is injective on prime-to-p torsion in Pic⁰
AlgebraicCurve.eq_zero_of_nsmul_eq_zero_of_pic0Map_eq_zero_of_smoothOfRelativeDimension_one_liesOverPrime743 below · cited by 1 · depth 27 - C-fixed differentials on a curve are logarithmic
AlgebraicCurve.exists_dlog_of_cartierOperator_fixed5 below · cited by 4 · depth 27 - Local ring at a smooth special point is a localisation
AlgebraicCurve.exists_formallySmooth_isLocalizationAtPrime_localRing_of_mem_smoothLocus0 below · cited by 1 · depth 27 - Regular differentials versus Weil differentials of divisor 0
AlgebraicCurve.exists_linearEquiv_regularDifferentials_omegaSpace_zero29 below · cited by 1 · depth 27 - Local coordinate at a smooth point of the special fibre
AlgebraicCurve.exists_maximalIdeal_localRing_eq_map_sup_span_of_ord_eq_one_of_mem_smoothLocus6 below · cited by 1 · depth 27 - Riemann–Roch interpolation: simple zero at one place, prescribed values
AlgebraicCurve.exists_mem_riemannRochSpace_ord_sub_eq_one_hasValue_of_isCurveOver_of_essFiniteType88 below · cited by 2 · depth 27 - Galois equivariance of reduction of places, fixed base
AlgebraicCurve.exists_monoidHom_semilinearAut_reduction_smul_eq_of_smoothOfRelativeDimension_one_fixedBase1 below · cited by 1 · depth 27 - Level-k Kummer relation forces a vertex coboundary
AlgebraicCurve.exists_residue_pow_mul_eq_of_sum_smul_single_sub_single_sub_sum_sub_sum_smul_quadruple_mem_principal_of_semistableCovering_of_discFibres_of_rankOne_of_lifts_of_charZero_of_semistableModel32 below · cited by 1 · depth 27 - Constant reduction at the generic point of the special fibre
AlgebraicCurve.exists_valuationSubring_residue_of_smoothOfRelativeDimension_one_liesOverPrime9 below · cited by 1 · depth 27 - ℓ-power torsion classes with prescribed depths along annuli
AlgebraicCurve.exists_zsmul_mk_eq_zero_eq_add_sum_single_pow_evalAt_param_eq_mul_of_semistableCovering_of_discFibres_of_rankOne_of_isUnit_of_charZero_of_semistableModel1,100 below · cited by 1 · depth 27 - Čech H⁰ of Ω¹ on a two-chart cover equals the genus
AlgebraicCurve.finite_and_finrank_kaehlerSections_H0_eq_genusFF_of_isAlgClosed109 below · cited by 1 · depth 27 - Finiteness over intermediate fields containing a transcendental element
AlgebraicCurve.finite_intermediateField_of_transcendental_mem0 below · cited by 1 · depth 27 - Sections over U lie in L_{placesOf(U)}(0)
AlgebraicCurve.germToFunctionField_mem_lSpaceOn_placesOf1 below · cited by 2 · depth 27 - Principal divisors ascend finite separable extensions
AlgebraicCurve.hasPrincipalDivisors_of_finiteDimensional_of_isSeparable_of_hasPrincipalDivisors4 below · cited by 2 · depth 27 - Injectivity of Ω_{F/K}→Ω_{FE/E} for a constant field extension
AlgebraicCurve.kaehlerDifferential_map_injective_of_constantFieldExtension0 below · cited by 2 · depth 27 - Reduction of divisors under constant reduction of a smooth curve
AlgebraicCurve.mapDomain_reduction_eq_ord_residue_of_smoothOfRelativeDimension_one_of_exists_smul_mem92 below · cited by 1 · depth 27 - Places centred in a preimage open set
AlgebraicCurve.placesOf_preimage_eq_preimage_restrictAlong_placesOf9 below · cited by 2 · depth 27 - Integral pull-back preserves regular differentials
AlgebraicCurve.pullbackAlong_mem_regularDifferentials_of_isIntegral5 below · cited by 1 · depth 27 - Residue theorem over a perfect constant field
AlgebraicCurve.residueTheorem_of_perfectField65 below · cited by 1 · depth 27 - Injectivity on places forces a trivial finite extension
AlgebraicCurve.surjective_algebraMap_of_injective_restrict_place_of_isAlgClosed25 below · cited by 1 · depth 27 - Weil reciprocity over an algebraically closed constant field
AlgebraicCurve.weilReciprocity_of_isAlgClosed79 below · cited by 2 · depth 27 - Unique centre on a proper curve for each place
AlgebraicCurve.existsUnique_centre_place_of_isProper0 below · cited by 1 · depth 28 - Closed points of a proper curve have finitely many, at least one, centring place
AlgebraicCurve.exists_centre_and_finite_setOf_centre_of_isClosed_singleton43 below · cited by 2 · depth 28 - Every field isomorphic to K(Y) has Y as a curve model
AlgebraicCurve.exists_curveModel_iso_of_algEquiv_functionField77 below · cited by 1 · depth 28 - Existence of the weight-m floor divisor on a curve
AlgebraicCurve.exists_divisor_forall_eq_weightFloor0 below · cited by 3 · depth 28 - Vanishing of two-chart Čech H¹ of nD for large n
AlgebraicCurve.exists_forall_subsingleton_cechH1_nsmul_of_degree_pos_of_riemannGenusReachedAt13 below · cited by 1 · depth 28 - Local equation of a horizontal section of a relative curve
AlgebraicCurve.exists_mem_ord_eq_one_ord_residue_eq_one_of_smoothOfRelativeDimension_one12 below · cited by 1 · depth 28 - Lifting ℓ-power torsion tropical positions to divisor classes
AlgebraicCurve.exists_pow_zsmul_mk_eq_zero_forall_mu_sub_eq_sum_smul_lap_of_semistableCovering_of_discFibres_of_rankOne_of_charZero_of_semistableModel1,098 below · cited by 1 · depth 28 - Sections over U lie in L_{S_U}(0)
AlgebraicCurve.germ_mem_lSpaceOn_setOf_exists_centre_zero0 below · cited by 1 · depth 28 - Vanishing δ_z implies the local ring at z is regular
AlgebraicCurve.isRegularLocalRing_stalk_of_lSpaceOn_setOf_centre_zero_subset_range0 below · cited by 2 · depth 28 - Properness and a curve function field preclude affineness
AlgebraicCurve.not_isAffine_of_isProper_of_isCurveOver0 below · cited by 2 · depth 28 - Order of a pulled-back differential at a place (tame case)
AlgebraicCurve.ordDifferential_map_eq6 below · cited by 2 · depth 28 - Order zero is preserved by reduction at a smooth special point
AlgebraicCurve.ord_residue_eq_zero_of_forall_ord_eq_zero_of_smoothOfRelativeDimension_one_dvrDescent_of_exists_smul_mem80 below · cited by 1 · depth 28 - Places centred in π⁻¹U restrict to places centred in U
AlgebraicCurve.placesOf_preimage_subset_preimage_restrictAlong_placesOf7 below · cited by 1 · depth 28 - Places restricting into Pl_X(U) are centred over π⁻¹U
AlgebraicCurve.preimage_restrictAlong_placesOf_subset_placesOf_preimage7 below · cited by 1 · depth 28 - Residue theorem for F from K(x) via trace–residue commutation
AlgebraicCurve.residueTheorem_of_residueTheorem_ratFunc_of_residueTraceCompletionCommute0 below · cited by 1 · depth 28 - Residue theorem for K(x) over a perfect field
AlgebraicCurve.residueTheorem_ratFunc_of_perfectField63 below · cited by 1 · depth 28 - Degree of the weight-m floor divisor without elliptic places
AlgebraicCurve.six_mul_degree_eq_mul_finrank_of_forall_eq_weightFloor_of_ord_eq_three_two62 below · cited by 2 · depth 28 - Global-to-local δ map on an affine chart: surjectivity and kernel
AlgebraicCurve.surjective_and_ker_pi_lSpaceOn_centre_quotient_of_isAffineOpen58 below · cited by 2 · depth 28 - Nonvanishing and maximality for λ_ω
AlgebraicCurve.weilOfKaehler_ne_zero_and_maximal0 below · cited by 1 · depth 28 - Weil reciprocity descends along finite separable extensions
AlgebraicCurve.weilReciprocity_algebraMap_of_isSeparable29 below · cited by 1 · depth 28 - Reduction of a smooth special point is a place
AlgebraicCurve.existsUnique_place_residue_localRing_surjective_of_mem_smoothLocus_of_valuationSubring6 below · cited by 1 · depth 29 - Finitely generated extensions of transcendence degree one are curves
AlgebraicCurve.isCurveOver_and_essFiniteType_and_exists_of_adjoin_finset_eq_top_of_isAlgebraic42 below · cited by 1 · depth 29 - Functions regular at all places centred at a closed point
AlgebraicCurve.lSpaceOn_setOf_centre_eq_span_integralClosure_mul43 below · cited by 1 · depth 29 - Holomorphy ring of an affine chart is the integral closure
AlgebraicCurve.lSpaceOn_setOf_exists_centre_eq_span_integralClosure43 below · cited by 1 · depth 29 - Polar differentials under constant field extension
AlgebraicCurve.map_mem_polarDifferentials_and_mem_span_image_of_constantFieldExtension_of_isAlgClosed130 below · cited by 1 · depth 29 - Few ℓ^K-torsion classes of tropical position zero
AlgebraicCurve.natCard_torsion_tropicalPositionZero_mul_pow_le_of_semistableCovering_of_discFibres_of_rankOne_of_charZero_of_semistableModel1,091 below · cited by 1 · depth 29 - Local–global decomposition of a finite over-ring on a chart
AlgebraicCurve.surjective_and_ker_pi_span_mul_quotient_of_finite3 below · cited by 1 · depth 29 - Unique residue at a place with at most a simple pole
AlgebraicCurve.existsUnique_hasSimpleResidue_of_hasSimplePoleAt2 below · cited by 13 · depth 30 - Separating generation passes to constant field extensions over a perfect field
AlgebraicCurve.exists_finiteDimensional_isSeparable_adjoin_of_constantFieldExtension_of_perfectField3 below · cited by 1 · depth 30 - Realising zero-sum residue data by a differential with simple poles
AlgebraicCurve.exists_mem_polarDifferentials_forall_hasSimpleResidue_of_sum_eq_zero65 below · cited by 3 · depth 30 - Trivial multidegree splits a degree-zero divisor across charts
AlgebraicCurve.exists_principal_support_subset_chartDomains_of_multidegree_eq_zero0 below · cited by 1 · depth 30 - Chartwise reduction of a principal divisor is nodal principal
AlgebraicCurve.isNodalPrincipal_mapDomain_placeMap_of_forall_ord_eq_zero_of_semistableCovering_of_rankOne32 below · cited by 1 · depth 30 - Regularity outside S ascends along a constant field extension
AlgebraicCurve.isRegularAt_map_of_forall_isRegularAt_of_constantFieldExtension_of_isAlgClosed15 below · cited by 1 · depth 30 - Linear disjointness of F and an algebraic K' over K
AlgebraicCurve.linearIndependent_algebraMap_of_constantsAreBase_of_isAlgebraic1 below · cited by 1 · depth 30 - Polar differentials under constant field extension: inclusion
AlgebraicCurve.map_mem_polarDifferentials_of_constantFieldExtension_of_isAlgClosed15 below · cited by 2 · depth 30 - Integral closure as intersection of the valuation rings of places
AlgebraicCurve.mem_integralClosure_iff_forall_place41 below · cited by 4 · depth 30 - Polar differentials span under constant field extension
AlgebraicCurve.mem_span_image_polarDifferentials_of_constantFieldExtension_of_isAlgClosed129 below · cited by 2 · depth 30 - Residue theorem for differentials with at most simple poles
AlgebraicCurve.sum_eq_zero_of_forall_hasSimpleResidue_of_mem_polarDifferentials10 below · cited by 4 · depth 30 - Prime-to-p torsion: nodal-principal reduction forces principality
AlgebraicCurve.sum_mem_principal_of_zsmul_mem_principal_of_isNodalPrincipal_mapDomain_placeMap_of_semistableModel_of_descent224 below · cited by 1 · depth 30 - Unique expansion g=sum_{i<p} aᵢᵖ tⁱ along t
AlgebraicCurve.existsUnique_pDigits_of_D_ne_zero3 below · cited by 3 · depth 31 - Cartier data for reduction-trivial divisors on a semistable model
AlgebraicCurve.exists_cartierData_eq_ord_and_pt_mem_iff_of_forall_mapDomain_placeMap_eq_zero_of_prod_evalAt_param_zpow_eq_one_add_of_semistableModel2 below · cited by 1 · depth 31 - Descent of Kummer Cartier data to a finite henselian level
AlgebraicCurve.exists_cartierData_kummer_finiteLevel_of_cartierData_of_divisor_of_semistableModel_of_descent93 below · cited by 1 · depth 31 - Vertical unit constants for Cartier data on a semistable model
AlgebraicCurve.exists_forall_smul_div_pow_mem_integers_of_cartierData_of_divisor_of_semistableModel8 below · cited by 1 · depth 31 - Residue map on differentials with simple poles along S
AlgebraicCurve.exists_linearMap_hasSimpleResidue_ker_eq_regular_range_eq_sum_zero_finrank_corner66 below · cited by 5 · depth 31 - Dimension of differentials with simple poles on S
AlgebraicCurve.finite_and_finrank_polarDifferentials_eq61 below · cited by 4 · depth 31 - Carrier hypotheses from end-slope data at node pairs
AlgebraicCurve.carrier_hypotheses_of_endSlopes_of_nodePairs0 below · cited by 1 · depth 32 - Rigidity: node-compatible functions on a glued curve are constant
AlgebraicCurve.exists_eq_algebraMap_of_hasValue_pair_of_generalPosition0 below · cited by 1 · depth 32 - Regular differentials descend along a constant-field extension
AlgebraicCurve.exists_mem_smul_D_of_map_mem_regularDifferentials_of_constantFieldExtension16 below · cited by 1 · depth 32 - Lifting a function field to a good constant reduction in characteristic zero
AlgebraicCurve.exists_charZero_constantReduction_isGood206 below · cited by 1 · depth 33 - Affine open with finite complement containing a finite set
AlgebraicCurve.exists_isAffineOpen_forall_mem_and_finite_compl9 below · cited by 1 · depth 33 - A character of S is evaluation at a rational place
AlgebraicCurve.exists_place_isRational_forall_evalAt_eq_of_algHom41 below · cited by 4 · depth 33 - Genus and Pic⁰ under renaming the constant field
AlgebraicCurve.genusFF_eq_and_nonempty_pic0_addEquiv_of_algebraMap_eq_comp0 below · cited by 3 · depth 33 - Logarithmic derivative vanishes iff p-th power
AlgebraicCurve.inv_smul_D_eq_zero_iff_exists_pow_eq18 below · cited by 2 · depth 33 - Algebraic elements over an algebraically closed base are constants
AlgebraicCurve.mem_range_algebraMap_of_isAlgebraic0 below · cited by 1 · depth 33 - Good constant reduction of a normal-form Witt lift
AlgebraicCurve.exists_constantReduction_isGood_of_wittVector_normalFormOrder153 below · cited by 1 · depth 34 - Finitely many points of a proper smooth curve lie in an affine open
AlgebraicCurve.exists_isAffineOpen_forall_mem_of_finset_of_field8 below · cited by 1 · depth 34 - Cartier-fixed regular differentials and algebraically closed constant-field extensions
AlgebraicCurve.exists_mem_regularDifferentials_cartier_fixed_map_eq_of_constantFieldExtension_of_isAlgClosed146 below · cited by 1 · depth 34 - Split Riemann–Roch normal form of weights (0,1^g,2^g)
AlgebraicCurve.exists_riemannRochSpace_nsmul_poleDivisor_normalForm117 below · cited by 1 · depth 34 - Lifting a split normal-form cover of P¹ to Witt vectors
AlgebraicCurve.exists_wittVector_lift_of_normalFormOrder9 below · cited by 1 · depth 34 - Cartier operator commutes with constant-field base change
AlgebraicCurve.cartier_map_eq_map_cartier_of_constantFieldExtension5 below · cited by 1 · depth 35 - A proper smooth curve admits a finite map to P¹_K
AlgebraicCurve.exists_isFinite_hom_proj_of_isProper5 below · cited by 1 · depth 35 - Lifting split normal-form structure constants along small extensions
AlgebraicCurve.exists_lift_normalForm_structureConstants_of_smallExtension8 below · cited by 1 · depth 35 - Normal form for L(kD) along a reduced pole divisor of degree 2g+1
AlgebraicCurve.exists_normalForm_of_reduced_poleDivisor105 below · cited by 1 · depth 35 - Rational place from a non-maximal prime with transcendental j
AlgebraicCurve.exists_place_ringHom_residue_eq_of_isPrime_le_of_forall_aeval_mem17 below · cited by 2 · depth 35 - Existence of a regular prolongation with residually transcendental f
AlgebraicCurve.exists_regularProlongation_of_transcendental1 below · cited by 1 · depth 35 - Existence of a function with m ≥ 2g+1 distinct simple poles
AlgebraicCurve.exists_transcendental_reduced_poleDivisor115 below · cited by 1 · depth 35 - Degree n over k(y) of the fraction field of a rank-n order
AlgebraicCurve.finrank_adjoin_eq_card_of_mul_mem_span_of_fractions0 below · cited by 1 · depth 35 - Genus bound for a function field in normal form
AlgebraicCurve.genusFF_le_of_normalForm76 below · cited by 1 · depth 35 - Nonzero discriminant for split normal-form algebras over K[X]
AlgebraicCurve.discr_ne_zero_of_normalForm_split0 below · cited by 1 · depth 36 - Fibres of a finite integrally closed K[X]-algebra are monogenic
AlgebraicCurve.exists_aeval_sub_mem_smul_of_isMaximal_of_isIntegrallyClosed0 below · cited by 1 · depth 36 - Patching local trivialisations into a bounded Hochschild 2-cochain
AlgebraicCurve.exists_bounded_two_cochain_of_local_coboundaries0 below · cited by 1 · depth 36 - Coboundary near infinity for a split normal-form deformation
AlgebraicCurve.exists_coboundary_at_infinity_of_smallExtension_of_split2 below · cited by 1 · depth 36 - Local vanishing of the associativity obstruction for monogenic fibres
AlgebraicCurve.exists_local_coboundary_of_smallExtension_of_monogenic_fibre1 below · cited by 2 · depth 36 - Symmetric Hochschild 2-cocycles are coboundaries up to a multiplier
AlgebraicCurve.exists_mul_eq_hochschild_coboundary_of_discr_ne_zero1 below · cited by 1 · depth 36 - Intermediate subfields of a one-variable function field are curves
AlgebraicCurve.isCurveOver_and_essFiniteType_of_le_of_transcendental_mem42 below · cited by 2 · depth 37 - Dimension bound for function pairs with matched residues
AlgebraicCurve.natCast_le_degree_add_degree_of_linearIndependent_of_mem_riemannRochSpace_of_hasValue0 below · cited by 3 · depth 41
AlgebraicCurve.AlgEquiv 1
- A K-automorphism fixing every place is trivial
AlgebraicCurve.AlgEquiv.eq_one_of_forall_smul_place_eq13 below · cited by 2 · depth 24
AlgebraicCurve.Annulus 37
- Twisted chord bounds and rigidity on a doubly attached annulus
AlgebraicCurve.Annulus.chord_bounds_and_rigid_of_isAttached_both_ends_of_twist4 below · cited by 2 · depth 21 - Two-end bound μ(π)ᵃ≤μ(c') on an annulus
AlgebraicCurve.Annulus.abv_modulus_zpow_ord_residue_le_abv_of_isAttached_both_ends2 below · cited by 28 · depth 22 - Far-end bound μ(c') μ(π)^{a'}≤ 1 on an annulus
AlgebraicCurve.Annulus.abv_mul_abv_modulus_zpow_ord_residue_le_one_of_isAttached_both_ends2 below · cited by 27 · depth 22 - Newton-polygon identities for an annulus attached at both ends
AlgebraicCurve.Annulus.sum_ord_mul_log_abv_param_eq_of_isAttached_both_ends1 below · cited by 46 · depth 22 - Two-end maximum principle on an annulus, with scaling constant
AlgebraicCurve.Annulus.abv_evalAt_le_max_of_isAttached_both_ends3 below · cited by 2 · depth 23 - Width-one two-end unit is an isometry on an attached annulus
AlgebraicCurve.Annulus.abv_evalAt_sub_eq_abv_param_sub_of_isAttached_both_ends_of_ord_residue_eq_one4 below · cited by 1 · depth 23 - Isometry of a two-end unit above the radius μ(c')
AlgebraicCurve.Annulus.abv_evalAt_sub_eq_abv_param_sub_of_ord_residue_eq_one_of_abv_le5 below · cited by 1 · depth 23 - Mirror law for distances strictly inside the critical circle
AlgebraicCurve.Annulus.exists_abv_evalAt_sub_mul_eq_of_ord_residue_eq_one_of_abv_lt4 below · cited by 1 · depth 23 - Annuli at distinct places separated by a cross-unit have distinct far rings
AlgebraicCurve.Annulus.integers_ne_of_crossUnit8 below · cited by 3 · depth 23 - Leading-coefficient transport across an annulus
AlgebraicCurve.Annulus.ord_residue_eq_neg_and_evalAt_residue_mul_zpow_eq_of_forall_ord_eq_zero_of_rankOne14 below · cited by 2 · depth 23 - Two-end zero count and radius product on an annulus
AlgebraicCurve.Annulus.sum_eq_ord_add_ord_and_prod_valuation_evalAt_zpow_eq_of_regularProlongation0 below · cited by 19 · depth 23 - Two-end law on a doubly attached annulus
AlgebraicCurve.Annulus.ord_residue_add_nonneg_and_abv_le_one_of_isAttached_both_ends1 below · cited by 18 · depth 24 - Equality case of the two-end law on an annulus
AlgebraicCurve.Annulus.ord_residue_eq_neg_of_abv_eq_abv_modulus_zpow_of_isAttached_both_ends2 below · cited by 1 · depth 24 - Values along a two-ended annulus reduce to the node value
AlgebraicCurve.Annulus.residue_evalAt_eq_evalAt_residue_of_isAttached_both_ends3 below · cited by 4 · depth 24 - Unit reduction on an annulus equals its value at the node
AlgebraicCurve.Annulus.residue_evalAt_eq_evalAt_residue_of_ord_residue_eq_zero_of_regularProlongation11 below · cited by 1 · depth 24 - Unit reduction on an annulus read at the second end
AlgebraicCurve.Annulus.residue_evalAt_eq_evalAt_residue_of_ord_residue_eq_zero_of_regularProlongation_modulus_div_param11 below · cited by 1 · depth 24 - Chart functions of positive order at a node are small on the attached annulus
AlgebraicCurve.Annulus.abv_evalAt_lt_one_of_isAttached_of_ord_residue_pos0 below · cited by 2 · depth 25 - Maximum principle at the two ends of an attached annulus
AlgebraicCurve.Annulus.ord_residue_nonneg_and_evalAt_residue_eq_of_isAttached_of_isAttached9 below · cited by 4 · depth 25 - Units from the annulus unit principle have constant residue
AlgebraicCurve.Annulus.valuation_sub_lt_one_of_forall_isUnit0 below · cited by 3 · depth 25 - Unique simple zero of a two-end function on an annulus
AlgebraicCurve.Annulus.exists_unique_zero_and_isUnit_evalAt_div_param_sub_of_ord_residue_eq_one5 below · cited by 4 · depth 26 - Leading term of a two-end unit at an outer place
AlgebraicCurve.Annulus.residue_evalAt_mul_zpow_param_eq_of_isAttached_both_ends_of_forall_abv_lt4 below · cited by 8 · depth 26 - Attachment passes to upper subannuli with the same parameter
AlgebraicCurve.Annulus.IsAttached.of_param_eq_of_forall_mem_dom_iff0 below · cited by 1 · depth 27 - Annulus modulus exponent read off two end units at one place
AlgebraicCurve.Annulus.eq_of_mul_eq_algebraMap_pow_of_isUnit_evalAt_mul_evalAt_param_zpow_neg_one9 below · cited by 1 · depth 27 - An open band of an annulus is an annulus
AlgebraicCurve.Annulus.exists_band_dom_eq_and_param_eq_and_modulus_eq0 below · cited by 1 · depth 27 - Circle charts inside an annulus over a valuation ring
AlgebraicCurve.Annulus.exists_componentChart_ratFunc_of_valuation_lt_of_exists_lt52 below · cited by 1 · depth 27 - Two annuli with the same domain: parameter ratio is a unit of constant residue
AlgebraicCurve.Annulus.exists_forall_isUnit_evalAt_param_mul_inv_residue_eq_of_dom_eq_of_isAttached_of_rankOne1 below · cited by 1 · depth 27 - Complementary parameter at the other end of an annulus
AlgebraicCurve.Annulus.exists_twoEnd_of_modulus_ne_zero0 below · cited by 1 · depth 27 - Opposite orders at the two ends of an annulus
AlgebraicCurve.Annulus.ord_residue_eq_neg_and_valuation_eq_of_isAttached_of_isAttached5 below · cited by 2 · depth 27 - Band-wise leading coefficient of a two-end annulus unit
AlgebraicCurve.Annulus.residue_evalAt_mul_zpow_param_mul_eq_of_isAttached_both_ends_of_forall_abv_ne4 below · cited by 1 · depth 27 - Two-end residue law for a doubly attached annulus
AlgebraicCurve.Annulus.exists_isUnit_residue_mul_evalAt_eq_evalAt_of_isAttached_of_isAttached15 below · cited by 3 · depth 28 - Reduction of the Gauss ring of an interior circle onto k(X)
AlgebraicCurve.Annulus.exists_ringHom_ratFunc_of_valuation_lt_of_exists_lt3 below · cited by 1 · depth 28 - Valuation ring of an interior circle of an annulus
AlgebraicCurve.Annulus.exists_valuationSubring_mem_iff_of_valuation_lt2 below · cited by 1 · depth 28 - Slopes across an interior circle of an annulus
AlgebraicCurve.Annulus.mapDomain_and_slope_of_valuation_lt_of_exists_lt47 below · cited by 1 · depth 28 - Equal-depth ratio on an annulus: divisor and end residues
AlgebraicCurve.Annulus.ord_sub_div_sub_and_residue_src_eq_one_and_residue_tgt_eq_of_depth_eq0 below · cited by 1 · depth 28 - Factorisation of a function on an annulus into translates of the parameter
AlgebraicCurve.Annulus.exists_eq_mul_prod_param_sub_zpow0 below · cited by 4 · depth 29 - Constant reduction of a unit along an interior circle
AlgebraicCurve.Annulus.residue_evalAt_eq_of_forall_isUnit_evalAt2 below · cited by 2 · depth 29 - Valuation of a translate of the annulus parameter at a place
AlgebraicCurve.Annulus.valuation_evalAt_param_sub_algebraMap0 below · cited by 3 · depth 29
AlgebraicCurve.CartierB 2
- Genus bound for p-torsion of Pic⁰
AlgebraicCurve.CartierB.finite_and_card_torsion_le_pow_genusFF19 below · cited by 1 · depth 17 - Cartier-fixed differentials independent mod p are K-independent
AlgebraicCurve.CartierB.linearIndependent_of_cartier_fixed0 below · cited by 1 · depth 19
AlgebraicCurve.CellDissection 11
- Periods of third-kind differentials along loops avoiding poles
AlgebraicCurve.CellDissection.exists_int_pathIntegral_eq_sum_periods_add_sum_residues15 below · cited by 1 · depth 18 - Path integrals modulo the lattice of edge jumps
AlgebraicCurve.CellDissection.exists_int_pathIntegral_sub_primitive_eq_sum_jump9 below · cited by 1 · depth 18 - Fundamental cycles realised by loops in the skeleton
AlgebraicCurve.CellDissection.exists_loop_pathIntegral_eq_sum_cycle_mul_edgeInt10 below · cited by 1 · depth 18 - Polygon boundary word of a cut cell dissection
AlgebraicCurve.CellDissection.exists_polygonWord0 below · cited by 1 · depth 18 - Normalised cell primitives of regular differentials on a dual tree
AlgebraicCurve.CellDissection.exists_primitives_jump_eq_zero_of_dualTree13 below · cited by 1 · depth 18 - Tree–cotree splitting of a cell dissection
AlgebraicCurve.CellDissection.exists_tree_cotree0 below · cited by 1 · depth 18 - Reversed copy of an edge integrates to minus the edge integral
AlgebraicCurve.CellDissection.intervalIntegral_bdryIntegrand_neg_eq_neg_edgeInt9 below · cited by 3 · depth 18 - Kirchhoff's law and the boundary-word formula for primitive jumps
AlgebraicCurve.CellDissection.jump_kirchhoff_and_wordFormula_of_primitives10 below · cited by 1 · depth 18 - Summed residue theorem over a cell dissection
AlgebraicCurve.CellDissection.two_pi_I_mul_sum_residue_mul_primitive_eq_sum_jump_mul_edgeInt11 below · cited by 1 · depth 18 - Kirchhoff's law and the word formula for arc jumps
AlgebraicCurve.CellDissection.kirchhoff_and_jump_formula_of_arc_values0 below · cited by 1 · depth 19 - Euler count of a grid dissection of an n-sheeted cover
AlgebraicCurve.CellDissection.euler_count_grid0 below · cited by 1 · depth 20
AlgebraicCurve.ComponentChart 17
- Inertia induces the identity on a Gauss-presented component chart
AlgebraicCurve.ComponentChart.inducesOnChart_arithmeticGalois_of_gaussPresentation_of_mem_inertiaSubgroupIn1 below · cited by 4 · depth 22 - Chordal proximity on a component chart equals the disc kernel
AlgebraicCurve.ComponentChart.prox_eq_of_chartData_of_minor2 below · cited by 2 · depth 22 - Transport of a fibre parameter along a field automorphism
AlgebraicCurve.ComponentChart.comap_fibreParam_laws0 below · cited by 1 · depth 23 - Centre dichotomy for valuation rings over a component chart
AlgebraicCurve.ComponentChart.eq_integers_or_existsUnique_isCentre_of_forall_tubeBounded_mem_of_regular0 below · cited by 3 · depth 23 - Rigidity of disc-fibred component charts
AlgebraicCurve.ComponentChart.exists_algEquiv_residue_eq_and_placeMap_eq_smul_of_integers_eq_of_dom_eq_of_hasDiscFibres265 below · cited by 3 · depth 23 - Positivity of ord_Q of a reduction is σ-invariant
AlgebraicCurve.ComponentChart.ord_residue_pos_iff_of_isCentre_of_comap_eq0 below · cited by 3 · depth 23 - Chart comparison for proximity of evaluation vectors under a bounded linear change
AlgebraicCurve.ComponentChart.chartComparison_of_chartData_of_mulVec7 below · cited by 2 · depth 24 - Transport of a component chart along a residually trivial semilinear automorphism
AlgebraicCurve.ComponentChart.exists_integers_eq_comap_of_semilinearAut_of_residue_eq1 below · cited by 3 · depth 24 - Residues of tube-bounded chart functions form a subalgebra
AlgebraicCurve.ComponentChart.exists_tubeBounded_residue_eq_of_mem_adjoin0 below · cited by 3 · depth 24 - Fibre coordinate from a reduction of order one
AlgebraicCurve.ComponentChart.fibreParam_laws_of_ord_residue_sub_eq_one1 below · cited by 1 · depth 24 - Component chart integers localise the tube-bounded ring
AlgebraicCurve.ComponentChart.forall_tubeBounded_mem_integers_and_exists_mul_eq_of_not_mem_dom92 below · cited by 3 · depth 24 - Reduction of a pole-free chart unit is regular off nodes
AlgebraicCurve.ComponentChart.ord_residue_nonneg_of_not_mem_nodes_of_forall_mem_dom_ord_nonneg0 below · cited by 24 · depth 24 - Two simple uniformisers on a residue fibre give equal distances
AlgebraicCurve.ComponentChart.abv_evalAt_sub_eq_of_ord_residue_eq_one0 below · cited by 1 · depth 25 - Chart units take unit values at some rational place
AlgebraicCurve.ComponentChart.exists_mem_dom_forall_isUnit_evalAt_of_forall_isUnit4 below · cited by 3 · depth 25 - Simple zero on the reduction forces a unique simple zero on the fibre
AlgebraicCurve.ComponentChart.ord_eq_of_fibre_of_ord_residue_eq_one0 below · cited by 1 · depth 25 - Transport of component-chart reduction data along a semilinear automorphism
AlgebraicCurve.ComponentChart.exists_equiv_placeMap_smul_eq_of_forall_mem_integers_iff_of_forall_mem_dom_iff_of_discFibres263 below · cited by 1 · depth 27 - ord_Q of a reduction vanishes above a zero-free class
AlgebraicCurve.ComponentChart.ord_residue_eq_zero_of_forall_ord_eq_zero0 below · cited by 2 · depth 27
AlgebraicCurve.ConstantReduction 13
- Surjectivity of the reduction map on Pic⁰
AlgebraicCurve.ConstantReduction.pic0Map_surjective1 below · cited by 1 · depth 13 - Surjectivity of place reduction for a constant reduction
AlgebraicCurve.ConstantReduction.placeMap_surjective0 below · cited by 2 · depth 14 - Reduction is injective on m-torsion of Pic⁰ under good constant reduction
AlgebraicCurve.ConstantReduction.eq_zero_of_nsmul_eq_zero_of_pic0Map_eq_zero_of_isAlgClosed206 below · cited by 4 · depth 15 - Existence of a doubly transcendental element in a constant reduction
AlgebraicCurve.ConstantReduction.exists_transcendental_residue0 below · cited by 3 · depth 16 - Good constant reduction is defectless at some transcendental element
AlgebraicCurve.ConstantReduction.exists_transcendental_residue_finrank_adjoin_eq_of_isGood99 below · cited by 2 · depth 16 - Deuring's degree inequality for a constant reduction
AlgebraicCurve.ConstantReduction.finiteDimensional_and_finrank_adjoin_residue_le0 below · cited by 3 · depth 16 - Constant reductions of function fields are pointwise
AlgebraicCurve.ConstantReduction.isPointwise_of_hasPrincipalDivisors0 below · cited by 5 · depth 22 - Unit values at rational places under constant reduction
AlgebraicCurve.ConstantReduction.isUnit_evalAt_of_ord_eq_zero_of_hasPrincipalDivisors1 below · cited by 1 · depth 23 - Residue linear independence lifts under constant reduction
AlgebraicCurve.ConstantReduction.linearIndependent_of_linearIndependent_residue0 below · cited by 1 · depth 23 - From place-level congruence to Pic⁰ under constant reduction
AlgebraicCurve.ConstantReduction.pic0Map_apply_eq_smul_add_smul_of_forall_mapDomain_placeMap_single0 below · cited by 1 · depth 25 - Pointwise compatibility of a constant reduction at a rational place
AlgebraicCurve.ConstantReduction.exists_residue_mem_evalAt_mem_algebraMap_residue_eq_of_forall_ord_neg_placeMap_ne1 below · cited by 4 · depth 28 - Poles reduce to poles under a degree-preserving constant reduction
AlgebraicCurve.ConstantReduction.mapDomain_placeMap_poleDivisor_eq_and_ord_residue_neg_and_placeMap_ne_of_degree_eq_of_deg_pos0 below · cited by 4 · depth 28 - Lifting m-torsion classes through a good constant reduction
AlgebraicCurve.ConstantReduction.exists_nsmul_eq_zero_and_pic0Map_eq_of_nsmul_eq_zero976 below · cited by 1 · depth 31
AlgebraicCurve.CurveModel 68
- Generic compatibility transports points to places along φ
AlgebraicCurve.CurveModel.pointEquivPlace_comp_eq_congrRingEquiv_of_fromSpecStalk_comp_eq0 below · cited by 4 · depth 12 - A function field with a smooth proper model has infinitely many places
AlgebraicCurve.CurveModel.infinite_place39 below · cited by 16 · depth 13 - Fibre multiplicities of a finite map of curve models
AlgebraicCurve.CurveModel.ker_comap_eq_prod_ker_pow_ramificationIndex1 below · cited by 9 · depth 13 - Formal unramifiedness near a rational point gives ramification index one
AlgebraicCurve.CurveModel.ramificationIndexAlong_pointEquivPlace_eq_one_of_formallyUnramified0 below · cited by 1 · depth 13 - Place-compatible finite morphism induces the given function field embedding
AlgebraicCurve.CurveModel.ffEquiv_symm_stalkMap_eq_algebraMap0 below · cited by 2 · depth 14 - Two-chart glued model: proper, smooth, closed points are places
AlgebraicCurve.CurveModel.isProper_smooth_places_affineCover_glued0 below · cited by 9 · depth 14 - A Frobenius endomorphism of a curve model twists places by g
AlgebraicCurve.CurveModel.placeOfPoint_eq_smul_of_fromSpecStalk_comp_eq_frobenius1 below · cited by 2 · depth 14 - Place at a rational point of the t-chart is centred
AlgebraicCurve.CurveModel.coe_mem_and_sub_algebraMap_mem_nonunits_of_range_stalk_eq_iota00 below · cited by 3 · depth 15 - Place at a rational point of the pole chart is centred
AlgebraicCurve.CurveModel.coe_mem_and_sub_algebraMap_mem_nonunits_of_range_stalk_eq_iotaInf0 below · cited by 3 · depth 15 - Divisor class map on a smooth proper curve model
AlgebraicCurve.CurveModel.exists_divisorClassMap116 below · cited by 3 · depth 15 - Finite function-field embeddings realised by finite flat morphisms of models
AlgebraicCurve.CurveModel.exists_finite_flat_hom_of_algHom0 below · cited by 3 · depth 15 - Finite flat morphism of curve models induced by a function-field embedding
AlgebraicCurve.CurveModel.exists_hom_of_algHom0 below · cited by 5 · depth 15 - Finite extensions of function fields induce finite flat morphisms of models
AlgebraicCurve.CurveModel.exists_hom_pointEquivPlace_restrict_eq1 below · cited by 4 · depth 15 - Germ at a K-point evaluates to its place value
AlgebraicCurve.CurveModel.ffEquiv_symm_mem_and_evalAt_pointEquivPlace_eq_stalkClosedPointTo0 below · cited by 24 · depth 15 - Chart functions evaluate to their residue at a place
AlgebraicCurve.CurveModel.hasValue_placeOfPoint_of_sub_algebraMap_mem0 below · cited by 2 · depth 15 - Value at a place detects the prime of a closed point
AlgebraicCurve.CurveModel.sub_algebraMap_mem_of_hasValue_placeOfPoint0 below · cited by 1 · depth 15 - Riemann–Roch genus on a smooth proper model equals genusFF
AlgebraicCurve.CurveModel.eq_genusFF_of_forall_ell_sub_ell_eq60 below · cited by 11 · depth 16 - Smooth proper model of κ(X) over algebraically closed κ
AlgebraicCurve.CurveModel.exists_curveModel_ratFunc1 below · cited by 4 · depth 16 - Uniqueness of smooth proper models, compatibly with places
AlgebraicCurve.CurveModel.exists_iso_comp_toBase_eq_placeOfPoint_congr_eq5 below · cited by 4 · depth 16 - Places of t-chart models restrict along a field homomorphism
AlgebraicCurve.CurveModel.placeOfPoint_ofGenerator_iota0_comap0 below · cited by 2 · depth 16 - Places at infinity restrict along a map of function fields
AlgebraicCurve.CurveModel.placeOfPoint_ofGenerator_iotaInf_comap0 below · cited by 2 · depth 16 - Jacobian, Abel–Jacobi map and Pic⁰ dictionary over algebraically closed fields
AlgebraicCurve.CurveModel.exists_representsRelSubPic_abelJacobi_of_isAlgClosed534 below · cited by 7 · depth 17 - Čech h⁰ of 𝒪(sum P-sum Q) equals ℓ of the divisor
AlgebraicCurve.CurveModel.finrank_H0_sectionsOf_invModule_prod_ker_tensor_module_prod_ker_eq_ell130 below · cited by 3 · depth 17 - Čech cohomology of mathcal O_C for a model of k(t)
AlgebraicCurve.CurveModel.finrank_H1_sectionsOf_unit_eq_zero_and_finrank_H0_eq_one_of_ratFunc135 below · cited by 11 · depth 17 - Chart rings of a smooth proper model of K(T)
AlgebraicCurve.CurveModel.range_sections_eq_map_eval2_polyPart_invPolyPart_of_coe_eq_compl6 below · cited by 3 · depth 17 - Units of the global sections of a proper model are constants
AlgebraicCurve.CurveModel.exists_eq_appTop_of_isUnit38 below · cited by 1 · depth 18 - Trivialitycriterion: invertible modules of trivial χ on a rational curve model
AlgebraicCurve.CurveModel.nonempty_iso_unit_of_eulerChar_sectionsOf_eq_of_ratFunc303 below · cited by 1 · depth 18 - Two affine lines glued by inversion form a curve model of κ(X)
AlgebraicCurve.CurveModel.exists_iso_of_twoAffineLineCover2 below · cited by 2 · depth 19 - Places of K-points on a curve model are rational
AlgebraicCurve.CurveModel.isRational_pointEquivPlace1 below · cited by 13 · depth 21 - Transport of a curve model along a K-isomorphism of function fields
AlgebraicCurve.CurveModel.exists_curveModel_iso_ffEquiv_symm_germToFunctionField_eq_of_algEquiv0 below · cited by 5 · depth 22 - Chart coordinates agree with point coordinates modulo the place
AlgebraicCurve.CurveModel.ffEquiv_symm_germToFunctionField_sub_algebraMap_mem_nonunits_pointEquivPlace_of_comp_eq_specMap_comp0 below · cited by 12 · depth 22 - Pull-back of a point twist is the conorm twist
AlgebraicCurve.CurveModel.nonempty_pullback_foldr_ofPoint_pow_iso_foldr_ofPoint_pointEquivPlace_symm_of_pointEquivPlace_comp_eq_restrictAlong81 below · cited by 1 · depth 22 - Pull-back of a point ideal sheaf is the conorm
AlgebraicCurve.CurveModel.nonempty_pullback_ofPoint_module_iso_foldr_pow_ramificationIndexAlong_of_pointEquivPlace_comp_eq_restrictAlong64 below · cited by 2 · depth 22 - Places determined by centres on a two-chart model
AlgebraicCurve.CurveModel.pointEquivPlace_eq_of_forall_sub_algebraMap_mem_nonunits_of_twoChartIntegralModel0 below · cited by 1 · depth 22 - Points and orders under an algebraically closed constant field extension
AlgebraicCurve.CurveModel.existsUnique_point_and_ord_eq_and_ord_eq_zero_of_iso_pullback_of_ffEquiv_symm_germToFunctionField_eq2 below · cited by 2 · depth 23 - Transport of a curve model along a K-algebra isomorphism
AlgebraicCurve.CurveModel.exists_curveModel_iso_of_algEquiv0 below · cited by 2 · depth 23 - Finiteness and flatness of a curve morphism inducing φ
AlgebraicCurve.CurveModel.isFinite_and_flat_and_locallyOfFinitePresentation_and_surjective_of_pointEquivPlace_comp_eq_restrictAlong48 below · cited by 1 · depth 23 - Smooth proper model over an infinite perfect field
AlgebraicCurve.CurveModel.nonempty_of_perfectField1 below · cited by 6 · depth 23 - Semilinear transport of places under a curve-model automorphism
AlgebraicCurve.CurveModel.pointEquivPlace_eq_smul_pointEquivPlace_of_fromSpecStalk_comp_eq_of_apply_closedPoint_eq0 below · cited by 5 · depth 23 - Finite surjection of curve models induces a function-field embedding
AlgebraicCurve.CurveModel.exists_algHom_finiteAlong_pointEquivPlace_restrictAlong_of_isFinite0 below · cited by 3 · depth 24 - Smoothness of relative dimension g for Pic⁰
AlgebraicCurve.CurveModel.smoothOfRelativeDimension_genusFF_of_representsRelSubPic540 below · cited by 1 · depth 24 - Degree-one places give K-rational points on a curve model
AlgebraicCurve.CurveModel.exists_comp_toBase_eq_id_and_base_closedPoint_eq_of_deg_eq_one0 below · cited by 1 · depth 25 - Curve automorphism over Specτ gives τ-semilinear function field automorphism
AlgebraicCurve.CurveModel.exists_semilinearAut_baseAut_eq_and_pointEquivPlace_eq_smul0 below · cited by 8 · depth 25 - Cotangent space of Pic⁰ at origin has dimension g
AlgebraicCurve.CurveModel.finrank_cotangentSpace_zeroSection_eq_genusFF_of_representsRelSubPic164 below · cited by 1 · depth 25 - Irreducible one-dimensional fibres from a curve model
AlgebraicCurve.CurveModel.irreducibleSpace_and_topologicalKrullDim_pullback_eq_one5 below · cited by 2 · depth 25 - Places are determined by their moduli point under an isomorphism of models
AlgebraicCurve.CurveModel.place_eq_of_pointEquivPlace_symm_comp_eq0 below · cited by 6 · depth 25 - Recognising a place from chart coordinates at a K-point
AlgebraicCurve.CurveModel.eq_pointEquivPlace_of_forall_ffEquiv_symm_germToFunctionField_sub_algebraMap_mem_nonunits_of_baseChange0 below · cited by 2 · depth 26 - Curve model of L on an iterated base change
AlgebraicCurve.CurveModel.exists_curveModel_hom_pullback_pullback_germ_eq_of_ringEquiv_functionField73 below · cited by 1 · depth 26 - Relative q-Frobenius as a semilinear automorphism of the model
AlgebraicCurve.CurveModel.exists_semilinear_iso_pointEquivPlace_eq_restrictAlong_frobenius7 below · cited by 1 · depth 26 - A place determined by its readings on an affine chart
AlgebraicCurve.CurveModel.eq_pointEquivPlace_of_forall_ffEquiv_symm_germToFunctionField_sub_algebraMap_mem_nonunits_of_isClosedImmersion0 below · cited by 1 · depth 27 - Two isomorphic curve models differ by a constant-field automorphism
AlgebraicCurve.CurveModel.exists_algEquiv_pointEquivPlace_comp_hom_eq_ofAlgAut_smul_pointEquivPlace0 below · cited by 1 · depth 27 - Unit sections have order zero at places of the curve model
AlgebraicCurve.CurveModel.ord_placeOfPoint_ffEquiv_symm_germToFunctionField_eq_zero_of_isUnit1 below · cited by 6 · depth 27 - Surjectivity and generic-or-closed points under a birational proper map
AlgebraicCurve.CurveModel.surjective_and_eq_genericPoint_or_isClosed_singleton_of_isIso_stalkMap2 below · cited by 8 · depth 27 - Birational ν hits the generic point and identifies function fields
AlgebraicCurve.CurveModel.apply_genericPoint_eq_and_nonempty_algEquiv_functionField_of_isIso_stalkMap0 below · cited by 4 · depth 28 - Value of a section at a K-point equals its residue
AlgebraicCurve.CurveModel.ffEquiv_symm_germToFunctionField_mem_and_sub_algebraMap_appLE_mem_nonunits_pointEquivPlace0 below · cited by 12 · depth 28 - Generator of mathfrak m_{C,x} has order one at the place of x
AlgebraicCurve.CurveModel.ord_placeOfPoint_ffEquiv_symm_algebraMap_eq_one_of_maximalIdeal_eq_span0 below · cited by 5 · depth 28 - Density of lifted K-points in a base-changed curve model
AlgebraicCurve.CurveModel.dense_range_isPullback_lift_specMap_comp_point0 below · cited by 2 · depth 29 - Function-field automorphism induces an automorphism of the curve model
AlgebraicCurve.CurveModel.exists_iso_fromSpecStalk_comp_eq_and_pointEquivPlace_symm_comp_eq_of_algEquiv1 below · cited by 2 · depth 29 - Reading rational functions of an R-model in a curve model's function field
AlgebraicCurve.CurveModel.exists_ringHom_functionField_ffEquiv_symm_stalkMap_eq_of_isIso_pullback1 below · cited by 3 · depth 29 - Finiteness of the integral closure of sections on an affine chart
AlgebraicCurve.CurveModel.finite_integralClosure_sections_of_isIso_stalkMap9 below · cited by 1 · depth 29 - Transport of a pinned function-field embedding along a lift
AlgebraicCurve.CurveModel.germ_app_eq_of_germ_eq_of_comp_eq_comp_of_fromSpecStalk_comp_eq1 below · cited by 2 · depth 29 - Point stabiliser equals place stabiliser times kerθ
AlgebraicCurve.CurveModel.natCard_stabilizer_pointEquivPlace_mul_natCard_ker_eq1 below · cited by 1 · depth 29 - Holomorphic maps from the analytic model are algebraic
AlgebraicCurve.CurveModel.existsUnique_hom_comp_eq_of_differentiableAt_appLE_of_isSeparated86 below · cited by 1 · depth 30 - Local holomorphic lifting of a curve map to ℂ^g
AlgebraicCurve.CurveModel.exists_differentiableOn_lift_pointEquiv_comp_of_differentiableOn_appLE_of_isSeparated18 below · cited by 1 · depth 30 - Finiteness of a birational morphism to a proper curve
AlgebraicCurve.CurveModel.isFinite_of_isIso_stalkMap4 below · cited by 1 · depth 30 - Bilinear relations transfer along a pinned correspondence
AlgebraicCurve.CurveModel.sum_mul_eq_zero_of_sum_mul_eq_zero_of_dense_of_germ_eq0 below · cited by 2 · depth 30 - Place-wise data yield a morphism on an open subscheme
AlgebraicCurve.CurveModel.exists_opens_hom_comp_eq_of_existsUnique_evalAt_eq_appLE8 below · cited by 1 · depth 31 - Regular sections are holomorphic in the place charts
AlgebraicCurve.CurveModel.isOpen_and_differentiableAt_appLE_pointEquivPlace_symm_of_meromorphicOrderAt_eq_ord11 below · cited by 1 · depth 31
AlgebraicCurve.Differential 9
- Correspondences preserve regular differentials
AlgebraicCurve.Differential.correspondence_mem_regularDifferentials0 below · cited by 9 · depth 10 - Trace of differentials at a geometric point is a sum over lifts
AlgebraicCurve.Differential.pullbackAlong_traceAlong_eq_sum_lifts0 below · cited by 1 · depth 11 - Differential form of Abel's theorem over a constant-field extension
AlgebraicCurve.Differential.sum_ord_smul_pullbackAlong_eq_zero81 below · cited by 1 · depth 11 - Constant field extension: regular differentials and correspondences
AlgebraicCurve.Differential.map_correspondence_regularDifferentials_of_constantFieldExtension88 below · cited by 1 · depth 20 - Swapping automorphism conjugates tr_αβ^* into tr_βα^*
AlgebraicCurve.Differential.pullbackAlong_traceAlong_pullbackAlong_eq_traceAlong_pullbackAlong_pullbackAlong_of_swap0 below · cited by 1 · depth 20 - Trace along a separable map agrees with traceDiff
AlgebraicCurve.Differential.traceAlong_eq_traceDiff1 below · cited by 2 · depth 20 - Functoriality of pull-back of Kähler differentials
AlgebraicCurve.Differential.pullbackAlong_comp0 below · cited by 2 · depth 27 - Naturality of the trace on differentials under isomorphisms of the pair
AlgebraicCurve.Differential.pullbackAlong_traceAlong_eq_traceAlong_pullbackAlong_of_algEquiv0 below · cited by 2 · depth 27 - Differential correspondence commutes with constant field extension
AlgebraicCurve.Differential.map_correspondence_eq_correspondence_map_of_separableAlong_of_constantFieldExtension5 below · cited by 1 · depth 30
AlgebraicCurve.Divisor 80
- Push–pull exchange for a linearly disjoint square of fields
AlgebraicCurve.Divisor.pullbackAlong_pushforwardAlong_eq_pushforwardAlong_pullbackAlong19 below · cited by 13 · depth 8 - Descent of n-divisibility of divisor classes along constant-field extension
AlgebraicCurve.Divisor.exists_natCast_dvd_ord_sub_of_constantFieldExtension123 below · cited by 2 · depth 9 - Push-forward of a principal divisor is the norm
AlgebraicCurve.Divisor.pushforwardNormFormula5 below · cited by 2 · depth 9 - Degree-zero divisors are principal in genus zero
AlgebraicCurve.Divisor.isPrincipal_of_genus_eq_zero14 below · cited by 5 · depth 10 - Divisor correspondences depend only on the two underlying maps
AlgebraicCurve.Divisor.correspondence_congr0 below · cited by 6 · depth 11 - Correspondence of a multiple of a single place
AlgebraicCurve.Divisor.correspondence_single0 below · cited by 11 · depth 11 - Collapse of a correspondence on a single place
AlgebraicCurve.Divisor.correspondence_single_of_forall_restrictAlong_eq1 below · cited by 3 · depth 11 - Degree of a divisor as a sum over its support
AlgebraicCurve.Divisor.degree_eq_sum_support0 below · cited by 7 · depth 11 - Every divisor descends to a finite constant field
AlgebraicCurve.Divisor.exists_finite_constantField_form_pullbackConstants_eq50 below · cited by 2 · depth 11 - Composite of two divisor correspondences through a roof
AlgebraicCurve.Divisor.correspondence_correspondence5 below · cited by 3 · depth 12 - Degree of the pole divisor of x equals [F:K(x)]
AlgebraicCurve.Divisor.degree_eq_finrank_adjoin_of_eq_max_neg_ord13 below · cited by 12 · depth 12 - Degree of a divisor as a sum over its support
AlgebraicCurve.Divisor.degree_eq_sum0 below · cited by 8 · depth 12 - Multiplicativity of f ↦ f(D) in the divisor
AlgebraicCurve.Divisor.evalFun_add0 below · cited by 11 · depth 12 - Multiplicativity of f ↦ f(D) at rational places
AlgebraicCurve.Divisor.evalFun_mul1 below · cited by 10 · depth 12 - Non-vanishing of f(D) when f avoids the support of D
AlgebraicCurve.Divisor.evalFun_ne_zero0 below · cited by 10 · depth 12 - Evaluation of f at a two-point divisor (v₁)-(v₂)
AlgebraicCurve.Divisor.evalFun_single_sub_single1 below · cited by 7 · depth 12 - Evaluation at a divisor is multiplicative in f ↦ fⁿ
AlgebraicCurve.Divisor.evalFun_zpow_left4 below · cited by 8 · depth 12 - Evaluation at an integer multiple of a divisor
AlgebraicCurve.Divisor.evalFun_zsmul0 below · cited by 8 · depth 12 - Functoriality of divisor push-forward along composites
AlgebraicCurve.Divisor.pushforwardAlong_pushforwardAlong2 below · cited by 24 · depth 12 - Pole divisor degree at most [F:K(x)]
AlgebraicCurve.Divisor.degree_le_finrank_adjoin_of_eq_max_neg_ord3 below · cited by 1 · depth 13 - Descent of n-torsion divisor classes under constant field extension
AlgebraicCurve.Divisor.exists_torsion_descent_of_constantFieldExtension81 below · cited by 3 · depth 13 - Descent of n-torsion divisor classes along constant field extensions
AlgebraicCurve.Divisor.exists_torsion_descent_of_constantFieldExtension_of_finite51 below · cited by 1 · depth 13 - Lower bound [F:K(x)] ≤ deg of the pole divisor
AlgebraicCurve.Divisor.finrank_adjoin_le_degree_of_eq_max_neg_ord10 below · cited by 1 · depth 13 - Principal divisors descend along a constant-field extension
AlgebraicCurve.Divisor.isPrincipal_of_constantFieldExtension17 below · cited by 4 · depth 13 - Specialisation principle for principal divisors under constant reduction
AlgebraicCurve.Divisor.isPrincipal_of_forall_isPrincipal_mapDomain_placeReduction118 below · cited by 1 · depth 13 - Constant reduction commutes with a correspondence and its base change
AlgebraicCurve.Divisor.mapDomain_placeReduction_correspondence75 below · cited by 2 · depth 13 - Reduction compatible with conorm on degree-zero divisors
AlgebraicCurve.Divisor.mapDomain_pullbackAlong_eq_and_restrictAlong_eq_of_degZero0 below · cited by 2 · depth 13 - Transitivity of divisor pull-back along composed embeddings
AlgebraicCurve.Divisor.pullbackAlong_pullbackAlong2 below · cited by 15 · depth 13 - Pull-back after push-forward equals the sum over deck transformations
AlgebraicCurve.Divisor.pullbackAlong_pushforwardAlong_eq_sum_ofAlgAut_smul_of_forall_comp_eq153 below · cited by 1 · depth 13 - Functoriality of divisor push-forward along composites
AlgebraicCurve.Divisor.pushforwardAlong_comp0 below · cited by 1 · depth 13 - Pushforward after pullback along a surjection is the identity
AlgebraicCurve.Divisor.pushforwardAlong_pullbackAlong_of_surjective8 below · cited by 5 · depth 13 - Push-forward norm formula for finite separable extensions
AlgebraicCurve.Divisor.pushforwardNormFormula_of_isSeparable6 below · cited by 9 · depth 13 - Effective divisors have non-negative degree
AlgebraicCurve.Divisor.degree_nonneg_of_nonneg0 below · cited by 2 · depth 14 - Projection formula for evaluation of functions at divisors
AlgebraicCurve.Divisor.evalFun_algebraMap_pushforward6 below · cited by 9 · depth 14 - Projection formula for pull-back of divisors and norms
AlgebraicCurve.Divisor.evalFun_pullback17 below · cited by 8 · depth 14 - Deuring–Roquette: invariance of ℓ(D) under good constant reduction
AlgebraicCurve.Divisor.exists_finset_finrank_riemannRochSpace_mapDomain_placeReduction_eq116 below · cited by 1 · depth 14 - Riemann–Roch spaces as kernels of a uniform symmetric-value system
AlgebraicCurve.Divisor.exists_symmValue_rows_kernel_iff20 below · cited by 1 · depth 14 - Abel's theorem: sufficiency of the period condition
AlgebraicCurve.Divisor.isPrincipal_of_abelJacobiDiv_mem_pathPeriodLattice187 below · cited by 5 · depth 14 - Pull-back along a K-automorphism is the semilinear action
AlgebraicCurve.Divisor.pullbackAlong_algEquiv_eq_ofAlgAut_smul0 below · cited by 3 · depth 14 - Push–pull formula for a fibre square splitting into two components
AlgebraicCurve.Divisor.pullbackAlong_pushforwardAlong_eq_add_of_adjoin_eq_top7 below · cited by 1 · depth 14 - Push–pull formula over a split fibre product of coverings
AlgebraicCurve.Divisor.pullbackAlong_pushforwardAlong_eq_sum_of_decomposition4 below · cited by 1 · depth 14 - Pushforward of a principal divisor is the divisor of the norm
AlgebraicCurve.Divisor.pushforward_div6 below · cited by 2 · depth 14 - Exchanged legs force a divisor correspondence to be self-transpose
AlgebraicCurve.Divisor.correspondence_comm_of_exchange15 below · cited by 1 · depth 15 - Schmidt descent for G-invariant divisors of a constant field extension
AlgebraicCurve.Divisor.existsUnique_pullbackConstants_eq_of_forall_smul_eq8 below · cited by 1 · depth 15 - Generic upper bound for constant reduction of divisors
AlgebraicCurve.Divisor.exists_finset_finrank_riemannRochSpace_mapDomain_placeReduction_le113 below · cited by 1 · depth 15 - Pull-back of an unramified place: φ^*[w₀]=[W₀]+sumⱼ eⱼ[Wⱼ]
AlgebraicCurve.Divisor.exists_pullbackAlong_single_restrictAlong_eq_single_add_sum_of_ramificationIndexAlong_eq_one7 below · cited by 1 · depth 15 - Frobenius-fixed divisor classes contain Frobenius-fixed divisors
AlgebraicCurve.Divisor.exists_smul_eq_and_isPrincipal_sub_of_frobeniusSemilinear0 below · cited by 1 · depth 15 - Semicontinuity of divisor dimension under constant reduction
AlgebraicCurve.Divisor.finrank_riemannRochSpace_le_finrank_riemannRochSpace_mapDomain_placeReduction77 below · cited by 1 · depth 15 - Integral residues and periods in 2π iℤ give a principal divisor
AlgebraicCurve.Divisor.isPrincipal_of_forall_pathIntegral_eq_two_pi_I_mul76 below · cited by 1 · depth 15 - Descent of principal divisors along a constant field extension
AlgebraicCurve.Divisor.isPrincipal_of_isPrincipal_pullbackConstants_of_isConstantFieldExtension4 below · cited by 2 · depth 15 - Support of a pulled-back divisor lies over its support
AlgebraicCurve.Divisor.support_pullback_subset0 below · cited by 2 · depth 15 - Support of a push-forward divisor lies in restricted support
AlgebraicCurve.Divisor.support_pushforward_subset0 below · cited by 1 · depth 15 - Deuring reduction of a principal divisor along A
AlgebraicCurve.Divisor.mapDomain_placeReduction_eq_ord_of_retraction63 below · cited by 3 · depth 16 - Pull-back of a push-forward across a two-component fibre product
AlgebraicCurve.Divisor.pullbackAlong_pushforwardAlong_eq_add_of_normFormulaAlong4 below · cited by 1 · depth 16 - Conjugating a correspondence by cross-intertwining automorphisms transposes it
AlgebraicCurve.Divisor.ofAlgAut_smul_correspondence_eq_correspondence_ofAlgAut_smul_of_comp_eq_comp15 below · cited by 2 · depth 17 - Push-forward of a principal divisor is the divisor of the norm
AlgebraicCurve.Divisor.pushforwardNormFormula_of_finiteDimensional2 below · cited by 2 · depth 17 - Triviality of the stabiliser of W_{g-1}
AlgebraicCurve.Divisor.mem_principal_of_forall_ell_pos_add0 below · cited by 1 · depth 18 - Push-forward of a degree-one place over a degree-one place
AlgebraicCurve.Divisor.pushforwardAlong_single_one0 below · cited by 4 · depth 18 - Correspondences on divisors and logarithmic differentials agree
AlgebraicCurve.Divisor.correspondence_eq_ord_norm_and_dlog_norm_eq_traceAlong_pullbackAlong3 below · cited by 3 · depth 19 - Correspondence acts on a prime divisor by the degree
AlgebraicCurve.Divisor.correspondence_single_eq_finrankAlong_smul9 below · cited by 2 · depth 19 - Unique degree-zero conorm along a constant-field extension
AlgebraicCurve.Divisor.degZero.existsUnique_conorm_of_constantFieldExtension_of_isAlgClosed3 below · cited by 2 · depth 19 - Galois trace of a divisor equals π^*π_*
AlgebraicCurve.Divisor.sum_galois_smul_eq_pullback_pushforward11 below · cited by 3 · depth 19 - Injectivity of the dlog recipe in characteristic p
AlgebraicCurve.Divisor.exists_eq_pow_and_eq_ord_of_inv_smul_D_eq_zero0 below · cited by 2 · depth 20 - Well-definedness of Serre's dlog map on p-torsion
AlgebraicCurve.Divisor.inv_smul_D_eq_inv_smul_D_of_isPrincipal_sub0 below · cited by 2 · depth 20 - Divisor pull-back commutes with algebraically closed constant-field extension
AlgebraicCurve.Divisor.pullbackAlong_liesOver_of_liesOver3 below · cited by 1 · depth 20 - Push-forward of divisors commutes with constant extension
AlgebraicCurve.Divisor.pushforwardAlong_liesOver_of_liesOver3 below · cited by 1 · depth 20 - Degree-zero divisors generated by differences of two points
AlgebraicCurve.Divisor.degZero_le_closure_single_sub_single_of_surjective0 below · cited by 4 · depth 21 - Pushforward of a pullback divisor is multiplication by [F':F]
AlgebraicCurve.Divisor.pushforward_pullback_of_finite0 below · cited by 6 · depth 22 - Support of a correspondence on a prime divisor, and its transpose
AlgebraicCurve.Divisor.mem_support_correspondence_single_iff_exists_and_iff_mem_support_correspondence_single2 below · cited by 6 · depth 23 - Correspondence through an intermediate field scales by [F:E]
AlgebraicCurve.Divisor.correspondence_eq_finrankAlong_smul_correspondence_of_comp_eq15 below · cited by 2 · depth 24 - Zero divisor of x-a has degree [F:K(x)]
AlgebraicCurve.Divisor.degree_eq_finrank_adjoin_of_eq_max_ord_sub_algebraMap14 below · cited by 1 · depth 24 - Pull-back of a degree-one place as a sum of n places
AlgebraicCurve.Divisor.exists_pullbackAlong_single_one_eq_sum0 below · cited by 3 · depth 24 - Push-forward of a place over an algebraically closed base
AlgebraicCurve.Divisor.pushforwardAlong_single_one_of_isAlgClosed1 below · cited by 5 · depth 24 - Coefficient of the correspondence ψ_*φ^* at a single place
AlgebraicCurve.Divisor.correspondence_single_one_apply_eq_sum_and_eq_finsum0 below · cited by 1 · depth 25 - Division of a principal divisor by n
AlgebraicCurve.Divisor.exists_degZero_ord_eq_mul_of_dvd_ord0 below · cited by 1 · depth 27 - Leg-exchange symmetry for supports of divisorial correspondences
AlgebraicCurve.Divisor.smul_mem_support_correspondence_single_smul_of_mem_support_of_comp_eq19 below · cited by 1 · depth 27 - Evaluation of f on a pullback divisor equals evaluation of N(f)
AlgebraicCurve.Divisor.evalFun_pullback_of_isPurelyInseparable4 below · cited by 1 · depth 28 - Push-forward of div f is div N_{F'/F}(f), separable case
AlgebraicCurve.Divisor.pushforward_div_of_isSeparable7 below · cited by 2 · depth 28 - Divisor surgery: moving support into the good places
AlgebraicCurve.Divisor.exists_isPrincipal_degree_eq_zero_forall_mem_support_add_of_surgery0 below · cited by 1 · depth 38 - Degree of pushed-forward parts adds to degree, degree-one places
AlgebraicCurve.Divisor.degree_mapDomain_filter_add_degree_mapDomain_filter_eq_degree_of_deg_eq_one0 below · cited by 1 · depth 41
AlgebraicCurve.DivisorialWeilPairingData 9
- Left non-degeneracy of the divisorial Weil pairing, assuming divisibility
AlgebraicCurve.DivisorialWeilPairingData.toHom_injective_of_divisible76 below · cited by 10 · depth 13 - Correspondence adjointness of the divisorial Weil pairing
AlgebraicCurve.DivisorialWeilPairingData.pair_correspondence_eq_pair_correspondence26 below · cited by 5 · depth 15 - Semilinear equivariance of the divisorial Weil pairing
AlgebraicCurve.DivisorialWeilPairingData.pair_semilinearSmul0 below · cited by 9 · depth 21 - Non-degeneracy of divisorial Weil pairings on Pic⁰[n]
AlgebraicCurve.DivisorialWeilPairingData.toHom_injective_of_isCurveOver292 below · cited by 2 · depth 21 - Tower compatibility of divisorial Weil pairings at levels m and mn
AlgebraicCurve.DivisorialWeilPairingData.pair_eq_pair_of_coe_eq_nsmul0 below · cited by 3 · depth 22 - Adjunction for divisorial Weil pairings along a finite map
AlgebraicCurve.DivisorialWeilPairingData.pair_pullbackAlong_eq_pair_pushforwardAlongHom24 below · cited by 3 · depth 26 - Weil pairing adjunction along a purely inseparable map
AlgebraicCurve.DivisorialWeilPairingData.pair_pullbackAlong_eq_pair_pushforwardAlongHom_of_isPurelyInseparable13 below · cited by 1 · depth 26 - Perfectness of a divisorial Weil pairing on Pic⁰[n]
AlgebraicCurve.DivisorialWeilPairingData.perfect_of_divisible_coprime_of_isAlgClosed90 below · cited by 1 · depth 26 - Injectivity of the divisorial Weil pairing, n invertible
AlgebraicCurve.DivisorialWeilPairingData.toHom_injective_of_divisible_coprime_of_isAlgClosed88 below · cited by 1 · depth 27
AlgebraicCurve.FunctionField 2
- Rational function field embeds via a non-constant element
AlgebraicCurve.FunctionField.exists_ratFuncAlgHom_apply_X_eq0 below · cited by 6 · depth 14 - Finiteness of a function field over K(g) for any embedding
AlgebraicCurve.FunctionField.finite_of_ratFuncAlgHom0 below · cited by 2 · depth 15
AlgebraicCurve.GluedPic0 10
- Place differences generate the glued degree-zero class group
AlgebraicCurve.GluedPic0.closure_setOf_mk_single_sub_single_eq_top1 below · cited by 6 · depth 12 - n-divisibility passes to the glued degree-zero class group
AlgebraicCurve.GluedPic0.exists_nsmul_eq_of_forall_pic07 below · cited by 2 · depth 12 - Middle exactness of glued Pic⁰ at rational glued places
AlgebraicCurve.GluedPic0.ker_toPic0Pair_eq_range_nodeUnit0 below · cited by 25 · depth 12 - Node units vanish in GluedPic⁰ only for constants
AlgebraicCurve.GluedPic0.nodeUnit_eq_zero_iff_of_constantsAreBase0 below · cited by 6 · depth 12 - Surjectivity of the glued Picard group onto the pair
AlgebraicCurve.GluedPic0.toPic0Pair_surjective5 below · cited by 6 · depth 12 - Order of the m-torsion of ker(toPic0Pair)
AlgebraicCurve.GluedPic0.natCard_ker_toPic0Pair_inf_torsionBy2 below · cited by 5 · depth 14 - Lifting n-torsion through the glued degree-zero class group
AlgebraicCurve.GluedPic0.exists_zsmul_eq_zero_and_toPic0Pair_eq6 below · cited by 2 · depth 24 - No p-power torsion among node-unit classes in characteristic p
AlgebraicCurve.GluedPic0.eq_zero_of_mem_range_nodeUnit_of_pow_char_smul_eq_zero1 below · cited by 2 · depth 28 - Rosenlicht generation of the glued degree-zero class group
AlgebraicCurve.GluedPic0.mem_closure_mk_pair_of_riemannRoch11 below · cited by 1 · depth 35 - No p-power torsion in the kernel of GPic⁰ → Pic⁰ × Pic⁰
AlgebraicCurve.GluedPic0.eq_zero_of_pow_char_smul_eq_zero_of_toPic0Pair_eq_zero2 below · cited by 1 · depth 36
AlgebraicCurve.GluingData 1
- Glued principal data push forward along finite separable maps
AlgebraicCurve.GluingData.isGluedPrincipal_pushforwardMap_of_separableAlong12 below · cited by 1 · depth 19
AlgebraicCurve.IsConfluentPattern 1
- Orders in a confluent pattern exhaust {0,…,nᵥ-1}
AlgebraicCurve.IsConfluentPattern.exists_eq_of_lt_jetMult0 below · cited by 2 · depth 18
AlgebraicCurve.IsCurveOver 3
- Existence of a separating transcendental element on a curve
AlgebraicCurve.IsCurveOver.exists_separating_transcendental0 below · cited by 97 · depth 10 - Finiteness over K(t) for every transcendental t
AlgebraicCurve.IsCurveOver.finiteDimensional_adjoin_simple_of_transcendental_of_essFiniteType1 below · cited by 42 · depth 22 - A curve over a perfect field has transcendence degree one
AlgebraicCurve.IsCurveOver.trdeg_eq_one2 below · cited by 2 · depth 27
AlgebraicCurve.IsFrobeniusEndo 3
- Degree of a Frobenius endomorphism: [F:φⁿ(F)]=(p^r)ⁿ
AlgebraicCurve.IsFrobeniusEndo.finrankAlong_pow_eq2 below · cited by 1 · depth 19 - Relative Frobenius is totally ramified at every place
AlgebraicCurve.IsFrobeniusEndo.ramificationIndexAlong_eq0 below · cited by 2 · depth 19 - Frobenius endomorphisms are radicial on places
AlgebraicCurve.IsFrobeniusEndo.restrictAlong_injective0 below · cited by 1 · depth 22
AlgebraicCurve.KummerCover 1
- Splitting field of Xᵖ - f has degree p
AlgebraicCurve.KummerCover.finrank_eq0 below · cited by 2 · depth 11
AlgebraicCurve.KwPke 1
- Separability over F^ℓ(t) from separability over K(t)
AlgebraicCurve.KwPke.kw_pke_hsep_of_isSeparable_adjoin0 below · cited by 4 · depth 15
AlgebraicCurve.NodalPic0 1
- Torsion bound for the degree-zero class group of a nodal curve
AlgebraicCurve.NodalPic0.natCard_torsion_mul_pow_le_pow_of_isAlgClosed980 below · cited by 1 · depth 30
AlgebraicCurve.NodeAnnulusEngine 25
- Node regularity and the slope law from layered crossing presentations
AlgebraicCurve.NodeAnnulusEngine.ord_residue_nonneg_and_finsum_ord_eq_ord_residue_add_of_ringEquiv_uvCrossingModel_layers108 below · cited by 9 · depth 23 - Node places separated by the value of the coordinate y
AlgebraicCurve.NodeAnnulusEngine.eq_of_mem_of_evalAt_eq0 below · cited by 1 · depth 24 - Admissible values of a node coordinate are attained
AlgebraicCurve.NodeAnnulusEngine.exists_mem_and_evalAt_eq0 below · cited by 1 · depth 24 - Node places count horizontal zeros in a uv-crossing model
AlgebraicCurve.NodeAnnulusEngine.finite_and_finsum_ord_eq_finsum_finrank_mul_length_of_ringEquiv_uvCrossingModel99 below · cited by 1 · depth 24 - Order at the first end is minus the largest dominant index
AlgebraicCurve.NodeAnnulusEngine.ord_residue_smul_eq_neg_sSup_dominantIndices_of_ringEquiv_uvCrossingModel38 below · cited by 1 · depth 24 - Order at the U-end as least dominant index
AlgebraicCurve.NodeAnnulusEngine.ord_residue_smul_eq_sInf_dominantIndices_of_ringEquiv_uvCrossingModel38 below · cited by 1 · depth 24 - Node coordinate minus its value is a uniformiser
AlgebraicCurve.NodeAnnulusEngine.ord_sub_evalAt_eq_one0 below · cited by 1 · depth 24 - Places over a horizontal prime counted by W-ranks of branches
AlgebraicCurve.NodeAnnulusEngine.finite_and_ncard_eq_finsum_finrank_of_forall_iff_evalAt_eq_zero95 below · cited by 1 · depth 25 - Order at a node place equals length on the crossing model
AlgebraicCurve.NodeAnnulusEngine.toNat_ord_eq_length_localizedModule_of_forall_iff_evalAt_eq_zero58 below · cited by 1 · depth 25 - Counting places above a horizontal prime by length(mathcal N₀/(𝔭+varpi))
AlgebraicCurve.NodeAnnulusEngine.finite_and_ncard_eq_length_of_forall_iff_evalAt_eq_zero76 below · cited by 1 · depth 26 - Total W-rank of branches over a horizontal prime
AlgebraicCurve.NodeAnnulusEngine.finsum_finrank_quotient_eq_length_of_comap_eq51 below · cited by 1 · depth 26 - Finiteness of a horizontal residue field over the constants
AlgebraicCurve.NodeAnnulusEngine.finiteDimensional_fractionRing_quotient_of_not_mem2 below · cited by 2 · depth 27 - Reducedness of R/𝔭R for horizontal primes of the node ring
AlgebraicCurve.NodeAnnulusEngine.isReduced_quotient_map_of_ne_bot_of_not_mem31 below · cited by 1 · depth 27 - Linear disjointness over C and L-generation of F
AlgebraicCurve.NodeAnnulusEngine.linearDisjoint_and_exists_sum_smul_div_of_isAlgClosedIn0 below · cited by 1 · depth 27 - Places over a horizontal prime as C-points of mathcal N₀/𝔭
AlgebraicCurve.NodeAnnulusEngine.nonempty_equiv_ringHom_quotient_of_forall_iff_evalAt_eq_zero69 below · cited by 1 · depth 27 - Orders at both branches of a monomial germ wV^e
AlgebraicCurve.NodeAnnulusEngine.residue_eq_zero_and_ord_residue_eq_and_ord_residue_smul_eq_neg_of_eq_mul_V_pow32 below · cited by 3 · depth 27 - A place of S over a horizontal prime is determined by its values on mathcal N₀
AlgebraicCurve.NodeAnnulusEngine.eq_of_forall_evalAt_eq_of_forall_iff_evalAt_eq_zero60 below · cited by 1 · depth 28 - Every C-point of mathcal N₀/𝔭 in A is an evaluation
AlgebraicCurve.NodeAnnulusEngine.exists_mem_and_forall_evalAt_eq_coe_apply_of_ringHom_quotient66 below · cited by 1 · depth 28 - Evaluation at a place gives a C-point mathcal N₀/𝔭 → A
AlgebraicCurve.NodeAnnulusEngine.exists_ringHom_quotient_forall_coe_apply_eq_evalAt0 below · cited by 1 · depth 28 - Base change of a rational node ring to a larger constant layer
AlgebraicCurve.NodeAnnulusEngine.exists_localizedBaseChange_of_layer_pos_localization2 below · cited by 4 · depth 29 - Maximum principle at a node: integrality over the node ring
AlgebraicCurve.NodeAnnulusEngine.isIntegral_and_evalAt_mem_of_mem_ends_of_forall_mem_toValuationSubring49 below · cited by 2 · depth 29 - Residue order equals U-order at a crossing-presented end
AlgebraicCurve.NodeAnnulusEngine.exists_isUnit_sub_mul_U_pow_mem_of_ord_residue_eq_of_ringEquiv_uvCrossingModel0 below · cited by 2 · depth 30 - Residue order equals V-order modulo (π,U)
AlgebraicCurve.NodeAnnulusEngine.exists_isUnit_sub_mul_V_pow_mem_of_ord_residue_eq_of_ringEquiv_uvCrossingModel0 below · cited by 2 · depth 30 - Proper valuation rings over mathcal N₀ with varpi invertible come from S
AlgebraicCurve.NodeAnnulusEngine.exists_mem_and_toValuationSubring_eq_of_forall_mem_of_not_mem_nonunits2 below · cited by 1 · depth 30 - Crossing presentation of a layer compatible with the base
AlgebraicCurve.NodeAnnulusEngine.exists_ringEquiv_adicCompletion_uvCrossingModel_of_layer_compatibleUV36 below · cited by 2 · depth 30
AlgebraicCurve.NodeRingLayers 2
- Normality and branch valuation rings of a node-ring layer
AlgebraicCurve.NodeRingLayers.mem_of_isIntegral_and_exists_valuationSubring_localization_of_uvCrossingModel51 below · cited by 1 · depth 24 - Node ring base-changed to a layer of constants
AlgebraicCurve.NodeRingLayers.isMaximal_and_exists_isNoetherianRing_isLocalRing_localization_closure_union_layer0 below · cited by 2 · depth 25
AlgebraicCurve.Pic0 107
- pⁿ-torsion of Pic⁰ has order p^{2gn}
AlgebraicCurve.Pic0.abelJacobiCard_genus297 below · cited by 12 · depth 10 - Order of a divisor class divides m when mD is principal
AlgebraicCurve.Pic0.addOrderOf_mk_dvd_of_isPrincipal4 below · cited by 10 · depth 10 - Divisor-level commuting correspondences commute on Pic⁰
AlgebraicCurve.Pic0.correspondence_correspondence_comm0 below · cited by 3 · depth 10 - Torsion of Pic⁰ over a locally finite constant field
AlgebraicCurve.Pic0.exists_nsmul_eq_zero_of_charP_of_forall_pow_eq_self57 below · cited by 2 · depth 10 - Principality of degree-zero divisors transports along base-compatible isomorphisms
AlgebraicCurve.Pic0.forall_isPrincipal_of_ringEquiv0 below · cited by 2 · depth 10 - Relations on Pic⁰ pass to regular differentials
AlgebraicCurve.Pic0.freeAlgebra_lift_differential_eq_zero_of_lift_correspondence_eq_zero148 below · cited by 4 · depth 10 - A class in Pic⁰ vanishes iff the divisor is principal
AlgebraicCurve.Pic0.mk_eq_zero_iff0 below · cited by 8 · depth 10 - Principality of all degree-zero divisors makes Pic⁰ trivial
AlgebraicCurve.Pic0.subsingleton_of_forall_isPrincipal0 below · cited by 5 · depth 10 - Torsion criterion in Pic⁰: mD principal kills m[D]
AlgebraicCurve.Pic0.zsmul_mk_eq_zero_of_isPrincipal2 below · cited by 12 · depth 10 - Divisibility of Pic⁰ of a curve over an algebraically closed field
AlgebraicCurve.Pic0.exists_nsmul_eq260 below · cited by 32 · depth 11 - Relations in Pic⁰ make geometric cycles principal
AlgebraicCurve.Pic0.exists_principal_geometricCycle_of_lift_correspondence_eq_zero140 below · cited by 1 · depth 11 - Finiteness of Pic⁰ over a finite constant field
AlgebraicCurve.Pic0.finite_of_finite27 below · cited by 5 · depth 11 - p-torsion of Pic⁰ has order p^{2g}
AlgebraicCurve.Pic0.natCard_torsion_prime_eq_pow_genus242 below · cited by 3 · depth 11 - Natural-number form: mD principal implies m[D]=0
AlgebraicCurve.Pic0.nsmul_mk_eq_zero_of_isPrincipal3 below · cited by 1 · depth 11 - Integer multiples commute with passage to divisor classes
AlgebraicCurve.Pic0.zsmul_mk0 below · cited by 2 · depth 11 - Base change of correspondence relations on Pic⁰
AlgebraicCurve.Pic0.freeAlgebra_lift_baseChange_correspondence_eq_zero129 below · cited by 2 · depth 12 - Invariance of Pic⁰ torsion under constant field extension
AlgebraicCurve.Pic0.natCard_torsion_eq_of_constantFieldExtension82 below · cited by 3 · depth 12 - Order of p-torsion in Pic⁰ over ℂ
AlgebraicCurve.Pic0.natCard_torsion_prime_eq_pow_genus_complex213 below · cited by 2 · depth 12 - Roof package along a surjective leg of function fields
AlgebraicCurve.Pic0.roof_package_of_surjective167 below · cited by 1 · depth 12 - Abel–Jacobi: Pic⁰ of a complex function field is ℂ^g/L
AlgebraicCurve.Pic0.exists_addEquiv_quotient_submodule_complex211 below · cited by 1 · depth 13 - Moving lemma for degree-zero divisor classes
AlgebraicCurve.Pic0.exists_mk_eq_forall_notMem_support4 below · cited by 8 · depth 13 - Frobenius fixed classes on Pic⁰ and resultants Res(Xⁿ-1,P)
AlgebraicCurve.Pic0.exists_monic_natCard_fixedPoints_iterate_eq_resultant_of_pushforwardAlong_frobenius133 below · cited by 6 · depth 13 - Torsion in Pic⁰ yields a function with divisor nD
AlgebraicCurve.Pic0.exists_ord_eq_mul_of_nsmul_mk_eq_zero0 below · cited by 6 · depth 13 - Finiteness of p^k-torsion of Pic⁰ in characteristic p
AlgebraicCurve.Pic0.finite_torsion_pow_char74 below · cited by 5 · depth 13 - Rational places generate Pic⁰ over a base point
AlgebraicCurve.Pic0.mem_closure_mk_single_sub_single0 below · cited by 11 · depth 13 - Generators of Pic⁰ avoiding a finite set of places
AlgebraicCurve.Pic0.mem_closure_mk_single_sub_single_of_notMem2 below · cited by 3 · depth 13 - Existence of a divisorial Weil pairing datum at level n
AlgebraicCurve.Pic0.nonempty_divisorialWeilPairingData83 below · cited by 17 · depth 13 - Abel–Jacobi: Pic⁰ of a complex curve as ℂ^g/L
AlgebraicCurve.Pic0.exists_addEquiv_quotient_submodule_of_chartedSpace_complex200 below · cited by 1 · depth 14 - Kummer witness for a nonzero n-torsion divisor class
AlgebraicCurve.Pic0.exists_ord_eq_mul_and_forall_pow_ne_of_ne_zero0 below · cited by 2 · depth 14 - Existence of the Weil pairing on Pic⁰[n]
AlgebraicCurve.Pic0.exists_weilPairing306 below · cited by 3 · depth 14 - Finiteness and n^{2g} bound for n-torsion of Pic⁰
AlgebraicCurve.Pic0.finite_and_card_torsion_le_of_natCast_ne_zero946 below · cited by 10 · depth 14 - Finiteness of p-torsion in Pic⁰ in characteristic p
AlgebraicCurve.Pic0.finite_torsion_char72 below · cited by 1 · depth 14 - Finiteness of Pic⁰[n] from prime-power torsion
AlgebraicCurve.Pic0.finite_torsion_of_forall_primePow0 below · cited by 8 · depth 14 - Frobenius-fixed divisor classes counted by the class number
AlgebraicCurve.Pic0.natCard_fixedPoints_eq_natCard_pic0_of_pushforwardAlong_frobenius66 below · cited by 1 · depth 14 - Weil pairing as a homomorphism into the character group
AlgebraicCurve.Pic0.torsion.exists_addMonoidHom_eval_eq_pairing17 below · cited by 4 · depth 14 - Moving torsion classes off a finite set, with rational support
AlgebraicCurve.Pic0.torsion.move_of_forall_isRational5 below · cited by 4 · depth 14 - Logarithmic differentials inject the p-torsion of Pic⁰
AlgebraicCurve.Pic0.exists_injective_addMonoidHom_torsion_dlog13 below · cited by 1 · depth 15 - Divisibility of Pic⁰ in characteristic p
AlgebraicCurve.Pic0.exists_nsmul_eq_of_charP601 below · cited by 8 · depth 15 - Prime torsion of Pic⁰ has order ℓ^{2g}
AlgebraicCurve.Pic0.natCard_torsion_prime_eq_pow_two_mul_genusFF_of_forall_pow_eq_self810 below · cited by 6 · depth 15 - Genus zero forces trivial degree-zero divisor class group
AlgebraicCurve.Pic0.subsingleton_of_genusFF_eq_zero83 below · cited by 3 · depth 15 - Witness function for an n-torsion divisor class
AlgebraicCurve.Pic0.torsion.exists_forall_ord_eq_mul0 below · cited by 3 · depth 15 - ℓ-power torsion of Pic⁰ for curves with Frobenius
AlgebraicCurve.Pic0.abelJacobiCard_genusFF_of_frobenius948 below · cited by 4 · depth 16 - Prime-to-p torsion of Pic⁰ over 𝔽̄_q
AlgebraicCurve.Pic0.abelJacobiCard_genusFF_of_frobenius_of_isAlgebraic674 below · cited by 2 · depth 16 - Existence of the Jacobian group scheme and Abel–Jacobi dictionary
AlgebraicCurve.Pic0.exists_relativeGroupLaw_equiv_of_curveModel535 below · cited by 1 · depth 16 - Injectivity of the conorm on Pic⁰ under constant field extension
AlgebraicCurve.Pic0.exists_injective_conorm_of_constantFieldExtension_of_isAlgClosed27 below · cited by 5 · depth 17 - Frobenius acts with finite orbits on places, divisors and Pic⁰
AlgebraicCurve.Pic0.exists_iterate_apply_eq_self_of_pushforwardAlong_frobenius_of_isAlgebraic59 below · cited by 1 · depth 17 - Push-forward on Pic⁰ is equivariant for intertwined semilinear automorphisms
AlgebraicCurve.Pic0.pushforwardAlongHom_smul4 below · cited by 4 · depth 18 - Conorm map on Pic⁰ intertwines correspondence operators
AlgebraicCurve.Pic0.conorm_correspondence_eq_correspondence_conorm5 below · cited by 1 · depth 19 - Equivariance of the conorm on Pic⁰ under compatible semilinear automorphisms
AlgebraicCurve.Pic0.conorm_smul_eq_smul_conorm_of_semilinearAut_compatible0 below · cited by 2 · depth 19 - Natural equivariant uniformisation of Jacobians of Mumford quotients
AlgebraicCurve.Pic0.exists_equivariantUniformization_family_natural_of_mumfordQuotient_of_v_card_stabilizer_eq_one296 below · cited by 1 · depth 19 - The dlog injection Pic⁰[p]↪Ω_{F/K}, pinned by its recipe
AlgebraicCurve.Pic0.exists_injective_addMonoidHom_torsion_apply_eq_inv_smul_D52 below · cited by 3 · depth 19 - Torsion classes in Pic⁰ come from n-divisible functions
AlgebraicCurve.Pic0.exists_mk_eq_and_dvd_ord_of_mem_torsion0 below · cited by 7 · depth 19 - Finiteness of n-torsion of Pic⁰ in characteristic zero
AlgebraicCurve.Pic0.finite_torsion_of_isAlgClosed_of_charZero242 below · cited by 6 · depth 19 - Faithfulness of the cotangent representation of correspondences
AlgebraicCurve.Pic0.freeAlgebra_lift_correspondence_eq_zero_of_lift_differential_eq_zero279 below · cited by 2 · depth 19 - Levelwise Lefschetz trace formula for Frobenius on Pic⁰[ℓ^m]
AlgebraicCurve.Pic0.trace_pow_torsion_eq_of_pushforwardAlong1,202 below · cited by 3 · depth 19 - Push-forward square for pinned Mumford uniformisations along φ
AlgebraicCurve.Pic0.eFull_comp_pullback_eq_mk_pushforwardAlong_of_mumfordQuotient_theta65 below · cited by 1 · depth 20 - Pullback compatibility of pinned Mumford uniformisations of Pic⁰
AlgebraicCurve.Pic0.eFull_comp_pushforward_eq_mk_pullbackAlong_of_mumfordQuotient_theta_of_v_card_stabilizer_eq_one138 below · cited by 1 · depth 20 - Equivariant Manin–Drinfeld uniformisation of Jacobians of Mumford quotients
AlgebraicCurve.Pic0.exists_equivariantUniformization_of_mumfordQuotient_theta_of_mem_valuationSubring_iff_of_v_card_stabilizer_eq_one247 below · cited by 1 · depth 20 - Conorm on Pic⁰ is injective and preserves n-torsion
AlgebraicCurve.Pic0.exists_injective_conorm_map_torsion_eq_of_isAlgClosed252 below · cited by 1 · depth 20 - Frobenius polynomial on Pic⁰: place counts and ℓ-primary kernels
AlgebraicCurve.Pic0.exists_monic_natCard_primaryComponent_ker_aeval_of_pushforwardAlong_frobenius902 below · cited by 1 · depth 20 - Correspondence relations descend from a constant-field extension
AlgebraicCurve.Pic0.freeAlgebra_lift_correspondence_eq_zero_of_baseChange97 below · cited by 1 · depth 20 - Faithfulness of the cotangent action of correspondences over ℂ
AlgebraicCurve.Pic0.freeAlgebra_lift_correspondence_eq_zero_of_lift_differential_eq_zero_complex221 below · cited by 1 · depth 20 - Adjointness of pullback and pushforward for Mumford period pairings
AlgebraicCurve.Pic0.periodPairing_pullback_eq_periodPairing_pushforward_of_mumfordQuotient_theta50 below · cited by 1 · depth 20 - Frobenius characteristic polynomial equals the zeta numerator
AlgebraicCurve.Pic0.eq_of_natCard_ker_aeval_eq_natAbs_resultant_of_natCard_fixedPoints_restrictAlong_eq900 below · cited by 1 · depth 21 - Antisymmetric Weil pairing on the torsion of Pic⁰
AlgebraicCurve.Pic0.exists_antisymmWeilPairing302 below · cited by 2 · depth 21 - Theta torus point lifting a two-point divisor and its push-forward
AlgebraicCurve.Pic0.exists_eFull_eq_mk_single_sub_single_and_eFull_comp_pullback_eq_mk_pushforwardAlong_of_mumfordQuotient_theta62 below · cited by 1 · depth 21 - A theta torus point compatible with degeneracy push-forward
AlgebraicCurve.Pic0.exists_eFull_eq_mk_single_sub_single_and_eFull_comp_pushforward_eq_mk_pullbackAlong_of_mumfordQuotient_theta_of_v_card_stabilizer_eq_one135 below · cited by 1 · depth 21 - Equivariant transport of Pic⁰ along a field isomorphism
AlgebraicCurve.Pic0.exists_equiv_addEquiv_mk_eq_and_smul_of_ringEquiv1 below · cited by 1 · depth 21 - Kernel orders of G(T) on Pic⁰ as |Res(G,P)|
AlgebraicCurve.Pic0.exists_monic_natCard_ker_aeval_eq_natAbs_resultant_of_pushforwardAlong_frobenius897 below · cited by 2 · depth 21 - Period datum of a Mumford quotient pinned to analytic periods
AlgebraicCurve.Pic0.exists_periodDatum_Q_mul_period_eq_one_of_mumfordQuotient81 below · cited by 1 · depth 21 - Theta uniformisation of Pic⁰ of a tame Mumford quotient
AlgebraicCurve.Pic0.exists_torusPoints_uniformization_of_periodDatum_of_mumfordQuotient_of_v_card_stabilizer_eq_one146 below · cited by 1 · depth 21 - Divisibility of Pic⁰ for function fields over K(X)
AlgebraicCurve.Pic0.exists_zsmul_eq_of_finiteDimensional_ratFunc263 below · cited by 6 · depth 21 - Glued n-torsion count under g+1 = 2h+#S
AlgebraicCurve.Pic0.finite_torsion_and_natCard_torsion_eq_natCard_gluedPic0_torsion_mul_of_genusFF_add_one_eq861 below · cited by 5 · depth 21 - Order of the ℓ^k-torsion of Pic⁰
AlgebraicCurve.Pic0.natCard_torsion_pow_eq_pow_two_mul_genusFF_mul_of_forall_pow_eq_self811 below · cited by 4 · depth 21 - Equivariance of the period pairing and the theta-pinned uniformisation
AlgebraicCurve.Pic0.periodDatum_equivariant_of_theta_pinned_uniformization_of_mumfordQuotient125 below · cited by 1 · depth 21 - π_*∘π^*=[F':F] on Pic⁰
AlgebraicCurve.Pic0.pushforwardHom_pullbackHom1 below · cited by 2 · depth 21 - δ on Pic⁰[p] intertwines ψ_*φ^* with tr_ψφ^*
AlgebraicCurve.Pic0.torsion_apply_eq_traceAlong_pullbackAlong_of_correspondence5 below · cited by 1 · depth 21 - Frobenius characteristic polynomial equals the zeta numerator
AlgebraicCurve.Pic0.eq_of_natCard_ker_aeval_eq_natAbs_resultant_of_natCard_fixedPoints_restrictAlong_eq_of_isAlgebraic172 below · cited by 1 · depth 22 - Points dictionary for any object representing rigidified Pic⁰
AlgebraicCurve.Pic0.exists_equiv_points_add_and_poincare_iso_ofPoint_of_representsRelSubPic_of_curveModel_of_isAlgClosed541 below · cited by 1 · depth 22 - Polynomiality of kernel orders of G(T) on Pic⁰
AlgebraicCurve.Pic0.exists_mvPolynomial_eval_eq_natCard_ker_aeval_of_pushforwardAlong_frobenius894 below · cited by 1 · depth 22 - Principal theta divisors on a Mumford quotient are periods
AlgebraicCurve.Pic0.exists_prod_theta_eq_period_of_isPrincipal_of_v_card_stabilizer_eq_one138 below · cited by 1 · depth 22 - Equal theta multipliers give a principal divisor on a Mumford quotient
AlgebraicCurve.Pic0.isPrincipal_sum_sub_sum_of_prod_theta_eq_of_v_card_stabilizer_eq_one119 below · cited by 1 · depth 22 - Surjectivity of G(T) on Pic⁰ when ker G(T) is finite
AlgebraicCurve.Pic0.surjective_aeval_of_finite_ker_of_pushforwardAlong_frobenius598 below · cited by 1 · depth 22 - Frobenius characteristic polynomial on Pic⁰ equals the zeta numerator
AlgebraicCurve.Pic0.eq_map_of_natCard_ker_aeval_eq_abs_resultant_of_natCard_fixedPoints_restrictAlong_eq893 below · cited by 1 · depth 23 - Abel–Jacobi dictionary for Pic⁰ of a curve
AlgebraicCurve.Pic0.exists_equiv_points_add_and_poincare_iso_ofPoint_of_representsRelSubPic_of_curveModel292 below · cited by 1 · depth 23 - Rational Tate module of Pic⁰ of a quotient curve
AlgebraicCurve.Pic0.exists_injective_linearMap_rationalTateModule_fixedField_range_eq_iInf_ker_and_comp_eq20 below · cited by 3 · depth 23 - Frobenius degree theory on Pic⁰: #ker G(T)=Res(G,P)
AlgebraicCurve.Pic0.exists_monic_natCard_ker_aeval_eq_resultant_map_of_pushforwardAlong_frobenius790 below · cited by 2 · depth 23 - Norm homomorphism of Pic⁰ inducing divisor push-forward
AlgebraicCurve.Pic0.exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic111 below · cited by 2 · depth 23 - Rational ℓ-adic Tate module of Pic⁰ has dimension 2g
AlgebraicCurve.Pic0.finrank_rationalTateModule_eq_two_mul_genusFF_of_charZero296 below · cited by 1 · depth 23 - Frobenius characteristic polynomial on Pic⁰ equals the zeta numerator
AlgebraicCurve.Pic0.eq_map_of_natCard_ker_aeval_eq_abs_resultant_of_natCard_fixedPoints_restrictAlong_eq_of_isAlgebraic172 below · cited by 1 · depth 24 - ℓ-adic Weil pairing on the Tate module of Pic⁰
AlgebraicCurve.Pic0.exists_weilPairing_tateModule_of_isCurveOver308 below · cited by 3 · depth 24 - Vanishing differential of the Frobenius endomorphism of a Jacobian
AlgebraicCurve.Pic0.map_maximalIdeal_le_sq_of_pushforwardAlong_frobenius_of_representsRelSubPic536 below · cited by 1 · depth 24 - Correspondence and its transpose are adjoint on T_ℓ Pic⁰
AlgebraicCurve.Pic0.weilPairing_tateModule_correspondence_eq_correspondence87 below · cited by 1 · depth 24 - Semilinear equivariance of the ℓ-adic Weil pairing
AlgebraicCurve.Pic0.weilPairing_tateModule_rep_semilinearAut85 below · cited by 2 · depth 24 - Frobenius push-forward realised by a semilinear Jacobian morphism
AlgebraicCurve.Pic0.exists_semilinear_hom_of_pushforwardAlong_frobenius_of_representsRelSubPic31 below · cited by 1 · depth 25 - Divisibility of Pic⁰ by integers invertible in K
AlgebraicCurve.Pic0.exists_zsmul_eq_of_finiteDimensional_ratFunc_of_forall_pow_eq_self545 below · cited by 1 · depth 26 - Divisorial Weil pairing datum on Pic⁰[n] over ̄ K
AlgebraicCurve.Pic0.nonempty_divisorialWeilPairingData_of_isAlgClosed90 below · cited by 5 · depth 26 - Image of δ is the Cartier-fixed regular differentials
AlgebraicCurve.Pic0.range_eq_setOf_cartier_fixed_and_isRegularDiff24 below · cited by 1 · depth 26 - Inertia fixes p-torsion of Pic⁰ under good reduction
AlgebraicCurve.Pic0.semilinearAut_smul_eq_self_of_mem_inertiaSubgroupIn_of_smoothOfRelativeDimension_one734 below · cited by 1 · depth 26 - Levelwise adjointness passes to the ℓ-adic Weil pairings
AlgebraicCurve.Pic0.weilPairing_tateModule_apply_eq_of_forall_pair_eq0 below · cited by 2 · depth 26 - Existence of a Jacobian pack over an algebraically closed field
AlgebraicCurve.Pic0.exists_jacobianPack_of_isAlgClosed537 below · cited by 1 · depth 27 - Divisibility of Pic⁰ by n invertible in K
AlgebraicCurve.Pic0.exists_nsmul_eq_of_jacobianPack_of_natCast_ne_zero2 below · cited by 1 · depth 27 - Injective maps into the character group of Pic⁰[n] are bijective
AlgebraicCurve.Pic0.torsion.bijective_of_injective0 below · cited by 1 · depth 27 - ℓ^k-torsion of Pic⁰ has order ℓ^{2gk} in characteristic zero
AlgebraicCurve.Pic0.natCard_torsion_pow_eq_pow_two_mul_genusFF_mul_of_charZero978 below · cited by 2 · depth 29 - Order ℓ^{2g} for the ℓ-torsion of Pic⁰
AlgebraicCurve.Pic0.natCard_torsion_prime_eq_pow_two_mul_genusFF_of_natCast_ne_zero977 below · cited by 1 · depth 30 - Lower bound ℓ^{2g} for ℓ-torsion of Pic⁰
AlgebraicCurve.Pic0.pow_two_mul_genusFF_le_natCard_torsion_prime_of_natCast_ne_zero975 below · cited by 1 · depth 32 - Vanishing in Pic⁰ iff f is an n-th power
AlgebraicCurve.Pic0.mk_eq_zero_iff_exists_pow0 below · cited by 2 · depth 33
AlgebraicCurve.Place 289
- Places over an algebraically closed field have degree one
AlgebraicCurve.Place.deg_eq_one_of_isAlgClosed0 below · cited by 18 · depth 9 - Unique unramified place above P in a constant-field extension
AlgebraicCurve.Place.exists_comap_algebraMap_eq_of_constantFieldExtension2 below · cited by 31 · depth 9 - Positivity of the inertia degree in a finite extension
AlgebraicCurve.Place.inertiaDeg_pos_of_finiteDimensional0 below · cited by 19 · depth 9 - Valuation ring membership iff non-negative order
AlgebraicCurve.Place.mem_iff_ord_nonneg0 below · cited by 189 · depth 9 - Order at a place computed by any valuation with the same ring
AlgebraicCurve.Place.ord_eq_neg_log_of_valuationSubring_eq1 below · cited by 15 · depth 9 - Local exchange identity for places in a linearly disjoint compositum
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_exchange18 below · cited by 1 · depth 9 - The identity r e f=[M:F'] for Galois extensions of places
AlgebraicCurve.Place.card_fiberOver_mul_ramificationIndex_mul_inertiaDeg9 below · cited by 8 · depth 10 - Degree one places: residue field generated by constants
AlgebraicCurve.Place.deg_eq_one_iff_surjective_algebraMap_residueField0 below · cited by 65 · depth 10 - A place centred at an affine point is the point's place
AlgebraicCurve.Place.eq_ofHeightOneSpectrum_of_XClass_mem_nonunits_of_YClass_mem_nonunits1 below · cited by 3 · depth 10 - Prescribed orders at finitely many places
AlgebraicCurve.Place.exists_forall_ord_eq0 below · cited by 18 · depth 10 - Every place admits an element of positive order
AlgebraicCurve.Place.exists_ord_pos0 below · cited by 5 · depth 10 - Every place extends to a finite separable extension
AlgebraicCurve.Place.exists_restrict_eq3 below · cited by 36 · depth 10 - Pullback of a valuation subring along φ gives a place
AlgebraicCurve.Place.exists_toValuationSubring_eq_comap_ringHom0 below · cited by 15 · depth 10 - Multiplicativity of the inertia degree in a tower
AlgebraicCurve.Place.inertiaDegAlong_comp0 below · cited by 5 · depth 10 - Positivity of the inertia degree f(w∣ v)
AlgebraicCurve.Place.inertiaDeg_pos1 below · cited by 13 · depth 10 - Equivalence with v's adic valuation from equal valuation rings
AlgebraicCurve.Place.isEquiv_adicValuation_of_valuationSubring_eq0 below · cited by 2 · depth 10 - A place is rational iff its degree is 1
AlgebraicCurve.Place.isRational_iff_deg_eq_one0 below · cited by 198 · depth 10 - Places of a function field over an algebraically closed constant field are rational
AlgebraicCurve.Place.isRational_of_isAlgClosed38 below · cited by 40 · depth 10 - Nonnegative order implies membership in the valuation ring
AlgebraicCurve.Place.mem_of_ord_nonneg0 below · cited by 107 · depth 10 - Functions with order zero at every place are constant
AlgebraicCurve.Place.mem_range_algebraMap_of_forall_ord_eq_zero_of_isAlgClosed11 below · cited by 45 · depth 10 - Valuation subrings containing j are integrally closed over K[j]
AlgebraicCurve.Place.mem_toValuationSubring_of_isIntegral_adjoin0 below · cited by 46 · depth 10 - ordᵥ(df) = ordᵥ(f) - 1 at zeros and poles
AlgebraicCurve.Place.ordDiff_D_eq_ord_sub_one16 below · cited by 7 · depth 10 - The two orders of a differential at a place agree
AlgebraicCurve.Place.ordDiff_eq_ordDifferential55 below · cited by 5 · depth 10 - Order zero at a place iff adic valuation one
AlgebraicCurve.Place.ord_eq_zero_iff_adicValuation_eq_one0 below · cited by 5 · depth 10 - Integral with integral inverse over K[j] implies order zero
AlgebraicCurve.Place.ord_eq_zero_of_isIntegral_adjoin1 below · cited by 12 · depth 10 - Elements of a place's valuation ring have nonnegative order
AlgebraicCurve.Place.ord_nonneg_of_mem0 below · cited by 108 · depth 10 - Order at a place is invariant under nonzero constants
AlgebraicCurve.Place.ord_smul_of_ne_zero0 below · cited by 24 · depth 10 - Multiplicativity of the ramification index in a tower
AlgebraicCurve.Place.ramificationIndexAlong_comp1 below · cited by 22 · depth 10 - e(w∣ v) equals the ramification index of the fibre centre
AlgebraicCurve.Place.ramificationIndex_eq_ramificationIdx_fiberCenter1 below · cited by 9 · depth 10 - Fundamental identity sum_w e_w f_w = [F':F] along an embedding
AlgebraicCurve.Place.sum_ramificationIndexAlong_mul_inertiaDegAlong6 below · cited by 24 · depth 10 - Fundamental identity sum_{w∣ v} e f = [F':F]
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg5 below · cited by 35 · depth 10 - Bi-fibre count over a linearly disjoint compositum
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_bifiber17 below · cited by 1 · depth 10 - Fundamental inequality sum_w e_w f_w ≤ [F':F]
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_le_finrank2 below · cited by 14 · depth 10 - dt≠ 0 for t of nonzero order at a place
AlgebraicCurve.Place.D_ne_zero_of_ord_ne_zero5 below · cited by 7 · depth 11 - Uniqueness of the coefficient against dt
AlgebraicCurve.Place.diffCoeff_smul_D_eq5 below · cited by 8 · depth 11 - The chosen coefficient reproduces ω: diffCoeff(t,ω) dt=ω
AlgebraicCurve.Place.diffCoeff_smul_D_of_ord_ne_zero5 below · cited by 5 · depth 11 - Nonvanishing of the value at a rational place of a function of order zero
AlgebraicCurve.Place.evalAt_ne_zero1 below · cited by 47 · depth 11 - Galois transitivity on places above a fixed place
AlgebraicCurve.Place.exists_algEquiv_smul_eq_of_restrict_eq0 below · cited by 16 · depth 11 - Places extend along finite separable extensions
AlgebraicCurve.Place.exists_comap_eq_toValuationSubring0 below · cited by 4 · depth 11 - Discrete order functions trivial on K come from places
AlgebraicCurve.Place.exists_of_orderMap0 below · cited by 17 · depth 11 - Ramification index: ord_w = ecdotordᵥ on F
AlgebraicCurve.Place.exists_ord_algebraMap_eq_mul_ord0 below · cited by 3 · depth 11 - Existence of a uniformiser at a place
AlgebraicCurve.Place.exists_ord_eq_one0 below · cited by 39 · depth 11 - Hahn-series embedding with bounded ramification yields a place
AlgebraicCurve.Place.exists_ord_mul_eq_order_of_hasRamBound0 below · cited by 9 · depth 11 - Restriction of a place along an integral extension
AlgebraicCurve.Place.exists_toValuationSubring_eq_comap0 below · cited by 1 · depth 11 - Finiteness of residue degree in a finite extension
AlgebraicCurve.Place.finite_residueField_of_finiteDimensional0 below · cited by 5 · depth 11 - Inertia degree is constant on a Galois fibre
AlgebraicCurve.Place.inertiaDeg_eq_of_restrict_eq2 below · cited by 3 · depth 11 - Maximal ideal of a place in terms of ordᵥ
AlgebraicCurve.Place.mk_mem_maximalIdeal_iff0 below · cited by 15 · depth 11 - Positivity of ramification indices along an integral map
AlgebraicCurve.Place.one_le_ramificationIndexAlong0 below · cited by 43 · depth 11 - Order of a differential is independent of the uniformiser
AlgebraicCurve.Place.ordDiff_eq_ord_diffCoeff16 below · cited by 3 · depth 11 - Strict ultrametric equality: ord(f+g)=ord f
AlgebraicCurve.Place.ord_add_eq_of_lt0 below · cited by 61 · depth 11 - Constants have order of vanishing zero at every place
AlgebraicCurve.Place.ord_algebraMap0 below · cited by 143 · depth 11 - Regularity of df/dt at a place with uniformiser t
AlgebraicCurve.Place.ord_diffCoeff_D_nonneg9 below · cited by 5 · depth 11 - Bounded Puiseux denominators bound ramification over a closed point
AlgebraicCurve.Place.ord_dvd_of_forall_hahnSeries_embedding_hasRamBound25 below · cited by 4 · depth 11 - Unramifiedness from integral Puiseux exponents at a simple root
AlgebraicCurve.Place.ord_eq_one_of_forall_hahnSeries_embedding_hasRamBound_one25 below · cited by 1 · depth 11 - Natural-number constants have order zero at a place
AlgebraicCurve.Place.ord_natCast1 below · cited by 3 · depth 11 - Invariance of ordᵥ under negation
AlgebraicCurve.Place.ord_neg1 below · cited by 23 · depth 11 - Valuation of a norm as a sum over the fibre
AlgebraicCurve.Place.ord_norm_eq_sum_fiberOver2 below · cited by 14 · depth 11 - Ramification index is constant on a Galois fibre
AlgebraicCurve.Place.ramificationIndex_eq_of_restrict_eq3 below · cited by 5 · depth 11 - Transitivity of restriction of places along a tower
AlgebraicCurve.Place.restrictAlong_restrictAlong0 below · cited by 22 · depth 11 - Surjectivity of restriction of places along a finite separable map
AlgebraicCurve.Place.restrictAlong_surjective4 below · cited by 30 · depth 11 - F'-automorphisms fix restrictions of places to F'
AlgebraicCurve.Place.restrict_ofAlgAut_smul0 below · cited by 8 · depth 11 - Restriction of places along a commuting square
AlgebraicCurve.Place.smul_restrictAlong0 below · cited by 15 · depth 11 - Fundamental equality sum e(w|v)f(w|v)=[F':F] for places
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_fiberOver2 below · cited by 13 · depth 11 - Nonzero order at a place forces transcendence
AlgebraicCurve.Place.transcendental_of_ord_ne_zero0 below · cited by 33 · depth 11 - Places with finite residue field over an algebraically closed base have degree one
AlgebraicCurve.Place.deg_eq_one_of_isAlgClosed_of_finite0 below · cited by 5 · depth 12 - Uniqueness of a place with ramification index exceeding half the degree
AlgebraicCurve.Place.eq_of_finrank_lt_two_mul_ramificationIndex4 below · cited by 2 · depth 12 - Evaluation of a constant at a place returns the constant
AlgebraicCurve.Place.evalAt_algebraMap0 below · cited by 118 · depth 12 - Values at a place agree for functions congruent modulo mathfrak mᵥ
AlgebraicCurve.Place.evalAt_congr1 below · cited by 46 · depth 12 - Value of an inverse at a rational place
AlgebraicCurve.Place.evalAt_inv2 below · cited by 25 · depth 12 - Evaluation at a rational place is multiplicative
AlgebraicCurve.Place.evalAt_mul0 below · cited by 81 · depth 12 - Evaluation at a rational place respects integer powers
AlgebraicCurve.Place.evalAt_zpow3 below · cited by 13 · depth 12 - Places of a constant-field extension centred at K-embeddings
AlgebraicCurve.Place.existsUnique_valuation_sub_lt_one_of_constantFieldExtension53 below · cited by 5 · depth 12 - Degree-one places arise from embeddings into K((T))
AlgebraicCurve.Place.exists_algHom_laurentSeries_order_eq_ord0 below · cited by 2 · depth 12 - Unique place above P in a constant field extension
AlgebraicCurve.Place.exists_comap_algebraMap_eq_of_constantFieldExtension_of_isAlgClosed7 below · cited by 16 · depth 12 - Finitely many places descend to a finite constant field
AlgebraicCurve.Place.exists_finite_constantField_form_fiberConstants_eq_singleton49 below · cited by 1 · depth 12 - Prescribed unit values at finitely many places
AlgebraicCurve.Place.exists_forall_mem_hasValue11 below · cited by 7 · depth 12 - Divisibility of [P-Q] modulo n on a curve
AlgebraicCurve.Place.exists_natCast_dvd_ord_sub_single_sub_single259 below · cited by 1 · depth 12 - Finitely many places of each degree over a finite field
AlgebraicCurve.Place.finite_setOf_deg_eq2 below · cited by 4 · depth 12 - The w-adic valuation is equivalent to its place's valuation
AlgebraicCurve.Place.isEquiv_adicValuation_ofHeightOneSpectrum0 below · cited by 1 · depth 12 - Membership in a place's valuation ring via v≤ 1
AlgebraicCurve.Place.mem_iff_adicValuation_le_one0 below · cited by 12 · depth 12 - Maximal ideal of a place: v-valuation <1
AlgebraicCurve.Place.mem_maximalIdeal_iff_adicValuation_lt_one0 below · cited by 4 · depth 12 - A function with no zeros or poles is constant
AlgebraicCurve.Place.mem_range_algebraMap_of_forall_ord_eq_zero11 below · cited by 9 · depth 12 - Ultrametric inequality for ord at a place
AlgebraicCurve.Place.min_ord_le_ord_add0 below · cited by 47 · depth 12 - Regularity of g dJ from ord(g⁶J⁴(J-1728)³)≥ 0
AlgebraicCurve.Place.ordDiff_smul_D_nonneg_of_ord_pow_six_mul_pow_four_mul_sub_1728_pow_three_nonneg20 below · cited by 4 · depth 12 - Puiseux bound d forces ord_W(p)∣ d
AlgebraicCurve.Place.ord_dvd_of_hahnSeries_embedding_of_isGalois18 below · cited by 1 · depth 12 - Pole orders forced by the modular polynomial's support
AlgebraicCurve.Place.ord_eq_mul_or_eq_mul_of_modular_support0 below · cited by 2 · depth 12 - Laurent embedding over a simple root forces ord_W p = 1
AlgebraicCurve.Place.ord_eq_one_of_hahnSeries_embedding_of_isGalois18 below · cited by 1 · depth 12 - Nonnegative order propagates to elements integral over K[j]
AlgebraicCurve.Place.ord_nonneg_of_isIntegral_adjoin_of_ord_nonneg3 below · cited by 17 · depth 12 - Order at the place of w is nonzero exactly on w
AlgebraicCurve.Place.ord_ofHeightOneSpectrum_ne_zero_iff0 below · cited by 4 · depth 12 - Unramifiedness of degree-p Kummer covers at rational places
AlgebraicCurve.Place.ramificationIndex_eq_one_of_forall_dvd_ord7 below · cited by 1 · depth 12 - Unramifiedness of places in a constant field extension
AlgebraicCurve.Place.ramificationIndex_forgetConstants_eq_one_of_isConstantFieldExtension2 below · cited by 20 · depth 12 - The adic valuation of a place is rank one discrete
AlgebraicCurve.Place.adicValuation_isRankOneDiscrete0 below · cited by 1 · depth 13 - The adic valuation of a place is trivial on the base field
AlgebraicCurve.Place.adicValuation_isTrivialOn0 below · cited by 1 · depth 13 - Valuation subring of a place's adic valuation
AlgebraicCurve.Place.adicValuation_valuationSubring0 below · cited by 1 · depth 13 - Places of a constant-field extension are determined by their centre
AlgebraicCurve.Place.eq_of_forall_valuation_sub_lt_one_of_constantFieldExtension52 below · cited by 1 · depth 13 - Weak approximation with prescribed exact error valuations
AlgebraicCurve.Place.exists_forall_adicValuation_sub_eq8 below · cited by 2 · depth 13 - Divisibility of the class of P-Q over ℂ
AlgebraicCurve.Place.exists_natCast_dvd_ord_sub_single_sub_single_complex199 below · cited by 1 · depth 13 - Nontrivial valuation subrings over the constants are places
AlgebraicCurve.Place.exists_of_valuationSubring0 below · cited by 15 · depth 13 - Places from K-algebra embeddings into Laurent series
AlgebraicCurve.Place.exists_ord_mul_eq_order_of_algHom_laurentSeries1 below · cited by 10 · depth 13 - A transcendental element has a pole at some place
AlgebraicCurve.Place.exists_ord_neg_of_transcendental10 below · cited by 12 · depth 13 - Pullback of a proper valuation subring is a place
AlgebraicCurve.Place.exists_toValuationSubring_eq_comap_ringHom_of_isSeparable0 below · cited by 14 · depth 13 - Elements of order one at a place are separating
AlgebraicCurve.Place.isSeparable_adjoin_of_ord_eq_one2 below · cited by 23 · depth 13 - Order of a differential scales: ordᵥ(gω)=ordᵥ g+ordᵥω
AlgebraicCurve.Place.ordDiff_smul7 below · cited by 5 · depth 13 - Additivity of ordDiff under scaling over a perfect field
AlgebraicCurve.Place.ordDiff_smul_of_perfectField8 below · cited by 12 · depth 13 - Order of the zero differential at a place vanishes
AlgebraicCurve.Place.ordDiff_zero_of_perfectField5 below · cited by 2 · depth 13 - At new places of a constant field extension, dx has order 0
AlgebraicCurve.Place.ordDifferential_D_eq_zero_of_constantFieldExtension_of_forall_mem5 below · cited by 1 · depth 13 - Differentiation by a uniformiser preserves the valuation ring
AlgebraicCurve.Place.ord_diffCoeff_D_nonneg_of_perfectField8 below · cited by 20 · depth 13 - Lower bound ordᵥ(f)-1leordDiffᵥ(df) over a perfect field
AlgebraicCurve.Place.ord_sub_one_le_ordDiff_D_of_perfectField15 below · cited by 7 · depth 13 - Ramification indices along a commuting square of places
AlgebraicCurve.Place.ramificationIndexAlong_smul0 below · cited by 5 · depth 13 - Restriction along σ is the action of σ⁻¹
AlgebraicCurve.Place.restrictAlong_algEquiv_eq_ofAlgAut_symm_smul0 below · cited by 8 · depth 13 - Restriction along a Frobenius-type endomorphism is a twist
AlgebraicCurve.Place.restrictAlong_eq_smul_of_forall_eq_inv_smul_pow0 below · cited by 5 · depth 13 - A uniformiser at a place has nonzero differential
AlgebraicCurve.Place.D_ne_zero_of_ord_eq_one4 below · cited by 10 · depth 14 - Places of a function field in one variable have nonzero degree
AlgebraicCurve.Place.deg_ne_zero_of_finiteDimensional_adjoin12 below · cited by 21 · depth 14 - Uniformiser coordinate of a differential over a perfect field
AlgebraicCurve.Place.diffCoeff_smul_D_eq_of_ord_eq_one4 below · cited by 10 · depth 14 - A uniformiser's differential spans Ω_{F/K} over a perfect field
AlgebraicCurve.Place.diffCoeff_smul_D_of_ord_eq_one4 below · cited by 15 · depth 14 - Laurent expansion at a place of degree one
AlgebraicCurve.Place.exists_algHom_laurentSeries_of_deg_eq_one0 below · cited by 2 · depth 14 - Places of a complex function field form a compact Riemann surface
AlgebraicCurve.Place.exists_chartedSpace_meromorphicOrderAt_evalAt_eq_ord_complex38 below · cited by 6 · depth 14 - Weak approximation at finitely many places of F/K
AlgebraicCurve.Place.exists_forall_adicValuation_sub_le7 below · cited by 2 · depth 14 - Prescribing orders at finitely many places
AlgebraicCurve.Place.exists_forall_ord_eq_finset3 below · cited by 10 · depth 14 - No poles above v implies integrality over mathcal Oᵥ
AlgebraicCurve.Place.exists_integralClosureAt_of_ord_fiber_nonneg6 below · cited by 13 · depth 14 - Rational place over a rational place has inertia degree one
AlgebraicCurve.Place.inertiaDeg_eq_one_of_isRational1 below · cited by 23 · depth 14 - Embeddings inducing a given place counted by the ramification index
AlgebraicCurve.Place.ncard_algHom_comp_eq_preimage_eq_ramificationIndexAlong47 below · cited by 1 · depth 14 - Differentiation with respect to a uniformiser preserves 𝒪ᵥ
AlgebraicCurve.Place.ord_diffCoeff_D_nonneg_of_isSeparable5 below · cited by 7 · depth 14 - Order of a norm equals order of the polynomial value
AlgebraicCurve.Place.ord_norm_sub_eq_ord_eval0 below · cited by 7 · depth 14 - Fibre degree formula for a constant-field extension
AlgebraicCurve.Place.sum_deg_fiberConstants_eq_deg_of_isCurveOver46 below · cited by 1 · depth 14 - Rational fibres: sum_{w∣ v} e(w∣ v) = [F':F]
AlgebraicCurve.Place.sum_ramificationIndex_eq_finrank8 below · cited by 12 · depth 14 - Regular at a place implies analytic in the chart
AlgebraicCurve.Place.analyticAt_evalAt_extChartAt_symm_of_mem0 below · cited by 18 · depth 15 - Connectedness of the place space of a complex function field
AlgebraicCurve.Place.connectedSpace_of_chartedSpace_of_meromorphicOrderAt_eq_ord_complex28 below · cited by 1 · depth 15 - Multiplicity of the fibre centre in (c) equals ord_w(c)
AlgebraicCurve.Place.count_normalizedFactors_span_singleton5 below · cited by 1 · depth 15 - Truncated Cauchy inverse system determines Taylor coefficients of s⁻¹
AlgebraicCurve.Place.eq_taylorCoeff_inv_of_forall_sum_antidiagonal_eq12 below · cited by 2 · depth 15 - Evaluation of a base function at a place and at its restriction
AlgebraicCurve.Place.evalAt_algebraMap_eq_evalAt_restrict0 below · cited by 25 · depth 15 - Value of a norm as a weighted product over the fibre
AlgebraicCurve.Place.evalAt_norm_eq_prod_fiber14 below · cited by 5 · depth 15 - Finitely many parameter discs cover every place
AlgebraicCurve.Place.exists_finset_forall_exists_localParam_eq_complex0 below · cited by 2 · depth 15 - Places extend along separable constant field extensions
AlgebraicCurve.Place.exists_forgetConstants_restrict_eq_of_isConstantFieldExtension2 below · cited by 17 · depth 15 - A common denominator clearing integral elements into d(j)bᵢ
AlgebraicCurve.Place.exists_gram_denominator_of_mulTable0 below · cited by 1 · depth 15 - Local analytic parametrisation at a place of a complex function field
AlgebraicCurve.Place.exists_localParam_meromorphicOrderAt_evalAt_eq_ord_complex4 below · cited by 2 · depth 15 - Existence of a uniformizing, separating linear form
AlgebraicCurve.Place.exists_uniformizing_separating_form1 below · cited by 1 · depth 15 - Vanishing of the first e Taylor coefficients means ordᵥ f ≥ e
AlgebraicCurve.Place.forall_lt_taylorCoeff_eq_zero_iff_le_ord1 below · cited by 7 · depth 15 - Values at places are transported by automorphisms
AlgebraicCurve.Place.hasValue_smul_iff0 below · cited by 6 · depth 15 - Uniqueness of the analytic branch through a complex place
AlgebraicCurve.Place.localParam_eventually_eq_comp_evalAt_complex0 below · cited by 1 · depth 15 - Taylor expansion commutes with multivariate polynomial expressions
AlgebraicCurve.Place.mk_taylorCoeff_aeval12 below · cited by 2 · depth 15 - Taylor expansion of z in the parameter z-z(v) is z(v)+T
AlgebraicCurve.Place.mk_taylorCoeff_eq_C_add_X3 below · cited by 8 · depth 15 - Order of a pulled-back differential at a tame place
AlgebraicCurve.Place.ordDiff_pullbackDiff_of_natCast_ramificationIndexAlong_ne_zero16 below · cited by 4 · depth 15 - Algebraic elements have order zero at every place
AlgebraicCurve.Place.ord_eq_zero_of_isAlgebraic0 below · cited by 18 · depth 15 - Hilbert decomposition over a rational place of K(t)
AlgebraicCurve.Place.ord_restrictAlong_eq_natCard_algHom_of_isGalois28 below · cited by 23 · depth 15 - No cancellation of K'-weighted F-sums at lifted places
AlgebraicCurve.Place.ord_sum_algebraMap_mul_le_ord_of_linearIndependent_of_constantFieldExtension2 below · cited by 1 · depth 15 - Fibres of places over distinct base places are disjoint
AlgebraicCurve.Place.pairwiseDisjoint_fiber0 below · cited by 2 · depth 15 - Chart reading of h dg as Rᵥh·(Rᵥg)'
AlgebraicCurve.Place.readDifferential_smul_D_eventuallyEq_chartRead_mul_deriv6 below · cited by 6 · depth 15 - Residue field of a place in a separable constant field extension
AlgebraicCurve.Place.residueField_eq_compositum_of_isConstantFieldExtension0 below · cited by 2 · depth 15 - Wild lower bound for the different exponent
AlgebraicCurve.Place.sub_one_add_pow_sub_one_le_ordDiff_D_of_isGalois40 below · cited by 3 · depth 15 - Bi-fibre degree count with graph defect for a compositum
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_bifiber_defect15 below · cited by 1 · depth 15 - Two-component local exchange identity for e and f
AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_exchange_add6 below · cited by 1 · depth 15 - Additivity of Taylor coefficients at a rational place
AlgebraicCurve.Place.taylorCoeff_add2 below · cited by 13 · depth 15 - Taylor coefficients of a constant at a place
AlgebraicCurve.Place.taylorCoeff_algebraMap2 below · cited by 10 · depth 15 - Cauchy product rule for Taylor coefficients at a place
AlgebraicCurve.Place.taylorCoeff_mul7 below · cited by 9 · depth 15 - Homogeneity of Taylor coefficients at a rational place
AlgebraicCurve.Place.taylorCoeff_smul2 below · cited by 8 · depth 15 - Formal Taylor branch satisfies the relation re-expanded at a place
AlgebraicCurve.Place.evalEval_C_add_X_mk_taylorCoeff_eq_zero11 below · cited by 6 · depth 16 - Places realising primes of an integral model over a valuation subring
AlgebraicCurve.Place.exists_of_isPrime_over_valuationSubring0 below · cited by 7 · depth 16 - Proper valuation subrings over K are discrete
AlgebraicCurve.Place.exists_of_valuationSubring_of_isSeparable0 below · cited by 5 · depth 16 - Places extend along an integral map of function fields
AlgebraicCurve.Place.exists_restrictAlong_eq_of_finiteDimensional_fieldRange4 below · cited by 7 · depth 16 - Inertia degree of a place equals that of its centre
AlgebraicCurve.Place.inertiaDeg_eq_inertiaDeg_fiberCenter1 below · cited by 2 · depth 16 - Order at w versus powers of the fibre prime
AlgebraicCurve.Place.le_ord_iff_mem_pow_fiberCenter3 below · cited by 2 · depth 16 - Taylor expansion commutes with bivariate polynomial evaluation
AlgebraicCurve.Place.mk_taylorCoeff_evalEval9 below · cited by 6 · depth 16 - Residue degree of a place under a finite extension is finite
AlgebraicCurve.Place.module_finite_residueField_restrict0 below · cited by 2 · depth 16 - Places above v in an algebraically closed constant extension
AlgebraicCurve.Place.natCard_setOf_comap_eq_eq_deg_of_linearDisjoint4 below · cited by 3 · depth 16 - Trace preserves pole-order bounds at a place
AlgebraicCurve.Place.neg_le_ord_trace_of_forall_le_ord2 below · cited by 4 · depth 16 - The zero differential has order 0 at every place
AlgebraicCurve.Place.ordDiff_zero5 below · cited by 3 · depth 16 - Norm has trivial order at v when f is a unit above v
AlgebraicCurve.Place.ord_norm_eq_zero_of_forall_fiber3 below · cited by 1 · depth 16 - A place restricts to w once its ring contains w's
AlgebraicCurve.Place.restrict_eq_of_forall_mem0 below · cited by 2 · depth 16 - Shift of Taylor coefficients along a uniformiser
AlgebraicCurve.Place.taylorCoeff_succ_eq_taylorCoeff_taylorRem_one1 below · cited by 3 · depth 16 - Additivity of Taylor remainders at a rational place
AlgebraicCurve.Place.taylorRem_add1 below · cited by 1 · depth 16 - Taylor remainder when the lower coefficients vanish
AlgebraicCurve.Place.taylorRem_eq_mul_inv_pow_of_forall_taylorCoeff_eq_zero0 below · cited by 3 · depth 16 - Taylor remainders along a uniformiser stay in the valuation ring
AlgebraicCurve.Place.taylorRem_mem_toValuationSubring0 below · cited by 5 · depth 16 - Taylor remainders at a rational place are K-homogeneous
AlgebraicCurve.Place.taylorRem_smul1 below · cited by 1 · depth 16 - Strict trace bound along places over a fixed place
AlgebraicCurve.Place.trace_eq_zero_or_neg_add_one_le_ord_trace_of_forall_le_ord0 below · cited by 2 · depth 16 - Root–fibre criterion for partial G/partial Y≠ 0, separable case
AlgebraicCurve.Place.derivative_evalEval_evalAt_ne_zero_of_ord_sub_eq_one_of_forall_evalAt_ne_of_isSeparable8 below · cited by 3 · depth 17 - Uniqueness of places above a rational place
AlgebraicCurve.Place.eq_of_comap_toValuationSubring_eq_of_isRational0 below · cited by 2 · depth 17 - Vanishing at a rational place means order at least one
AlgebraicCurve.Place.evalAt_eq_zero_iff_one_le_ord4 below · cited by 32 · depth 17 - Degree-one places: integers are constants modulo 𝔪
AlgebraicCurve.Place.exists_sub_algebraMap_mem_maximalIdeal1 below · cited by 4 · depth 17 - Separability of F/K(t) when ordᵥ t is tame
AlgebraicCurve.Place.isSeparable_adjoin_of_ord_ne_zero_of_cast_natAbs_ne_zero_divisorClassGroup3 below · cited by 1 · depth 17 - Trace at a place: Tr(g)(x)=sum_{y∣ x}e(y∣ x) g(y)
AlgebraicCurve.Place.mem_and_evalAt_trace_eq_sum_ramificationIndexAlong_smul_evalAt9 below · cited by 2 · depth 17 - Trace preserves pole-order bounds at a place
AlgebraicCurve.Place.neg_le_ord_trace_of_forall_le_ord_of_isCurveOver2 below · cited by 2 · depth 17 - Trace floor with the different gain at a place
AlgebraicCurve.Place.neg_le_ord_trace_of_forall_le_ord_sub_pred2 below · cited by 2 · depth 17 - Order at a place equals valuation at its fibre centre
AlgebraicCurve.Place.neg_log_valuation_fiberCenter_eq_ord2 below · cited by 1 · depth 17 - Over a perfect base field, ordDiffᵥ=ordDifferentialᵥ
AlgebraicCurve.Place.ordDiff_eq_ordDifferential_of_perfectField16 below · cited by 3 · depth 17 - Kummer covers are totally ramified when ord u is prime to n
AlgebraicCurve.Place.ramificationIndexAlong_eq_of_pow_eq_of_isCoprime_ord7 below · cited by 7 · depth 17 - Kummer covers are unramified at places where u is a unit
AlgebraicCurve.Place.ramificationIndexAlong_eq_one_of_pow_eq_of_mem_of_inv_mem1 below · cited by 6 · depth 17 - Places and orders transport along inclusions between equal intermediate fields
AlgebraicCurve.Place.restrictAlong_inclusion_of_le_of_le0 below · cited by 1 · depth 17 - Shift of Taylor remainders at a place
AlgebraicCurve.Place.taylorRem_succ_eq_taylorRem_taylorRem_one0 below · cited by 1 · depth 17 - Places of F/K over a Dedekind order are determined by their centre
AlgebraicCurve.Place.eq_of_forall_coe_mem_nonunits_iff_of_isDedekindDomain0 below · cited by 2 · depth 18 - Uniqueness of the order function of a place
AlgebraicCurve.Place.eq_ord_of_addHom_of_nonneg_iff1 below · cited by 1 · depth 18 - A function with a pole at exactly one place
AlgebraicCurve.Place.exists_not_mem_and_forall_ne_mem27 below · cited by 1 · depth 18 - Elements of tame nonzero order are separating
AlgebraicCurve.Place.isSeparable_adjoin_of_ord_ne_zero_of_cast_natAbs_ne_zero2 below · cited by 3 · depth 18 - Regularity of roots of a plane relation at rational places
AlgebraicCurve.Place.mem_toValuationSubring_of_evalEval_eq_zero_of_eval_leadingCoeff_ne_zero0 below · cited by 2 · depth 18 - Order of df at a place, tame case
AlgebraicCurve.Place.ordDiff_D_eq_ord_sub_algebraMap_sub_one_of_cast_natAbs_ne_zero16 below · cited by 2 · depth 18 - Valuation of a norm as a sum over places above v
AlgebraicCurve.Place.ord_norm_eq_sum_fiberOver_of_isSeparable3 below · cited by 4 · depth 18 - Rational places are unramified in separable Kummer extensions
AlgebraicCurve.Place.ramificationIndex_eq_one_of_forall_dvd_ord_of_isSeparable8 below · cited by 1 · depth 18 - First non-vanishing Taylor coefficient equals the value of f t^{-e}
AlgebraicCurve.Place.taylorCoeff_eq_evalAt_mul_inv_pow_of_forall_taylorCoeff_eq_zero1 below · cited by 2 · depth 18 - Non-vanishing of the leading Taylor coefficient at a place
AlgebraicCurve.Place.taylorCoeff_ord_ne_zero2 below · cited by 1 · depth 18 - Continuity of place restriction along an integral map
AlgebraicCurve.Place.continuous_restrictAlong5 below · cited by 2 · depth 19 - Chain rule: δ f = (df/dt) δ t for a uniformiser t
AlgebraicCurve.Place.derivation_apply_eq_diffCoeff_D_mul5 below · cited by 1 · depth 19 - Uniqueness of the formal branch through a simple root
AlgebraicCurve.Place.eq_map_mk_taylorCoeff_of_evalEval_C_add_X_eq_zero13 below · cited by 1 · depth 19 - Evaluation at a rational place commutes with constant field extension
AlgebraicCurve.Place.evalAt_map_eq_of_comap_eq0 below · cited by 6 · depth 19 - Values of a trace at a rational place
AlgebraicCurve.Place.evalAt_trace_eq_sum_fiber9 below · cited by 1 · depth 19 - Order-one combinations have an order-one summand
AlgebraicCurve.Place.exists_ord_eq_one_of_ord_sum_eq_one0 below · cited by 1 · depth 19 - Every constant value of a transcendental element is attained at some place
AlgebraicCurve.Place.exists_ord_sub_algebraMap_pos_of_transcendental2 below · cited by 5 · depth 19 - Realising a prescribed place of M/K under an embedding sending t ↦ j
AlgebraicCurve.Place.exists_valuationSubring_algHom_apply_eq_forall_mem_iff2 below · cited by 1 · depth 19 - Residue of t df/f equals ordᵥ f
AlgebraicCurve.Place.mul_diffCoeff_D_div_mem_and_evalAt_eq_intCast_ord13 below · cited by 2 · depth 19 - Valuation at w as zero order of the chart-read function
AlgebraicCurve.Place.ord_sub_algebraMap_eq_analyticOrderNatAt_chartRead1 below · cited by 1 · depth 19 - Transfer of a uniformiser across a separable plane relation
AlgebraicCurve.Place.ord_sub_algebraMap_evalAt_eq_one_of_derivative_evalEval_ne_zero_of_ord_sub_algebraMap_evalAt_eq_one2 below · cited by 1 · depth 19 - Simple-pole expansion of a differential in a chart at a place
AlgebraicCurve.Place.readDifferential_eventuallyEq_div_add_of_ordDifferential3 below · cited by 1 · depth 19 - Unramified reading of f gives an analytic local coordinate
AlgebraicCurve.Place.analyticCoord_of_agree1 below · cited by 1 · depth 20 - Analyticity of a local coordinate in every chart of its domain
AlgebraicCurve.Place.analyticCoord_of_center0 below · cited by 1 · depth 20 - Uniqueness of canonical local residue data at a rational place
AlgebraicCurve.Place.canonicalLocalResidueDataK_eq6 below · cited by 1 · depth 20 - Unramified, fibre-separating places give simple points of G
AlgebraicCurve.Place.derivative_evalEval_evalAt_ne_zero_of_ord_sub_eq_one_of_forall_evalAt_ne7 below · cited by 2 · depth 20 - Non-vanishing of a_d(z₀) from regularity on a fibre
AlgebraicCurve.Place.eval_leadingCoeff_ne_zero_of_forall_mem_toValuationSubring31 below · cited by 2 · depth 20 - Unique place centred at a smooth point of a plane curve
AlgebraicCurve.Place.existsUnique_sub_algebraMap_mem_nonunits_of_eval_pderiv_ne_zero0 below · cited by 2 · depth 20 - Moving lemma for places of a function field
AlgebraicCurve.Place.exists_isIntegral_adjoin_and_ord_eq_one_and_forall_ord_eq_zero0 below · cited by 1 · depth 20 - Degree-one places extend to a constant field extension
AlgebraicCurve.Place.exists_place_laurentBaseChange_of_deg_eq_one172 below · cited by 4 · depth 20 - Divisibility of y-y(v) by z-z(v) at a simple point
AlgebraicCurve.Place.exists_sub_algebraMap_evalAt_eq_mul_of_derivative_evalEval_ne_zero0 below · cited by 1 · depth 20 - HasValue criterion via integrality and order of g-c
AlgebraicCurve.Place.hasValue_iff_mem_and_eq_or_ord_sub_pos0 below · cited by 38 · depth 20 - Value of a norm at a place below a separable covering
AlgebraicCurve.Place.hasValue_norm_along_of_separableAlong5 below · cited by 1 · depth 20 - Value a at a rational place versus ordᵥ(f-a)>0
AlgebraicCurve.Place.mem_and_evalAt_eq_iff_ord_sub_algebraMap_pos0 below · cited by 13 · depth 20 - Differentials of functions regular at v are regular
AlgebraicCurve.Place.ordDiff_D_nonneg16 below · cited by 2 · depth 20 - Fibres of a transcendental function have degree [F:ℂ(x)]
AlgebraicCurve.Place.sum_fiber_ord_eq_finrank20 below · cited by 2 · depth 20 - Riemann–Hurwitz formula in terms of the values of x
AlgebraicCurve.Place.sum_ramification_evalAt_eq110 below · cited by 1 · depth 20 - Every K-differential of F is g dt when ordᵥ t ≠ 0
AlgebraicCurve.Place.exists_eq_smul_D_of_ord_ne_zero5 below · cited by 1 · depth 21 - Unramified fibre count equals the degree over ℂ(f)
AlgebraicCurve.Place.card_fiber_eq_finrank_adjoin_of_ord_eq_one10 below · cited by 1 · depth 22 - Transport of places along compatible isomorphisms K≃ K', F≃ F'
AlgebraicCurve.Place.exists_equiv_comap_eq_and_ord_eq_and_deg_eq_of_ringEquiv0 below · cited by 1 · depth 22 - A non-constant function with poles only at a prescribed place
AlgebraicCurve.Place.exists_not_mem_range_and_forall_ne_ord_nonneg28 below · cited by 10 · depth 22 - Rationality of the place at a k-point of an integral scheme
AlgebraicCurve.Place.isRational_of_range_stalk_section_eq0 below · cited by 2 · depth 22 - Places are determined by containment of their valuation rings
AlgebraicCurve.Place.eq_of_toValuationSubring_le0 below · cited by 3 · depth 23 - Equivariance of evaluation at a place under semilinear automorphisms
AlgebraicCurve.Place.evalAt_smul_smul_eq_baseAut_evalAt0 below · cited by 7 · depth 23 - Local subring at a rational place versus its localisation
AlgebraicCurve.Place.forall_localSubring_iff_forall_localization0 below · cited by 3 · depth 23 - Order zero for elements integral over K[1/j]_{(1/j)} and inverse
AlgebraicCurve.Place.ord_eq_zero_of_not_mem_of_eval_monic_eq_zero_of_coeff_eq_aeval_inv_div1 below · cited by 1 · depth 23 - Almost all places of a curve have full fibre
AlgebraicCurve.Place.finite_setOf_card_fiberAlong_ne_finrankAlong122 below · cited by 3 · depth 24 - Holomorphy rings off a non-empty finite set of places
AlgebraicCurve.Place.isDedekindDomain_iInf_toSubring_and_isMaximal_iff_of_finset_nonempty92 below · cited by 3 · depth 24 - Membership in a restricted valuation ring via ord
AlgebraicCurve.Place.mem_comap_iff_ord_nonneg0 below · cited by 2 · depth 24 - Ramification index along a cover equals ord_w(φ(X)-a)
AlgebraicCurve.Place.ramificationIndexAlong_eq_ord_sub_of_restrictAlong_eq_placeOfPoint38 below · cited by 1 · depth 24 - Two images with almost-everywhere separating restrictions generate F'
AlgebraicCurve.Place.subfieldClosure_range_union_eq_top_of_restrictAlong_injOn116 below · cited by 2 · depth 24 - Fibre size bounded by degree, with equality iff unramified
AlgebraicCurve.Place.card_fiberAlong_le_finrankAlong_and_iff8 below · cited by 1 · depth 25 - All but finitely many fibres of x are unramified
AlgebraicCurve.Place.exists_finset_forall_ord_sub_algebraMap_eq_one_of_ord_pos110 below · cited by 4 · depth 25 - Places over ℚ̄ have open Galois stabilisers
AlgebraicCurve.Place.exists_intermediateField_finiteDimensional_forall_smul_eq_of_descent5 below · cited by 1 · depth 25 - Adapted primitive element at a place of a curve
AlgebraicCurve.Place.exists_isIntegral_adjoin_eq_top_ord_sub_algebraMap_eq_one5 below · cited by 1 · depth 25 - Proper valuation subrings containing K are places
AlgebraicCurve.Place.exists_of_valuationSubring_of_finiteDimensional1 below · cited by 22 · depth 25 - Inertia degree one over an algebraically closed base field
AlgebraicCurve.Place.inertiaDegAlong_eq_one_of_isAlgClosed0 below · cited by 2 · depth 25 - A uniformizer has non-zero differential over a perfect constant field
AlgebraicCurve.Place.kaehlerD_ne_zero_of_ord_eq_one5 below · cited by 9 · depth 25 - Ramification in tame Kummer extensions of a function field
AlgebraicCurve.Place.ramificationIndex_eq_div_gcd_natAbs_ord_of_isSplittingField_X_pow_sub_C7 below · cited by 1 · depth 25 - Transport of places along an isomorphism commutes with restriction
AlgebraicCurve.Place.restrictAlong_congrEquiv_and_existsUnique_iff0 below · cited by 4 · depth 25 - Unique place above a rational valuation ring, and its rationality
AlgebraicCurve.Place.existsUnique_forall_mem_iff_adjoin_sup_of_isAlgebraic_of_forall_exists_sub_mem2 below · cited by 1 · depth 26 - Rationality of places with finite residue over an algebraically closed field
AlgebraicCurve.Place.exists_sub_algebraMap_mem_nonunits_of_isAlgClosed0 below · cited by 2 · depth 26 - Finiteness of poles and of places with prescribed values
AlgebraicCurve.Place.finite_setOf_not_mem_toValuationSubring_or_evalAt_mem0 below · cited by 2 · depth 26 - Ramification and inertia in a tower via place stabilisers
AlgebraicCurve.Place.ramificationIndex_mul_inertiaDeg_mul_natCard_stabilizer_eq_natCard_stabilizer1 below · cited by 1 · depth 26 - Canonical residue as t⁻¹-coefficient of a Laurent expansion
AlgebraicCurve.Place.algebraMap_coeff_neg_one_eq_localResidue_mul_differentialCoeff_D2 below · cited by 1 · depth 27 - Extending a valuation ring of F₁ to a place of F/L
AlgebraicCurve.Place.exists_forall_mem_iff_adjoin_sup_of_linearDisjoint0 below · cited by 1 · depth 27 - Residue of dt/t is 1 at any uniformiser
AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_inv_eq_one0 below · cited by 1 · depth 28 - Vanishing residue of partialᵥ(dt) t⁻⁽ⁿ⁺¹⁾ at a rational place
AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_pow_inv_eq_zero_of_surjective_algebraMap0 below · cited by 3 · depth 28 - Local subring of a function field is dominated by a place
AlgebraicCurve.Place.exists_forall_mem_and_mem_nonunits_iff_not_isUnit_of_isLocalRing2 below · cited by 4 · depth 28 - Ramification index n transported across a constant-field square
AlgebraicCurve.Place.exists_place_comap_eq_and_ord_eq_mul_ord_of_forall_smul_maximalIdeal_map_eq_pow8 below · cited by 1 · depth 28 - Division by sections bounds a pushed-forward divisor by residue order
AlgebraicCurve.Place.mapDomain_filter_apply_le_ord_of_sections0 below · cited by 1 · depth 28 - Places above P determined by their centre on 𝒪_P[κ']
AlgebraicCurve.Place.eq_of_comap_eq_of_forall_mem_nonunits_iff1 below · cited by 3 · depth 29 - Unique unramified extension of a rational place under algebraic constant extension
AlgebraicCurve.Place.exists_comap_algebraMap_eq_of_constantFieldExtension_of_deg_eq_one_of_isAlgebraic7 below · cited by 1 · depth 29 - Extension of a valuation subring to a place of E/κ
AlgebraicCurve.Place.exists_comap_toValuationSubring_eq_of_ne_top_of_isAlgebraic2 below · cited by 3 · depth 29 - Preimage of a place along a ring homomorphism of fields
AlgebraicCurve.Place.exists_toValuationSubring_eq_comap_of_ne_top0 below · cited by 7 · depth 29 - Ramification index preserved under separable constant extension
AlgebraicCurve.Place.ord_algebraMap_eq_mul_ord_of_constant_extension4 below · cited by 1 · depth 29 - Orders unchanged in a separable constant-field extension
AlgebraicCurve.Place.ord_algebraMap_eq_ord_of_comap_eq_of_isSeparable_of_adjoin_eq_top3 below · cited by 4 · depth 29 - Constant field extension at a place with separable residue field
AlgebraicCurve.Place.ord_eq_and_sum_deg_eq_deg_of_comap_eq_of_linearDisjoint3 below · cited by 1 · depth 29 - Norm has trivial order at v when f does on the fibre
AlgebraicCurve.Place.ord_norm_eq_zero_of_forall_fiber_of_isSeparable7 below · cited by 1 · depth 29 - Fixing a rational place above P forces inertia at P
AlgebraicCurve.Place.smul_eq_and_sub_mem_nonunits_of_smul_eq_of_comap_eq0 below · cited by 1 · depth 29 - Inertia elements fix places above after separable constant extension
AlgebraicCurve.Place.smul_eq_of_comap_eq_of_forall_sub_mem_nonunits2 below · cited by 1 · depth 29 - Restriction of a place along an integral extension is proper
AlgebraicCurve.Place.comap_algebraMap_ne_top0 below · cited by 1 · depth 30 - Cartier operator divides pole orders by p
AlgebraicCurve.Place.exists_eq_smul_dCoord_and_uniformizer_pow_mul_mem_of_cartierLaws15 below · cited by 1 · depth 30 - Sequential compactness of the place space of a complex function field
AlgebraicCurve.Place.exists_strictMono_forall_tendsto_evalAt_complex6 below · cited by 1 · depth 30 - Automorphism pull-back preserves regularity, simple poles and residues
AlgebraicCurve.Place.isRegularAt_and_hasSimplePoleAt_and_hasSimpleResidue_smul_pullbackAlong_of_algEquiv52 below · cited by 2 · depth 30 - Recognising a place through a K-homomorphism via a separating test family
AlgebraicCurve.Place.mem_toValuationSubring_iff_map_mem_of_forall_place_eq_of_testFamily2 below · cited by 5 · depth 30 - Separable constant extensions are unramified: e=1
AlgebraicCurve.Place.ramificationIndex_forgetConstants_eq_one_of_adjoin_range_eq_top2 below · cited by 1 · depth 30 - Algebraic coordinates of a holomorphic map to an affine open
AlgebraicCurve.Place.existsUnique_forall_mem_toValuationSubring_and_evalAt_eq_appLE_of_differentiableAt75 below · cited by 1 · depth 31 - Dichotomy for the places missing an affine open
AlgebraicCurve.Place.forall_not_le_preimage_or_finite_setOf_of_differentiableAt_appLE_of_isSeparated1 below · cited by 2 · depth 31 - Simple pole and residue ordᵥ(f) of df/f
AlgebraicCurve.Place.hasSimplePoleAt_inv_smul_D_and_hasSimpleResidue_intCast_ord18 below · cited by 4 · depth 31 - Regularity at w of differentials of functions from the smaller field
AlgebraicCurve.Place.isRegularAt_D_algebraMap_of_forall_algebraMap_mem_of_isAlgClosed4 below · cited by 2 · depth 31 - Cartier operator: regularity, simple poles, p-th roots of residues
AlgebraicCurve.Place.isRegularAt_and_hasSimplePoleAt_and_hasSimpleResidue_of_cartierLaws_of_finiteDimensional15 below · cited by 2 · depth 31 - Regular functions have regular derivative with respect to a uniformiser
AlgebraicCurve.Place.ord_nonneg_of_D_eq_smul_D_of_ord_eq_one10 below · cited by 6 · depth 31 - Logarithmic differential: simple poles with residue ordᵥ f
AlgebraicCurve.Place.hasPoleOrderLE_one_inv_smul_D_and_hasLogResidue_intCast_ord17 below · cited by 1 · depth 32 - Pull-back of differentials scales simple residues by e
AlgebraicCurve.Place.isRegularAt_and_hasSimplePoleAt_and_hasSimpleResidue_mul_pullbackAlong_restrictAlong52 below · cited by 1 · depth 32 - Trace of differentials preserves regularity, simple poles and residues
AlgebraicCurve.Place.isRegularAt_and_hasSimplePoleAt_and_hasSimpleResidue_sum_traceAlong_of_separableAlong_of_isAlgClosed55 below · cited by 1 · depth 32 - Uniformizer witnessing dCoord and the local pole predicates
AlgebraicCurve.Place.exists_irreducible_dCoord_eq_D_and_hasPoleOrderLE_iff0 below · cited by 1 · depth 33 - Extension of places with residue retraction along finite extensions
AlgebraicCurve.Place.exists_place_comap_algebraMap_eq_of_finite17 below · cited by 2 · depth 35 - Simple-root criterion for unramifiedness of a place
AlgebraicCurve.Place.ramificationIndexAlong_eq_one_of_ord_eval_derivative_eq_zero0 below · cited by 3 · depth 36 - Finiteness of poles and prescribed values of a non-constant function
AlgebraicCurve.Place.exists_finset_forall_notMem_toValuationSubring_or_ord_sub_algebraMap_pos_imp_mem76 below · cited by 1 · depth 39
AlgebraicCurve.RROpens 9
- Interpolation with prescribed non-zero values and one pole
AlgebraicCurve.RROpens.exists_forall_hasValue_forall_ord_nonneg8 below · cited by 4 · depth 13 - Degree-zero classes as effective divisors minus g[P]
AlgebraicCurve.RROpens.exists_effective_sub_add_smul_single_mem_principal0 below · cited by 1 · depth 14 - Riemann–Roch descent: ℓ(G-T)=0 for ℓ(G) places in a prescribed infinite set
AlgebraicCurve.RROpens.exists_finset_subset_ell_sub_sum_single_eq_zero0 below · cited by 3 · depth 14 - General position of r-g degree-one places from a finite pool
AlgebraicCurve.RROpens.exists_injective_ell_sub_sum_single_eq_one_of_le_card1 below · cited by 3 · depth 15 - A divisor of degree g-1 with no sections
AlgebraicCurve.RROpens.exists_degree_eq_sub_one_and_ell_eq_zero3 below · cited by 1 · depth 18 - Block general position with exact drop of ℓ by e
AlgebraicCurve.RROpens.exists_injective_forall_forall_mem_ell_sub_sum_single_add_eq_ell_of_lt_card1 below · cited by 1 · depth 18 - Block form of simultaneous general position for ℓ(D-sum vⱼ)=1
AlgebraicCurve.RROpens.exists_injective_forall_forall_mem_ell_sub_sum_single_eq_one_of_lt_card1 below · cited by 1 · depth 19 - Interpolation in L(D-mP₀) with prescribed leading coefficient
AlgebraicCurve.RROpens.exists_mem_riemannRochSpace_sub_hasValue_mul_zpow_neg_forall_hasValue1 below · cited by 2 · depth 37 - Dimension bound for two Riemann–Roch spaces glued with a twist
AlgebraicCurve.RROpens.finrank_le_of_forall_mem_riemannRochSpace_sub_and_hasValue_nodes_and_hasValue_leading2 below · cited by 1 · depth 38
AlgebraicCurve.RadialRegion 9
- Star-convex open neighbourhood of a radial region
AlgebraicCurve.RadialRegion.exists_isOpen_starConvex_subset0 below · cited by 3 · depth 18 - Pairing, reparametrisation and vertex data for a square grid
AlgebraicCurve.RadialRegion.exists_grid_geometry2 below · cited by 1 · depth 20 - Pullback of a radial region along u↦ q+u^e, with laps
AlgebraicCurve.RadialRegion.exists_pow_pullback_laps0 below · cited by 1 · depth 20 - Inversion of a radial region centred at the origin
AlgebraicCurve.RadialRegion.exists_recip0 below · cited by 1 · depth 20 - A rectangle is a radial region with six arcs
AlgebraicCurve.RadialRegion.exists_rect_sixArcs0 below · cited by 2 · depth 20 - Reparametrising a shared straight edge of two radial regions
AlgebraicCurve.RadialRegion.exists_reparam_across_edge0 below · cited by 2 · depth 20 - Same-side reparametrisation of a shared straight side of radial regions
AlgebraicCurve.RadialRegion.exists_reparam_same_side0 below · cited by 2 · depth 20 - Grid window as a radial region with perimeter arcs
AlgebraicCurve.RadialRegion.exists_window_perimeter2 below · cited by 1 · depth 20 - Exact refinement of a radial region by inserted angles
AlgebraicCurve.RadialRegion.exists_refine_exact0 below · cited by 1 · depth 21
AlgebraicCurve.RationalFunctionField 47
- Places of K(t) have nonzero degree
AlgebraicCurve.RationalFunctionField.deg_ne_zero10 below · cited by 4 · depth 10 - The place at infinity of K(t) has degree one
AlgebraicCurve.RationalFunctionField.deg_placeInfty8 below · cited by 27 · depth 10 - Divisors of rational functions on P¹ have degree zero
AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_forall_eq_ord20 below · cited by 13 · depth 10 - Places of K(t): the finite places and ∞
AlgebraicCurve.RationalFunctionField.eq_ofHeightOneSpectrum_or_eq_placeInfty3 below · cited by 30 · depth 10 - Finite-dimensionality of L(0) when the constants are K
AlgebraicCurve.RationalFunctionField.finiteDimensional_lSpace_zero_of_constantsAreBase23 below · cited by 26 · depth 10 - A nonzero rational function has finitely many zeros and poles
AlgebraicCurve.RationalFunctionField.finite_setOf_ord_ne_zero0 below · cited by 14 · depth 10 - Principal divisors on the rational function field
AlgebraicCurve.RationalFunctionField.hasPrincipalDivisors21 below · cited by 12 · depth 10 - Existence of a place for a finite separable extension of K(X)
AlgebraicCurve.RationalFunctionField.nonempty_place_of_ratFunc_tower4 below · cited by 4 · depth 10 - At most one place of K(t)/K is not finite
AlgebraicCurve.RationalFunctionField.subsingleton_setOf_forall_ne_ofHeightOneSpectrum0 below · cited by 19 · depth 10 - The place at infinity of K(t) has degree one
AlgebraicCurve.RationalFunctionField.deg_eq_one_of_forall_ne_ofHeightOneSpectrum6 below · cited by 4 · depth 11 - Places of K(t) over an algebraically closed K have degree one
AlgebraicCurve.RationalFunctionField.deg_eq_one_of_isAlgClosed37 below · cited by 4 · depth 11 - Degree of a finite place of K(t) equals deg p
AlgebraicCurve.RationalFunctionField.deg_ofHeightOneSpectrum3 below · cited by 3 · depth 11 - Divisors cut out by a polynomial have degree zero
AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_forall_eq_ord_algebraMap18 below · cited by 2 · depth 11 - The infinite place of K(t) characterised among all places
AlgebraicCurve.RationalFunctionField.eq_placeInfty_iff_forall_ne_ofHeightOneSpectrum2 below · cited by 3 · depth 11 - Places of K(t) for K algebraically closed: P¹(K)
AlgebraicCurve.RationalFunctionField.eq_placeOfPoint_or_eq_placeInfty4 below · cited by 32 · depth 11 - Existence of a place of K(t)/K that is not a finite place
AlgebraicCurve.RationalFunctionField.exists_forall_ne_ofHeightOneSpectrum0 below · cited by 4 · depth 11 - Degree-zero divisors on P¹ are principal
AlgebraicCurve.RationalFunctionField.isPrincipal_of_degree_eq_zero18 below · cited by 2 · depth 11 - At the infinite place of K(X), ord = -deg
AlgebraicCurve.RationalFunctionField.ord_eq_neg_intDegree_of_forall_ne_ofHeightOneSpectrum6 below · cited by 5 · depth 11 - Order at the infinite place is minus the degree
AlgebraicCurve.RationalFunctionField.ord_placeInfty8 below · cited by 25 · depth 11 - Order at infinity of a polynomial is -deg q
AlgebraicCurve.RationalFunctionField.ord_placeInfty_algebraMap9 below · cited by 32 · depth 11 - Order at the place t=a equals root multiplicity
AlgebraicCurve.RationalFunctionField.ord_placeOfPoint_algebraMap37 below · cited by 53 · depth 11 - The place at infinity of K(t) is not a finite place
AlgebraicCurve.RationalFunctionField.placeInfty_ne_ofHeightOneSpectrum0 below · cited by 6 · depth 11 - Existence of the genus for separable extensions of K(X)
AlgebraicCurve.RationalFunctionField.stichtenothGenusExists23 below · cited by 3 · depth 11 - A pole of X forces the place at infinity on K(X)
AlgebraicCurve.RationalFunctionField.eq_placeInfty_of_ord_X_neg5 below · cited by 10 · depth 12 - A generator of a height-one prime of K[X] has order 1
AlgebraicCurve.RationalFunctionField.ord_ofHeightOneSpectrum_of_span3 below · cited by 17 · depth 12 - Places of K(t)/K other than the finite ones lie at infinity
AlgebraicCurve.RationalFunctionField.toValuationSubring_eq_of_forall_ne_ofHeightOneSpectrum3 below · cited by 2 · depth 12 - Order at a finite place of K(t) equals the adic valuation
AlgebraicCurve.RationalFunctionField.ord_ofHeightOneSpectrum_eq_neg_log2 below · cited by 2 · depth 13 - ordᵤ(X)≥ 0 for every place u≠∞ of K(X)
AlgebraicCurve.RationalFunctionField.ord_X_nonneg_of_ne_placeInfty23 below · cited by 1 · depth 14 - ord_∞(X) = -1 on the rational function field
AlgebraicCurve.RationalFunctionField.ord_placeInfty_X23 below · cited by 7 · depth 14 - Principal divisors on K(t) have degree zero
AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_isPrincipal21 below · cited by 1 · depth 15 - Reduction of principal divisors on the projective line
AlgebraicCurve.RationalFunctionField.mapDomain_eq_ord_div_map_of_primitive14 below · cited by 2 · depth 15 - Divisor of X-b on the rational function field
AlgebraicCurve.RationalFunctionField.ord_X_sub_C37 below · cited by 17 · depth 15 - The place t=a differs from the place at infinity
AlgebraicCurve.RationalFunctionField.placeOfPoint_ne_placeInfty1 below · cited by 25 · depth 15 - Weil reciprocity for the rational function field
AlgebraicCurve.RationalFunctionField.weilReciprocity37 below · cited by 2 · depth 15 - Double transitivity of k(t)-automorphisms on rational places
AlgebraicCurve.RationalFunctionField.exists_algEquiv_congrRingEquiv_placeInfty_eq_placeOfPoint_zero_eq53 below · cited by 2 · depth 16 - Order at infinity under reduction of E(X) to K(X)
AlgebraicCurve.RationalFunctionField.ord_placeInfty_eq_ord_placeInfty_add_sum_ord_placeOfPoint_of_reduction51 below · cited by 1 · depth 17 - Riemann's theorem over a rational function subfield
AlgebraicCurve.RationalFunctionField.stichtenothGenusExists_of_ratFunc_tower23 below · cited by 4 · depth 17 - Riemann–Roch for K(t) with genus 0 and -2[∞]
AlgebraicCurve.RationalFunctionField.ell_sub_ell_eq_genus_zero25 below · cited by 1 · depth 18 - Rational functions with trivial divisor are constant
AlgebraicCurve.RationalFunctionField.exists_algebraMap_of_forall_ord_eq_zero37 below · cited by 1 · depth 19 - The q-Frobenius on K(X) has exactly q+1 fixed places
AlgebraicCurve.RationalFunctionField.finite_fixedPoints_restrictAlong_and_natCard_eq_of_map_X_eq_X_pow0 below · cited by 1 · depth 21 - Finite index of speciality for function fields over K(X)
AlgebraicCurve.RationalFunctionField.indexOfSpecialtyFinite_of_ratFunc_tower23 below · cited by 1 · depth 23 - Polynomials lie in every place other than ∞
AlgebraicCurve.RationalFunctionField.algebraMap_polynomial_mem_of_ne_placeInfty23 below · cited by 1 · depth 25 - Value at infinity: f(∞)=c when deg(f-c)<0
AlgebraicCurve.RationalFunctionField.evalAt_placeInfty_eq37 below · cited by 1 · depth 25 - Evaluation at the place t=a of K(t) is q ↦ q(a)
AlgebraicCurve.RationalFunctionField.evalAt_placeOfPoint_algebraMap37 below · cited by 1 · depth 25 - Simple-pole residue cancellation for c dX/p on P¹
AlgebraicCurve.RationalFunctionField.trace_localResidue_finitePlace_add_trace_localResidue_placeInfty_eq_zero0 below · cited by 1 · depth 29 - Traceless residues of higher poles at finite places of K(X)
AlgebraicCurve.RationalFunctionField.trace_localResidue_finitePlace_div_pow_eq_zero60 below · cited by 1 · depth 29 - Vanishing of the residue of Xⁿ dX at infinity
AlgebraicCurve.RationalFunctionField.trace_localResidue_placeInfty_X_pow_eq_zero1 below · cited by 1 · depth 29
AlgebraicCurve.RegularProlongation 91
- Rigidity of q-th roots under good regular reduction (rank one)
AlgebraicCurve.RegularProlongation.exists_pow_eq_of_residue_eq_pow_of_finrank_eq_of_krullDimLE_one179 below · cited by 2 · depth 10 - Uniqueness of a regular prolongation from its trace on L(x)
AlgebraicCurve.RegularProlongation.eq_integers_of_forall_mem_adjoin_iff0 below · cited by 31 · depth 11 - Existence and uniqueness of the reduced place on a chart
AlgebraicCurve.RegularProlongation.existsUnique_place_forall_residue_sub_mem_nonunits9 below · cited by 9 · depth 11 - Monic equation with Gauss degree bounds over a regular prolongation
AlgebraicCurve.RegularProlongation.exists_monic_coeff_natDegree_le_of_forall_valuationSubring7 below · cited by 13 · depth 11 - Reduction map on places from surjectivity on both charts
AlgebraicCurve.RegularProlongation.exists_placeMap_mapDomain_eq_ord_of_residue_integralClosure_surjective32 below · cited by 5 · depth 11 - Kummer splitting of a regular prolongation in degree q
AlgebraicCurve.RegularProlongation.exists_prolongation_of_card_roots_eq5 below · cited by 2 · depth 11 - Equal genera force surjective reduction onto affine charts
AlgebraicCurve.RegularProlongation.residue_integralClosure_surjective_of_genusFF_eq61 below · cited by 10 · depth 11 - Deuring's genus inequality for a complete family of prolongations
AlgebraicCurve.RegularProlongation.sum_genusFF_le_of_sum_finrank_eq_of_krullDimLE_one127 below · cited by 3 · depth 11 - Deuring's reduction of div(f) at a finite place
AlgebraicCurve.RegularProlongation.sum_ord_eq_ord_residue_of_residue_integralClosure_surjective31 below · cited by 7 · depth 11 - Minimal polynomial over L[x] has coefficients in 𝒪
AlgebraicCurve.RegularProlongation.coe_minpoly_adjoin_coeff_mem_integers3 below · cited by 4 · depth 12 - Gauss-norm integrality at a residually transcendental point
AlgebraicCurve.RegularProlongation.coeff_mem_of_aeval_mem_integers0 below · cited by 9 · depth 12 - Eventual dimension count for joint residue spans
AlgebraicCurve.RegularProlongation.exists_forall_finrank_residueSpan_inf_add_card_le98 below · cited by 1 · depth 12 - Connectedness of constant reduction over a rank-one valuation ring
AlgebraicCurve.RegularProlongation.exists_forall_residue_eq_algebraMap_of_mem_residueSpan_inf_of_krullDimLE_one33 below · cited by 1 · depth 12 - Finiteness of each residue extension over k(̄ fᵢ)
AlgebraicCurve.RegularProlongation.finiteDimensional_adjoin_residue_of_sum_finrank_eq2 below · cited by 5 · depth 12 - Joint residue image of V has k-dimension dim_L V
AlgebraicCurve.RegularProlongation.finrank_span_pi_residue_eq_finrank_of_sum_finrank_eq2 below · cited by 3 · depth 12 - Reduction preserves dimension of finite-dimensional linear systems
AlgebraicCurve.RegularProlongation.finrank_span_residue_eq_finrank0 below · cited by 6 · depth 12 - Pole bound for residues under a regular prolongation
AlgebraicCurve.RegularProlongation.mul_min_ord_residue_le_of_monic0 below · cited by 6 · depth 12 - Deuring's multiplicity inequality on the finite chart
AlgebraicCurve.RegularProlongation.ord_residue_le_sum_ord_of_isIntegral_adjoin7 below · cited by 2 · depth 12 - Residues of L(M· D) lie in both chart spans
AlgebraicCurve.RegularProlongation.span_residue_lSpace_le_residueSpan_inf2 below · cited by 2 · depth 12 - Deuring reduction: equal multiplicity totals on the finite chart
AlgebraicCurve.RegularProlongation.sum_ord_eq_sum_ord_residue_of_isIntegral_adjoin30 below · cited by 1 · depth 12 - Transcendence lifts from the residue field of a regular prolongation
AlgebraicCurve.RegularProlongation.transcendental_of_residue_transcendental0 below · cited by 4 · depth 12 - Finite generation of the overlap module over a valuation ring
AlgebraicCurve.RegularProlongation.exists_forall_mul_eq_sum_add_sum_inv_pow_mul_of_sum_finrank_eq27 below · cited by 2 · depth 13 - Zero joint residue forces division by a non-unit of A
AlgebraicCurve.RegularProlongation.exists_nonunit_smul_eq_of_forall_residue_eq_zero_of_sum_finrank_eq4 below · cited by 2 · depth 13 - Joint residues in ρ(T)∩ρ(T') have constant components
AlgebraicCurve.RegularProlongation.forall_exists_residue_eq_algebraMap_of_mem_residueSpan_inf13 below · cited by 1 · depth 13 - Joint residues of f-integral functions: integrality and monic denominators
AlgebraicCurve.RegularProlongation.forall_ord_residueSpan_nonneg_and_exists_monic_of_isAlgClosed25 below · cited by 2 · depth 13 - Fundamental inequality for several regular prolongations
AlgebraicCurve.RegularProlongation.sum_finrank_adjoin_residue_le1 below · cited by 22 · depth 13 - An A-generating family with independent joint residues
AlgebraicCurve.RegularProlongation.exists_basis_mem_integers_piResidue_linearIndependent_of_sum_finrank_eq3 below · cited by 6 · depth 14 - Completeness of a defectless family of regular prolongations
AlgebraicCurve.RegularProlongation.exists_eq_integers_of_forall_mem_adjoin_iff_of_sum_finrank_eq_of_isAlgClosed4 below · cited by 8 · depth 14 - Simultaneous residues at distinct regular prolongations
AlgebraicCurve.RegularProlongation.exists_forall_residue_eq1 below · cited by 2 · depth 14 - Gauss basis with residual generation for complete regular prolongations
AlgebraicCurve.RegularProlongation.exists_gaussBasis_forall_eq_sum_aeval_add_mul_of_sum_finrank_eq22 below · cited by 1 · depth 14 - Residues stay integral over k[̄ f]
AlgebraicCurve.RegularProlongation.isIntegral_adjoin_residue_of_exists_monic_bivariate_eval_eq_zero0 below · cited by 4 · depth 14 - Gauss prolongations agree on L(f) for transcendental residue
AlgebraicCurve.RegularProlongation.mem_adjoin_iff_mem_integers_iff_of_transcendental_residue1 below · cited by 10 · depth 14 - Gauss basis of the common integers, with independent residues
AlgebraicCurve.RegularProlongation.exists_gaussBasis_mem_integralClosure_piResidue_uniqueRepr_of_sum_finrank_eq6 below · cited by 4 · depth 15 - Valuation rings sharing the trace of a regular prolongation
AlgebraicCurve.RegularProlongation.exists_of_forall_mem_adjoin_iff_of_isAlgebraic1 below · cited by 1 · depth 15 - Deuring's good-place lemma for regular prolongations
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_existsUnique_integers_of_isAlgClosed4 below · cited by 1 · depth 16 - Genus preservation under good reduction, via Riemann–Roch data
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_forall_exists_degree_eq_and_ell_eq123 below · cited by 1 · depth 16 - Rigidity of q-th roots under good regular prolongations
AlgebraicCurve.RegularProlongation.exists_pow_eq_of_residue_eq_pow_of_finrank_eq_of_isAlgClosed190 below · cited by 1 · depth 16 - Genus does not drop: ℓ(m̄ D)≤ℓ(mD) for large m
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_forall_exists_forall_ell_nsmul_le121 below · cited by 1 · depth 17 - Uniqueness of the reduction map on places along a regular prolongation
AlgebraicCurve.RegularProlongation.placeMap_unique_of_forall_mapDomain_eq_ord47 below · cited by 2 · depth 17 - Deuring's genus inequality for defectless families of regular prolongations
AlgebraicCurve.RegularProlongation.sum_genusFF_le_of_sum_finrank_eq_of_isAlgClosed137 below · cited by 1 · depth 17 - Reduction of an adapted basis at almost all constant places
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_forall_lSpace_le_span_and_linearIndependent_residue80 below · cited by 1 · depth 18 - Eventual dimension count for residue spans of regular prolongations
AlgebraicCurve.RegularProlongation.exists_forall_finrank_residueSpan_inf_add_card_le_of_isAlgClosed98 below · cited by 1 · depth 18 - Constants of a complete family of regular prolongations
AlgebraicCurve.RegularProlongation.exists_forall_residue_eq_algebraMap_of_mem_residueSpan_inf_of_isAlgClosed43 below · cited by 1 · depth 18 - Integrality of coefficients from joint integrality of sum_τ r_τ(f)z_τ
AlgebraicCurve.RegularProlongation.coeff_mem_of_sum_aeval_mul_mem_of_unique_pi_residue_repr0 below · cited by 1 · depth 19 - Reduction of an integral basis spans the reduced regular functions
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_forall_eq_sum_aeval_residue_mul_residue_of_forall_ord_nonneg77 below · cited by 1 · depth 19 - Reduction of an integral basis at almost all constant places
AlgebraicCurve.RegularProlongation.exists_finset_forall_valuation_eq_one_forall_exists_mem_integers_residue_uniqueRepr_and_span12 below · cited by 2 · depth 19 - Constant reduction: joint residues regular on both charts are diagonal
AlgebraicCurve.RegularProlongation.exists_forall_residue_eq_algebraMap_of_mem_residueSpan_inf_of_ringKrullDim_lt_top38 below · cited by 1 · depth 19 - Descent of a complete family of regular prolongations
AlgebraicCurve.RegularProlongation.exists_regularProlongation_intermediateField_sum_finrank_adjoin_residue_eq4 below · cited by 1 · depth 19 - Rank-one induction step for constancy modulo a coarsening
AlgebraicCurve.RegularProlongation.exists_eq_algebraMap_add_mul_of_valuation_lt_one_of_krullDimLE_one35 below · cited by 1 · depth 20 - Traces commute with residues under a regular prolongation
AlgebraicCurve.RegularProlongation.exists_residue_trace_eq_trace_residue_of_finrank_eq9 below · cited by 3 · depth 20 - Coarsening a regular prolongation preserves residual transcendence
AlgebraicCurve.RegularProlongation.exists_regularProlongation_mem_integers_iff_of_le0 below · cited by 1 · depth 21 - Matching residues at a coarsened prolongation are constant
AlgebraicCurve.RegularProlongation.exists_residue_eq_algebraMap_of_le_of_forall_residue_eq15 below · cited by 1 · depth 21 - Reduction of a function field along a regular prolongation
AlgebraicCurve.RegularProlongation.isCurveOver_and_essFiniteType_of_exists_transcendental47 below · cited by 33 · depth 21 - Simultaneously good transcendental element for several regular prolongations
AlgebraicCurve.RegularProlongation.exists_forall_transcendental_residue1 below · cited by 1 · depth 22 - Residue degree bounded by degree for a regular prolongation
AlgebraicCurve.RegularProlongation.finiteDimensional_and_finrank_adjoin_residue_le0 below · cited by 3 · depth 22 - Residue-compatible isomorphism of reduced fields for equal prolongations
AlgebraicCurve.RegularProlongation.exists_algEquiv_residue_eq_of_integers_eq0 below · cited by 6 · depth 23 - Component chart from a regular prolongation and a disc family
AlgebraicCurve.RegularProlongation.exists_componentChart_of_discFamily0 below · cited by 15 · depth 23 - Incompatibility of a reciprocal annulus pair at a smooth point
AlgebraicCurve.RegularProlongation.false_of_annulus_attached_regularProlongation_of_smoothPointPackage0 below · cited by 3 · depth 23 - Étale chart with sections gives a residue disc
AlgebraicCurve.RegularProlongation.isResidueDisc_of_etaleChart_of_sections1 below · cited by 29 · depth 23 - Residue discs transfer along an isomorphism of reduced fields
AlgebraicCurve.RegularProlongation.isResidueDisc_of_integers_eq_of_algEquiv0 below · cited by 3 · depth 23 - Sections, valuation reading and locality at a smooth chart
AlgebraicCurve.RegularProlongation.disc_sections_locality_of_smoothPoint11 below · cited by 12 · depth 24 - Unique rational branch place with uniformiser ̄ y
AlgebraicCurve.RegularProlongation.exists_place_isRational_forall_mem_iff_exists_residue_eq_and_ord_eq_one_and_forall_eq_of_isNoetherianRing_range0 below · cited by 5 · depth 24 - Smooth-point package ascends a directed tower of constant fields
AlgebraicCurve.RegularProlongation.exists_smoothPointPackage_of_directed_subfieldTower_and_forall_disc_eq3 below · cited by 12 · depth 24 - Smooth-point package ascends a directed tower of constants
AlgebraicCurve.RegularProlongation.exists_smoothPointPackage_of_directed_subfieldTower_of_discPlaces3 below · cited by 6 · depth 24 - Pole preservation for residues under a regular prolongation
AlgebraicCurve.RegularProlongation.mul_min_ord_residue_le_of_forall_valuationSubring_mem9 below · cited by 2 · depth 24 - Unique place of the disc inducing a given section
AlgebraicCurve.RegularProlongation.existsUnique_mem_disc_forall_evalAt_eq_of_section9 below · cited by 1 · depth 25 - Regular prolongations with equal valuation rings have k-isomorphic residue fields
AlgebraicCurve.RegularProlongation.exists_algEquiv_of_integers_eq0 below · cited by 3 · depth 25 - Reduction of a local subring as a rational place
AlgebraicCurve.RegularProlongation.exists_place_eq_of_residue_image0 below · cited by 1 · depth 25 - Disc places are cut out by primes of S₁
AlgebraicCurve.RegularProlongation.exists_prime_forall_evalAt_eq_zero_iff_dvd_of_mem_disc0 below · cited by 3 · depth 25 - Smooth-point stalk propagated up a tower of constant fields
AlgebraicCurve.RegularProlongation.exists_smoothPointStalks_tower_of_base20 below · cited by 3 · depth 25 - Places of a package disc read units of S
AlgebraicCurve.RegularProlongation.forall_mem_and_isUnit_iff_of_mem_packageDisc0 below · cited by 1 · depth 25 - Uniqueness of the near end of an attached annulus
AlgebraicCurve.RegularProlongation.integers_le_of_annulus_attached_of_forall_mem_of_param_mem_units4 below · cited by 1 · depth 25 - Divisor locality on the disc at a smooth point
AlgebraicCurve.RegularProlongation.locality_of_disc_of_smoothPoint4 below · cited by 1 · depth 25 - A package disc is contained in a residue disc
AlgebraicCurve.RegularProlongation.packageDisc_subset_of_isResidueDisc35 below · cited by 1 · depth 25 - A residue disc lies in any smooth-point package disc
AlgebraicCurve.RegularProlongation.subset_packageDisc_of_isResidueDisc35 below · cited by 2 · depth 25 - A place of the disc centred at each non-varpi prime
AlgebraicCurve.RegularProlongation.exists_mem_disc_forall_evalAt_eq_zero_iff_dvd_of_prime2 below · cited by 2 · depth 26 - Transport of a regular prolongation along an L-automorphism
AlgebraicCurve.RegularProlongation.exists_mem_integers_iff_map_mem_and_residue_eq_of_algEquiv0 below · cited by 1 · depth 26 - Powers of a non-unit correct any function to a unit
AlgebraicCurve.RegularProlongation.exists_pow_mul_zpow_mem_integers_residue_ne_zero_of_forall_residue_eq_zero4 below · cited by 3 · depth 26 - Base change of a smooth-point stalk package along a finite constants layer
AlgebraicCurve.RegularProlongation.exists_smoothPointStalk_baseChange_layer10 below · cited by 13 · depth 26 - Algebraic smooth-point block over a henselian discrete valuation ring
AlgebraicCurve.RegularProlongation.smoothPointStalk_algebraicBlock_of_formallyEtale_of_henselian17 below · cited by 13 · depth 26 - Centre, residue character and uniformiser for a read local subring
AlgebraicCurve.RegularProlongation.comap_maximalIdeal_eq_span_and_residue_eq_and_ord_eq_one_of_reads_of_constants0 below · cited by 3 · depth 27 - A chart element separating a place from the node
AlgebraicCurve.RegularProlongation.exists_mem_nonunit_ord_residue_eq_zero_of_ne_centre0 below · cited by 3 · depth 27 - Krull dimension at most one passes to a regular prolongation
AlgebraicCurve.RegularProlongation.krullDimLE_one_integers0 below · cited by 2 · depth 27 - Nonnegative ord_Q of a reduction from effectivity on D
AlgebraicCurve.RegularProlongation.ord_residue_nonneg_of_degreeOn_of_forall_ord_nonneg0 below · cited by 3 · depth 27 - Factoring a unit-normalised element by a power of varpi
AlgebraicCurve.RegularProlongation.exists_mul_eq_pow_mul_of_residue_smul_ne_zero_of_ringEquiv_uvCrossingModel31 below · cited by 1 · depth 28 - Common normalising constant for two regular prolongations
AlgebraicCurve.RegularProlongation.exists_smul_mem_integers_and_residue_ne_zero_or0 below · cited by 1 · depth 29 - Ascent of positive reduction order to an enlarged local ring
AlgebraicCurve.RegularProlongation.forall_ord_residue_pos_of_maximalIdeal_le_map_sup_span0 below · cited by 1 · depth 30 - A defectless family of prolongations exhausts all extensions
AlgebraicCurve.RegularProlongation.exists_eq_integers_of_forall_mem_adjoin_iff_of_sum_finrank_eq3 below · cited by 1 · depth 32 - Norms commute with reduction for compatible regular prolongations
AlgebraicCurve.RegularProlongation.exists_norm_mem_integers_and_residue_norm_eq_norm_residue0 below · cited by 1 · depth 32 - Recognising the residue field of a regular prolongation
AlgebraicCurve.RegularProlongation.exists_algEquiv_apply_eq_residue_of_transcendental2 below · cited by 1 · depth 35 - Perturbing a simple pole to residue orders (-1,0) or (0,-1)
AlgebraicCurve.RegularProlongation.exists_mem_riemannRochSpace_add_single_ord_eq_neg_one_and_ord_residue_pair_of_jointLaw0 below · cited by 1 · depth 41
AlgebraicCurve.RiemannGenusReachedAt 1
- deg D-ℓ(D) is constant above a genus-realising divisor
AlgebraicCurve.RiemannGenusReachedAt.eq_of_ge0 below · cited by 8 · depth 12
AlgebraicCurve.SemilinearAut 18
- Correspondences on Pic⁰ commute with intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.pic0_correspondence_smul9 below · cited by 6 · depth 10 - Divisor correspondences commute with intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.correspondence_smul8 below · cited by 5 · depth 11 - Invariance of inertia degree under intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.inertiaDeg_smul0 below · cited by 2 · depth 12 - Equivariance of divisor pull-back along an embedding
AlgebraicCurve.SemilinearAut.pullbackAlong_smul4 below · cited by 14 · depth 12 - Equivariance of divisor pushforward along an embedding
AlgebraicCurve.SemilinearAut.pushforwardAlong_smul3 below · cited by 4 · depth 12 - Ramification index is invariant under intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.ramificationIndex_smul1 below · cited by 3 · depth 12 - Order at a transported place of an element from the subfield
AlgebraicCurve.SemilinearAut.ord_algebraMap_smul0 below · cited by 1 · depth 13 - Divisor pullback commutes with intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.pullback_smul3 below · cited by 1 · depth 13 - Divisor pushforward commutes with intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.pushforward_smul2 below · cited by 1 · depth 13 - Extending constant-field automorphisms to F-fixing semilinear automorphisms
AlgebraicCurve.SemilinearAut.exists_baseAut_eq_of_constantFieldExtension1 below · cited by 1 · depth 14 - Compatibility of the two actions of Aut(F/K) on places
AlgebraicCurve.SemilinearAut.ofAlgAut_smul_place0 below · cited by 5 · depth 14 - Equivariance of restriction of places under intertwined semilinear automorphisms
AlgebraicCurve.SemilinearAut.restrict_smul0 below · cited by 11 · depth 14 - Unique homomorphic extension of a semilinear S-action to a constant-field extension
AlgebraicCurve.SemilinearAut.existsUnique_monoidHom_baseAut_eq_smul_algebraMap_eq_of_constantFieldExtension2 below · cited by 1 · depth 19 - Semilinear automorphisms extend uniquely along a constant field extension
AlgebraicCurve.SemilinearAut.existsUnique_baseAut_eq_smul_algebraMap_eq_of_constantFieldExtension1 below · cited by 1 · depth 20 - Stability of a section-cut residue disc under a semilinear automorphism
AlgebraicCurve.SemilinearAut.mem_iff_smul_mem_of_forall_mem_iff_sections0 below · cited by 9 · depth 22 - Swapping the legs of a correspondence on Pic⁰
AlgebraicCurve.SemilinearAut.pic0_correspondence_swap_smul0 below · cited by 4 · depth 22 - A ℚ̄-linear automorphism is determined by its action on places
AlgebraicCurve.SemilinearAut.eq_of_forall_smul_place_eq14 below · cited by 4 · depth 23 - Semilinear automorphisms determined by base action and places
AlgebraicCurve.SemilinearAut.eq_of_baseAut_eq_of_forall_smul_place_eq15 below · cited by 5 · depth 25
AlgebraicCurve.SemistableCovering 21
- Incidence graph of a semistable covering is connected
AlgebraicCurve.SemistableCovering.exists_src_mem_iff_tgt_notMem_of_discFibres_of_rankOne112 below · cited by 5 · depth 23 - Width-one refinement of a semistable covering, reindexed
AlgebraicCurve.SemistableCovering.exists_widthOne_covering_equiv_of_discFibres_of_rankOne99 below · cited by 1 · depth 25 - Principality of glued chart divisors along a semistable covering
AlgebraicCurve.SemistableCovering.sum_mem_principal_of_zsmul_mem_principal_of_forall_evalAt_eq_of_discFibres_of_rankOne_of_charZero_of_semistableModel238 below · cited by 1 · depth 25 - Refining a semistable covering into width-one annuli
AlgebraicCurve.SemistableCovering.exists_circleCharts_and_bands_width_one_of_discFibres_of_rankOne97 below · cited by 1 · depth 26 - Graded Deuring lifting on a width-one semistable covering
AlgebraicCurve.SemistableCovering.exists_forall_residue_smul_eq_of_forall_ord_ge_of_forall_evalAt_mul_eq_of_width_one111 below · cited by 1 · depth 26 - Lifting glued chart sections across one twisted annulus
AlgebraicCurve.SemistableCovering.exists_mem_riemannRochSpace_forall_residue_eq_of_glued_of_rankOne120 below · cited by 2 · depth 26 - Equal-depth annulus points differ by chart-supported degree-zero divisors
AlgebraicCurve.SemistableCovering.exists_ne_zero_ord_eq_single_sub_single_of_depth_eq_of_rankOne124 below · cited by 2 · depth 26 - No zeros on width-one annuli; Laplacian chart degrees
AlgebraicCurve.SemistableCovering.forall_ord_eq_zero_and_sum_eq_lap_of_ord_residue_smul_eq_of_width_one4 below · cited by 1 · depth 26 - Chart residues of a Riemann–Roch space are independent
AlgebraicCurve.SemistableCovering.exists_linearIndependent_pi_residue_of_mem_riemannRochSpace_of_rankOne8 below · cited by 2 · depth 27 - Weighted reductions of a Riemann–Roch space are independent
AlgebraicCurve.SemistableCovering.exists_linearIndependent_pi_residue_smul_of_mem_riemannRochSpace_of_rankOne8 below · cited by 1 · depth 27 - Principal chartwise push-forwards for [P]-[P'] on one annulus
AlgebraicCurve.SemistableCovering.exists_ne_zero_ord_eq_single_sub_single_mapDomain_placeMap_mem_principal_of_valuation_eq_of_rankOne127 below · cited by 1 · depth 27 - Dimension of the graded glued Riemann–Roch space equals ℓ(D)
AlgebraicCurve.SemistableCovering.finiteDimensional_and_finrank_graded_glued_riemannRochSpace_eq_finrank_of_width_one101 below · cited by 1 · depth 27 - Divisor of g restricted to the annuli is [P']-[P]
AlgebraicCurve.SemistableCovering.ord_eq_single_sub_single_of_forall_residue_evalAt_ne_zero_of_rankOne2 below · cited by 1 · depth 27 - Graded reductions on a width-one semistable covering are compatible
AlgebraicCurve.SemistableCovering.ord_residue_smul_ge_and_evalAt_mul_eq_of_forall_smul_mem_integers_of_width_one10 below · cited by 1 · depth 27 - ℓ-power depth divisors are Laplacian potentials after subdivision
AlgebraicCurve.SemistableCovering.exists_pow_mul_single_eq_sum_smul_lap_of_modEq0 below · cited by 1 · depth 28 - Degree-zero vectors are torsion modulo graph Laplacian rows
AlgebraicCurve.SemistableCovering.isOfFinAddOrder_mk_quotient_lap_of_sum_eq_zero_of_forall_exists_src_mem_iff0 below · cited by 1 · depth 29 - N-torsion bound for the Jacobian of a subdivided multigraph
AlgebraicCurve.SemistableCovering.natCard_torsion_quotient_lap_mul_pow_le_pow_of_forall_exists_src_mem_iff0 below · cited by 1 · depth 29 - Refinement of a subdivided multigraph: Laplacian rows under extension by zero
AlgebraicCurve.SemistableCovering.refine_lap_mem_closure_range_lap_of_forall_eq_mul0 below · cited by 1 · depth 29 - Reduction of L(D) onto node-matched tuples on the components
AlgebraicCurve.SemistableCovering.exists_forall_residue_eq_of_forall_evalAt_eq_of_discFibres_of_rankOne102 below · cited by 1 · depth 31 - Dimension of glued Riemann–Roch spaces on a semistable covering
AlgebraicCurve.SemistableCovering.finiteDimensional_and_finrank_glued_riemannRochSpace_eq_finrank_of_rankOne92 below · cited by 1 · depth 32 - Chart reductions of a Riemann–Roch function match along the annuli
AlgebraicCurve.SemistableCovering.residue_mem_riemannRochSpace_and_evalAt_eq_of_forall_mem_integers_of_rankOne10 below · cited by 1 · depth 32
AlgebraicCurve.SemistableModel 20
- Étale coordinate at a smooth closed point of the special fibre
AlgebraicCurve.SemistableModel.exists_etaleCoordinate_localRing_of_mem_smoothLocus2 below · cited by 1 · depth 24 - Local rings along a specialisation, as subrings of F
AlgebraicCurve.SemistableModel.localRing_le_and_exists_mem_localRing_mul_eq_of_specializes0 below · cited by 5 · depth 24 - Relative dimension n forces dim_F Ω_{F/L} = n
AlgebraicCurve.SemistableModel.finrank_kaehlerDifferential_eq_of_smoothOfRelativeDimension0 below · cited by 5 · depth 25 - Function field of a descended model: F=F₀· L
AlgebraicCurve.SemistableModel.Descent.exists_finset_mem_subfieldClosure_union_image_algebraMap0 below · cited by 2 · depth 26 - Base change of a descended semistable model along A₁
AlgebraicCurve.SemistableModel.Descent.exists_isIntegral_pullback_isIntegrallyClosed_stalk_and_subfield_equiv_functionField_of_range_eq_inter2 below · cited by 2 · depth 26 - Global sections of a semistable model are A
AlgebraicCurve.SemistableModel.bijective_appTop_toBase13 below · cited by 2 · depth 27 - Descent of the congruence u≡ 1 to a flat level
AlgebraicCurve.SemistableModel.exists_eq_one_add_baseToFunctionField_mul_of_level3 below · cited by 2 · depth 27 - Descent of saturated open covers along a closed surjection
AlgebraicCurve.SemistableModel.exists_opens_preimage_eq_of_isClosedMap_of_saturated0 below · cited by 2 · depth 27 - A single constant works on every component
AlgebraicCurve.SemistableModel.exists_smul_div_pow_mem_integers_of_isPreconnected_of_fintype_mem_range0 below · cited by 2 · depth 27 - Fibres of a semistable model over a finite level
AlgebraicCurve.SemistableModel.fibre_shapes_of_level1 below · cited by 2 · depth 27 - Stalks of a semistable model are integrally closed
AlgebraicCurve.SemistableModel.isIntegrallyClosed_stalk0 below · cited by 1 · depth 27 - Standard opens of a semistable model and their points
AlgebraicCurve.SemistableModel.isOpen_compl_closures_and_mem_iff0 below · cited by 2 · depth 27 - Smooth closed points of integral models: special fibre a DVR
AlgebraicCurve.SemistableModel.isPrincipalIdealRing_stalk_quotient_map_maximalIdeal_of_mem_smoothLocus5 below · cited by 2 · depth 27 - Closed fibre of a semistable model is reduced at finite level
AlgebraicCurve.SemistableModel.isReduced_pullback_residue_of_level3 below · cited by 2 · depth 27 - Regularity descends along a flat level of a semistable model
AlgebraicCurve.SemistableModel.mem_localRing_iff_mem_range_of_level1 below · cited by 3 · depth 27 - Units of mathcal O_{X,x} from Gauss units and order data
AlgebraicCurve.SemistableModel.smul_div_pow_mem_localRing_of_forall_ord_eq0 below · cited by 2 · depth 27 - Étale coordinate at a smooth closed special point over a DVR
AlgebraicCurve.SemistableModel.exists_etaleCoordinate_residueChar_localRing_of_mem_smoothLocus_of_isDiscreteValuationRing9 below · cited by 1 · depth 28 - Stalk mod uniformiser at a smooth closed special point
AlgebraicCurve.SemistableModel.isDiscreteValuationRing_stalk_quotient_span_of_mem_smoothLocus_of_isDiscreteValuationRing4 below · cited by 1 · depth 28 - Stalks of a semistable model are reduced modulo mathfrak m_A
AlgebraicCurve.SemistableModel.isReduced_stalk_quotient_maximalIdeal_of_base_eq_closedPoint0 below · cited by 1 · depth 28 - Parameters (varpi,t) at a smooth closed point of the special fibre
AlgebraicCurve.SemistableModel.exists_maximalIdeal_stalk_eq_span_pair_of_mem_smoothLocus_of_isDiscreteValuationRing4 below · cited by 1 · depth 29
AlgebraicCurve.TranscendenceTower 2
- Degree of the pulled-back pole divisor equals [F:E]
AlgebraicCurve.TranscendenceTower.degree_poleDivisor_eq_finrank6 below · cited by 1 · depth 14 - Coefficients of the pole divisor π^*(v)
AlgebraicCurve.TranscendenceTower.poleDivisor_apply6 below · cited by 1 · depth 14
AlgebraicCurve.TwoChartIntegralModel 127
- Finite chart is dense in a fibre without isolated points
AlgebraicCurve.TwoChartIntegralModel.dense_range_chart_pullback_of_not_isOpen_singleton0 below · cited by 1 · depth 13 - Functoriality of the two-chart integral model along a finite extension
AlgebraicCurve.TwoChartIntegralModel.exists_hom_isFinite_surjective_chartMap_finite_of_algHom0 below · cited by 6 · depth 13 - Affine neighbourhood of a finite set in an open, over an affine base open
AlgebraicCurve.TwoChartIntegralModel.exists_isAffineOpen_le_preimage_forall_mem_of_finset6 below · cited by 1 · depth 13 - Retraction of finite-chart algebras yields a closed-immersion section of the fibre
AlgebraicCurve.TwoChartIntegralModel.exists_isClosedImmersion_comp_eq_id_of_retraction4 below · cited by 1 · depth 13 - Transport of the two-chart integral model along an R-algebra isomorphism
AlgebraicCurve.TwoChartIntegralModel.exists_iso_of_algEquiv_apply_eq0 below · cited by 16 · depth 13 - Comparable two-chart integral models are isomorphic over R
AlgebraicCurve.TwoChartIntegralModel.exists_iso_of_mem_chartAlgFin_of_forall_exists_mul_mem1 below · cited by 9 · depth 13 - Automorphism of the two-chart model induced by σ
AlgebraicCurve.TwoChartIntegralModel.exists_iso_toBase_eq_and_iotaFin_comp_eq_of_algEquiv4 below · cited by 6 · depth 13 - The two charts form a two-affine open cover with affine overlap
AlgebraicCurve.TwoChartIntegralModel.exists_twoAffineOpenCover_U0_eq_chartFinOpen0 below · cited by 14 · depth 13 - Finite type of the two chart rings of the integral model
AlgebraicCurve.TwoChartIntegralModel.finiteType_chartAlgFin_and_chartAlgInf0 below · cited by 148 · depth 13 - Finiteness of the two chart rings over R[X]
AlgebraicCurve.TwoChartIntegralModel.finite_polynomial_chartAlgFin_and_chartAlgInf0 below · cited by 68 · depth 13 - Flatness of the two-chart integral model over a Dedekind base
AlgebraicCurve.TwoChartIntegralModel.flat_toBase0 below · cited by 25 · depth 13 - Two components cover the fibre and are distinct
AlgebraicCurve.TwoChartIntegralModel.forall_mem_range_or_mem_range_comp_and_range_ne_of_minimalPrimes_eq0 below · cited by 1 · depth 13 - Pinned automorphisms of the two-chart model: commuting and squaring
AlgebraicCurve.TwoChartIntegralModel.hom_comm_and_hom_comp_hom_eq_of_chartPins1 below · cited by 1 · depth 13 - Rigidity of maps from the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.hom_ext_of_iotaFin_comp_eq1 below · cited by 5 · depth 13 - The chart ring A_R(S) has fraction field F
AlgebraicCurve.TwoChartIntegralModel.isFractionRing_chartAlg0 below · cited by 85 · depth 13 - The two-chart integral model is integral
AlgebraicCurve.TwoChartIntegralModel.isIntegral0 below · cited by 34 · depth 13 - Sections over affine opens of the two-chart integral model are integrally closed
AlgebraicCurve.TwoChartIntegralModel.isIntegrallyClosed_sections_of_isAffineOpen4 below · cited by 5 · depth 13 - Stalks of the two-chart integral model are integrally closed
AlgebraicCurve.TwoChartIntegralModel.isIntegrallyClosed_stalk2 below · cited by 9 · depth 13 - Properness of the two-chart integral model over a Noetherian base
AlgebraicCurve.TwoChartIntegralModel.isProper_toBase0 below · cited by 16 · depth 13 - Reduced fibres of a two-chart integral model over ℤ₍ₚ₎
AlgebraicCurve.TwoChartIntegralModel.isReduced_pullback_toBase_of_isReduced_chartAlg_quotient_span_natCast5 below · cited by 1 · depth 13 - Two-chart integral model is locally of finite presentation
AlgebraicCurve.TwoChartIntegralModel.locallyOfFinitePresentation_toBase0 below · cited by 18 · depth 13 - Image point's place is the restriction along Φ
AlgebraicCurve.TwoChartIntegralModel.pointEquivPlace_eq_restrictAlong_of_chartPin0 below · cited by 9 · depth 13 - Characteristic-zero base changes of the two-chart model are smooth curves
AlgebraicCurve.TwoChartIntegralModel.smoothOfRelativeDimension_one_pullback_snd_toBase_of_charZero1 below · cited by 11 · depth 13 - Localisation of the base commutes with chart algebras
AlgebraicCurve.TwoChartIntegralModel.exists_algEquiv_tensor_chartAlg0 below · cited by 8 · depth 14 - Functoriality and finiteness of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_hom_isFinite_surjective_of_algHom0 below · cited by 3 · depth 14 - Comparison morphism between two-chart integral models
AlgebraicCurve.TwoChartIntegralModel.exists_hom_of_mem_chartAlgFin_of_forall_pow_mul_mem0 below · cited by 1 · depth 14 - Finite sets of points of the two-chart integral model lie in an affine open
AlgebraicCurve.TwoChartIntegralModel.exists_isAffineOpen_forall_mem_of_finset3 below · cited by 3 · depth 14 - Base change of a two-chart integral model is the glued curve
AlgebraicCurve.TwoChartIntegralModel.exists_iso_glued_pullback_toBase_of_algEquiv_chartAlg_chartRing0 below · cited by 5 · depth 14 - Pole chart of j' lies in a unipotent localisation of A_∞(j)
AlgebraicCurve.TwoChartIntegralModel.forall_mem_chartAlgInf_exists_one_add_mul_and_mul_mem0 below · cited by 4 · depth 14 - Reduction of chart rings at a Gauss-type place stays a domain
AlgebraicCurve.TwoChartIntegralModel.isDomain_tensorProduct_chartAlg_of_finrank_le_finrank_adjoin_range14 below · cited by 4 · depth 14 - Integrality of a base change of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.isIntegral_pullback_toBase_of_isDomain_tensorProduct_chartAlg0 below · cited by 7 · depth 14 - Chart rings of the two-chart model are integrally closed
AlgebraicCurve.TwoChartIntegralModel.isIntegrallyClosed_chartAlg1 below · cited by 58 · depth 14 - Two-chart integral model is locally of finite type
AlgebraicCurve.TwoChartIntegralModel.locallyOfFiniteType_toBase0 below · cited by 9 · depth 14 - Minimal primes over varpi are the centres of two branch rings
AlgebraicCurve.TwoChartIntegralModel.mem_minimalPrimes_span_iff_of_valuationSubring_pair6 below · cited by 36 · depth 14 - Fibrewise smoothness criterion for a two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.smoothOfRelativeDimension_one_toBase_ratLocalizedAt_of_forall_pullback_snd1 below · cited by 1 · depth 14 - Zariski connectedness for the two-chart integral model over ℤ_{(ℓ)}
AlgebraicCurve.TwoChartIntegralModel.connectedSpace_pullback_toBase_specMap_ratLocalizedAt5 below · cited by 1 · depth 15 - Two-chart integral model and localisation of the base
AlgebraicCurve.TwoChartIntegralModel.exists_isPullback_toBase_of_isLocalization4 below · cited by 3 · depth 15 - Base change of the two-chart integral model along a localisation
AlgebraicCurve.TwoChartIntegralModel.exists_iso_baseChange_baseChange_of_isLocalization4 below · cited by 4 · depth 15 - DVRs from minimal primes of varpi in the finite chart
AlgebraicCurve.TwoChartIntegralModel.exists_valuationSubring_of_mem_minimalPrimes_chartAlgFin4 below · cited by 55 · depth 15 - Integrality of (1+j⁻¹a)j'⁻¹ gives the two-chart visibility condition
AlgebraicCurve.TwoChartIntegralModel.forall_mem_chartAlgInf_exists_one_add_mul_and_mul_mem_of_isIntegral_mul1 below · cited by 1 · depth 15 - Branch valuation rings as localisations of the finite chart
AlgebraicCurve.TwoChartIntegralModel.le_and_height_eq_one_and_exists_div_of_valuationSubring_of_transcendental4 below · cited by 63 · depth 15 - varpi generates the maximal ideal at minimal primes of (varpi)
AlgebraicCurve.TwoChartIntegralModel.map_span_eq_maximalIdeal_localization_atPrime_of_forall_valuationSubring_mul_inv_mem6 below · cited by 1 · depth 15 - Integrality of j at a place forces the finite chart
AlgebraicCurve.TwoChartIntegralModel.mem_range_iotaFin_of_ffEquiv_symm_germ_mem_placeOfPoint0 below · cited by 3 · depth 15 - Section of the two-chart integral model from an algebra map
AlgebraicCurve.TwoChartIntegralModel.nonempty_schemeHomOver_id_toBase_of_algHom0 below · cited by 2 · depth 15 - Smoothness over k of the two-chart model read off the charts
AlgebraicCurve.TwoChartIntegralModel.smoothOfRelativeDimension_pullback_snd_toBase_of_tensor_charts0 below · cited by 1 · depth 15 - Chart ring modulo varpi at a unique Gauss valuation
AlgebraicCurve.TwoChartIntegralModel.span_singleton_isPrime_and_mem_iff_mem_nonunits_of_valuationSubring8 below · cited by 3 · depth 15 - Chart base change commutes with chart inclusions
AlgebraicCurve.TwoChartIntegralModel.chartIncl_comp_chartBaseChange0 below · cited by 2 · depth 16 - Chart rings localise with the base ring
AlgebraicCurve.TwoChartIntegralModel.isLocalization_chartAlg0 below · cited by 9 · depth 16 - Finite chart of the two-chart model localises on the base
AlgebraicCurve.TwoChartIntegralModel.isPullback_chartFin1 below · cited by 2 · depth 16 - The pole chart base-changes along a localisation
AlgebraicCurve.TwoChartIntegralModel.isPullback_chartInf1 below · cited by 2 · depth 16 - Uniqueness of the valuation above an integral special fibre
AlgebraicCurve.TwoChartIntegralModel.valuationSubring_eq_of_isPrime_span_of_forall_aeval_mem4 below · cited by 5 · depth 16 - Finite chart of a base-changed two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_isOpenImmersion_spec_tensor_chartAlgFin0 below · cited by 17 · depth 17 - Points off the finite chart are poles of j
AlgebraicCurve.TwoChartIntegralModel.exists_jInvChartInf_mem_and_iotaInf_eq_of_not_mem_range_iotaFin0 below · cited by 8 · depth 19 - Pole-chart points in the finite chart: criterion via 1/j
AlgebraicCurve.TwoChartIntegralModel.iotaInf_mem_range_iotaFin_iff0 below · cited by 3 · depth 19 - Function field of the base-changed two-chart model is generated by finite chart and base
AlgebraicCurve.TwoChartIntegralModel.subfieldClosure_range_germToFunctionField_union_range_eq_top0 below · cited by 2 · depth 21 - Semilinear Γ-action on the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_hom_comp_toBase_eq_and_iotaFin_comp_eq_of_mulSemiringAction_of_smul_eq0 below · cited by 4 · depth 22 - Stalks of the base-changed two-chart model over the finite chart
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_tensor_chartAlgFin0 below · cited by 1 · depth 22 - Geometric integrality of the two-chart model's generic fibre
AlgebraicCurve.TwoChartIntegralModel.geometricallyIntegral_baseChange_toBase_of_intermediateField_laurentSeries1 below · cited by 2 · depth 22 - Germs of j and j⁻¹ multiply to 1
AlgebraicCurve.TwoChartIntegralModel.germToFunctionField_jChartFin_mul_germToFunctionField_jInvChartInf0 below · cited by 1 · depth 22 - Characteristic-zero fibres of the two-chart model are reduced
AlgebraicCurve.TwoChartIntegralModel.isReduced_pullback_toBase_of_charZero0 below · cited by 1 · depth 22 - Generic-fibre regularity of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.isRegularLocalRing_stalk_of_asIdeal_eq_bot2 below · cited by 1 · depth 22 - Dimension two at a crossing of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.ringKrullDim_stalk_eq_two_of_not_subsingleton_minimalPrimes1 below · cited by 3 · depth 22 - Closed immersion of a glued two-chart curve into a base change
AlgebraicCurve.TwoChartIntegralModel.exists_isClosedImmersion_glued_pullback_of_surjective0 below · cited by 1 · depth 23 - Finiteness of crossing points in the special fibre
AlgebraicCurve.TwoChartIntegralModel.finite_setOf_not_subsingleton_minimalPrimes_span_germ2 below · cited by 3 · depth 23 - Reducedness of the geometric special fibre of a two-chart model
AlgebraicCurve.TwoChartIntegralModel.isReduced_pullback_toBase_of_forall_map_span_eq_maximalIdeal11 below · cited by 2 · depth 23 - Generic fibre of the j-finite chart is regular of dimension one
AlgebraicCurve.TwoChartIntegralModel.isRegularLocalRing_localization_fractionRing_tensor_chartAlgFin0 below · cited by 3 · depth 23 - Regularity of the pole chart over the fraction field
AlgebraicCurve.TwoChartIntegralModel.isRegularLocalRing_localization_fractionRing_tensor_chartAlgInf0 below · cited by 1 · depth 23 - Stalk dimension bound for the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.ringKrullDim_stalk_le_ringKrullDim_add_one3 below · cited by 8 · depth 23 - Generic fibre of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_iso_glued_pullback_toBase_of_isFractionRing2 below · cited by 2 · depth 24 - Stalk of the two-chart integral model at a finite-chart point
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_chartAlgFin0 below · cited by 45 · depth 24 - 1/j lies in no minimal prime over varpi in the pole chart
AlgebraicCurve.TwoChartIntegralModel.jInvChartInf_not_mem_of_mem_minimalPrimes_span5 below · cited by 2 · depth 24 - Maximality of chart primes over a maximal ideal
AlgebraicCurve.TwoChartIntegralModel.isMaximal_of_map_le_of_aeval_mem0 below · cited by 4 · depth 25 - Stalks of the two-chart integral model over a Noetherian base
AlgebraicCurve.TwoChartIntegralModel.isNoetherianRing_stalk_and_essFiniteType_and_isDomain_and_injective0 below · cited by 24 · depth 25 - Components of the special fibre counted by minimal primes of varpi
AlgebraicCurve.TwoChartIntegralModel.ncard_irreducibleComponents_pullback_toBase_eq_ncard_minimalPrimes_of_surjective1 below · cited by 2 · depth 25 - Krull dimension ≥ 2 at closed special-fibre points of the finite chart
AlgebraicCurve.TwoChartIntegralModel.two_le_ringKrullDim_stalk_of_isMaximal_of_mem1 below · cited by 7 · depth 25 - Global sections of the two-chart model equal R iff R integrally closed in F
AlgebraicCurve.TwoChartIntegralModel.bijective_algebraMap_globalSections_iff_isIntegrallyClosedIn2 below · cited by 1 · depth 26 - Maximal chart points over varpi are closed in the model
AlgebraicCurve.TwoChartIntegralModel.eq_of_specializes_of_isMaximal_of_mem_chart8 below · cited by 7 · depth 26 - Chart rings are G-stable with invariants the lower chart ring
AlgebraicCurve.TwoChartIntegralModel.exists_mulSemiringAction_chartAlg_and_isInvariant_of_isInvariant0 below · cited by 10 · depth 26 - Function field of the two-chart integral model is F
AlgebraicCurve.TwoChartIntegralModel.exists_ringEquiv_functionField_apply_eq_algebraMap_germ3 below · cited by 2 · depth 26 - Function field of the two-chart model: finite-chart normalisation
AlgebraicCurve.TwoChartIntegralModel.exists_ringEquiv_functionField_apply_eq_algebraMap_germ_iotaFin1 below · cited by 2 · depth 26 - Stalk at a finite-chart point of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_atPrime_of_iotaFin_apply_eq0 below · cited by 22 · depth 26 - Stalk of the two-chart model on the j⁻¹-chart
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_chartAlgInf0 below · cited by 6 · depth 26 - Stalk map of a comparison morphism of two-chart models
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_comp_stalkMap_eq_localRingHom0 below · cited by 5 · depth 26 - Discrete valuation through a point of the special fibre
AlgebraicCurve.TwoChartIntegralModel.exists_valuationSubring_forall_mem_nonunits_mem_asIdeal_of_mem_toBase5 below · cited by 1 · depth 26 - Chart criterion for the local ring and maximal ideal at a point
AlgebraicCurve.TwoChartIntegralModel.forall_iff_mem_localRing_and_forall_iff_exists_mem_maximalIdeal2 below · cited by 2 · depth 26 - Primes over varpi in the finite chart: minimal or maximal
AlgebraicCurve.TwoChartIntegralModel.mem_minimalPrimes_or_isMaximal_of_mem_chartAlgFin0 below · cited by 12 · depth 26 - Points over the generic point lie in the smooth locus
AlgebraicCurve.TwoChartIntegralModel.mem_smoothLocus_toBase_of_asIdeal_eq_bot_of_charZero5 below · cited by 1 · depth 26 - Smoothness of the two-chart integral model over its base
AlgebraicCurve.TwoChartIntegralModel.smooth_toBase_iff_smooth_chartAlgFin_and_chartAlgInf0 below · cited by 3 · depth 26 - Integral over R[j] and over R[j⁻¹] implies integral over R
AlgebraicCurve.TwoChartIntegralModel.chartAlgFin_inf_chartAlgInf_eq_integralClosure0 below · cited by 1 · depth 27 - Global sections of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_algEquiv_globalSections_chartAlgFin_inf_chartAlgInf0 below · cited by 1 · depth 27 - Transport of primes of the finite chart ring along an automorphism
AlgebraicCurve.TwoChartIntegralModel.exists_bijective_primes_chartAlgFin_localization_iff_of_forall_mem_iff0 below · cited by 3 · depth 27 - Transport of cusp primes by an automorphism fixing j'
AlgebraicCurve.TwoChartIntegralModel.exists_bijective_primes_chartAlgInf_localization_iff_of_algEquiv_apply_eq0 below · cited by 3 · depth 27 - Function with exact pole order along coordinate branches
AlgebraicCurve.TwoChartIntegralModel.exists_forall_mem_localization_chartAlg_and_mul_pow_jChartFin_isUnit_of_branchData75 below · cited by 1 · depth 27 - Glued two-chart curve mapping to the base-changed integral model
AlgebraicCurve.TwoChartIntegralModel.exists_glued_hom_pullback_of_compatible0 below · cited by 1 · depth 27 - Finite sets of the base change lie in one affine open
AlgebraicCurve.TwoChartIntegralModel.exists_isAffineOpen_subset_of_finite_of_isClosed_baseChange4 below · cited by 1 · depth 27 - Étale coordinate and residue character at a smooth special point
AlgebraicCurve.TwoChartIntegralModel.exists_subring_etaleCoordinate_residueChar_iff_charts_of_smooth_of_isDiscreteValuationRing60 below · cited by 3 · depth 27 - Finiteness of crossings in a field base change of the two-chart model
AlgebraicCurve.TwoChartIntegralModel.finite_setOf_pullback_fst_mem_not_subsingleton_minimalPrimes_span_germ4 below · cited by 1 · depth 27 - Gauss-point valuations of g are the A_{mathfrak q_i}
AlgebraicCurve.TwoChartIntegralModel.forall_over_gauss_iff_exists_forall_mem_iff_of_mul_pow_isUnit_of_forall_mem_localization11 below · cited by 1 · depth 27 - Birationality of the glued curve onto the base-changed integral model
AlgebraicCurve.TwoChartIntegralModel.isIso_stalkMap_genericPoint_glued_hom_of_ker_mem_minimalPrimes3 below · cited by 1 · depth 27 - Chart-wise image of the glued curve as a kernel zero locus
AlgebraicCurve.TwoChartIntegralModel.mem_range_glued_hom_iff_ker_le_of_chart_eq1 below · cited by 1 · depth 27 - Chart compatibility at j⁻¹ follows from that at j
AlgebraicCurve.TwoChartIntegralModel.ringEquiv_functionField_apply_eq_algebraMap_germ_iotaInf_of_iotaFin0 below · cited by 1 · depth 27 - Positive-degree divisor bounding chart sections of the two-chart model
AlgebraicCurve.TwoChartIntegralModel.exists_divisor_degree_pos_lSpaceOn_nsmul_le_of_branchData6 below · cited by 1 · depth 28 - Polynomials in a pole function are units of A_q
AlgebraicCurve.TwoChartIntegralModel.forall_aeval_mem_and_inv_mem_of_mul_pow_mul_eq_of_le_of_isMaximal0 below · cited by 1 · depth 28 - Transport of chart algebras along a field isomorphism
AlgebraicCurve.TwoChartIntegralModel.mem_chartAlg_iff_mem_chartAlg_image_of_ringEquiv0 below · cited by 9 · depth 28 - A localisation regular along neighbouring primes is not the Gauss point
AlgebraicCurve.TwoChartIntegralModel.not_forall_aeval_mem_and_inv_mem_of_forall_lt_of_forall_jInvChartInf_mem0 below · cited by 1 · depth 28 - Germ of a constant in y is a non-unit on the model
AlgebraicCurve.TwoChartIntegralModel.exists_asIdeal_eq_and_germ_mem_maximalIdeal_stalk_of_isMaximal_of_mem1 below · cited by 8 · depth 29 - Frame isomorphisms induce isomorphisms of two-chart integral models
AlgebraicCurve.TwoChartIntegralModel.exists_iso_of_ringEquiv_of_ringEquiv_apply_eq0 below · cited by 6 · depth 29 - Proper varpi-torsion-free chart ideal is centred at a place
AlgebraicCurve.TwoChartIntegralModel.exists_place_forall_mem_and_forall_mem_nonunits_of_forall_mul_eq_zero_imp3 below · cited by 1 · depth 29 - Places containing j detect the j-chart up to varpi-powers
AlgebraicCurve.TwoChartIntegralModel.exists_pow_mul_mem_chartAlgFin_of_forall_place2 below · cited by 1 · depth 29 - No monic value p(j) in mathfrak m_V at a non-maximal centre
AlgebraicCurve.TwoChartIntegralModel.forall_monic_aeval_not_mem_maximalIdeal_of_not_isMaximal_centre0 below · cited by 9 · depth 29 - Integrally closed R[j]-integral subalgebras equal the finite chart algebra
AlgebraicCurve.TwoChartIntegralModel.eq_chartAlgFin_of_isIntegral_of_integrallyClosed0 below · cited by 2 · depth 30 - Base change of the integral closure along a map of constants
AlgebraicCurve.TwoChartIntegralModel.exists_algEquiv_tensorProduct_chartAlg_adjoin_coeffMap_of_isIntegrallyClosed2 below · cited by 2 · depth 30 - Completed stalk of the two-chart model at a dominated point
AlgebraicCurve.TwoChartIntegralModel.exists_ringEquiv_adicCompletion_stalk_adicCompletion_comap_of_dominates1 below · cited by 2 · depth 30 - Flatness of the chart algebra over a Bézout domain
AlgebraicCurve.TwoChartIntegralModel.flat_chartAlg0 below · cited by 6 · depth 30 - Geometric normality of the chart ring L ⊗_R C
AlgebraicCurve.TwoChartIntegralModel.isDomain_and_isIntegrallyClosed_tensorProduct_chartAlgFin_of_le_laurentSeries13 below · cited by 2 · depth 30 - Supersingular closed points contract along maps of j-chart rings
AlgebraicCurve.TwoChartIntegralModel.isMaximal_comap_and_mem_comap_and_mem_ssJSet_of_ringHom_chartAlgFin27 below · cited by 2 · depth 30 - Power series expansion of the pole chart ring at ∞
AlgebraicCurve.TwoChartIntegralModel.exists_ringHom_powerSeries_chartAlgInf_coe_eq_and_algebraMap_eq_coeff_zero0 below · cited by 10 · depth 31 - Stalks of the base-changed model at pole-chart points
AlgebraicCurve.TwoChartIntegralModel.exists_stalk_iso_localization_tensor_chartAlgInf0 below · cited by 1 · depth 31 - Smoothness of the generic fibre of the j-finite chart ring
AlgebraicCurve.TwoChartIntegralModel.smooth_tensorProduct_chartAlgFin_of_charZero7 below · cited by 1 · depth 31 - Base change of the pole chart of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_isOpenImmersion_spec_tensor_chartAlgInf0 below · cited by 3 · depth 33 - Residue fields of the finite chart over algebraically closed κ
AlgebraicCurve.TwoChartIntegralModel.exists_sub_algebraMap_mem_of_isMaximal_chartAlgFin0 below · cited by 6 · depth 33 - Germs of a constant agree through chart and base
AlgebraicCurve.TwoChartIntegralModel.germ_iotaFin_algebraMap_eq_germ_top_toBase_appTop1 below · cited by 9 · depth 33 - Centre of V is a minimal prime over varpi in the finite chart
AlgebraicCurve.TwoChartIntegralModel.exists_isPrime_mem_iff_mem_nonunits_mem_minimalPrimes_span_of_valuationSubring0 below · cited by 2 · depth 35 - Finite free base change of the two-chart integral model
AlgebraicCurve.TwoChartIntegralModel.exists_isPullback_toBase_specMap_and_iotaFin_comp_eq_of_isPushout_of_chartAlg_eq_span0 below · cited by 1 · depth 35 - Functoriality and integrality of the finite chart ring along φ
AlgebraicCurve.TwoChartIntegralModel.exists_ringHom_chartAlgFin_coe_eq_and_isIntegral_and_forall_exists_of_algHom0 below · cited by 2 · depth 35 - Chart rings and power bases of unit discriminant
AlgebraicCurve.TwoChartIntegralModel.mem_chartAlg_image_iff_exists_eq_sum_mul_pow_of_powerBasis_of_isUnit_discr0 below · cited by 1 · depth 35 - Chart algebra at a transcendental j is finite over R[j]
AlgebraicCurve.TwoChartIntegralModel.moduleFinite_adjoin_jChartFin_chartAlgFin0 below · cited by 4 · depth 35
AlgebraicCurve.WeilDatum 6
- Multiplicativity of the Weil pairing in the first argument
AlgebraicCurve.WeilDatum.addLeft_pairing6 below · cited by 1 · depth 15 - Invariance of the Weil datum pairing under linear equivalence
AlgebraicCurve.WeilDatum.pairing_eq_of_isPrincipal_sub14 below · cited by 1 · depth 15 - Non-vanishing of the Weil datum pairing
AlgebraicCurve.WeilDatum.pairing_ne_zero3 below · cited by 1 · depth 15 - Antisymmetry of the pairing of a Weil datum
AlgebraicCurve.WeilDatum.symm_pairing0 below · cited by 2 · depth 15 - Pairing invariance under purely inseparable pullback and pushforward
AlgebraicCurve.WeilDatum.pairing_eq_pairing_of_pullbackAlong_of_pushforwardAlong_of_isPurelyInseparable12 below · cited by 1 · depth 27 - Weil pairing adjunction along a finite separable map
AlgebraicCurve.WeilDatum.pairing_eq_pairing_of_pullbackAlong_of_pushforwardAlong_of_separableAlong23 below · cited by 1 · depth 27