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Namespace CerednikDrinfeld 2,376 theorems

— 237 · BruhatTits 17 · CSTower 5 · CartierLift 1 · CosetGraph 40 · EdgeFamily 3 · FormalODModule 225 · FormalOmega 222 · FormalQuotientDatum 2 · GradedCartierModuleData 49 · HeckeTower 5 · JPrimeTorsionDatum 2 · LevelU 3 · Mumford 50 · Omega 178 · OmegaNr 1 · Onr 3 · QM 1063 · ShimuraCurveModel 10 · SpecialFormal 197 · SpecialFormalODModule 54 · SpecialModule 3 · TwoPlaceTorsionDatum 3 · UnramQuad 3

directly in CerednikDrinfeld 237

• Nonvanishing 𝔪-torsion for doubly-new eigenform ideals CerednikDrinfeld.mTorsionNeBot_of_isTwoNewEigenformIdeal 0 below · cited by 1 · depth 10
• Existence of a two-place p-torsion datum with laws CerednikDrinfeld.exists_twoPlaceTorsionDatum_laws_of_ssLevelDatum_of_squarefree_of_six_mul_dvd_of_neZero 10,433 below · cited by 1 · depth 14
• Class-set Hecke laws for an Eichler order and its meet order CerednikDrinfeld.classSetHeckeLaws_of_isEichlerOrder_meetOrder 47 below · cited by 2 · depth 15
• Two-level Deuring transport of class sets to supersingular places CerednikDrinfeld.exists_equiv_classSet_ssPlaces_degeneracy_hecke_comm 1,356 below · cited by 2 · depth 15
• Two-place p-torsion datum from Čerednik–Drinfeld over class-set graphs CerednikDrinfeld.exists_twoPlaceTorsionDatum_laws_classSet_of_squarefree_of_six_mul_dvd_of_neZero 10,428 below · cited by 1 · depth 15
• Matching class-set Hecke data with supersingular Hecke data CerednikDrinfeld.nonempty_matching_classSetHeckeData_heckeData 1 below · cited by 1 · depth 15
• Right translation by n corresponds to Atkin–Lehner on supersingular places CerednikDrinfeld.autOnPlaces_eq_of_isAtkinLehnerLevelAut_of_forall_toValuationSubring_eq_comap_moduliPlace 575 below · cited by 1 · depth 16
• Cartier anchors for toric monodromy, with witnesses identified CerednikDrinfeld.exists_cartierAnchors_degeneracyDuality_jZero_ssPlaces_correspondence_arithFrobC_restrictAlong_placeWidthChar 3,361 below · cited by 1 · depth 16
• Eichler class set bijects with level-N supersingular places CerednikDrinfeld.exists_classSet_equiv_ssPlaces_forall_toValuationSubring_eq_comap_moduliPlace_ker 726 below · cited by 1 · depth 16
• Joint degeneracy push-forward is onto degree-zero divisors CerednikDrinfeld.exists_mem_characterLattice_mulVec_eq_pair_of_connected_of_not_bipartite 0 below · cited by 1 · depth 16
• Shimura curve model with toric uniformisations at q and q' CerednikDrinfeld.exists_shimuraCurveModel_goodReduction_and_toricUniformization_pair_of_six_mul_dvd_of_neZero 10,423 below · cited by 1 · depth 16
• Degeneracy pushforwards intertwine edge and vertex Hecke matrices away from q CerednikDrinfeld.jointDelta_classSetEdgeHecke_mulVecLin_eq_classSetVertexHecke_mulVecLin_jointDelta_of_ne 20 below · cited by 1 · depth 16
• U_q preserves the joint kernel of the class-set degeneracy maps CerednikDrinfeld.jointDelta_classSetEdgeHecke_mulVecLin_eq_zero_of_forall_jointDelta_eq_zero_of_mem_primeHeckeSet 39 below · cited by 1 · depth 16
• Bi-invariance of the level Hecke set under the idelic stabiliser CerednikDrinfeld.mul_mem_levelHeckeUSet_and_mul_mem_levelHeckeUSet_of_mem_finiteIdeleStabilizer 5 below · cited by 5 · depth 16
• Bi-invariance of the mathcal U_q Hecke set under stabiliser units CerednikDrinfeld.mul_mem_uHeckeSet_and_mul_mem_uHeckeSet_of_mem_finiteIdeleStabilizer_meetOrder 5 below · cited by 2 · depth 16
• Degeneracy inclusion is compatible with the two class-set dictionaries CerednikDrinfeld.restrictAlong_levelAlphaC_eq_of_forall_toValuationSubring_eq_comap_moduliPlace_of_prime 513 below · cited by 1 · depth 16
• Matched Hecke data: isometric equivariant ribbon kernels CerednikDrinfeld.ribbon_kernelEquiv 0 below · cited by 1 · depth 16
• Frobenius matrix on supersingular places equals the prime Hecke matrix CerednikDrinfeld.ssFrobMatrixC_apply_eq_classSetHeckeMatrix_primeHeckeSet_of_forall_toValuationSubring_eq_comap_moduliPlace 528 below · cited by 1 · depth 16
• Supersingular U_ℓ matrix equals class-set Hecke matrix, ℓ ∣ N CerednikDrinfeld.ssHeckeMatrixC_apply_eq_classSetHeckeMatrix_levelHeckeUSet_of_dvd_of_forall_toValuationSubring_eq_comap_moduliPlace_of_five_le 1,082 below · cited by 1 · depth 16
• Supersingular Hecke matrix equals the Brandt matrix at ℓ CerednikDrinfeld.ssHeckeMatrixC_apply_eq_classSetHeckeMatrix_primeHeckeSet_of_forall_toValuationSubring_eq_comap_moduliPlace 593 below · cited by 1 · depth 16
• Place width equals class weight at level Nq CerednikDrinfeld.toPNat_placeWidth_eq_classWeight_of_forall_toValuationSubring_eq_comap_moduliPlace 623 below · cited by 1 · depth 16
• Two descriptions of the Uₛ-set on the meet order agree CerednikDrinfeld.uHeckeSet_eq_levelHeckeUSet_meetOrder_of_mem_primeHeckeSet 34 below · cited by 1 · depth 16
• Deuring correspondence: equal idèle classes iff isomorphic curves CerednikDrinfeld.classSet_mk_eq_iff_nonempty_variableChange_of_kernelIdealSet 115 below · cited by 6 · depth 17
• Level-one Deuring correspondence with Brandt matrices CerednikDrinfeld.exists_classSet_equiv_ssPlaces_one_kernelIdealSet_of_rationalEndSubring 721 below · cited by 1 · depth 17
• Kernel-ideal realisation of xg by an isogeny from W CerednikDrinfeld.exists_dualPair_image_kernelIdealSet_comp_eq_star_smul_ofFiniteIdele_mul_of_mem_finiteAdeleBox 100 below · cited by 2 · depth 17
• Type-m sublattices of Iₓ and cyclic N-subgroups of W CerednikDrinfeld.exists_equiv_ofFiniteIdele_mul_isAddCyclic_forall_ker_eq_of_kernelIdealSet_comp_eq 194 below · cited by 1 · depth 17
• Fibre of the R-class set over a Λ₁-class CerednikDrinfeld.exists_fibre_classSetForget_equiv_quot_ofFiniteIdele_mul_of_eq_inf_conjByFiniteIdele 12 below · cited by 1 · depth 17
• Realisation of finite idele classes by cyclic N-isogenies CerednikDrinfeld.exists_kernelIdealSet_realisation_isAddCyclic_ker_of_inf_conjByFiniteIdele 173 below · cited by 7 · depth 17
• Shimura curve model with period uniformisation at both q,q' CerednikDrinfeld.exists_shimuraCurveModel_goodReduction_and_periodUniformization_pair_of_six_mul_dvd_of_neZero 10,403 below · cited by 1 · depth 17
• Unit translate of lattices iff isomorphic pairs (W,kerψ) CerednikDrinfeld.exists_smul_eq_iff_exists_ker_eq_map_of_comp_eq_smul_id_of_card_ker_eq 182 below · cited by 1 · depth 17
• Two-level joint semistable specialisation with pinned Hecke transport CerednikDrinfeld.exists_twoLevelSemistableSpecialization_jointConstruction_ssPlaces_heckeTransport_canonical_levelPrimeIntertwine_correspondence_arithFrobC_restrictAlong_placeWidthChar 3,351 below · cited by 1 · depth 17
• Joint two-level semistable specialisation with degeneracy and Hecke transport CerednikDrinfeld.exists_twoLevelSemistableSpecialization_jointConstruction_ssPlaces_heckeTransport_correspondence_restrictAlong_degeneracyComp_placeWidthChar 3,338 below · cited by 1 · depth 17
• Kernel ideal of an intermediate quotient, coprime case CerednikDrinfeld.image_kernelIdealSet_comp_eq_of_ker_eq_div_nsmul_ker_of_coprime 11 below · cited by 2 · depth 17
• Kernel-ideal transport along a Hecke idele at q CerednikDrinfeld.image_kernelIdealSet_comp_eq_star_smul_ofFiniteIdele_mul_and_exists_dualPair_ker_eq_map_of_meetOrder_eq_of_conjByFiniteIdele_eq 113 below · cited by 1 · depth 17
• Frobenius twist shifts the kernel ideal by P CerednikDrinfeld.image_kernelIdealSet_ratPointHom_frobenius_comp_eq_star_smul_ofFiniteIdele_mul 50 below · cited by 1 · depth 17
• Edge Hecke operator at q: b_* U_q = q a_* CerednikDrinfeld.jointDelta_one_classSetEdgeHecke_mulVecLin_eq_natCast_smul_jointDelta_zero_of_mem_primeHeckeSet 33 below · cited by 1 · depth 17
• Degeneracy relation a_*U_q=T_qa_*-b_* on class sets CerednikDrinfeld.jointDelta_zero_classSetEdgeHecke_mulVecLin_eq_classSetVertexHecke_mulVecLin_sub_of_mem_primeHeckeSet 33 below · cited by 1 · depth 17
• Units of a conjugated Eichler order count automorphisms preserving kerψ CerednikDrinfeld.natCard_isUnitOf_conjByFiniteIdele_eq_natCard_rationalAut_map_ker_eq_of_image_kernelIdealSet_comp_eq 216 below · cited by 1 · depth 17
• Squared isogeny degree equals the relative index of kernel ideals CerednikDrinfeld.natCard_ker_sq_eq_relIndex_ofFiniteIdele_mul_of_image_kernelIdealSet_comp_eq 106 below · cited by 3 · depth 17
• Brandt U_ℓ count at a prime dividing the level CerednikDrinfeld.natCard_ofFiniteIdele_levelHeckeUSet_eq_natCard_subgroup_dualPair_ker_of_dvd_of_inf_conjByFiniteIdele 191 below · cited by 1 · depth 17
• Level-N Brandt count equals enhanced supersingular ℓ-isogeny count CerednikDrinfeld.natCard_ofFiniteIdele_primeHeckeSet_eq_natCard_subgroup_dualPair_ker_of_inf_conjByFiniteIdele 176 below · cited by 1 · depth 17
• Degeneracy push-forwards intertwine the U_ℓ class-set matrices CerednikDrinfeld.pushforward_classSetHeckeMatrix_levelHeckeUSet_meetOrder_mulVecLin 18 below · cited by 1 · depth 17
• Hecke matrices commute with both class-set degeneracy push-forwards CerednikDrinfeld.pushforward_classSetHeckeMatrix_primeHeckeSet_meetOrder_mulVecLin 11 below · cited by 1 · depth 17
• Component-group duality: dim_k Ψ[𝔪]=dim_k Ψ^{dagger}/𝔪Ψ^{dagger} CerednikDrinfeld.ribbon_finrank_torsion_eq_finrank_quotient_componentGroup 0 below · cited by 2 · depth 17
• Conjugation by a q-Hecke idèle preserves the level-ℓ Hecke set CerednikDrinfeld.conj_mem_levelHeckeUSet_iff_of_mem_primeHeckeSet 14 below · cited by 1 · depth 18
• Every finite idèle class arises as a kernel ideal CerednikDrinfeld.exists_kernelIdealSet_eq_star_smul_ofFiniteIdele 128 below · cited by 2 · depth 18
• Kernel ideal of ψ∘χ as an adelic lattice CerednikDrinfeld.exists_mem_finiteAdeleBox_image_kernelIdealSet_comp_eq_star_smul_ofFiniteIdele_mul_of_dualPair 130 below · cited by 1 · depth 18
• Iwahori U_q-elements times the q-idele lie in qwidehatR^× CerednikDrinfeld.exists_mem_finiteIdeleStabilizer_mul_eq_natCast_smul_of_mem_uHeckeSet 27 below · cited by 2 · depth 18
• Deuring's isomorphism criterion with markings and level transport CerednikDrinfeld.exists_mem_rationalHomSet_comp_eq_id_forall_mem_ker_of_image_kernelIdealSet_eq_image_mul 115 below · cited by 4 · depth 18
• Endomorphism ring of an isogenous curve as left order CerednikDrinfeld.exists_ringHom_range_eq_conjByFiniteIdele_forall_apply_eq_mul_of_image_kernelIdealSet_eq 213 below · cited by 1 · depth 18
• Čerednik–Drinfeld equivariant uniformisation at both ramified primes CerednikDrinfeld.exists_shimuraCurveModel_goodReduction_and_equivariantUniformization_pair_of_six_mul_dvd_of_neZero 10,401 below · cited by 1 · depth 18
• Joint two-level semistable specialisation with widths and arithmetic Frobenius CerednikDrinfeld.exists_twoLevelSemistableSpecialization_jointConstruction_ssPlaces_heckeTransport_correspondence_restrictAlong_degeneracyComp_placeWidthChar_frobArithFrobC 3,338 below · cited by 1 · depth 18
• Isomorphic targets give kernel ideals differing by a unit CerednikDrinfeld.exists_units_image_kernelIdealSet_eq_image_mul_of_exists_variableChange 18 below · cited by 5 · depth 18
• Isomorphic enhanced curves give kernel-ideal pairs differing by a unit CerednikDrinfeld.exists_units_image_kernelIdealSet_pair_eq_image_mul_of_comp_eq_id_forall_mem_ker 176 below · cited by 1 · depth 18
• One prime Hecke step in the kernel-ideal dictionary CerednikDrinfeld.forall_exists_natCard_eq_image_setOf_comp_eq_star_smul_ofFiniteIdele_mul_of_mem_primeHeckeSet 29 below · cited by 1 · depth 18
• U_q is minus the shift on the ribbon kernel CerednikDrinfeld.heckeKernelMap_classSetHeckeData_apply_eq_neg_comp_classSetShift 43 below · cited by 2 · depth 18
• Deuring: Frobenius kernel ideal is the prime above q' CerednikDrinfeld.image_kernelIdealSet_ratPointHom_frobenius_eq_setOf_padicValRat_nrd 28 below · cited by 1 · depth 18
• Cyclic kernel versus primitivity of the idele g CerednikDrinfeld.isAddCyclic_ker_iff_forall_inv_smul_not_mem_finiteAdeleBox_of_image_kernelIdealSet_comp_eq 106 below · cited by 1 · depth 18
• Primitivity of g forces cyclic kernel for ψ CerednikDrinfeld.isAddCyclic_ker_of_forall_inv_smul_not_mem_finiteAdeleBox_of_image_kernelIdealSet_comp_eq 26 below · cited by 1 · depth 18
• Level-ℓ Hecke sets of nested orders agreeing at ℓ CerednikDrinfeld.mem_levelHeckeUSet_iff_mem_levelHeckeUSet_of_forall_localBox_eq 14 below · cited by 1 · depth 18
• Prime-ℓ Hecke sub-ideals versus dual-pair ℓ-isogeny kernels CerednikDrinfeld.natCard_subideal_primeHeckeSet_eq_natCard_subgroup_dualPair_of_kernelIdealSet 168 below · cited by 1 · depth 18
• Iwahori Hecke set meets exactly q cosets CerednikDrinfeld.ncard_setOf_exists_mem_uHeckeSet_quotientMk_eq_of_mem_primeHeckeSet 29 below · cited by 1 · depth 18
• Iwahori cosets at q biject with Hecke cosets avoiding n CerednikDrinfeld.uHeckeSet_quotient_bijOn_primeHeckeSet_quotient_diff_of_prime 29 below · cited by 1 · depth 18
• The U_q Hecke set is contained in the T_q Hecke set CerednikDrinfeld.uHeckeSet_subset_primeHeckeSet 1 below · cited by 1 · depth 18
• Kernel ideal of Frobenius is the prime above q' CerednikDrinfeld.exists_injective_mem_rationalHomSet_kernelIdealSet_eq_nrd_dvd 28 below · cited by 1 · depth 19
• Quotient-graph presentation of the Mumford side at q' CerednikDrinfeld.exists_quotientPresentation_classSet_hecke_of_descentIntertwining_one_zero 3,953 below · cited by 1 · depth 19
• Quotient-graph presentation at q with class set and Hecke CerednikDrinfeld.exists_quotientPresentation_classSet_hecke_of_descentIntertwining_zero_one 3,951 below · cited by 1 · depth 19
• Shimura curve model, Hecke tower and Čerednik interchange pair CerednikDrinfeld.exists_shimuraCurveModel_heckeTower_and_interchangeData_pair_of_six_mul_dvd_of_neZero 10,249 below · cited by 1 · depth 19
• Symmetry group with compatible semilinear actions over q' CerednikDrinfeld.exists_symmetryGroup_semilinearAction_invariantFieldOf_of_descentIntertwining_one_zero 43 below · cited by 1 · depth 19
• Symmetry group of the Čerednik–Drinfeld tower at q CerednikDrinfeld.exists_symmetryGroup_semilinearAction_invariantFieldOf_of_descentIntertwining_zero_one 39 below · cited by 1 · depth 19
• Invariant fields of the level groups at q' are curves CerednikDrinfeld.isCurveOver_invariantFieldOf_levelGroups_of_descentIntertwining_one_zero 3,838 below · cited by 1 · depth 19
• Mumford fields of the level groups are curves over C CerednikDrinfeld.isCurveOver_invariantFieldOf_levelGroups_of_descentIntertwining_zero_one 3,838 below · cited by 1 · depth 19
• Exactly q local cosets for the prime Hecke set at q CerednikDrinfeld.ncard_setOf_quotientMk_stabilizer_localBox_meetOrder_eq_of_mem_primeHeckeSet 25 below · cited by 1 · depth 19
• Iwahori U_q-cosets as unit translates of an Atkin–Lehner idele CerednikDrinfeld.uHeckeSet_cosets_eq_finiteIdeleStabilizer_mul_of_conjByFiniteIdele_meetOrder_eq 40 below · cited by 1 · depth 19
• Level-group vertex stabilisers have order prime to q' CerednikDrinfeld.valuation_natCard_stabilizer_vertex_levelGroups_eq_one_of_descentIntertwining_one_zero 3,792 below · cited by 1 · depth 19
• Vertex stabilisers of the level groups have order prime to q CerednikDrinfeld.valuation_natCard_stabilizer_vertex_levelGroups_eq_one_of_descentIntertwining_zero_one 3,792 below · cited by 1 · depth 19
• Realisation-independent permutation actions on quotients of the Bruhat–Tits tree CerednikDrinfeld.exists_perm_quotVert_quotEdge_realisation_independent_of_descentIntertwining_one_zero 3,952 below · cited by 1 · depth 20
• Realisation-independent permutation actions on Bruhat–Tits tree quotients CerednikDrinfeld.exists_perm_quotVert_quotEdge_realisation_independent_of_descentIntertwining_zero_one 3,950 below · cited by 1 · depth 20
• Isomorphic degeneracy data have isometric ribbon kernels CerednikDrinfeld.exists_ribbonKernel_linearEquiv_ribbonGram_eq_of_equiv 0 below · cited by 4 · depth 20
• Signed permutation action on a ribbon kernel by isometries CerednikDrinfeld.exists_ribbonKernel_monoidHom_apply_perm_eq_sgn_mul 0 below · cited by 4 · depth 20
• Čerednik interchange at q and q' for X^{qq'}₀(N) CerednikDrinfeld.exists_shimuraCurveModel_heckeTower_and_cerednikInterchange_pair_of_six_mul_dvd_of_neZero 10,236 below · cited by 1 · depth 20
• Global norm-ℓ element realising the U-Hecke idele at ℓ ∣ N CerednikDrinfeld.exists_units_finiteIdele_levelHeckeUSet_meetOrder_eq_tmul_one_of_dvd 75 below · cited by 1 · depth 20
• Global quaternion of reduced norm ℓ away from q CerednikDrinfeld.exists_units_finiteIdele_primeHeckeSet_meetOrder_eq_tmul_one_of_not_dvd 45 below · cited by 1 · depth 20
• No p-torsion for p≥ 5 in finite subgroups of ρ(H^×) CerednikDrinfeld.not_dvd_natCard_of_le_map_quaternion_units_of_prime_of_five_le 0 below · cited by 5 · depth 20
• Push-forward after pull-back is the class-set Hecke map at ℓ CerednikDrinfeld.pushforward_pullback_eq_heckeKernelMap_of_mapE_comp_eq 98 below · cited by 4 · depth 20
• Unfolding a degeneracy datum onto two vertex sheets changes nothing CerednikDrinfeld.ribbonKernel_unfold_eq_and_ribbonGram_eq_and_heckeKernelMap_eq 0 below · cited by 4 · depth 20
• Iwahori U_q-cosets versus T_q-cosets for an Eichler order CerednikDrinfeld.uHeckeSet_cosetDictionary_of_mem_primeHeckeSet 24 below · cited by 1 · depth 20
• Fixing the Δ-invariant field forces ρ(g)∈ρ(Δ) CerednikDrinfeld.apply_mem_map_of_forall_smul_invariantFieldOf_eq_of_relIndex_ne_zero_of_frame 3,899 below · cited by 2 · depth 21
• Elements fixing the Mumford invariant field lie in ρ(Δ) CerednikDrinfeld.apply_mem_map_of_forall_smul_invariantFieldOf_eq_of_relIndex_ne_zero_of_frame_one_zero 3,899 below · cited by 2 · depth 21
• Forgetful class-set map computed on idelic representatives CerednikDrinfeld.classSetForget_mk_of_le 0 below · cited by 1 · depth 21
• Class of ̄ w x equals the varpi'-shift of x CerednikDrinfeld.classSet_mk_eq_classSetShift_mk_of_finiteAdeleEvalAt_eq_mul_of_nrd_eq 41 below · cited by 1 · depth 21
• Single-place shift by s on the quaternionic class set CerednikDrinfeld.classSet_mk_eq_mk_mul_of_finiteAdeleEvalAt_eq_inv_mul 27 below · cited by 2 · depth 21
• Čerednik descent intertwining: base level implies all levels CerednikDrinfeld.descentIntertwining_of_base_one_zero 3,911 below · cited by 1 · depth 21
• Čerednik–Drinfeld descent intertwining: all levels from the base level CerednikDrinfeld.descentIntertwining_of_base_zero_one 3,910 below · cited by 1 · depth 21
• Forget and shift degeneracy morphisms of class-set graphs, degree ℓ+1 CerednikDrinfeld.exists_finiteHom_classSetDegeneracyData_meetOrder_forget_and_shift_degTotal_eq_add_one 40 below · cited by 1 · depth 21
• Two degree-ℓ morphisms of class-set degeneracy data CerednikDrinfeld.exists_finiteHom_classSetDegeneracyData_meetOrder_forget_and_shift_degTotal_eq_of_dvd 67 below · cited by 1 · depth 21
• Atkin–Lehner rigidity at a split Hecke prime CerednikDrinfeld.exists_mul_self_eq_finiteIdeleDiagonal_mul_and_mem_finiteIdeleStabilizer_meetOrder_of_conjByFiniteIdele_mul_eq 22 below · cited by 5 · depth 21
• Čerednik interchange and Hecke tower for X^{qq'}₀(N) CerednikDrinfeld.exists_shimuraCurveModel_heckeTower_and_cerednikInterchangeBase_pair_of_six_mul_dvd_of_neZero 10,217 below · cited by 1 · depth 21
• Oriented level-ℓ Hecke set is one double coset of ̂ R^× CerednikDrinfeld.levelHeckeUSet_eq_doubleCoset_finiteIdeleStabilizer_of_dvd_of_squarefree 52 below · cited by 3 · depth 21
• Meet order with an idele trivial at v has the same local box CerednikDrinfeld.localBox_meetOrder_eq_of_forall_finiteAdeleEvalAt_eq_one 27 below · cited by 1 · depth 21
• Completion at the ramified place is unchanged by meeting with a conjugate CerednikDrinfeld.localBox_meetOrder_eq_of_isDefiniteRamifiedExactlyAt 21 below · cited by 2 · depth 21
• Local box of R ∩ n̂ R n⁻¹ at a place where n is a local unit CerednikDrinfeld.localBox_meetOrder_eq_of_map_finiteAdeleEvalAt_mem_localBoxUnits 27 below · cited by 2 · depth 21
• Prime Hecke set at q meets exactly q+1 unit cosets CerednikDrinfeld.natCard_setOf_exists_mem_primeHeckeSet_quotientMk_eq_eq_succ_of_prime 17 below · cited by 3 · depth 21
• Push–pull identity for U_ℓ on class-set edge divisors CerednikDrinfeld.pushforward_comp_pullbackFun_eq_classSetHeckeMatrix_levelHeckeUSet_mulVec_of_dvd 59 below · cited by 1 · depth 21
• Push–pull along class-set degeneracy maps computes T_ℓ CerednikDrinfeld.pushforward_comp_pullbackFun_eq_classSetHeckeMatrix_primeHeckeSet_mulVec_of_degTotal_eq 28 below · cited by 1 · depth 21
• Index ℓ of an idelic stabiliser of a meet order CerednikDrinfeld.relIndex_finiteIdeleStabilizer_meetOrder_eq_of_mem_levelHeckeUSet_meetOrder_of_dvd 53 below · cited by 3 · depth 21
• Index ℓ of the stabiliser of a meet order in U_R CerednikDrinfeld.relIndex_finiteIdeleStabilizer_meetOrder_eq_of_mem_levelHeckeUSet_meetOrder_of_mem_primeHeckeSet 53 below · cited by 2 · depth 21
• Index ℓ+1 for the stabiliser of a prime Hecke meet order CerednikDrinfeld.relIndex_finiteIdeleStabilizer_meetOrder_eq_succ_of_mem_primeHeckeSet 26 below · cited by 3 · depth 21
• Shift by a normalising idele acts by right multiplication CerednikDrinfeld.classSetShift_mk_of_conjByFiniteIdele_eq 27 below · cited by 2 · depth 22
• Descent intertwining base above q' from an oriented moduli witness CerednikDrinfeld.exists_descentIntertwiningBase_of_rigidOrientedModuliWitness_one_zero_of_two_mul_dvd_of_neZero 9,893 below · cited by 1 · depth 22
• Descent intertwining base at q from a rigid oriented moduli witness CerednikDrinfeld.exists_descentIntertwiningBase_of_rigidOrientedModuliWitness_zero_one_of_two_mul_dvd_of_neZero 9,892 below · cited by 1 · depth 22
• Rigid oriented moduli witness, Eichler–Shimura relation, Hecke tower CerednikDrinfeld.exists_shimuraCurveModel_rigidOrientedModuliWitness_heckeTower_of_six_mul_dvd_of_neZero 6,271 below · cited by 1 · depth 22
• Truncated norm-p idele lies in the prime Hecke set CerednikDrinfeld.mem_primeHeckeSet_of_nrd_eq_of_forall_finiteAdeleEvalAt_eq 17 below · cited by 1 · depth 22
• Connectedness of the Eichler class-set graph CerednikDrinfeld.classSet_eq_empty_or_eq_univ_of_forall_mem_iff_of_squarefree 3,733 below · cited by 1 · depth 23
• Transport of descent intertwining data along a tower isomorphism CerednikDrinfeld.descentIntertwiningBase_of_towerIso 0 below · cited by 2 · depth 23
• Čerednik descent datum at q' from a moduli tower witness CerednikDrinfeld.exists_descentIntertwiningBase_of_moduliTowerWitness_one_zero_of_two_mul_dvd 9,818 below · cited by 1 · depth 23
• Descent intertwining datum at q from a moduli tower witness CerednikDrinfeld.exists_descentIntertwiningBase_of_moduliTowerWitness_zero_one_of_two_mul_dvd 9,817 below · cited by 1 · depth 23
• Canonical model, moduli witness and Hecke tower for X₀^{qq'}(N) CerednikDrinfeld.exists_shimuraCurveModel_rigidModuliWitness_heckeTower_of_six_mul_dvd_of_neZero 6,040 below · cited by 1 · depth 23
• Adic base data at a place of ℚ̄ above q CerednikDrinfeld.exists_adicBase_ratClosure 6 below · cited by 2 · depth 24
• Geometric fibre model of a smooth proper curve over ℤ[1/D] CerednikDrinfeld.exists_barFunctionField_curveModel_of_smoothProperCurve_of_two_mul_dvd 79 below · cited by 1 · depth 24
• Mumford embedding from Čerednik–Drinfeld uniformisation CerednikDrinfeld.exists_mumfordEmbedding_of_cerednikDrinfeld_uniformization_one_zero_of_two_mul_dvd 8,616 below · cited by 1 · depth 24
• Mumford embedding from Čerednik–Drinfeld uniformisation CerednikDrinfeld.exists_mumfordEmbedding_of_cerednikDrinfeld_uniformization_zero_one_of_two_mul_dvd 8,615 below · cited by 1 · depth 24
• Algebraic core of the Eichler–Shimura congruence relation CerednikDrinfeld.apply_apply_sub_apply_add_smul_eq_zero_of_reduction 0 below · cited by 1 · depth 25
• Central, odd and even elements of the away-unit group CerednikDrinfeld.awayUnits_central_odd_even_feed_one_zero_of_two_mul_dvd 34 below · cited by 1 · depth 25
• Parity of vdet describes Γ₂ at all levels CerednikDrinfeld.awayUnits_central_odd_even_feed_zero_one_of_two_mul_dvd 34 below · cited by 1 · depth 25
• Scalar re-alignment of a Frobenius twist in Čerednik–Drinfeld descent CerednikDrinfeld.cerednikDrinfeld_realign_of_frobTwist_eq_on_fixed 1 below · cited by 1 · depth 25
• Smooth, geometrically connected generic fibres of the coarse models CerednikDrinfeld.coarseModuli_smooth_geometricallyConnected_feed_one_zero_of_two_mul_dvd 5,772 below · cited by 1 · depth 25
• Smoothness and geometric connectedness of the generic fibres CerednikDrinfeld.coarseModuli_smooth_geometricallyConnected_feed_zero_one_of_two_mul_dvd 5,772 below · cited by 1 · depth 25
• Discreteness and cocompactness of Γ₁ on the lattice tree CerednikDrinfeld.evenAwayUnits_finite_stabilizer_finite_orbits_feed_one_zero_of_two_mul_dvd 3,794 below · cited by 1 · depth 25
• Finite stabilisers and finitely many orbits for Γ₂ on the tree CerednikDrinfeld.evenAwayUnits_finite_stabilizer_finite_orbits_feed_zero_one_of_two_mul_dvd 3,793 below · cited by 1 · depth 25
• Tame vertex stabilisers for Γ₁ on the q' side CerednikDrinfeld.evenAwayUnits_v_card_stabilizer_eq_one_one_zero_of_two_mul_dvd 3,793 below · cited by 1 · depth 25
• Tame vertex stabilisers for the away-unit groups Γ₂ CerednikDrinfeld.evenAwayUnits_v_card_stabilizer_eq_one_zero_one_of_two_mul_dvd 3,793 below · cited by 1 · depth 25
• Virtual torsion-freeness of the even away-unit groups CerednikDrinfeld.evenAwayUnits_virtuallyTorsionFree_feed_one_zero_of_two_mul_dvd 37 below · cited by 1 · depth 25
• Virtually torsion-free even away-unit groups at q CerednikDrinfeld.evenAwayUnits_virtuallyTorsionFree_feed_zero_one_of_two_mul_dvd 37 below · cited by 1 · depth 25
• Function field of a Čerednik–Drinfeld quotient as Γ'-invariant meromorphic functions CerednikDrinfeld.exists_ringEquiv_functionField_pullback_invariantFieldOf_smul_level_of_cerednikDrinfeld_quotient_of_tame_of_virtuallyTorsionFree_of_smooth 721 below · cited by 2 · depth 25
• Integers prime to r are units in a π-adically complete domain CerednikDrinfeld.isUnit_natCast_of_not_dvd_of_card_quotient 0 below · cited by 32 · depth 25
• Mumford embedding of the Shimura tower over ℚ_{q'} CerednikDrinfeld.mumfordEmbedding_assembly_of_functionField_equiv_one_zero_of_two_mul_dvd 5,797 below · cited by 1 · depth 25
• Assembling the Mumford embedding from the function-field identification CerednikDrinfeld.mumfordEmbedding_assembly_of_functionField_equiv_zero_one_of_two_mul_dvd 5,797 below · cited by 1 · depth 25
• Divisibility from Fr^m acting trivially on Frᵈ-fixed elements CerednikDrinfeld.dvd_of_forall_frobenius_zpow_apply_eq_of_fixed 0 below · cited by 1 · depth 26
• Integral points of a Čerednik–Drinfeld quotient as Γ'-orbits of adic points CerednikDrinfeld.exists_adicPoint_to_sections_of_cerednikDrinfeld_quotient 424 below · cited by 1 · depth 26
• Function field of a Čerednik–Drinfeld quotient embeds into invariant functions CerednikDrinfeld.exists_ringHom_functionField_invariantFieldOf_eval_of_cerednikDrinfeld_quotient_of_smooth 531 below · cited by 1 · depth 26
• Function field embedding into the Čerednik–Drinfeld model over C CerednikDrinfeld.exists_ringHom_functionField_pullback_completion_of_moduliTowerWitness_one_zero_of_two_mul_dvd 5,777 below · cited by 1 · depth 26
• Equivariant embedding of ̄ F into the completed function field CerednikDrinfeld.exists_ringHom_functionField_pullback_completion_of_moduliTowerWitness_zero_one_of_two_mul_dvd 5,777 below · cited by 1 · depth 26
• Complex uniformisation of the Shimura curve of an Eichler order CerednikDrinfeld.exists_uniformizedHeckeCurve_fuchsianGroup 85 below · cited by 2 · depth 26
• Finiteness of affinoid points sent off a generic open CerednikDrinfeld.finite_affinoid_toOmega_not_le_preimage_of_cerednikDrinfeld_quotient_of_smooth 15 below · cited by 1 · depth 26
• Integrality of the C-fibre of a Čerednik–Drinfeld quotient CerednikDrinfeld.isIntegral_pullback_of_cerednikDrinfeld_quotient_of_smooth 10 below · cited by 1 · depth 26
• Mumford embedding read off from the function-field identification CerednikDrinfeld.mumfordEmbedding_readoff_of_functionField_equiv_of_ringHom_one_zero_of_two_mul_dvd 894 below · cited by 1 · depth 26
• Reading off the Mumford embedding from Čerednik–Drinfeld data CerednikDrinfeld.mumfordEmbedding_readoff_of_functionField_equiv_of_ringHom_zero_one_of_two_mul_dvd 894 below · cited by 1 · depth 26
• Equivariance of Čerednik–Drinfeld evaluation under an intertwining morphism CerednikDrinfeld.ringHom_functionField_germ_app_eq_inv_smul_of_eval_of_cerednikDrinfeld_quotient 18 below · cited by 1 · depth 26
• Frobenius on X acts as w⁻¹ under evaluation CerednikDrinfeld.ringHom_functionField_germ_app_eq_inv_smul_of_frobenius_of_eval_of_cerednikDrinfeld_quotient 18 below · cited by 1 · depth 26
• Galois equivariance of the Čerednik–Drinfeld evaluation embedding CerednikDrinfeld.ringHom_functionField_germ_app_eq_zpow_smul_fracMap_of_isometricAut_of_eval_of_cerednikDrinfeld_quotient 226 below · cited by 1 · depth 26
• Evaluation onto invariant meromorphic functions is surjective CerednikDrinfeld.surjective_ringHom_functionField_invariantFieldOf_of_eval_of_tame_of_cerednikDrinfeld_quotient_of_virtuallyTorsionFree_of_smooth 678 below · cited by 1 · depth 26
• Determinant valuation of a split quaternion unit equals v_q(nrd) CerednikDrinfeld.vdet_unitsMap_eq_padicValRat_nrd 5 below · cited by 4 · depth 26
• Uniform denominator clearing for sections on a Čerednik–Drinfeld quotient CerednikDrinfeld.exists_cover_sections_ne_zero_mul_eq_sum_of_cerednikDrinfeld_quotient 3 below · cited by 2 · depth 27
• Non-constant rational function on the generic fibre of a Čerednik–Drinfeld quotient CerednikDrinfeld.exists_functionField_ne_const_of_cerednikDrinfeld_quotient 25 below · cited by 1 · depth 27
• Hecke multiset families for Eichler orders and Fuchsian descent CerednikDrinfeld.exists_heckeFamily_map_orbit_eq_of_isEichlerOrder 6 below · cited by 1 · depth 27
• Invariant chartwise meromorphic pullbacks of sections along the uniformisation CerednikDrinfeld.exists_invariant_chartwiseMeromorphic_pullback_of_cerednikDrinfeld_quotient_of_eval_of_smooth 487 below · cited by 1 · depth 27
• Γ'-invariant chartwise meromorphic pullbacks on a Čerednik–Drinfeld quotient CerednikDrinfeld.exists_invariant_chartwiseMeromorphic_pullback_of_cover_clearing_of_cerednikDrinfeld_quotient 41 below · cited by 2 · depth 27
• Comparison of the two coarse models over the q'-adic completion CerednikDrinfeld.exists_iso_pullback_completion_of_moduliTowerWitness_one_zero_of_two_mul_dvd 5,594 below · cited by 1 · depth 27
• The two models agree over the completion at q CerednikDrinfeld.exists_iso_pullback_completion_of_moduliTowerWitness_zero_one_of_two_mul_dvd 5,594 below · cited by 1 · depth 27
• Pull-backs of sections holomorphic on edge-region pieces lying over V CerednikDrinfeld.exists_linearPieces_le_preimage_holOn_apply_toOmega_eq_of_cerednikDrinfeld_quotient 441 below · cited by 2 · depth 27
• An away-from-r unit of det-valuation one at level ℓ CerednikDrinfeld.exists_mem_inf_levelSubgroup_vdet_eq_one_of_isEichlerOrder_meetOrder 91 below · cited by 2 · depth 27
• Function field embeds into Γ'-invariants via chartwise meromorphic pull-backs CerednikDrinfeld.exists_ringHom_functionField_invariantFieldOf_eval_of_chartwiseMeromorphic 77 below · cited by 1 · depth 27
• Equivariant embedding of ̄ F into K(mathcal X_{0,C}) CerednikDrinfeld.exists_ringHom_functionField_of_iso_pullback_completion_one_zero_of_two_mul_dvd 5,768 below · cited by 1 · depth 27
• Čerednik–Drinfel'd embedding of ̄ F into K(mathcal X_{0,C}) CerednikDrinfeld.exists_ringHom_functionField_of_iso_pullback_completion_zero_one_of_two_mul_dvd 5,768 below · cited by 1 · depth 27
• Sections near an adic point as quotients of integral sections CerednikDrinfeld.exists_sections_ne_zero_mul_eq_sum_of_cerednikDrinfeld_quotient 3 below · cited by 2 · depth 27
• Frobenius parity of the decomposition group action via ψ₀ CerednikDrinfeld.exists_smul_psi_eq_psi_frobenius_pow_iff_parity_of_decompositionSubgroup 0 below · cited by 2 · depth 27
• Finitely many affinoid points miss a given generic neighbourhood CerednikDrinfeld.finite_affinoid_toOmega_of_not_le_preimage_of_cerednikDrinfeld_quotient_of_smooth 14 below · cited by 4 · depth 27
• Adic points of a Čerednik–Drinfeld quotient and their twisted fibres CerednikDrinfeld.forall_exists_adicPoint_and_theta_eq_iff_of_cerednikDrinfeld_quotient 420 below · cited by 1 · depth 27
• Hypotheses of the complex uniformisation for Γ(R,ι) CerednikDrinfeld.fuchsianGroup_discrete_neg_mem_and_exists_isCompact 10 below · cited by 4 · depth 27
• Residue ring at π is algebraically closed of characteristic r CerednikDrinfeld.isAlgClosed_and_charP_quotient_of_isMaximal_of_forall_monic 0 below · cited by 9 · depth 27
• Tree-lattice facts for the even part of a Čerednik–Drinfeld group CerednikDrinfeld.map_evenPart_le_typePreserving_and_graphAction_and_finite_of_cerednikDrinfeld_group 15 below · cited by 2 · depth 27
• Fuchsian groups from rational division quaternion algebras have no cusps CerednikDrinfeld.not_isCusp_fuchsianGroup_of_forall_isUnit 0 below · cited by 1 · depth 27
• Twisting a Čerednik–Drinfeld adic point by a base automorphism CerednikDrinfeld.specPoint_eq_specMap_comp_of_map_pt_eq_act_pt_of_cerednikDrinfeld_quotient 208 below · cited by 1 · depth 27
• Central vdet = 2, odd and even away units CerednikDrinfeld.awayUnits_exists_central_vdet_two_and_exists_vdet_one_and_exists_even 0 below · cited by 4 · depth 28
• Čerednik–Drinfeld points see coefficients only through Fr²-invariants CerednikDrinfeld.cerednikDrinfeld_apply_eq_of_forall_fr_fr_eq 205 below · cited by 1 · depth 28
• Real embeddings carry the reduced norm to the determinant CerednikDrinfeld.det_map_eq_nrd 0 below · cited by 4 · depth 28
• Discreteness of the Fuchsian group of a rational quaternion order CerednikDrinfeld.discreteTopology_fuchsianGroup 0 below · cited by 1 · depth 28
• Values of pulled-back sections at a base-changed point CerednikDrinfeld.eval_app_pullback_fst_eq_algebraMap_eval_app 0 below · cited by 3 · depth 28
• Finite vertex stabilisers and finitely many vertex orbits CerednikDrinfeld.evenAwayUnits_finite_stabilizer_vertex_and_exists_finset_orbits_of_not_dvd 61 below · cited by 3 · depth 28
• Finitely many vertex orbits for the even level-ℓ group CerednikDrinfeld.evenAwayUnits_inf_levelSubgroup_exists_finset_orbits 63 below · cited by 1 · depth 28
• Geometric function field and curve model of X/ℤ[1/Nqq'] CerednikDrinfeld.exists_barFunctionField_curveModel_of_smoothProperCurve 79 below · cited by 1 · depth 28
• Local quotient form of a pulled-back section on Ω_C CerednikDrinfeld.exists_disc_holOn_mul_pullback_eq_of_cover_clearing_of_cerednikDrinfeld_quotient 13 below · cited by 1 · depth 28
• Finite edge-chart cover over an open of the Čerednik–Drinfeld quotient CerednikDrinfeld.exists_finset_chartUnitLocus_cover_of_cerednikDrinfeld_quotient 12 below · cited by 1 · depth 28
• Finitely many norm-one unit classes of given reduced norm CerednikDrinfeld.exists_finset_forall_nrd_eq_exists_mul_unit 2 below · cited by 2 · depth 28
• Finitely many Γ-orbits of fixed points of reduced norm ν CerednikDrinfeld.exists_finset_forall_smul_eq_of_nrd_eq_of_not_isSquare 12 below · cited by 1 · depth 28
• Formal quotient datum with coefficient and adic-point laws CerednikDrinfeld.exists_formalQuotientDatum_coeff_adicFib 203 below · cited by 3 · depth 28
• Affinoid meromorphy of pulled-back sections on Čerednik–Drinfeld quotients CerednikDrinfeld.exists_holOn_affinoid_mul_pullback_eq_of_cover_clearing_of_cerednikDrinfeld_quotient 37 below · cited by 1 · depth 28
• Global quotient presentation of a Γ'-invariant affinoid-meromorphic function CerednikDrinfeld.exists_holRing_forall_finite_mul_eq_of_invariant_of_cerednikDrinfeld_group 65 below · cited by 1 · depth 28
• Γ'-invariant pull-back of a section to Drinfeld's upper half plane CerednikDrinfeld.exists_invariant_pullback_apply_toOmega_eq_of_cerednikDrinfeld_quotient 7 below · cited by 1 · depth 28
• Closedness of the uniformised locus in the special fibre CerednikDrinfeld.exists_isClosed_iff_exists_theta_eq_of_cerednikDrinfeld_quotient 207 below · cited by 2 · depth 28
• Cocompactness of the Fuchsian group of an order in a rational quaternion division algebra CerednikDrinfeld.exists_isCompact_forall_exists_fuchsianGroup_smul_mem 7 below · cited by 2 · depth 28
• Elements of the Fuchsian group come from norm-one units CerednikDrinfeld.exists_isUnitOf_nrd_eq_one_of_mem_fuchsianGroup 1 below · cited by 5 · depth 28
• Edge-chart unit locus as a finite union of linear pieces CerednikDrinfeld.exists_linearPieces_eq_chartUnitLocus_of_cerednikDrinfeld_quotient 13 below · cited by 1 · depth 28
• Holomorphy of pull-backs of regular functions on edge-chart loci CerednikDrinfeld.exists_mem_holOn_apply_toOmega_eq_of_chartMap_of_cerednikDrinfeld_quotient 7 below · cited by 1 · depth 28
• A non-generic point on a Čerednik–Drinfeld generic fibre CerednikDrinfeld.exists_ne_genericPoint_pullback_of_cerednikDrinfeld_quotient 23 below · cited by 1 · depth 28
• Edge charts covering a formal Čerednik–Drinfeld quotient CerednikDrinfeld.exists_opens_chartMorphism_of_cerednikDrinfeld_quotient 422 below · cited by 1 · depth 28
• Absolutely unramified complete DVR with residue field 𝔽ᵣ is ℤᵣ CerednikDrinfeld.exists_ringEquiv_padicInt_algebraMap_apply_eq_of_isAdicComplete_of_natCard_quotient_eq 1 below · cited by 3 · depth 28
• Complete unramified r-rings are Witt rings, compatibly with Frobenius CerednikDrinfeld.exists_ringEquiv_wittVector_apply_frobenius_eq_of_isAdicComplete_of_isMaximal 1 below · cited by 1 · depth 28
• Germ-pinned function-field embedding of a curve model over a completion CerednikDrinfeld.exists_ringHom_functionField_germ_eq_of_curveModel_of_iso_pullback_completion 3 below · cited by 2 · depth 28
• Level compatibility of the pinned function-field embedding CerednikDrinfeld.exists_ringHom_functionField_level_germ_app_degeneracy_eq_of_germ_eq_of_iso_pullback_completion_one_zero_of_two_mul_dvd 5,748 below · cited by 1 · depth 28
• Degeneracy compatibility of the pinned function-field embedding CerednikDrinfeld.exists_ringHom_functionField_level_germ_app_degeneracy_eq_of_germ_eq_of_iso_pullback_completion_zero_one_of_two_mul_dvd 5,748 below · cited by 1 · depth 28
• Finiteness of a Φ-fibre of coordinates inside an affinoid CerednikDrinfeld.finite_affinoid_toOmega_fibre_of_cerednikDrinfeld_quotient 7 below · cited by 1 · depth 28
• Atkin–Lehner equivariance of the pinned function-field embedding CerednikDrinfeld.germ_app_atkinLehner_eq_of_germ_eq_of_iso_pullback_completion_one_zero_of_two_mul_dvd 742 below · cited by 1 · depth 28
• Atkin–Lehner lifts act on germs through W₀ and W₁ CerednikDrinfeld.germ_app_atkinLehner_eq_of_germ_eq_of_iso_pullback_completion_zero_one_of_two_mul_dvd 742 below · cited by 1 · depth 28
• Decomposition-group equivariance of the pinned function field embedding CerednikDrinfeld.germ_app_decomposition_eq_of_germ_eq_of_iso_pullback_completion_one_zero_of_two_mul_dvd 25 below · cited by 1 · depth 28
• Decomposition-group equivariance of the pinned function-field embedding CerednikDrinfeld.germ_app_decomposition_eq_of_germ_eq_of_iso_pullback_completion_zero_one_of_two_mul_dvd 25 below · cited by 1 · depth 28
• Adic points in a chart unit locus factor through V CerednikDrinfeld.le_preimage_of_toOmega_mem_chartUnitLocus_of_cerednikDrinfeld_quotient 7 below · cited by 1 · depth 28
• Level-independence of chart preimages of an open of X CerednikDrinfeld.basicOpen_le_preimage_chartMorphism_of_level_zero_of_cerednikDrinfeld_quotient 3 below · cited by 1 · depth 29
• Homomorphisms GL₂(ℚᵣ)→ℤ are multiples of v∘det CerednikDrinfeld.exists_forall_apply_eq_zpow_of_monoidHom_generalLinearGroup_padic 0 below · cited by 4 · depth 29
• A prime Hecke element at ℓ, supported at ℓ CerednikDrinfeld.exists_forall_finiteAdeleEvalAt_eq_one_and_mem_primeHeckeSet_and_isEichlerOrder_meetOrder_of_not_dvd 30 below · cited by 1 · depth 29
• Formal quotient datum from a Mumford tower and a tower quotient CerednikDrinfeld.exists_formalQuotientDatum_coeff_adicFib_of_mumfordTower_of_towerQuotientDatum 85 below · cited by 1 · depth 29
• Elements normalising an Eichler order normalise its Fuchsian group CerednikDrinfeld.exists_gl_conj_fuchsianGroup_iff_of_forall_mem_iff_conj_mem 2 below · cited by 1 · depth 29
• Minkowski step for orders in an indefinite quaternion algebra CerednikDrinfeld.exists_isCompact_finset_forall_sl 4 below · cited by 1 · depth 29
• Finite-index Schottky subgroup inside the image of the even part CerednikDrinfeld.exists_isSchottky_le_map_normal_relIndex_ne_zero_of_even 22 below · cited by 1 · depth 29
• Unramified complete base realised as the closure of ℚ CerednikDrinfeld.exists_liesOverPrime_ringEquiv_ratClosure_range_iff_of_isAdicComplete_of_natCard_quotient_eq 7 below · cited by 1 · depth 29
• Valuation of the determinant as a homomorphism on GL_m(K₀) CerednikDrinfeld.exists_monoidHom_generalLinearGroup_eq_ofAdd_iff_of_irreducible 0 below · cited by 4 · depth 29
• Chart data transported from a formal quotient datum to X CerednikDrinfeld.exists_opens_chartMorphism_of_formalQuotientDatum_of_isIso 0 below · cited by 1 · depth 29
• Polynomial clearing of a pulled-back section on an edge region CerednikDrinfeld.exists_polynomial_ne_zero_mul_pullback_mem_holOn_edgeRegion_of_cover_clearing_of_cerednikDrinfeld_quotient 26 below · cited by 1 · depth 29
• Existence of a formal quotient datum for Drinfeld's half plane CerednikDrinfeld.nonempty_formalQuotientDatum 204 below · cited by 2 · depth 29
• Bilinear relations between the two degeneracy legs transfer generically CerednikDrinfeld.sum_mul_eq_zero_of_sum_phi_mul_phi_eq_zero_of_germ_eq_degeneracy_of_iso_pullback_completion_one_zero_of_two_mul_dvd 749 below · cited by 1 · depth 29
• Tower relations transfer to the degeneracy maps on function fields CerednikDrinfeld.sum_mul_eq_zero_of_sum_phi_mul_phi_eq_zero_of_germ_eq_degeneracy_of_iso_pullback_completion_zero_one_of_two_mul_dvd 749 below · cited by 1 · depth 29
• Density of place-indexed points on the level-ℓ curve CerednikDrinfeld.dense_setOf_exists_place_comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_iso_pullback_completion_one_zero_of_two_mul_dvd 747 below · cited by 1 · depth 30
• Density of place-defined level points on the ℓ-level model CerednikDrinfeld.dense_setOf_exists_place_comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_iso_pullback_completion_zero_one_of_two_mul_dvd 747 below · cited by 1 · depth 30
• Finitely many affinoid coordinates where a non-zero section vanishes CerednikDrinfeld.finite_affinoid_toOmega_eval_eq_zero_of_ne_zero_of_cerednikDrinfeld_quotient 0 below · cited by 1 · depth 30
• Orientation clause of U_ℓ forces sR₁s⁻¹not⊆Λ₁[1/r] CerednikDrinfeld.exists_mem_forall_pow_smul_mul_mul_star_ne_smul_of_mem_levelHeckeUSet_meetOrder_of_dvd 5 below · cited by 2 · depth 31
• Both degeneracy images of a C-point come from one place CerednikDrinfeld.exists_place_comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_comp_eq_pointEquivPlace_symm_of_iso_pullback_completion_one_zero_of_two_mul_dvd 744 below · cited by 1 · depth 31
• Lifting a C-point of the level-ℓ curve to a place CerednikDrinfeld.exists_place_comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_comp_eq_pointEquivPlace_symm_of_iso_pullback_completion_zero_one_of_two_mul_dvd 744 below · cited by 1 · depth 31
• Both degeneracies carry a level point to its restricted places CerednikDrinfeld.comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_withExtraLevel_isPullback_repT_of_iso_pullback_completion_one_zero_of_two_mul_dvd 24 below · cited by 1 · depth 32
• Degeneracy maps on level points and restricted places CerednikDrinfeld.comp_degeneracy_eq_pointEquivPlace_symm_restrictAlong_of_withExtraLevel_isPullback_repT_of_iso_pullback_completion_zero_one_of_two_mul_dvd 24 below · cited by 1 · depth 32
• Two coefficient maps agree on Frᵈ-fixed elements up to Fr^j CerednikDrinfeld.exists_forall_apply_eq_apply_frobenius_zpow_of_fixed 0 below · cited by 2 · depth 32
• Lifting a ℤ_{p²}-structure along a p-torsion-free surjection CerednikDrinfeld.exists_torsionFree_surjective_comp_eq_and_comp_eq_of_surjective 0 below · cited by 2 · depth 33
• Unit-reflecting p-torsion-free factorisation of a ring surjection CerednikDrinfeld.exists_torsionFree_surjective_comp_eq_forall_isUnit 0 below · cited by 2 · depth 33
• Localisation away from an element is functorial; basic opens pull back CerednikDrinfeld.exists_ringHom_away_comp_eq_and_not_mem_iff 0 below · cited by 2 · depth 34
• Factoring a ring map through a p-torsion-free surjection CerednikDrinfeld.exists_torsionFree_surjective_comp_eq 0 below · cited by 2 · depth 35
• Ring maps W(k)→κ[ε] are fixed by κ-algebra endomorphisms CerednikDrinfeld.ringHom_comp_eq_of_fstHom_comp_eq 0 below · cited by 1 · depth 38
• Period variation is a multiple of x₀ only on window vectors CerednikDrinfeld.not_exists_forall_period_variation_eq_mul_of_forall_ne_window 0 below · cited by 1 · depth 39

CerednikDrinfeld.BruhatTits 17

• Type-preserving quaternion units have even reduced-norm valuation CerednikDrinfeld.BruhatTits.mem_typePreserving_iff_even_padicValRat_nrd 10 below · cited by 9 · depth 19
• Finite vertex stabilisers from finite dart stabilisers on the Bruhat–Tits tree CerednikDrinfeld.BruhatTits.finite_stabilizer_vertex_of_finite_stabilizer_dart 6 below · cited by 16 · depth 20
• Type preservation on the Bruhat–Tits tree and parity of v(det) CerednikDrinfeld.BruhatTits.mem_typePreserving_iff_even_of_det_eq_mul_zpow 8 below · cited by 4 · depth 20
• Connectedness and bipartiteness of the Bruhat–Tits lattice graph CerednikDrinfeld.BruhatTits.tree_connected_and_colorable_two 0 below · cited by 31 · depth 20
• The Bruhat–Tits lattice graph of GL₂ is a tree CerednikDrinfeld.BruhatTits.tree_isTree 0 below · cited by 21 · depth 20
• Primitive integral g with det g=uvarpi^k moves [R²] exactly k CerednikDrinfeld.BruhatTits.dist_stdVertex_smul_stdVertex_eq_of_isInteger_of_det_eq_of_isUnit 6 below · cited by 3 · depth 21
• Finiteness of balls in the Bruhat–Tits tree CerednikDrinfeld.BruhatTits.finite_setOf_dist_le 4 below · cited by 7 · depth 21
• Neighbours of the standard vertex of the Bruhat–Tits tree CerednikDrinfeld.BruhatTits.adj_stdVertex_iff_eq_smul_of_uniformizer 0 below · cited by 9 · depth 22
• Integral g with det g = uvarpi^k moves the standard vertex by at most k CerednikDrinfeld.BruhatTits.dist_stdVertex_smul_stdVertex_le_of_isInteger_of_det_eq 3 below · cited by 1 · depth 22
• Transport of Bruhat–Tits tree and PGL₂-action under base change CerednikDrinfeld.BruhatTits.exists_iso_tree_mulEquiv_projGenLinGroup_baseChange 1 below · cited by 2 · depth 22
• Faithful action of PGL₂(K) on lattice vertices CerednikDrinfeld.BruhatTits.faithfulSMul_projGenLinGroup_vertex 0 below · cited by 1 · depth 22
• Functoriality of the Bruhat–Tits tree in the pair (R,K) CerednikDrinfeld.BruhatTits.exists_iso_tree_baseChange 0 below · cited by 1 · depth 23
• Bounded exponent for torsion in a tree lattice in PGL₂ CerednikDrinfeld.BruhatTits.exists_pos_forall_isOfFinOrder_pow_eq_one 8 below · cited by 1 · depth 23
• Bounded exponent of vertex stabilisers on the Bruhat–Tits tree CerednikDrinfeld.BruhatTits.exists_pos_forall_mem_stabilizer_pow_eq_one 5 below · cited by 1 · depth 24
• Finitely many vertex orbits implies finitely many dart orbits CerednikDrinfeld.BruhatTits.finite_quotEdge_of_finite_quotVert 5 below · cited by 5 · depth 24
• Every dart of the Bruhat–Tits tree is a GL₂-translate of a standard dart CerednikDrinfeld.BruhatTits.exists_smul_stdVertex_eq_fst_and_mul_smul_stdVertex_eq_snd 2 below · cited by 3 · depth 25
• Even part maps to a type-preserving tree lattice in PGL₂ CerednikDrinfeld.BruhatTits.treeLattice_facts_map_evenPart 14 below · cited by 5 · depth 27

CerednikDrinfeld.CSTower 5

• U_ℓ storey of the class-set tower at ℓ ∣ N CerednikDrinfeld.CSTower.isEichlerOrder_meetOrder_and_exists_storey_of_mem_levelHeckeUSet_of_evalAt_eq_one 63 below · cited by 7 · depth 20
• Level-ℓ storey of the class-set tower for a given shift CerednikDrinfeld.CSTower.isEichlerOrder_meetOrder_and_exists_storey_of_mem_primeHeckeSet_of_evalAt_eq_one 43 below · cited by 5 · depth 20
• Meet order at an admissible T_ℓ-shift is Eichler of level Nℓ CerednikDrinfeld.CSTower.isEichlerOrder_meetOrder_of_finiteIdeleDiagonal_mul_inv_mem_primeHeckeSet_meetOrder 42 below · cited by 6 · depth 20
• Level-raising idele at a prime ℓ dividing N CerednikDrinfeld.CSTower.exists_finiteIdeleDiagonal_mul_inv_mem_levelHeckeUSet_meetOrder_isEichlerOrder_of_dvd 45 below · cited by 1 · depth 21
• Meet order R∩ swidehat Rs⁻¹ is Eichler of level Nℓ CerednikDrinfeld.CSTower.isEichlerOrder_meetOrder_of_finiteIdeleDiagonal_mul_inv_mem_levelHeckeUSet_meetOrder 54 below · cited by 4 · depth 21

CerednikDrinfeld.CartierLift 1

• Universal digits forcing the relation p fᵢ=[p]fᵢ+sum_k V^k(d_{k,i}f) CerednikDrinfeld.CartierLift.exists_digits_forall_smul_eq_teichmuller_smul_add_sum_verschiebungInt 3 below · cited by 1 · depth 36

CerednikDrinfeld.CosetGraph 40

• Atkin–Lehner relations for the Čerednik level groups CerednikDrinfeld.CosetGraph.atkinLehner_relations_levelGroups_place 38 below · cited by 12 · depth 20
• Away units of R∩ s_f̂ R s_f⁻¹ at a near-global idele CerednikDrinfeld.CosetGraph.awayUnits_meetOrder_eq_inf_map_conj_of_finiteAdeleEvalAt_eq 27 below · cited by 13 · depth 20
• Finite-index subgroup of away units with torsion-free image CerednikDrinfeld.CosetGraph.exists_le_awayUnits_inf_typePreserving_relIndex_ne_zero_forall_isOfFinOrder_eq_one 36 below · cited by 4 · depth 20
• An r-unit of an Eichler order with reduced norm r CerednikDrinfeld.CosetGraph.exists_mem_awayUnits_nrd_eq 42 below · cited by 2 · depth 20
• Class-set dictionary for the Bruhat–Tits quotient at r CerednikDrinfeld.CosetGraph.exists_quot_equiv_classSet_shift_forget_of_mumfordSideFrame 3,794 below · cited by 4 · depth 20
• Global element of reduced norm q normalising the completed orders CerednikDrinfeld.CosetGraph.exists_units_nrd_eq_ramifiedPrime_forall_mem_localBoxUnits_and_normalizes 44 below · cited by 1 · depth 20
• Mumford frame at r: tame stabilisers, finite quotients, class sets CerednikDrinfeld.CosetGraph.finite_stabilizer_and_finite_quot_and_exists_equiv_classSet_of_mumfordSideFrame 3,783 below · cited by 13 · depth 20
• Local shape at v of a normalising prime Hecke element CerednikDrinfeld.CosetGraph.mul_self_mem_level_and_not_mem_level_and_mem_inf_conj_iff_of_mem_primeHeckeSet 21 below · cited by 4 · depth 20
• Index of Γ∩ sΓ s⁻¹ equals the Hecke arrow degree CerednikDrinfeld.CosetGraph.relIndex_inf_map_conj_eq_arrowDegree_of_hecke 95 below · cited by 7 · depth 20
• Index of Γ∩ sΓ s⁻¹ equals the degeneracy degree CerednikDrinfeld.CosetGraph.relIndex_inf_map_conj_eq_arrowDegree_of_hecke_one_zero 96 below · cited by 4 · depth 20
• Index of Γ∩ s⁻¹Γ s in Γ equals the Hecke arrow degree CerednikDrinfeld.CosetGraph.relIndex_inf_map_conj_inv_eq_arrowDegree_of_hecke 97 below · cited by 3 · depth 20
• Index of Γ∩ s⁻¹Γ s equals degeneracy degree CerednikDrinfeld.CosetGraph.relIndex_inf_map_conj_inv_eq_arrowDegree_of_hecke_one_zero 98 below · cited by 2 · depth 20
• Away units of the meet order R∩ sRs⁻¹ CerednikDrinfeld.CosetGraph.awayUnits_meetOrder_finiteIdeleDiagonal_eq_inf_map_conj 27 below · cited by 1 · depth 21
• Coset graph at a split prime versus Bruhat–Tits tree, equivariantly CerednikDrinfeld.CosetGraph.exists_iso_tree_ratClosure_smul_eq_and_natCard_stabilizer_mapDart_eq 23 below · cited by 4 · depth 21
• Torsion-free normal subgroup of finite index in ProjAwayUnits CerednikDrinfeld.CosetGraph.exists_normal_finiteIndex_forall_isOfFinOrder_imp_eq_one 17 below · cited by 1 · depth 21
• Coset graph quotient at r is the class-set degeneracy datum CerednikDrinfeld.CosetGraph.exists_quotVert_equiv_classSet_and_quotEdge_equiv_classSet_of_isEichlerOrder 3,743 below · cited by 2 · depth 21
• Coset graph modulo r-units versus class-set degeneracy datum CerednikDrinfeld.CosetGraph.exists_quotVert_equiv_classSet_and_quotEdge_equiv_classSet_of_isEichlerOrder_of_level 3,754 below · cited by 1 · depth 21
• Element of reduced norm ℓ in R[1/r], unit away from ℓ CerednikDrinfeld.CosetGraph.exists_units_nrd_eq_prime_forall_mem_localBox 43 below · cited by 1 · depth 21
• Dart stabiliser order is half the unit count of the conjugated meet order CerednikDrinfeld.CosetGraph.natCard_stabilizer_dart_eq_natCard_isUnitOf_conjByFiniteIdele_meetOrder_div_two 26 below · cited by 4 · depth 21
• Reversing a dart realises the class-set shift by n CerednikDrinfeld.CosetGraph.quotEdge_symm_eq_classSetShift_of_forall_eq_classSet_mk 29 below · cited by 1 · depth 21
• Fixing every coset-graph vertex forces a rational scalar CerednikDrinfeld.CosetGraph.exists_coe_eq_smul_one_of_forall_smul_vert_eq 12 below · cited by 2 · depth 22
• Coset graph at a split prime is the Bruhat–Tits tree CerednikDrinfeld.CosetGraph.exists_iso_tree_apply_coe_eq_smul_stdVertex 4 below · cited by 1 · depth 22
• A congruence subgroup of finite index in the r-unit group CerednikDrinfeld.CosetGraph.exists_normal_finiteIndex_awayUnits_forall_eq_one_add_smul 2 below · cited by 1 · depth 22
• Rational scalars that are products of Γ-conjugates lie in Γ CerednikDrinfeld.CosetGraph.mem_of_eq_algebraMap_of_eq_mul_conj 19 below · cited by 1 · depth 22
• Rational scalars built from Γ and a conjugate lie in Γ CerednikDrinfeld.CosetGraph.mem_of_eq_algebraMap_of_eq_mul_conj_one_zero 19 below · cited by 1 · depth 22
• r-integrality of a norm-q local unit of an Eichler order CerednikDrinfeld.CosetGraph.exists_pow_smul_mem_of_nrd_eq_of_forall_mem_localBoxUnits_of_forall_conj_mem_localBox_iff 14 below · cited by 2 · depth 24
• Conjugation preserving local boxes preserves the away-from-v units CerednikDrinfeld.CosetGraph.mul_mul_inv_mem_awayUnits_of_forall_localBox_iff 0 below · cited by 2 · depth 24
• Tame finite vertex stabilisers on the Bruhat–Tits tree CerednikDrinfeld.CosetGraph.finite_stabilizer_vertex_and_not_dvd_natCard_of_mumfordSideFrame 3,792 below · cited by 4 · depth 26
• An away-from-v unit of reduced norm r CerednikDrinfeld.CosetGraph.exists_mem_awayUnits_nrd_eq_of_le_isMaximalOrder 46 below · cited by 4 · depth 27
• Away-unit groups are monotone in the lattice CerednikDrinfeld.CosetGraph.awayUnits_mono 0 below · cited by 1 · depth 28
• Transport of away-from-v units under conjugation by γ₀ CerednikDrinfeld.CosetGraph.mem_awayUnits_iff_conj_mem_awayUnits_of_conjByFiniteIdele_finiteIdeleDiagonal_mul_eq 4 below · cited by 2 · depth 28
• Reduced norms of away-from-v units are ± powers of r CerednikDrinfeld.CosetGraph.exists_nrd_mul_pow_eq_pow_of_mem_awayUnits 1 below · cited by 2 · depth 29
• Away-unit groups are commensurable with their conjugates CerednikDrinfeld.CosetGraph.finiteIndex_subgroupOf_inf_map_conj_awayUnits 2 below · cited by 1 · depth 29
• Finiteness of the vertex orbit space for away-units CerednikDrinfeld.CosetGraph.finite_quotVert_projAwayUnits_of_isOrder 11 below · cited by 1 · depth 29
• Finiteness of vertex stabilisers in the projective away-unit group CerednikDrinfeld.CosetGraph.finite_stabilizer_projAwayUnits_vert_of_isOrder 10 below · cited by 1 · depth 29
• Units away from v detected by r-power denominators in R₀ CerednikDrinfeld.CosetGraph.mem_awayUnits_iff_exists_pow_smul_mem_of_forall_localBox_eq 4 below · cited by 5 · depth 29
• Reduced norms of away-units are units away from r CerednikDrinfeld.CosetGraph.padicValRat_nrd_eq_zero_of_mem_awayUnits 0 below · cited by 5 · depth 29
• r-integrality of ̄ sγ s forces γ∈ sΓ̃ s⁻¹ CerednikDrinfeld.CosetGraph.mem_map_conj_of_mem_awayUnits_of_exists_pow_smul_star_mul_mul_eq_smul 44 below · cited by 1 · depth 30
• Away-from-v units of an Eichler order span the algebra CerednikDrinfeld.CosetGraph.span_val_image_awayUnits_eq_top_of_isDefiniteRamifiedExactlyAt 27 below · cited by 1 · depth 30
• Non-commuting away-from-v units in a definite quaternion order CerednikDrinfeld.CosetGraph.exists_mem_awayUnits_mul_ne_mul_of_isDefiniteRamifiedExactlyAt 26 below · cited by 1 · depth 31

CerednikDrinfeld.EdgeFamily 3

• Reducedness of the edge chart ring over W(k)/p CerednikDrinfeld.EdgeFamily.isReduced_edgeRingCharP_wittVector_quotient_of_isAlgClosed 0 below · cited by 1 · depth 40
• Maps out of the edge chart ring killing ξ,η factor through the node CerednikDrinfeld.EdgeFamily.edgeRingCharP.eq_comp_of_apply_xi_eq_zero_of_apply_eta_eq_zero 0 below · cited by 1 · depth 43
• Branch-generic point of the edge chart dominating node points CerednikDrinfeld.EdgeFamily.edgeRingCharP.exists_ker_le_and_forall_ker_le_of_apply_eq_zero 0 below · cited by 2 · depth 43

CerednikDrinfeld.FormalODModule 225

• Kernel degree is invariant under a coefficient-ring isomorphism CerednikDrinfeld.FormalODModule.HasKernelOfDegree.map_of_bijective 0 below · cited by 8 · depth 28
• Right cancellation of [q^k] on formal mathcal O_D-modules CerednikDrinfeld.FormalODModule.Hom.eq_of_comp_act_pow_eq_of_hasKernelOfDegree 3 below · cited by 4 · depth 28
• Speciality of a formal mathcal O_D-module is stable under base change CerednikDrinfeld.FormalODModule.IsSpecial.map 0 below · cited by 28 · depth 28
• Cancellation by pᵈ in End(X) for finite height CerednikDrinfeld.FormalODModule.eq_of_natCast_pow_mul_eq_of_hasHeight 3 below · cited by 3 · depth 28
• Even kernel degree for a factor of [r^M] on a special formal module CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_pow_two_mul_of_mul_eq_natCast_of_isSpecial_of_hasHeight 44 below · cited by 3 · depth 28
• Quasi-inverse isogeny and transported embedding of 𝒪_D-endomorphisms CerednikDrinfeld.FormalODModule.exists_isODHom_comp_eq_act_pow_and_ringHom_centralizer_injective_of_isIsogenyOfHeight_of_isMaximal 35 below · cited by 5 · depth 28
• A power of p factors through an isogeny of formal 𝒪_D-modules CerednikDrinfeld.FormalODModule.exists_isODHom_comp_eq_act_pow_of_isIsogenyOfHeight_of_field 33 below · cited by 4 · depth 28
• Transport of an endomorphism embedding along a quasi-invertible isogeny CerednikDrinfeld.FormalODModule.exists_ringHom_centralizer_injective_forall_exists_toPowerSeries_eq_comp_of_comp_eq_act_pow 0 below · cited by 3 · depth 28
• Base change preserves kernels of degree d CerednikDrinfeld.FormalODModule.hasKernelOfDegree_map 2 below · cited by 45 · depth 28
• Injectivity of substitution along φ with finite kernel algebra CerednikDrinfeld.FormalODModule.subst_injective_of_hasKernelOfDegree 2 below · cited by 14 · depth 28
• Kernel degree p^{hm} of [p^m] in height h CerednikDrinfeld.FormalODModule.HasHeight.hasKernelOfDegree_act_pow 23 below · cited by 14 · depth 29
• Kernel degrees multiply under composition of series CerednikDrinfeld.FormalODModule.HasKernelOfDegree.comp 21 below · cited by 30 · depth 29
• Degree of the kernel under Xᵢ ↦ Xᵢ^q CerednikDrinfeld.FormalODModule.HasKernelOfDegree.comp_X_pow 0 below · cited by 5 · depth 29
• Uniqueness of the exponent in a p-power kernel degree CerednikDrinfeld.FormalODModule.HasKernelOfDegree.eq_of_pow_of_pow 0 below · cited by 9 · depth 29
• Degree of the outer factor of a composite CerednikDrinfeld.FormalODModule.HasKernelOfDegree.le_and_of_comp_pow 27 below · cited by 7 · depth 29
• Nilpotence of coordinates descends along an isogeny CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_of_X_pow_mem_span_comp 4 below · cited by 4 · depth 29
• Nilpotence of the variables modulo a kernel ideal of constant degree CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_of_hasKernelOfDegree 0 below · cited by 22 · depth 29
• A power of p kills a nilpotently supported closed subscheme CerednikDrinfeld.FormalODModule.exists_forall_act_pow_mem_span_of_isNilpotent_of_X_pow_mem 5 below · cited by 4 · depth 29
• Drinfeld's standard special formal mathcal O_D-module of height four CerednikDrinfeld.FormalODModule.exists_forall_isSpecial_map_and_hasHeight_four_map_of_isNilpotent 3 below · cited by 2 · depth 29
• Existence of a kernel degree over a field CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_of_X_pow_mem_span_of_field 15 below · cited by 4 · depth 29
• A height homomorphism GL₂(ℚᵣ)→ℤ from quaternionic endomorphisms CerednikDrinfeld.FormalODModule.exists_monoidHom_generalLinearGroup_finrank_kerAlgebra_eq_pow_of_hasHeight 19 below · cited by 4 · depth 29
• Frobenius series has kernel of degree r^{2k} CerednikDrinfeld.FormalODModule.hasKernelOfDegree_frobSeries 15 below · cited by 10 · depth 29
• Base change preserves kernel algebras of degree d CerednikDrinfeld.FormalODModule.hasKernelOfDegree_map_of_X_pow_mem 0 below · cited by 2 · depth 29
• Degree d kernel algebra from infinitesimal n-torsion CerednikDrinfeld.FormalODModule.hasKernelOfDegree_of_isFormalCoordinates_of_forall_isInfinitesimal 2 below · cited by 2 · depth 29
• Kernel degree from a presenting ideal with constant fibre rank CerednikDrinfeld.FormalODModule.hasKernelOfDegree_of_span_range_eq 0 below · cited by 7 · depth 29
• Cancelling an isogeny: β is an mathcal O_D-homomorphism CerednikDrinfeld.FormalODModule.isODHom_of_comp_eq_act_pow_of_subst_injective 0 below · cited by 2 · depth 29
• Cancelling unit factors in an identity of mathcal O_D-module series CerednikDrinfeld.FormalODModule.nthSeries_pow_comp_comp_act_inv_eq_of_nthSeries_mul_comp_eq 0 below · cited by 2 · depth 29
• Injectivity of substitution when κ[[X]]/(φ) is finite CerednikDrinfeld.FormalODModule.subst_injective_of_finite_kerAlgebra_of_field 0 below · cited by 2 · depth 29
• Injectivity of substitution detected on residue fields CerednikDrinfeld.FormalODModule.subst_injective_of_finite_kerAlgebra_of_residueFields 0 below · cited by 2 · depth 29
• Multiplicativity of kernel degrees under composition with a series over a field CerednikDrinfeld.FormalODModule.HasKernelOfDegree.comp_map_of_field 9 below · cited by 11 · depth 30
• Cancelling a finite locally free kernel of degree d CerednikDrinfeld.FormalODModule.HasKernelOfDegree.of_comp 24 below · cited by 2 · depth 30
• Rigidity for formal mathcal O_D-modules from p-adic density CerednikDrinfeld.FormalODModule.existsUnique_hom_apply_eq_adicEval_of_natural_of_dense_of_isNilpotent 2 below · cited by 2 · depth 30
• Verschiebung of a height-4 formal mathcal O_D-module CerednikDrinfeld.FormalODModule.exists_isODHom_map_iterateFrobenius_comp_X_pow_eq_act_pow 11 below · cited by 5 · depth 30
• Finiteness and projectivity of R[[x,y]]/(φ₁,φ₂) CerednikDrinfeld.FormalODModule.finite_and_projective_kerAlgebra_of_X_pow_mem 14 below · cited by 12 · depth 30
• The identity series has kernel of degree one CerednikDrinfeld.FormalODModule.hasKernelOfDegree_id 0 below · cited by 6 · depth 30
• ℤ/2-grading of the Cartier module of a formal mathcal O_D-module CerednikDrinfeld.FormalODModule.isCompl_gradedPiece_zero_one_of_isCompl_lieZero_lieOne 7 below · cited by 12 · depth 30
• Frobenius as a homomorphism between Frobenius-twisted base changes CerednikDrinfeld.FormalODModule.isODHom_frobSeries_map_of_forall_eq_pow 0 below · cited by 4 · depth 30
• Rigidity of mathcal O_D-linear maps after [p^{nμ}] CerednikDrinfeld.FormalODModule.act_pow_comp_eq_of_map_eq_and_isODHom_act_pow_comp_of_ker_pow_eq_bot 1 below · cited by 5 · depth 31
• Formal mathcal O_D-module homomorphisms from natural maps on nilpotent points CerednikDrinfeld.FormalODModule.existsUnique_hom_apply_eq_adicEval_of_natural_of_isNilpotent 1 below · cited by 1 · depth 31
• Lattice action on a formal group yields a formal mathcal O_D-module CerednikDrinfeld.FormalODModule.exists_F_eq_and_addVia_act_eq_of_isOrderCoord_of_isNilpotent 2 below · cited by 1 · depth 31
• A canonical ℤₚ²-parametrisation of η_{Φ,0} CerednikDrinfeld.FormalODModule.exists_addMonoidHom_bijOn_etaPiece_zero_of_isSpecial_of_hasHeight 137 below · cited by 1 · depth 31
• Constant kernel degree over a ring with connected spectrum CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_of_X_pow_mem_span_of_forall_isIdempotentElem 15 below · cited by 1 · depth 31
• Kernel degree of a sandwiched mathcal O_D-homomorphism is an even power of r CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_pow_two_mul_of_isODHom_of_comp_eq_act_pow_of_isAlgClosed 46 below · cited by 1 · depth 31
• Canonical L-map over W(k)/p, k algebraically closed CerednikDrinfeld.FormalODModule.exists_isCanonicalLMap_toGradedCartierModuleData_of_isSpecial_of_isAlgClosed 81 below · cited by 27 · depth 31
• An order in M₂(ℚₚ) acting compatibly with a rigidification CerednikDrinfeld.FormalODModule.exists_ringHom_centralizer_matrix_injective_and_rigidification_compat 154 below · cited by 2 · depth 31
• Complementary graded Cartier pieces over W(k)/p CerednikDrinfeld.FormalODModule.isCompl_gradedPiece_of_isSpecial_wittVector_quotient 6 below · cited by 1 · depth 31
• A graded piece of LieΦ lies in kervarpī CerednikDrinfeld.FormalODModule.lieZero_le_ker_lieVarpi_or_lieOne_le_ker_lieVarpi_of_isSpecial_wittVector_quotient 1 below · cited by 1 · depth 31
• The kernel ideal of ρ is an mathcal O_D-stable subgroup CerednikDrinfeld.FormalODModule.span_act_pow_le_span_and_subst_mem_span_of_isODHom_of_comp_eq_act_pow 1 below · cited by 2 · depth 31
• Speciality transports along isomorphisms of formal 𝒪_D-modules CerednikDrinfeld.FormalODModule.IsSpecial.of_isODHom_of_comp_eq_id 0 below · cited by 2 · depth 32
• Equal Frobenius twist and balanced r-heights CerednikDrinfeld.FormalODModule.eq_and_add_eq_add_of_hasKernelOfDegree_of_comp_act_pow_eq_comp 29 below · cited by 1 · depth 32
• Descent of mathcal O_D-homomorphisms along Pi, with Frobenius twist CerednikDrinfeld.FormalODModule.existsUnique_isODHom_frobTwist_comp_varpi_eq_of_mem_span_of_hasKernelOfDegree 28 below · cited by 1 · depth 32
• Degree-zero η-piece additively bijective to ℤₚ² CerednikDrinfeld.FormalODModule.exists_addMonoidHom_bijOn_etaPiece_zero_of_isCanonicalLMap 51 below · cited by 1 · depth 32
• Existence of a canonical L-map for formal mathcal O_D-modules CerednikDrinfeld.FormalODModule.exists_isCanonicalLMap_toGradedCartierModuleData 73 below · cited by 48 · depth 32
• Homogeneous V-basis for a special formal mathcal O_D-module with free Lie lines CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_of_isSpecial_of_free 8 below · cited by 6 · depth 32
• Quotients with equal kernel ideals are isomorphic CerednikDrinfeld.FormalODModule.exists_isODHom_comp_eq_of_span_range_eq_of_hasKernelOfDegree 28 below · cited by 2 · depth 32
• Endomorphisms of a special formal module as p-adic matrices CerednikDrinfeld.FormalODModule.exists_ringHom_centralizer_matrix_smul_eq_map_and_nsmul_apply_rigidification_eq 84 below · cited by 2 · depth 32
• Kernel degree p² for Pi under isogeny from a special module CerednikDrinfeld.FormalODModule.hasKernelOfDegree_varpi_sq_of_isIsogenyOfHeight_map_of_isSpecial_of_hasHeight 30 below · cited by 1 · depth 32
• Faithfulness and near-fullness of the matrix representation E CerednikDrinfeld.FormalODModule.injective_and_exists_pow_smul_map_eq_of_ringHom_centralizer_rigidification_compat 150 below · cited by 2 · depth 32
• Splitting of the Cartier module into graded pieces 0 and 1 CerednikDrinfeld.FormalODModule.isCompl_gradedPiece_zero_one_of_isNilpotent 5 below · cited by 43 · depth 32
• Speciality of the graded Cartier datum over W(k)/p CerednikDrinfeld.FormalODModule.isSpecialCartierModule_toGradedCartierModuleData_wittVector_quotient 27 below · cited by 3 · depth 32
• Speciality and height 4 lift along Artinian thickenings CerednikDrinfeld.FormalODModule.isSpecial_and_hasHeight_four_of_isIso_of_isArtinianRing 11 below · cited by 6 · depth 32
• Isomorphisms of formal mathcal O_D-modules induce graded Cartier isomorphisms CerednikDrinfeld.FormalODModule.Hom.bijective_map_and_forall_map_eq_of_isIso 0 below · cited by 1 · depth 33
• λ maps ηₙ bijectively onto the varpi=V locus CerednikDrinfeld.FormalODModule.bijOn_lambda_etaPiece_of_isCanonicalLMap_of_forall_exists 1 below · cited by 2 · depth 33
• Frobenius-fixed scalars act through W(j)∘θ on Cartier modules CerednikDrinfeld.FormalODModule.endAct_actEnd_eq_map_smul_of_frobenius_eq_of_isNilpotent 3 below · cited by 1 · depth 33
• Kernel degrees of 𝒪_D-endomorphisms of the base formal module CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_eq_four_mul_add_two_mul_vdet_of_centralizer_apply_eq_zpow_smul_heightNormalised_eq 47 below · cited by 1 · depth 33
• Structure constants of a homogeneous V-basis, with a₀₀a₀₁=p CerednikDrinfeld.FormalODModule.exists_hasStructureConstants_mul_eq_of_isHomogeneousVBasis 18 below · cited by 6 · depth 33
• Formal mathcal O_D-modules lift to the universal p-torsion-free base CerednikDrinfeld.FormalODModule.exists_liftRing_isHomogeneousVBasis_hasStructureConstants_liftConstants_and_isIso_of_isHausdorff 59 below · cited by 1 · depth 33
• Image of E contains p^mM₂(ℤₚ) CerednikDrinfeld.FormalODModule.exists_pow_smul_map_eq_of_ringHom_centralizer_rigidification_compat 63 below · cited by 1 · depth 33
• Height 4 is preserved by isogenies of formal 𝒪_D-modules CerednikDrinfeld.FormalODModule.hasHeight_four_of_isIsogenyOfHeight 26 below · cited by 2 · depth 33
• Kernels of degree d descend along nilpotent surjections CerednikDrinfeld.FormalODModule.hasKernelOfDegree_of_map_of_surjective_of_isNilpotent_ker 17 below · cited by 4 · depth 33
• Faithfulness of a rigidification-compatible matrix representation of End(Φ) CerednikDrinfeld.FormalODModule.injective_of_ringHom_centralizer_rigidification_compat 123 below · cited by 1 · depth 33
• Base change of the graded Cartier datum of X CerednikDrinfeld.FormalODModule.isBaseChangeAlong_toGradedCartierModuleData_baseChange 18 below · cited by 40 · depth 33
• Canonical L-map on a critical graded piece CerednikDrinfeld.FormalODModule.isCanonicalLMap_apply_eq_nMk_of_verschiebungInt_eq_endAct_varpiEnd 2 below · cited by 4 · depth 33
• Homogeneous V-basis splits the Cartier module into graded pieces CerednikDrinfeld.FormalODModule.isCompl_gradedPiece_zero_one_of_isHomogeneousVBasis 21 below · cited by 6 · depth 33
• Homogeneous V-basis makes the graded Cartier datum special CerednikDrinfeld.FormalODModule.isSpecialCartierModule_toGradedCartierModuleData 19 below · cited by 51 · depth 33
• The η-piece at a critical index, and injectivity CerednikDrinfeld.FormalODModule.mem_etaPiece_iff_of_isCanonicalLMap_apply_eq_nMk 40 below · cited by 4 · depth 33
• Kernel degree of the outer factor divides that of a composite CerednikDrinfeld.FormalODModule.HasKernelOfDegree.dvd_of_comp 21 below · cited by 1 · depth 34
• Cancelling the outer factor in kernel degrees CerednikDrinfeld.FormalODModule.HasKernelOfDegree.of_comp_left 21 below · cited by 3 · depth 34
• Tangent vectors of a homogeneous V-basis grade LieX CerednikDrinfeld.FormalODModule.IsHomogeneousVBasis.tangent_mem_and_existsUnique_smul_of_isNilpotent 0 below · cited by 10 · depth 34
• An mathcal O_D-linear isogeny of degree p^h divides [p^h] CerednikDrinfeld.FormalODModule.act_pow_mem_span_and_exists_isODHom_comp_eq_act_pow_of_hasKernelOfDegree 61 below · cited by 2 · depth 34
• No varpi-torsion in the Cartier module of a special formal mathcal O_D-module CerednikDrinfeld.FormalODModule.eq_zero_of_endAct_varpiEnd_eq_zero_of_isSpecial_of_hasHeight 39 below · cited by 2 · depth 34
• Kernel degree of a special formal module endomorphism via v_{det} CerednikDrinfeld.FormalODModule.exists_hasKernelOfDegree_eq_four_mul_add_two_mul_vdet_of_centralizer_apply_eq_zpow_smul_of_isSpecial 24 below · cited by 1 · depth 34
• Matching homogeneous V-bases give an isomorphism of formal mathcal O_D-modules CerednikDrinfeld.FormalODModule.exists_hom_isIso_forall_map_eq_of_hasStructureConstants 37 below · cited by 5 · depth 34
• Prescribed structure constants are realised by a formal mathcal O_D-module CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_and_hasStructureConstants_of_mul_eq 31 below · cited by 6 · depth 34
• Lifting a formal mathcal O_D-module by lifting its structure constants CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_hasStructureConstants_and_isIso_map_of_forall_apply_eq 59 below · cited by 2 · depth 34
• Zariski-local homogeneous V-bases for special formal 𝒪_D-modules CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_map_of_isSpecial_of_isNilpotent 29 below · cited by 3 · depth 34
• Condition [C2] for η-invariants over an algebraically closed field CerednikDrinfeld.FormalODModule.exists_nVarpi_eq_of_mem_etaPiece_of_toLieQuot_eq_of_isAlgClosed 45 below · cited by 1 · depth 34
• N-span of η_{Φ,0} and Piη_{Φ,0} up to pᵃ CerednikDrinfeld.FormalODModule.exists_pow_smul_eq_sum_smul_add_sum_smul_nVarpi_of_bijOn_etaPiece_zero_of_isAlgClosed 122 below · cited by 2 · depth 34
• Base change of the Lie eigenspace decomposition CerednikDrinfeld.FormalODModule.isCompl_lieZero_lieOne_map_and_eq_span_image 0 below · cited by 8 · depth 34
• Transport of homogeneous V-bases and structure constants along an isomorphism CerednikDrinfeld.FormalODModule.isHomogeneousVBasis_map_and_hasStructureConstants_map_of_hom_of_isIso 0 below · cited by 2 · depth 34
• Homogeneous V-bases have unit tangent determinant CerednikDrinfeld.FormalODModule.isHomogeneousVBasis_of_isHomogeneousVBasis_toGradedCartierModuleData 3 below · cited by 34 · depth 34
• Abstract homogeneous V-basis is a law-level V-basis CerednikDrinfeld.FormalODModule.isHomogeneousVBasis_of_toGradedCartierModuleData_of_algebra_padicInt 20 below · cited by 3 · depth 34
• Tangent classes of η(L) span both Lie pieces over 𝔽̄ₚ CerednikDrinfeld.FormalODModule.lieZero_le_span_tangent_and_lieOne_le_span_tangent_of_mem_etaPiece_of_isAlgClosed 42 below · cited by 1 · depth 34
• Base change of the graded pieces of Lie CerednikDrinfeld.FormalODModule.lieZero_lieOne_map_eq_span_image 0 below · cited by 4 · depth 34
• Order-zero structure constants give the tangent action of varpi CerednikDrinfeld.FormalODModule.linearPart_varpi_mulVec_tangent_eq_smul_of_hasStructureConstants 0 below · cited by 8 · depth 34
• Frobenius twist of the labelling: N, pieces, η, canonicity CerednikDrinfeld.FormalODModule.nMap_id_bijective_and_nPiece_and_eta_and_isCanonicalLMap_comp_frobenius 1 below · cited by 2 · depth 34
• Classes modulo VM have equal tangent vectors CerednikDrinfeld.FormalODModule.tangent_eq_of_mkQ_eq 0 below · cited by 2 · depth 34
• Tangent map sends Cartier graded pieces into Lie pieces CerednikDrinfeld.FormalODModule.tangent_mem_lieZero_and_lieOne_of_mem_gradedPiece_of_isNilpotent 0 below · cited by 13 · depth 34
• Invariant frame yields ℤₚ-basis and injective endomorphism matrix CerednikDrinfeld.FormalODModule.CritChart.exists_basis_coe_eq_and_injective_endMatrix_of_forall_existsUnique_of_isCompl 9 below · cited by 1 · depth 35
• Graded pieces are preserved by 𝒪_D-linear homomorphisms CerednikDrinfeld.FormalODModule.IsODHom.map_mem_gradedPiece 0 below · cited by 10 · depth 35
• Kernel of degree p^h is killed by p^h CerednikDrinfeld.FormalODModule.act_pow_mem_span_of_isODHom_of_hasKernelOfDegree 33 below · cited by 1 · depth 35
• λ identifies ηₙ with the Pi=V locus CerednikDrinfeld.FormalODModule.bijOn_lambda_etaPiece_of_isCanonicalLMap_of_charP 1 below · cited by 4 · depth 35
• Injectivity of varpi on Cartier modules over reduced bases CerednikDrinfeld.FormalODModule.eq_zero_of_endAct_varpiEnd_eq_zero_of_isReduced 33 below · cited by 16 · depth 35
• Equal Pi-structure constants force an isomorphism of Cartier modules CerednikDrinfeld.FormalODModule.exists_addMonoidHom_bijective_map_eq_of_hasStructureConstants 28 below · cited by 1 · depth 35
• Digit shape of a homogeneous V-basis over k[ε] CerednikDrinfeld.FormalODModule.exists_eq_sum_verschiebungInt_iterate_homothety_baseChange_of_baseChangeEq_eq 11 below · cited by 2 · depth 35
• Local freeness of the Lie eigenlines of a special formal mathcal O_D-module CerednikDrinfeld.FormalODModule.exists_free_lieZero_map_and_free_lieOne_map_of_isSpecial 1 below · cited by 1 · depth 35
• Frobenius lands in VM when a_{0,i_0}=0 CerednikDrinfeld.FormalODModule.exists_frobenius_eq_verschiebungInt_of_hasStructureConstants_of_apply_zero_eq_zero 0 below · cited by 2 · depth 35
• Cartier-module isomorphisms come from formal mathcal O_D-module isomorphisms CerednikDrinfeld.FormalODModule.exists_hom_isIso_forall_map_eq_of_bijective 28 below · cited by 1 · depth 35
• Universal formal mathcal O_D-module with homogeneous V-basis CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_and_hasStructureConstants_liftVar 30 below · cited by 1 · depth 35
• Existence of a homogeneous V-basis for special formal 𝒪_D-modules CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_of_isSpecial_of_free_of_isNilpotent 25 below · cited by 2 · depth 35
• Odd η-classes with tangent class Pi m₀ come from even ones CerednikDrinfeld.FormalODModule.exists_nVarpi_eq_of_mem_etaPiece_one_of_toLieQuot_eq 0 below · cited by 1 · depth 35
• Lifting a 1-critical η₁-section through Pi CerednikDrinfeld.FormalODModule.exists_nVarpi_eq_of_mem_etaPiece_one_of_toLieQuot_eq_of_critical_one 0 below · cited by 1 · depth 35
• Even η-classes with tangent class in varpi Lie₁ come from η₁ CerednikDrinfeld.FormalODModule.exists_nVarpi_eq_of_mem_etaPiece_zero_of_toLieQuot_eq 0 below · cited by 1 · depth 35
• Pi-preimage in η₁(L) at a 1-critical point CerednikDrinfeld.FormalODModule.exists_nVarpi_eq_of_mem_etaPiece_zero_of_toLieQuot_eq_of_critical_one 0 below · cited by 1 · depth 35
• First-order varpi-equivariance: cocycle pullback modulo coboundary CerednikDrinfeld.FormalODModule.exists_subst_varpi_eq_sum_linearPart_smul_add_addCoboundary_and_coeff_eq_snd_linearPart 0 below · cited by 1 · depth 35
• ℤ_{q²}-equivariance of a first-order cocycle tuple CerednikDrinfeld.FormalODModule.exists_sum_linearPart_act_smul_eq_subst_act_add_addCoboundary 0 below · cited by 1 · depth 35
• varpi-equivariance of the cocycle tuple, modulo coboundaries CerednikDrinfeld.FormalODModule.exists_sum_linearPart_varpi_smul_eq_subst_varpi_add_addCoboundary 0 below · cited by 1 · depth 35
• First-order structure constants of a reshaped V-basis over k[ε] CerednikDrinfeld.FormalODModule.hasStructureConstants_dualNumber_apply_eq_of_eq_sum_verschiebungInt_iterate_homothety 3 below · cited by 2 · depth 35
• Canonical L-maps on critical graded pieces in characteristic p CerednikDrinfeld.FormalODModule.isCanonicalLMap_apply_eq_nMk_of_charP 2 below · cited by 25 · depth 35
• Canonicity of L-maps under the σ-shift of the grading CerednikDrinfeld.FormalODModule.isCanonicalLMap_iff_isCanonicalLMap_comp_of_comp_frobenius 0 below · cited by 1 · depth 35
• Homogeneous V-basis implies the formal mathcal O_D-module is special CerednikDrinfeld.FormalODModule.isSpecial_of_isHomogeneousVBasis 0 below · cited by 4 · depth 35
• Pull-back of symmetric 2-cocycles along the action series CerednikDrinfeld.FormalODModule.isSymmTwoCocycle_subst_act_and_subst_act_add 0 below · cited by 2 · depth 35
• The η-piece at a critical index in characteristic p CerednikDrinfeld.FormalODModule.mem_etaPiece_iff_of_isCanonicalLMap_apply_eq_nMk_of_charP 0 below · cited by 13 · depth 35
• Reduction mod p is bijective on η-invariants CerednikDrinfeld.FormalODModule.nMap_bijOn_eta_of_eq_baseChangeEq_mk 96 below · cited by 9 · depth 35
• Kernel degree pins the central exponent: 2m' = 4(k-c) CerednikDrinfeld.FormalODModule.two_mul_eq_four_mul_sub_of_map_eq_pow_smul_inv_of_hasKernelOfDegree 25 below · cited by 2 · depth 35
• Connecting sum Theta: linearity, coboundaries, equivariance, peeling CerednikDrinfeld.FormalODModule.connectingSum_smul_add_and_addCoboundary_and_subst_act_and_addCoboundary_eq_subst_nthSeries_and_mem_span 0 below · cited by 1 · depth 36
• Absence of p-torsion in η(L) over Noetherian bases CerednikDrinfeld.FormalODModule.eq_zero_of_nsmul_eq_zero_of_mem_eta 114 below · cited by 2 · depth 36
• Graded finite V-adic expansion in a homogeneous V-basis CerednikDrinfeld.FormalODModule.existsUnique_eq_sum_verschiebung_iterate_homothety_add_of_mem_gradedPiece 9 below · cited by 4 · depth 36
• Zariski-local freeness and two invariant generators of a subgroup ideal CerednikDrinfeld.FormalODModule.exists_cover_free_and_span_range_eq_map_of_subgroup_ideal 28 below · cited by 2 · depth 36
• Zariski-local quotient of a formal mathcal O_D-module by a subgroup ideal CerednikDrinfeld.FormalODModule.exists_cover_isIsogenyOfHeight_span_range_eq_map_of_subgroup_ideal 46 below · cited by 1 · depth 36
• Universal structure constants for Frobenius in a homogeneous V-basis CerednikDrinfeld.FormalODModule.exists_forall_hasStructureConstants_frobenius_eq_sum 4 below · cited by 1 · depth 36
• Quotient by a finite free subgroup ideal is a Hopf algebra CerednikDrinfeld.FormalODModule.exists_hopfAlgebra_ker_eq_of_subgroup_ideal 1 below · cited by 2 · depth 36
• Speciality of a formal mathcal O_D-module is cut out by an idempotent CerednikDrinfeld.FormalODModule.exists_idempotent_isSpecial_map_iff 6 below · cited by 2 · depth 36
• Glueing formal 𝒪_D-modules along a fibre product of rings CerednikDrinfeld.FormalODModule.exists_map_pullbackFst_eq_and_isIso_map_pullbackSnd_of_isIso 2 below · cited by 1 · depth 36
• Representability of mathcal O_D-stable subgroup ideals by a projective scheme CerednikDrinfeld.FormalODModule.exists_scheme_represents_subgroup_ideal_and_isClosedImmersion_toProjSpace 24 below · cited by 1 · depth 36
• Pull-back along varpi twists the type by Frobenius CerednikDrinfeld.FormalODModule.forall_exists_subst_act_subst_varpi_eq_smul_of_type 0 below · cited by 1 · depth 36
• Speciality of formal 𝒪_D-modules is Zariski-local CerednikDrinfeld.FormalODModule.isSpecial_of_forall_isSpecial_map_away 7 below · cited by 1 · depth 36
• Speciality descends along surjections with nilpotent kernel CerednikDrinfeld.FormalODModule.isSpecial_of_isSpecial_map_of_surjective_of_isNilpotent 3 below · cited by 1 · depth 36
• Ideal generated by an mathcal O_D-homomorphism is stable CerednikDrinfeld.FormalODModule.subst_mem_span_of_isODHom 1 below · cited by 2 · depth 36
• Lie-level vanishing gives index-1 criticality after base change CerednikDrinfeld.FormalODModule.CritChart.isCritical_map_one_of_lieOne_le_ker_lieVarpi 4 below · cited by 10 · depth 37
• Lie-level vanishing yields index-0 criticality after base change CerednikDrinfeld.FormalODModule.CritChart.isCritical_map_zero_of_lieZero_le_ker_lieVarpi 4 below · cited by 12 · depth 37
• No p-torsion in η(L) over reduced Noetherian bases of characteristic p CerednikDrinfeld.FormalODModule.eq_zero_of_nsmul_eq_zero_of_mem_eta_of_isReduced 111 below · cited by 1 · depth 37
• Zariski-local invariant coordinates for an 𝒪_D-stable subgroup ideal CerednikDrinfeld.FormalODModule.exists_cover_invariant_coordinates_of_subgroup_ideal 39 below · cited by 1 · depth 37
• Base change of the coordinate presentation of Φ[p^N] CerednikDrinfeld.FormalODModule.exists_family_algHom_tensorProduct_adicEval_surjective_ker_eq_span_act_pow_natural 2 below · cited by 1 · depth 37
• Eta piece in degree one is a ℤₚ-lattice on invariants CerednikDrinfeld.FormalODModule.exists_forall_mem_etaPiece_one_iff_eq_nMk_sum_smul_of_isCritical_of_isAlgClosed 42 below · cited by 2 · depth 37
• η₀(L) as a ℤₚ-lattice at a critical index CerednikDrinfeld.FormalODModule.exists_forall_mem_etaPiece_zero_iff_eq_nMk_sum_smul_of_isCritical_of_isAlgClosed 42 below · cited by 5 · depth 37
• Φ[p^N] as a Hopf algebra with mathcal O_D-coaction CerednikDrinfeld.FormalODModule.exists_hopfAlgebra_ker_eq_span_act_pow_and_forall_bialgHom_subst_act 2 below · cited by 1 · depth 37
• Quotient formal mathcal O_D-module along invariant coordinates CerednikDrinfeld.FormalODModule.exists_isIsogenyOfHeight_of_invariant_coordinates 6 below · cited by 1 · depth 37
• Two invariant generators on a chart near a maximal ideal CerednikDrinfeld.FormalODModule.exists_notMem_free_and_span_range_eq_map_of_subgroup_ideal_of_isMaximal 26 below · cited by 1 · depth 37
• Eigenvalue splitting of Lie Y over a p-nilpotent base CerednikDrinfeld.FormalODModule.isCompl_lieZero_lieOne_and_eq_ker_of_isNilpotent 0 below · cited by 1 · depth 37
• Pi has colength one on each graded Cartier piece CerednikDrinfeld.FormalODModule.length_gradedSubmodule_quotient_map_varpiLinear_eq_one_of_isSpecial_of_hasHeight 49 below · cited by 1 · depth 37
• Isogeny of height 2h: colength h on each graded piece CerednikDrinfeld.FormalODModule.length_gradedSubmodule_quotient_range_mapLinear_eq_of_isIsogenyOfHeight_two_mul_of_isSpecial 47 below · cited by 5 · depth 37
• Base change of the ideal of a finite mathcal O_D-stable subgroup CerednikDrinfeld.FormalODModule.subgroup_ideal_map 0 below · cited by 2 · depth 37
• Critical index criterion on the Lie algebra for special formal mathcal O_D-modules CerednikDrinfeld.FormalODModule.CritChart.isCritical_iff_le_ker_lieVarpi_of_isSpecial 11 below · cited by 2 · depth 38
• Critical index and its invariants under algebraically closed base change CerednikDrinfeld.FormalODModule.CritChart.isCritical_map_and_surjOn_baseChange_invariants_of_isAlgClosed 44 below · cited by 1 · depth 38
• Subgroup ideals of a formal group law are invariantly generated CerednikDrinfeld.FormalODModule.eq_span_setOf_invariant_of_subgroup_ideal_of_free 12 below · cited by 1 · depth 38
• Vanishing in N(M) detected by jointly injective base changes CerednikDrinfeld.FormalODModule.eq_zero_of_forall_nMap_baseChange_eq_zero 20 below · cited by 1 · depth 38
• No p-torsion in η(L) over an algebraically closed field CerednikDrinfeld.FormalODModule.eq_zero_of_nsmul_eq_zero_of_mem_eta_of_isAlgClosed 35 below · cited by 1 · depth 38
• Invariant series are uniquely series in invariant generators CerednikDrinfeld.FormalODModule.existsUnique_subst_eq_of_invariant_of_span_range_eq_of_free 10 below · cited by 1 · depth 38
• Extra variables preserve unique expressibility in u CerednikDrinfeld.FormalODModule.existsUnique_subst_of_invariant_of_forall_existsUnique_subst 0 below · cited by 1 · depth 38
• Homogeneous V-basis for special formal mathcal O_D-modules over a field CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_of_isSpecial_field 9 below · cited by 6 · depth 38
• Descent of the mathcal O_D-action along invariant coordinates CerednikDrinfeld.FormalODModule.exists_isODHom_of_isLawHom_of_invariant_coordinates 0 below · cited by 1 · depth 38
• Descent of the formal group law along invariant coordinates CerednikDrinfeld.FormalODModule.exists_mvFormalGroup_isComm_isLawHom_of_invariant_coordinates 2 below · cited by 1 · depth 38
• Two invariant elements generate J modulo (x)J and n CerednikDrinfeld.FormalODModule.exists_pair_invariant_forall_exists_coeff_sub_mem_of_isMaximal 9 below · cited by 1 · depth 38
• Tangent line of η_{i_0} over κ[ε] and its period equation CerednikDrinfeld.FormalODModule.exists_tangent_eq_smul_and_forall_fst_snd_eq_of_mem_etaPiece_of_hasStructureConstants_dualNumber 32 below · cited by 2 · depth 38
• Homomorphisms from X.F determined on a homogeneous V-basis CerednikDrinfeld.FormalODModule.hom_eq_of_forall_map_apply_eq_of_isHomogeneousVBasis 9 below · cited by 1 · depth 38
• Base change preserves J-invariant power series CerednikDrinfeld.FormalODModule.map_mem_setOf_invariant_of_mem 0 below · cited by 2 · depth 38
• Tangent variation on η_{i_0} is no rescaling outside windows CerednikDrinfeld.FormalODModule.not_exists_forall_period_variation_eq_mul_of_mem_etaPiece_of_hasStructureConstants_dualNumber 159 below · cited by 1 · depth 38
• Compatibility of Theta with τ and V-divisibility on a critical piece CerednikDrinfeld.FormalODModule.apply_mkQ_eq_mkQ_and_mem_vRange_iff_of_apply_eq_nMk_of_isCritical_of_isAlgClosed 41 below · cited by 1 · depth 39
• Teichmüller equivariance implies full ℤ_{p²}-equivariance of Cartier modules CerednikDrinfeld.FormalODModule.endAct_actEnd_comp_eq_of_forall_teichmuller_of_isNilpotent 1 below · cited by 1 · depth 39
• Digit relations for a Pi=V invariant over dual numbers CerednikDrinfeld.FormalODModule.exists_digits_tangent_eq_and_fst_snd_eq_of_varpiEnd_eq_verschiebungInt_of_hasStructureConstants_dualNumber 25 below · cited by 1 · depth 39
• Coaction of the subgroup Hopf algebra on the levels B[[x]]/I^N CerednikDrinfeld.FormalODModule.exists_levelCoaction_of_subgroup_ideal_of_ne_zero 0 below · cited by 3 · depth 39
• Extending an invariant bijection to the critical graded pieces CerednikDrinfeld.FormalODModule.exists_linearMap_bijOn_gradedPiece_apply_eq_of_bijective_invariants_of_isCritical_of_isAlgClosed 43 below · cited by 1 · depth 39
• Two η_{i_0}-elements with 𝔽ₚ-independent tangent parts over κ[ε] CerednikDrinfeld.FormalODModule.exists_mem_etaPiece_tangent_eq_smul_forall_dvd_of_isAlgClosed_dualNumber 154 below · cited by 1 · depth 39
• One-step lift of an invariant element modulo I^{N+1} CerednikDrinfeld.FormalODModule.exists_mem_pow_and_invariant_succ 4 below · cited by 1 · depth 39
• Two invariant generators suffice over a field CerednikDrinfeld.FormalODModule.exists_span_range_eq_of_le_span_setOf_invariant_of_field 6 below · cited by 1 · depth 39
• First-order invariant representatives modulo (x) I CerednikDrinfeld.FormalODModule.exists_sub_mem_and_firstOrder_invariant 5 below · cited by 1 · depth 39
• η at a critical index over a base of characteristic p CerednikDrinfeld.FormalODModule.mem_etaPiece_iff_exists_varpiEnd_eq_verschiebungInt_of_charP 3 below · cited by 2 · depth 39
• Rigidity of η along a square-zero thickening, graded form CerednikDrinfeld.FormalODModule.nMap_bijOn_etaPiece_of_eq_baseChangeEq_of_surjective_of_mul_eq_zero 82 below · cited by 2 · depth 39
• Lie quotient M/VM identified with the tangent space CerednikDrinfeld.FormalODModule.exists_addEquiv_lieQuot_forall_apply_mkQ_eq_tangent 7 below · cited by 1 · depth 40
• Freeness of the formal plane over invariant generators CerednikDrinfeld.FormalODModule.exists_eq_sum_subst_mul_and_subst_eq_zero_of_invariant 0 below · cited by 1 · depth 40
• Special formal mathcal O_D-module of height 4 over the edge chart CerednikDrinfeld.FormalODModule.exists_isHomogeneousVBasis_hasStructureConstants_edgeRingConstants_isSpecial_hasHeight_of_isAlgClosed 101 below · cited by 1 · depth 40
• Two η-elements at a critical index with 𝔽ₚ-independent tangents CerednikDrinfeld.FormalODModule.exists_nMk_mem_etaPiece_tangent_eq_smul_forall_dvd_of_isAlgClosed 66 below · cited by 1 · depth 40
• Rigidification of the explicit edge family with standard Drinfeld lines CerednikDrinfeld.FormalODModule.forall_exists_isAdmissible_forall_isCartierQuadruple_map_line_eq_of_hasStructureConstants_edgeRingConstants 399 below · cited by 1 · depth 40
• Explicit height-4 edge isogeny between special formal mathcal O_D-modules CerednikDrinfeld.FormalODModule.exists_hom_map_eq_sub_verschiebungInt_and_isIsogenyOfHeight_of_hasStructureConstants_edgeConstants 64 below · cited by 1 · depth 41
• Normalised node isogeny onto the edge family's node fibre CerednikDrinfeld.FormalODModule.exists_isIsogenyOfHeight_map_node_rigidNum_single_eq 199 below · cited by 1 · depth 41
• Cartier quadruples at geometric points of the edge family CerednikDrinfeld.FormalODModule.forall_isCartierQuadruple_map_line_eq_of_rigidNum_single_eq_of_edge_isogeny 346 below · cited by 1 · depth 41
• Height four from the edge structure constants CerednikDrinfeld.FormalODModule.hasHeight_four_of_hasStructureConstants_edgeRingConstants_of_isAlgClosed 92 below · cited by 1 · depth 41
• Bijectivity of λ on η(L)ₙ in characteristic p CerednikDrinfeld.FormalODModule.bijOn_lambda_etaPiece_of_isCanonicalLMap_of_forall_exists_of_charP 1 below · cited by 3 · depth 42
• Closed Witt form of the edge structure constants CerednikDrinfeld.FormalODModule.endAct_varpiEnd_eq_teichmuller_sub_smul_add_verschiebungInt_of_hasStructureConstants_edgeConstants 4 below · cited by 15 · depth 42
• Edge structure constants force nilpotent coordinates modulo [p] CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_act_of_hasStructureConstants_edgeConstants 48 below · cited by 2 · depth 42
• Inverting an integral matrix by an 𝒪_D-linear endomorphism CerednikDrinfeld.FormalODModule.exists_centralizer_mul_map_eq_pow_smul_one_and_hasKernelOfDegree_of_det_eq 157 below · cited by 1 · depth 42
• Explicit homomorphism of special formal modules from Witt edge relations CerednikDrinfeld.FormalODModule.exists_hom_map_eq_sub_verschiebungInt_of_endAct_varpiEnd_eq_teichmuller 33 below · cited by 1 · depth 42
• Cartier quadruple of the edge family at a node point CerednikDrinfeld.FormalODModule.exists_isCartierQuadruple_map_line_eq_of_edge_isogeny_of_apply_xi_eq_zero_of_apply_eta_eq_zero 331 below · cited by 1 · depth 42
• Geometric fibre of the edge family on the η-branch CerednikDrinfeld.FormalODModule.exists_isCartierQuadruple_map_line_eq_of_edge_isogeny_of_apply_xi_eq_zero_of_apply_eta_ne_zero 319 below · cited by 1 · depth 42
• Cartier quadruple and Deligne lines at a point with y(ξ)≠ 0 CerednikDrinfeld.FormalODModule.exists_isCartierQuadruple_map_line_eq_of_edge_isogeny_of_apply_xi_ne_zero 320 below · cited by 1 · depth 42
• Degree p⁴ for [p] on the edge-family formal 𝒪_D-module CerednikDrinfeld.FormalODModule.finrank_kerAlgebra_map_act_eq_pow_four_of_hasStructureConstants_edgeRingConstants_of_isAlgClosed 90 below · cited by 1 · depth 42
• Finite locally free kernel from geometric fibre degrees CerednikDrinfeld.FormalODModule.hasKernelOfDegree_of_X_pow_mem_of_forall_finrank_eq_of_isAlgClosed 15 below · cited by 1 · depth 42
• Explicit edge homomorphism is an isogeny of height 4 CerednikDrinfeld.FormalODModule.isIsogenyOfHeight_four_of_map_eq_sub_verschiebungInt_edgeRingCharP 61 below · cited by 1 · depth 42
• Nilpotent coordinates on X[p] for a pure edge branch CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_act_of_hasStructureConstants_edgeConstants_zero 47 below · cited by 1 · depth 43
• Edge-family Cartier module is free of rank 4 on γ, Vγ CerednikDrinfeld.FormalODModule.exists_basis_cartierModule_eq_of_hasStructureConstants_edgeConstants 25 below · cited by 3 · depth 43
• Isogeny matrix on degree-zero Cartier pieces has determinant u p^h CerednikDrinfeld.FormalODModule.exists_det_eq_mul_pow_of_mapLinear_eq_sum_smul_of_isIsogenyOfHeight_two_mul 50 below · cited by 1 · depth 43
• Dual edge homomorphism ρᵈagger: X→ Y on Cartier modules CerednikDrinfeld.FormalODModule.exists_hom_map_eq_add_verschiebungInt_of_endAct_varpiEnd_eq_teichmuller 34 below · cited by 2 · depth 43
• Transporting homogeneous V-bases with Pi = V CerednikDrinfeld.FormalODModule.exists_hom_map_eq_of_endAct_varpiEnd_eq_verschiebungInt 33 below · cited by 2 · depth 43
• Node case: Cartier quadruple with node Deligne lines CerednikDrinfeld.FormalODModule.forall_isCartierQuadruple_map_node_line_eq_of_rigidNum_single_eq 309 below · cited by 1 · depth 43
• Edge structure constants force height four CerednikDrinfeld.FormalODModule.hasHeight_four_of_hasStructureConstants_edgeConstants_of_perfectRing 89 below · cited by 1 · depth 43
• Stalk kernels at an η-branch point of the edge family CerednikDrinfeld.FormalODModule.ker_eq_span_of_lattice_eq_of_isCartierQuadruple_map_edge_of_apply_xi_eq_zero_of_apply_eta_ne_zero 120 below · cited by 1 · depth 43
• Kernels of u₀ and u₁ at a ξ-point CerednikDrinfeld.FormalODModule.ker_eq_span_of_lattice_eq_of_isCartierQuadruple_map_edge_of_apply_xi_ne_zero 121 below · cited by 1 · depth 43
• η-branch: both Drinfeld lattices equal p⁻¹ diag(p,1) ℤₚ² CerednikDrinfeld.FormalODModule.lattice_eq_of_isCartierQuadruple_map_edge_of_apply_xi_eq_zero_of_apply_eta_ne_zero 89 below · cited by 1 · depth 43
• Lattices of a Cartier quadruple at a ξ-point CerednikDrinfeld.FormalODModule.lattice_eq_of_isCartierQuadruple_map_edge_of_apply_xi_ne_zero 90 below · cited by 1 · depth 43
• Nilpotence of coordinates passes to composites CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_comp_of_X_pow_mem_span 0 below · cited by 1 · depth 44
• Power-of-coordinate membership is invariant under formal coordinate change CerednikDrinfeld.FormalODModule.exists_X_pow_mem_span_of_X_pow_mem_span_comp_of_comp_eq_id 1 below · cited by 1 · depth 44
• Degree-one η-sections with tangent ratio -y(η) CerednikDrinfeld.FormalODModule.exists_isEtaSection_one_tangent_eq_neg_mul_of_edge_isogeny_of_apply_xi_eq_zero_of_apply_eta_ne_zero 114 below · cited by 1 · depth 44
• Degree-one eta-sections on the ξ-branch of the edge family CerednikDrinfeld.FormalODModule.exists_isEtaSection_one_tangent_eq_neg_mul_of_edge_isogeny_of_apply_xi_ne_zero 115 below · cited by 1 · depth 44
• Degree-zero η-sections on the η-branch: tangent ratio -y(η) CerednikDrinfeld.FormalODModule.exists_isEtaSection_zero_tangent_eq_neg_mul_of_edge_isogeny_of_apply_xi_eq_zero_of_apply_eta_ne_zero 115 below · cited by 1 · depth 44
• Degree-zero η-sections of the edge family where y(ξ)≠ 0 CerednikDrinfeld.FormalODModule.exists_isEtaSection_zero_tangent_eq_neg_mul_of_edge_isogeny_of_apply_xi_ne_zero 115 below · cited by 1 · depth 44
• Height 4 from a rank-4 Cartier module and nilpotent [p]-coordinates CerednikDrinfeld.FormalODModule.hasHeight_four_of_basis_cartierModule_of_X_pow_mem_span 38 below · cited by 1 · depth 44
• Node stalks of a Cartier quadruple: lattices and kernel lines CerednikDrinfeld.FormalODModule.lattice_eq_and_ker_eq_span_of_isCartierQuadruple_map_node_of_rigidNum_single_eq 97 below · cited by 1 · depth 44
• Rigidification numerator of the edge family at an arbitrary base point CerednikDrinfeld.FormalODModule.smul_rigidNum_map_single_eq_smul_baseChange_of_rigidNum_single_eq_of_edge_isogeny 0 below · cited by 6 · depth 44
• Rank of the Cartier module equals the height CerednikDrinfeld.FormalODModule.eq_four_of_basis_cartierModule_of_finrank_eq_pow 26 below · cited by 1 · depth 45
• Node kernels of a Cartier quadruple are coordinate lines CerednikDrinfeld.FormalODModule.ker_eq_span_of_lattice_eq_of_isCartierQuadruple_map_node_of_rigidNum_single_eq 94 below · cited by 1 · depth 45
• Node lattices of the Cartier quadruple of the normalised node triple CerednikDrinfeld.FormalODModule.lattice_eq_of_isCartierQuadruple_map_node_of_rigidNum_single_eq 91 below · cited by 1 · depth 45
• Degree-one η-sections at the node of the standard edge CerednikDrinfeld.FormalODModule.exists_isEtaSection_one_tangent_eq_neg_mul_map_node_of_rigidNum_single_eq 90 below · cited by 1 · depth 46
• Degree-zero η-sections at the node of the standard edge CerednikDrinfeld.FormalODModule.exists_isEtaSection_zero_tangent_eq_neg_mul_map_node_of_rigidNum_single_eq 90 below · cited by 1 · depth 46
• Node normalisation of the rigidification numerator propagates under base change CerednikDrinfeld.FormalODModule.smul_rigidNum_map_node_single_eq_smul_baseChange_of_rigidNum_single_eq 0 below · cited by 3 · depth 46

CerednikDrinfeld.FormalOmega 222

• Descent of fixed-coefficient geometric fibres along a Γ-quotient CerednikDrinfeld.FormalOmega.AlgFunctor.fibre_descent_of_fixed_fst 0 below · cited by 1 · depth 25
• Descent of a formal categorical quotient through p CerednikDrinfeld.FormalOmega.AlgFunctor.formalQuotient_descent 0 below · cited by 2 · depth 25
• Action laws for the twisted GL₂(K₀)-relation on Ω CerednikDrinfeld.FormalOmega.OmegaNr.isTwistedAct_laws 0 below · cited by 20 · depth 25
• Valuation ring of a complete rank-one field is an adic frame CerednikDrinfeld.FormalOmega.isAdicFrame_of_injective_of_forall_le_one_iff_mem_range 0 below · cited by 1 · depth 26
• Twisted Γ-action on adic points versus translation by Γ' CerednikDrinfeld.FormalOmega.AdicPoint.exists_isTwistedAct_iff_exists_eq_act 1 below · cited by 2 · depth 27
• Transport of an adic point along a base automorphism CerednikDrinfeld.FormalOmega.AdicPoint.exists_pt_eq_map_and_toOmega_eq_of_algEquiv 0 below · cited by 1 · depth 27
• GL₂(K)-equivariance of the adic coordinate map CerednikDrinfeld.FormalOmega.AdicPoint.toOmega_act 0 below · cited by 13 · depth 27
• Injectivity of the coordinate map on adic points of widehatΩ CerednikDrinfeld.FormalOmega.AdicPoint.toOmega_injective 0 below · cited by 8 · depth 27
• Every point of Ω comes from an adic point CerednikDrinfeld.FormalOmega.AdicPoint.toOmega_surjOn 5 below · cited by 12 · depth 27
• Proper 𝒪-schemes: R-points agree with C-points CerednikDrinfeld.FormalOmega.IsAdicFrame.injective_comp_and_exists_comp_eq_of_isProper 0 below · cited by 1 · depth 27
• The adic frame ring is local with non-zero reduction mod π CerednikDrinfeld.FormalOmega.IsAdicFrame.isLocalRing_and_nontrivial_modPow 0 below · cited by 2 · depth 27
• Scheme points over a π-adically complete local ring CerednikDrinfeld.FormalOmega.existsUnique_hom_comp_eq_of_compatible_modPow 1 below · cited by 5 · depth 27
• Points over algebraically closed π-nilpotent 𝒪-algebras CerednikDrinfeld.FormalOmega.nonempty_corep_and_nonempty_omega_of_isAlgClosed 3 below · cited by 2 · depth 27
• Standard-position points of Ω come from adic points CerednikDrinfeld.FormalOmega.AdicPoint.exists_toOmega_eq_of_mem_affinoid_zero_or_lt 0 below · cited by 1 · depth 28
• A Deligne datum in an edge chart is determined by its two lines CerednikDrinfeld.FormalOmega.DeligneDatum.eq_of_inEdgeChart_of_line_eq 0 below · cited by 23 · depth 28
• Deligne datum in the standard edge chart: existence of chart point CerednikDrinfeld.FormalOmega.DeligneDatum.exists_algHom_chartERing_line_eq_of_inEdgeChart_of_finite 0 below · cited by 11 · depth 28
• Deligne data depend only on the ideal (π) CerednikDrinfeld.FormalOmega.DeligneDatum.exists_equiv_of_span_singleton_eq 0 below · cited by 1 · depth 28
• Local covering of a Deligne datum by standard edge charts CerednikDrinfeld.FormalOmega.DeligneDatum.exists_finite_cover_isPullback_inEdgeChart_of_finite 7 below · cited by 6 · depth 28
• Values of K^× in C are powers of v(varpi) CerednikDrinfeld.FormalOmega.IsAdicFrame.exists_v_eq_zpow 0 below · cited by 1 · depth 28
• Drinfeld upper half plane is exhausted by the standard affinoids CerednikDrinfeld.FormalOmega.IsAdicFrame.isExhausted 0 below · cited by 1 · depth 28
• Homotheties act trivially on Drinfeld's formal upper half plane CerednikDrinfeld.FormalOmega.Omega.action_scalarGL 0 below · cited by 13 · depth 28
• Transport of the widehatΩ⊗widehat𝒪^{nr} datum along a frame isomorphism CerednikDrinfeld.FormalOmega.OmegaNr.exists_equiv_of_ringEquiv_frame 0 below · cited by 1 · depth 28
• Descent of a twisted-action relation to all π-nilpotent algebras CerednikDrinfeld.FormalOmega.OmegaNr.forall_eq_of_isTwistedAct_of_forall_isNoetherianRing_of_forall_isIdempotentElem 3 below · cited by 4 · depth 28
• Unique extension of a natural family on Noetherian connected test algebras CerednikDrinfeld.FormalOmega.existsUnique_extension_of_isNoetherianRing_of_forall_isIdempotentElem 22 below · cited by 4 · depth 28
• Unique natural extension of a family on connected Noetherian test algebras CerednikDrinfeld.FormalOmega.existsUnique_extension_prod_const_of_isNoetherianRing_of_forall_isIdempotentElem 23 below · cited by 4 · depth 28
• Descent of a Frobenius-invariant family to the fixed subalgebra CerednikDrinfeld.FormalOmega.existsUnique_factor_corep_fixedPoints_of_frobTwist_eq 6 below · cited by 5 · depth 28
• Edge chart points yield Deligne data with prescribed lines CerednikDrinfeld.FormalOmega.exists_deligneDatum_line_eq_inEdgeChart_of_isNilpotent 2 below · cited by 27 · depth 28
• Exhaustion of corep(O^{nr})×Ω by Noetherian subalgebras CerednikDrinfeld.FormalOmega.exists_finset_map_adjoin_eq_prod_corep_omega_of_irreducible 18 below · cited by 7 · depth 28
• Noetherian square-zero lifting suffices for Theta formal étaleness CerednikDrinfeld.FormalOmega.forall_existsUnique_lift_of_forall_isNoetherianRing_existsUnique_lift 28 below · cited by 2 · depth 28
• Edge chart membership via two vertex tubes and the edge tube CerednikDrinfeld.FormalOmega.AdicPoint.inEdgeChart_iff_toOmega_mem 2 below · cited by 2 · depth 29
• Edge-chart membership of an adic point is level-independent CerednikDrinfeld.FormalOmega.AdicPoint.inEdgeChart_pt_iff_inEdgeChart_pt_zero 0 below · cited by 2 · depth 29
• The coordinate of an adic point avoids K CerednikDrinfeld.FormalOmega.AdicPoint.toOmega_mem_upperHalfPlane 0 below · cited by 3 · depth 29
• Edge nondegeneracy transports under pull-back by h CerednikDrinfeld.FormalOmega.DeligneDatum.edgeNondegAt_pullback_act_inv 0 below · cited by 8 · depth 29
• Local equation for coincidence of two Deligne data CerednikDrinfeld.FormalOmega.DeligneDatum.exists_finset_forall_map_eq_iff_of_map_eq 1 below · cited by 2 · depth 29
• Edge diagrams extend to Deligne data on widehatΩ CerednikDrinfeld.FormalOmega.DeligneDatum.exists_inEdgeChart_and_line_eq 0 below · cited by 3 · depth 29
• Pull-back of a Deligne datum into the standard edge chart CerednikDrinfeld.FormalOmega.DeligneDatum.exists_isPullback_inEdgeChart_of_isLocalRing 6 below · cited by 3 · depth 29
• Twisted Mumford tower over unramified quadratic coefficients CerednikDrinfeld.FormalOmega.MumfordTower.exists_twistedTower 5 below · cited by 1 · depth 29
• Deligne data glue along a principal affine cover CerednikDrinfeld.FormalOmega.Omega.existsUnique_glue_of_span_eq_top 2 below · cited by 5 · depth 29
• Deligne data inject along injective algebra maps CerednikDrinfeld.FormalOmega.Omega.map_injective_of_injective 0 below · cited by 3 · depth 29
• Nondegeneracy of the standard edge at primes containing π CerednikDrinfeld.FormalOmega.edgeNondegAt_stdEdge_of_isUnit 0 below · cited by 2 · depth 29
• Frobenius twists act freely on coefficient legs CerednikDrinfeld.FormalOmega.eq_of_frobTwist_eq_frobTwist_of_isNilpotent_of_nontrivial 0 below · cited by 15 · depth 29
• Unique descent of morphisms along the Frobenius twist of Spec(B⊗𝒪̂^{nr}) CerednikDrinfeld.FormalOmega.existsUnique_specMap_includeLeft_comp_eq_of_specMap_frobenius_comp_eq 5 below · cited by 1 · depth 29
• Existence of lifts along arbitrary square-zero thickenings CerednikDrinfeld.FormalOmega.exists_lift_of_forall_isNoetherianRing_existsUnique_lift 26 below · cited by 1 · depth 29
• Uniqueness of lifts beyond the Noetherian case CerednikDrinfeld.FormalOmega.lift_eq_lift_of_forall_isNoetherianRing_existsUnique_lift 26 below · cited by 1 · depth 29
• Existence of a Mumford tower for a type-preserving Schottky group CerednikDrinfeld.FormalOmega.nonempty_mumfordTower_of_isSchottky 96 below · cited by 1 · depth 29
• Vertex nondegeneracy implies edge nondegeneracy at adjacent lattices CerednikDrinfeld.FormalOmega.DeligneDatum.edgeNondegAt_of_vertexNondegAt 0 below · cited by 3 · depth 30
• Zariski sheaf property for Deligne data over a ring CerednikDrinfeld.FormalOmega.DeligneDatum.eq_of_forall_map_eq_and_exists_forall_map_eq_of_span_eq_top 3 below · cited by 3 · depth 30
• Edge condition at the maximal ideal holds at every prime CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_of_edgeNondegAt_maximalIdeal 0 below · cited by 1 · depth 30
• Drinfeld data and Deligne data correspond bijectively CerednikDrinfeld.FormalOmega.DrinfeldDatum.forall_existsUnique_isQuadrupleOf_and_forall_exists_and_isIsomorphic_iff_of_isNilpotent 57 below · cited by 7 · depth 30
• Quotient map from the chart presentation of Mumford's scheme CerednikDrinfeld.FormalOmega.MumfordGlue.exists_quotientMap 13 below · cited by 1 · depth 30
• Properness and affine neighbourhoods in the Mumford glue tower CerednikDrinfeld.FormalOmega.MumfordGlue.isProper_and_affineNbhd 54 below · cited by 1 · depth 30
• Γ-action on a Mumford tower when Ntrianglelefteqρ(Γ) CerednikDrinfeld.FormalOmega.MumfordTower.exists_monoidHom_aut_forall_q_eq_q_comp_of_le 1 below · cited by 1 · depth 30
• Geometric and adic fibres of the descended quotient map CerednikDrinfeld.FormalOmega.descendedQuotientMap_fib_adicFib 15 below · cited by 1 · depth 30
• Universal property of the descended formal quotient tower CerednikDrinfeld.FormalOmega.descendedQuotientMap_univ 20 below · cited by 1 · depth 30
• Unramified layer of the descended formal quotient tower CerednikDrinfeld.FormalOmega.descendedQuotientMap_unramifiedLayer 49 below · cited by 1 · depth 30
• Descent along Spec(B ⊗ 𝒪̂^{nr}) → Spec B: epimorphism CerednikDrinfeld.FormalOmega.eq_of_specMap_includeLeft_comp_eq_of_isNilpotent 2 below · cited by 2 · depth 30
• Unique infinitesimal lifting of 𝒪-algebra maps out of 𝒪^{nr} CerednikDrinfeld.FormalOmega.existsUnique_algHom_comp_eq_of_surjective_of_isNilpotent 5 below · cited by 5 · depth 30
• Edge transitivity of GL₂(K) on full lattices CerednikDrinfeld.FormalOmega.exists_act_stdFullLattice_eq_and_act_act_eq_of_lt_of_lt 1 below · cited by 5 · depth 30
• Descent of the period maps to the finite-group quotient tower CerednikDrinfeld.FormalOmega.exists_descendedQuotientMap 0 below · cited by 1 · depth 30
• A finite quotient of Γ with even determinant valuation CerednikDrinfeld.FormalOmega.exists_finite_quotient_and_even_vdet_of_mem 0 below · cited by 1 · depth 30
• Every full lattice has a neighbour between π M and M CerednikDrinfeld.FormalOmega.exists_fullLattice_lt_and_lt 1 below · cited by 2 · depth 30
• Descent of T-points along B → B ⊗ 𝒪̂^{nr} CerednikDrinfeld.FormalOmega.exists_specMap_includeLeft_comp_eq_of_specMap_frobenius_comp_eq 4 below · cited by 1 · depth 30
• Mumford glue datum exists for a type-preserving Schottky group CerednikDrinfeld.FormalOmega.nonempty_mumfordGlue_of_isSchottky 45 below · cited by 1 · depth 30
• Deligne data are determined on a finite Zariski cover CerednikDrinfeld.FormalOmega.DeligneDatum.eq_of_forall_map_eq_of_span_eq_top 0 below · cited by 3 · depth 31
• Every Deligne datum comes from a Drinfeld datum CerednikDrinfeld.FormalOmega.DeligneDatum.exists_drinfeldDatum_isQuadrupleOf 43 below · cited by 1 · depth 31
• Zariski gluing of Deligne data over a finite cover CerednikDrinfeld.FormalOmega.DeligneDatum.exists_forall_map_eq_of_span_eq_top 1 below · cited by 2 · depth 31
• Uniqueness of the Deligne datum of a Drinfeld quadruple CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.deligneDatum_unique 1 below · cited by 5 · depth 31
• Drinfeld quadruples over a Deligne datum are unique up to isomorphism CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.isQuadrupleOf_iff_isIsomorphic 13 below · cited by 5 · depth 31
• Invariance of the quadruple relation under isomorphism CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.of_isIsomorphic 0 below · cited by 6 · depth 31
• Every Drinfeld datum arises from a Deligne datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isQuadrupleOf 11 below · cited by 2 · depth 31
• Drinfeld data exist over p-nilpotent W(k)-algebras CerednikDrinfeld.FormalOmega.DrinfeldDatum.nonempty_of_isNilpotent_of_isAlgClosed 0 below · cited by 1 · depth 31
• Affine neighbourhoods propagate up a Mumford glue tower CerednikDrinfeld.FormalOmega.MumfordGlue.affineNbhd_of_affineNbhd_zero 4 below · cited by 1 · depth 31
• Finite sets in the special level lie in an affine open CerednikDrinfeld.FormalOmega.MumfordGlue.affineNbhd_zero 48 below · cited by 1 · depth 31
• Properness of the glued Mumford levels over 𝒪/πⁿ⁺¹ CerednikDrinfeld.FormalOmega.MumfordGlue.isProper_zb 19 below · cited by 2 · depth 31
• Relation and overlap laws for glue core charts CerednikDrinfeld.FormalOmega.MumfordGlueCore.zeta_rel_and_zeta_overlap 22 below · cited by 1 · depth 31
• Chartwise universal property of a Mumford glue core CerednikDrinfeld.FormalOmega.MumfordGlueCore.zeta_univ_law 3 below · cited by 1 · depth 31
• Universal property of a base-changed level of a Mumford tower CerednikDrinfeld.FormalOmega.MumfordTower.existsUnique_hom_pullback_of_natural_of_invariant 3 below · cited by 1 · depth 31
• Normaliser action on the levels of a Mumford tower CerednikDrinfeld.FormalOmega.MumfordTower.exists_action_forall_q_eq_q_comp 0 below · cited by 1 · depth 31
• The twisted Mumford tower admits an unramified presentation CerednikDrinfeld.FormalOmega.MumfordTower.nonempty_nrPresentation 10 below · cited by 1 · depth 31
• Formal rigidity for Ω̂: chart-wise representable functors CerednikDrinfeld.FormalOmega.Omega.bijective_of_algFunctor_of_forall_existsUnique_lift_of_forall_bijective_of_forall_represents_inEdgeChart 43 below · cited by 3 · depth 31
• Label-ℓ locus maps bijectively to Ω̂ CerednikDrinfeld.FormalOmega.Omega.injective_surjective_labelPiece_of_algFunctor_of_forall_represents_inEdgeChart 45 below · cited by 2 · depth 31
• Residue ring of O^{nr}: characteristic r and r-th powers CerednikDrinfeld.FormalOmega.charP_residue_and_forall_exists_pow_eq 0 below · cited by 1 · depth 31
• Twisted equivariance of the descended family ρ₂ CerednikDrinfeld.FormalOmega.descendedFamily_comp_frobenius_zpow_eq_of_isPullback 1 below · cited by 1 · depth 31
• Adic points and fibres of the descended quotient map CerednikDrinfeld.FormalOmega.descendedQuotientMap_adicFib 11 below · cited by 1 · depth 31
• Geometric fibres of the descended Mumford-tower quotient map CerednikDrinfeld.FormalOmega.descendedQuotientMap_fib 5 below · cited by 1 · depth 31
• Charts of the unramified layer of the descended Čerednik–Drinfeld tower CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrCharts 10 below · cited by 1 · depth 31
• Chart functions on the unramified layer: injectivity and descent criterion CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions 17 below · cited by 1 · depth 31
• Natural families into a scheme are determined on split coefficient maps CerednikDrinfeld.FormalOmega.eq_of_natural_of_forall_eq_of_comp_mk_eq_comp_val 3 below · cited by 1 · depth 31
• Descent of a twisted Γ-invariant family to the Fr²-fixed subring CerednikDrinfeld.FormalOmega.existsUnique_factor_corep_fixedPoints_frobenius_sq_of_forall_isTwistedAct_eq 8 below · cited by 1 · depth 31
• Frobenius descent for π-nilpotent algebras CerednikDrinfeld.FormalOmega.existsUnique_tmul_one_eq_of_map_frobenius_eq 1 below · cited by 3 · depth 31
• Descent of σ-stable quasi-compact opens to Spec B CerednikDrinfeld.FormalOmega.exists_opens_preimage_eq_of_isCompact_of_preimage_frobenius_eq 2 below · cited by 1 · depth 31
• Witt vectors of k surject onto O^{nr}/π^N with kernel r^N CerednikDrinfeld.FormalOmega.exists_ringHom_wittVector_surjective_and_ker_eq_of_ker_eq_span 1 below · cited by 1 · depth 31
• Cyclic Galois levels over the fixed ring of Fr^m CerednikDrinfeld.FormalOmega.fixedPoints_frobenius_levels 0 below · cited by 3 · depth 31
• Frobenius twists trivial after a square-zero quotient CerednikDrinfeld.FormalOmega.frobTwist_eq_of_comp_frobTwist_eq_comp_of_squareZero 0 below · cited by 1 · depth 31
• Existence of a Mumford gluing core for a type-preserving Schottky group CerednikDrinfeld.FormalOmega.nonempty_mumfordGlueCore_of_isSchottky 25 below · cited by 1 · depth 31
• Uniqueness of ring maps from an absolutely unramified r-adic base CerednikDrinfeld.FormalOmega.ringHom_ext_of_isNilpotent_natCast 0 below · cited by 1 · depth 31
• Unique lifting mod π from square-zero thickenings CerednikDrinfeld.FormalOmega.AlgFunctor.existsUnique_lift_quotient_of_forall_existsUnique_lift_of_ker_sq_eq_bot 0 below · cited by 1 · depth 32
• Local-to-global for points of a scheme in a point functor CerednikDrinfeld.FormalOmega.AlgFunctor.exists_pt_eq_of_forall_isLocalization_atPrime 1 below · cited by 1 · depth 32
• Edge nondegeneracy of a Deligne datum under base change CerednikDrinfeld.FormalOmega.DeligneDatum.edgeNondegAt_map_iff_edgeNondegAt_comap 0 below · cited by 3 · depth 32
• Standard edge chart represents the edge subfunctor of Ω̂ CerednikDrinfeld.FormalOmega.DeligneDatum.existsUnique_algHom_chartERing_line_eq_and_natural_of_inEdgeChart 5 below · cited by 6 · depth 32
• Zariski-local reduction of Deligne data to the standard edge chart CerednikDrinfeld.FormalOmega.DeligneDatum.exists_cover_pullback_map_inEdgeChart_stdEdge_line_eq 7 below · cited by 3 · depth 32
• Drinfeld data glue along a finite cover of Spec B CerednikDrinfeld.FormalOmega.DeligneDatum.exists_drinfeldDatum_isQuadrupleOf_of_forall_away 28 below · cited by 1 · depth 32
• Drinfeld quadruple for a Deligne datum in an edge chart CerednikDrinfeld.FormalOmega.DeligneDatum.exists_drinfeldDatum_isQuadrupleOf_of_inEdgeChart 5 below · cited by 1 · depth 32
• Finite cover putting a Deligne datum in normalised edge charts CerednikDrinfeld.FormalOmega.DeligneDatum.exists_finite_cover_inEdgeChart_hasDetIndex_of_isNilpotent 11 below · cited by 1 · depth 32
• Edge charts transport along the GL₂(K) pull-back of Deligne data CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_act_of_isPullback 0 below · cited by 5 · depth 32
• Edge-chart membership is insensitive to local base change CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_iff_of_isBaseChange_of_isLocalHom 0 below · cited by 3 · depth 32
• Drinfeld lattices determined by the underlying Deligne datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.N_eq_of_isQuadrupleOf 8 below · cited by 1 · depth 32
• Uniqueness of a Drinfeld datum over a Deligne datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.isIsomorphic_of_N_eq 2 below · cited by 1 · depth 32
• Naturality in the test algebra of the quadruple–point comparison CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.map_of_isBaseChangeAlong 0 below · cited by 3 · depth 32
• Drinfeld quadruples transport along even translates CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.pullback_of_isTranslateEven 0 below · cited by 2 · depth 32
• Odd translates preserve the quadruple–Deligne datum correspondence CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.pullback_of_isTranslateOdd 0 below · cited by 2 · depth 32
• Finitely generated model for the kernel lines near a point CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_fg_forall_lineBaseChange_eq 7 below · cited by 1 · depth 32
• Stalkwise Deligne datum attached to a Drinfeld datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_localDeligneDatum 1 below · cited by 2 · depth 32
• Adic frame rings are complete valuation rings with fraction field C CerednikDrinfeld.FormalOmega.IsAdicFrame.isAdicComplete_and_exists_valuationRing_isFractionRing_isAlgClosed 0 below · cited by 1 · depth 32
• Components of the Mumford special fibre are smooth integral curves CerednikDrinfeld.FormalOmega.MumfordGlue.exists_isClosedImmersion_isIntegral_smoothOfRelativeDimension_one_of_mem_irreducibleComponents_zero 7 below · cited by 1 · depth 32
• Valuative lifting of points of the glued Mumford levels CerednikDrinfeld.FormalOmega.MumfordGlue.exists_lift_of_valuationRing 17 below · cited by 1 · depth 32
• Mumford levels are locally of finite type and quasi-compact CerednikDrinfeld.FormalOmega.MumfordGlue.locallyOfFiniteType_and_quasiCompact 0 below · cited by 3 · depth 32
• Level zero of a Mumford glue: reduced, of dimension ≤ 1, infinite components CerednikDrinfeld.FormalOmega.MumfordGlue.specialLevel_isField_isReduced_dim_infinite 5 below · cited by 1 · depth 32
• Equal chart points give Zariski-locally N-related Deligne data CerednikDrinfeld.FormalOmega.MumfordGlueCore.exists_finite_cover_isPullback_of_zeta_comp_eq 19 below · cited by 1 · depth 32
• Chart points equal iff Deligne data are N-equivalent (local case) CerednikDrinfeld.FormalOmega.MumfordGlueCore.zeta_comp_eq_iff_exists_isPullback_of_isLocalRing 16 below · cited by 3 · depth 32
• N-related Deligne data give the same chart point CerednikDrinfeld.FormalOmega.MumfordGlueCore.zeta_comp_eq_of_exists_isPullback 18 below · cited by 1 · depth 32
• Cartesian transition between consecutive Mumford gluing levels CerednikDrinfeld.FormalOmega.MumfordGlueLevel.exists_transition 12 below · cited by 1 · depth 32
• Adic points lift Mumford-tower families over a local base CerednikDrinfeld.FormalOmega.MumfordTower.exists_adicPoint_forall_q_eq_of_isLocalRing_of_finite_stabilizer 5 below · cited by 1 · depth 32
• Existence of the morphism induced by a natural N-invariant family CerednikDrinfeld.FormalOmega.MumfordTower.exists_hom_pullback_of_natural_of_invariant 1 below · cited by 1 · depth 32
• Every k-point of a Mumford tower level comes from widehatΩ CerednikDrinfeld.FormalOmega.MumfordTower.exists_q_eq_of_isAlgClosed 0 below · cited by 1 · depth 32
• Chart-local points are jointly epimorphic on a tower level CerednikDrinfeld.FormalOmega.MumfordTower.hom_ext_of_comp_eq_of_q 0 below · cited by 1 · depth 32
• Translated edge chart of Ω̂ is affinely representable CerednikDrinfeld.FormalOmega.Omega.exists_natural_injective_inEdgeChart_act_iff_spec_tensorProduct_chartERing 7 below · cited by 2 · depth 32
• Descending equality of Deligne data from B[1/e][1/f] to B[1/c] CerednikDrinfeld.FormalOmega.Omega.exists_notMem_map_away_eq_of_map_away_away_eq 0 below · cited by 1 · depth 32
• Rigidity of mathcal O_{2,n}-points in a local algebra CerednikDrinfeld.FormalOmega.algHom_eq_or_eq_comp_frobTwo_of_isLocalRing 1 below · cited by 2 · depth 32
• Base change for the edge chart ring Aₑ CerednikDrinfeld.FormalOmega.chartERing.existsUnique_isPushout_baseChange 1 below · cited by 4 · depth 32
• Truncated edge chart rings commute with base change CerednikDrinfeld.FormalOmega.chartERing.existsUnique_isPushout_quotient_baseChange 1 below · cited by 1 · depth 32
• Unramified edge charts compute the descended quotient point CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrCharts_comp_rY 0 below · cited by 1 · depth 32
• Chart images in the descended unramified quotient: open and covering CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrCharts_isOpen_and_cover 1 below · cited by 1 · depth 32
• Transition law and images of the unramified-layer quotient charts CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrCharts_transition 6 below · cited by 2 · depth 32
• Compatible sections of the unramified quotient are determined by chart values CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions_inj 1 below · cited by 1 · depth 32
• Chart values of compatible sections are Γ'-invariant CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions_inv_of_sections 1 below · cited by 1 · depth 32
• Lifting an invariant compatible family of chart functions to sections CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions_sections_of_inv 12 below · cited by 1 · depth 32
• Extension of natural families from Noetherian to all π-nilpotent algebras CerednikDrinfeld.FormalOmega.exists_extension_natural_agree_forall_isTwistedAct_eq_of_isNoetherianRing 26 below · cited by 2 · depth 32
• Involutivity of Fr₂ and a generator of mathcal O_{2,n} CerednikDrinfeld.FormalOmega.frobTwo_frobTwo_and_exists_generator 0 below · cited by 3 · depth 32
• Existence of a level-n Mumford gluing datum CerednikDrinfeld.FormalOmega.nonempty_mumfordGlueLevel_of_isSchottky 16 below · cited by 1 · depth 32
• Edge charts cover a Deligne datum on a finite basic cover CerednikDrinfeld.FormalOmega.DeligneDatum.exists_finset_span_eq_top_forall_inEdgeChart_map_away 0 below · cited by 1 · depth 33
• Deligne data over Frac V descend to V CerednikDrinfeld.FormalOmega.DeligneDatum.exists_isBaseChange_of_valuationRing_of_map_eq_zero 3 below · cited by 1 · depth 33
• Two edge-nondegenerate lattice pairs of a Deligne datum share a vertex CerednikDrinfeld.FormalOmega.DeligneDatum.exists_latticeMap_scalarGL_eq_of_edgeNondegAt_of_edgeNondegAt 2 below · cited by 6 · depth 33
• Vertex nondegeneracy pins the lattice to a given edge CerednikDrinfeld.FormalOmega.DeligneDatum.exists_latticeMap_scalarGL_eq_or_of_vertexNondegAt_of_edgeNondegAt 0 below · cited by 6 · depth 33
• Homothety invariance of the edge-chart condition CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_act_scalarGL_iff 0 below · cited by 2 · depth 33
• Re-orienting an edge chart: from (M',M) to (π M,M') CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_swap 0 below · cited by 2 · depth 33
• Homothety invariance of the vertex nondegeneracy condition CerednikDrinfeld.FormalOmega.DeligneDatum.vertexNondegAt_act_scalarGL_iff 0 below · cited by 4 · depth 33
• Lattice squeeze along a base change of Drinfeld data CerednikDrinfeld.FormalOmega.DrinfeldDatum.N_eq_of_le_of_mem_stratum_iff 3 below · cited by 1 · depth 33
• A Drinfeld datum is locally given by a Deligne datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_deligneDatum_away_forall_map 6 below · cited by 1 · depth 33
• Zariski gluing of Drinfeld data along a basic open cover CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_forall_isBaseChangeAlong_away_of_overlap 12 below · cited by 1 · depth 33
• Drinfeld data descend along localisations of the base CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isBaseChangeAlong_of_isLocalization 0 below · cited by 3 · depth 33
• Base change along f is invariant under isomorphism of the target datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.isBaseChangeAlong_of_isBaseChangeAlong_of_isIsomorphic 0 below · cited by 2 · depth 33
• Being the quadruple of a Deligne datum is Zariski-local CerednikDrinfeld.FormalOmega.DrinfeldDatum.isQuadrupleOf_of_forall_isBaseChangeAlong_away 0 below · cited by 1 · depth 33
• The two strata of a Drinfeld datum cover Spec B CerednikDrinfeld.FormalOmega.DrinfeldDatum.mem_stratum0_or_mem_stratum1 1 below · cited by 1 · depth 33
• Strata of Drinfeld data pull back along semilinear comparisons CerednikDrinfeld.FormalOmega.DrinfeldDatum.mem_stratum_iff_of_semilinear 1 below · cited by 1 · depth 33
• Homothetic lattices have determinant indices of equal parity CerednikDrinfeld.FormalOmega.HasDetIndex.even_sub_of_latticeMap_scalarGL 0 below · cited by 4 · depth 33
• Existence and uniqueness of the chart-law quotient family CerednikDrinfeld.FormalOmega.MumfordGlue.existsUnique_quotientFamily_of_chartLaw 9 below · cited by 1 · depth 33
• Equal chart points over a local base give N-related Deligne data CerednikDrinfeld.FormalOmega.MumfordGlueCore.exists_isPullback_of_zeta_comp_eq_of_isLocalRing 14 below · cited by 1 · depth 33
• N-related Deligne data give equal chart points over local rings CerednikDrinfeld.FormalOmega.MumfordGlueCore.zeta_comp_eq_of_exists_isPullback_of_isLocalRing 11 below · cited by 1 · depth 33
• Reduction commutes with edge-chart transports CerednikDrinfeld.FormalOmega.MumfordGlueLevel.factor_comp_alpha_eq_alpha_comp_factor 4 below · cited by 1 · depth 33
• Level compatibility of the vertex inclusion in Mumford gluing data CerednikDrinfeld.FormalOmega.MumfordGlueLevel.factor_comp_iota_eq_iota_comp_factor 0 below · cited by 1 · depth 33
• Level compatibility of vertex-chart transports τ_g CerednikDrinfeld.FormalOmega.MumfordGlueLevel.factor_comp_tau_eq_tau_comp_factor 3 below · cited by 1 · depth 33
• Consecutive Mumford gluing levels form a cartesian square CerednikDrinfeld.FormalOmega.MumfordGlueLevel.isPullback_zb_of_forall_zeta_comp_eq 8 below · cited by 1 · depth 33
• Natural N-invariant families agree when the q-images agree CerednikDrinfeld.FormalOmega.MumfordTower.eq_of_q_eq_of_natural_of_invariant 0 below · cited by 1 · depth 33
• Edge non-degeneracy of a natural family cuts out an open set CerednikDrinfeld.FormalOmega.Omega.exists_opens_forall_edgeNondegAt_iff_mem_of_natural 1 below · cited by 1 · depth 33
• Edge transport fixing or reversing the ends of an edge CerednikDrinfeld.FormalOmega.associated_algEquiv_chartERing_xi_eta_of_isPullback 4 below · cited by 6 · depth 33
• Flatness over the base of the edge chart ring CerednikDrinfeld.FormalOmega.chartERing.flat 0 below · cited by 1 · depth 33
• Each component of the special fibre of the edge chart is infinite CerednikDrinfeld.FormalOmega.chartERing.infinite_setOf_le_of_mem_minimalPrimes_quotient_level_zero 0 below · cited by 1 · depth 33
• Level-zero edge chart: primes maximal or minimal CerednikDrinfeld.FormalOmega.chartERing.isMaximal_or_mem_minimalPrimes_quotient_level_zero 0 below · cited by 1 · depth 33
• Reducedness of the level-zero quotient of the edge chart ring CerednikDrinfeld.FormalOmega.chartERing.isReduced_quotient_level_zero 0 below · cited by 2 · depth 33
• Each branch of the level-zero edge chart is smooth of dimension one CerednikDrinfeld.FormalOmega.chartERing.smoothOfRelativeDimension_one_specMap_quotient_of_mem_minimalPrimes_level_zero 3 below · cited by 1 · depth 33
• Unramified charts commute with the X'-tower transitions CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrCharts_kappa_transition 5 below · cited by 2 · depth 33
• Chart overlap agreement for invariant functions on the unramified layer CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions_overlap_agreement 8 below · cited by 1 · depth 33
• Unique edge-chart automorphism from an edge-stabilising g CerednikDrinfeld.FormalOmega.existsUnique_algEquiv_chartERing_isPullback_of_act_stdEdge 4 below · cited by 1 · depth 33
• Unique vertex-chart automorphism transporting Deligne data CerednikDrinfeld.FormalOmega.existsUnique_algEquiv_chartVRing_isPullback_of_act_stdFullLattice_eq 4 below · cited by 1 · depth 33
• The vertex chart as a localisation of the edge chart mod πⁿ⁺¹ CerednikDrinfeld.FormalOmega.exists_algHom_chartERing_chartVRing_isLocalization_away 0 below · cited by 1 · depth 33
• Equality of base-changed lines spreads to a basic open CerednikDrinfeld.FormalOmega.exists_not_mem_forall_lineBaseChange_eq_of_lineBaseChange_localization_eq 0 below · cited by 1 · depth 33
• Separatedness of a Mumford glued level over 𝒪/πⁿ⁺¹ CerednikDrinfeld.FormalOmega.isSeparated_of_mumfordGlueLaws 7 below · cited by 1 · depth 33
• A homothety preserving the determinant index fixes the lattice CerednikDrinfeld.FormalOmega.latticeMap_scalarGL_eq_self_of_hasDetIndex 0 below · cited by 4 · depth 33
• Zariski descent for the formal upper half-plane moduli package CerednikDrinfeld.FormalOmega.omegaPackage_isZariskiSheaf 4 below · cited by 1 · depth 33
• Fibre-square gluing for the p-adic Ω̂ package, Noetherian case CerednikDrinfeld.FormalOmega.omegaPackage_padic_existsUnique_map_pullbackFst_eq_and_map_pullbackSnd_eq_of_isNoetherianRing 1 below · cited by 1 · depth 33
• Deligne's edge condition for saturated lines over a valuation ring CerednikDrinfeld.FormalOmega.DeligneDatum.exists_edge_nondeg_saturation_of_valuationRing 2 below · cited by 1 · depth 34
• Openness of the edge nondegeneracy locus in Spec B CerednikDrinfeld.FormalOmega.DeligneDatum.isOpen_setOf_edgeNondegAt 0 below · cited by 2 · depth 34
• Zariski-local lifting of Pi-coordinates with α'β'=π CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.exists_cover_forall_exists_mul_eq_and_map_eq_of_isBaseChange 42 below · cited by 1 · depth 34
• Off the opposite stratum, N₀ = pN₁ or N₀ = N₁ CerednikDrinfeld.FormalOmega.DrinfeldDatum.N0_eq_of_not_mem_stratum 1 below · cited by 1 · depth 34
• Drinfeld datum near a point: Pi-compatible pair of linear maps CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_compatible_linearMap_pair_mk_tmul_eq_smul 1 below · cited by 1 · depth 34
• Gluing the modules of Drinfeld data along basic opens CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_gluedModules_of_baseChangeAlong_overlap 10 below · cited by 1 · depth 34
• Three local types of lattice pairs near a point of a Drinfeld datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isOpen_forall_lattice_eq_or_bijective_map 0 below · cited by 1 · depth 34
• Equal chart images on N-equivalent edges give N-related data CerednikDrinfeld.FormalOmega.MumfordGlueCore.exists_isPullback_of_zeta_comp_eq_of_edge_rel 11 below · cited by 1 · depth 34
• Equal chart points with η invertible give N-related Deligne data CerednikDrinfeld.FormalOmega.MumfordGlueCore.exists_isPullback_of_zeta_comp_eq_of_isUnit_eta 6 below · cited by 1 · depth 34
• Equal chart points with unit ξ give N-related data CerednikDrinfeld.FormalOmega.MumfordGlueCore.exists_isPullback_of_zeta_comp_eq_of_isUnit_xi 3 below · cited by 2 · depth 34
• Transition morphisms pull each Mumford chart back to its counterpart CerednikDrinfeld.FormalOmega.MumfordGlueLevel.preimage_opensRange_zeta_eq_of_forall_zeta_comp_eq 5 below · cited by 1 · depth 34
• Edge-chart incidence for translated Deligne data CerednikDrinfeld.FormalOmega.act_stdVertex_or_isUnit_of_isPullback_of_line_eq_chartERing 5 below · cited by 2 · depth 34
• Minimal primes of the level-zero edge chart fibre CerednikDrinfeld.FormalOmega.chartERing.eq_span_xi_or_eq_span_eta_of_mem_minimalPrimes_level_zero 0 below · cited by 1 · depth 34
• Level-zero branch of the edge chart is a localised line CerednikDrinfeld.FormalOmega.chartERing.exists_ringEquiv_quotient_level_zero_span_ofPoly_X_localizationAway 0 below · cited by 1 · depth 34
• Locally Γ'-related Deligne data at translated chart points CerednikDrinfeld.FormalOmega.descendedQuotientMap_nrFunctions_related_locally 1 below · cited by 1 · depth 34
• Gluing submodule-valued functions along a basic open cover of Spec B CerednikDrinfeld.FormalOmega.exists_forall_pointUnder_eq_and_isOpen_setOf_mem_of_span_eq_top 0 below · cited by 1 · depth 34
• Special fibre of Drinfeld's formal upper half plane is a scheme CerednikDrinfeld.FormalOmega.exists_scheme_locallyOfFiniteType_isSeparated_isReduced_equiv_omegaObj_of_isNoetherianRing 32 below · cited by 2 · depth 34
• Cartesian reduction square for truncated edge-chart rings CerednikDrinfeld.FormalOmega.isPullback_Spec_map_factor_chartERing 0 below · cited by 1 · depth 34
• Adjacent edge charts generate the vertex chart ring CerednikDrinfeld.FormalOmega.range_sup_range_comp_eq_top_of_isPullback_chartVRing 4 below · cited by 1 · depth 34
• Composition of base-change witnesses for Drinfeld data CerednikDrinfeld.FormalOmega.DrinfeldDatum.BaseChangeAlong.exists_comp 0 below · cited by 1 · depth 35
• Uniqueness of base change of a Drinfeld datum along g CerednikDrinfeld.FormalOmega.DrinfeldDatum.BaseChangeAlong.exists_iso 0 below · cited by 2 · depth 35
• Uniqueness of base-change comparison maps τ₀,τ₁ CerednikDrinfeld.FormalOmega.DrinfeldDatum.BaseChangeAlong.tau_unique 2 below · cited by 1 · depth 35
• Lifting a quadruple's (α,β) with αβ=π CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.exists_mul_eq_and_map_eq_of_isBaseChange_of_inEdgeChart 29 below · cited by 1 · depth 35
• Existence of even or odd GL₂(K)-translates of a Drinfeld datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isTranslateEven_or_exists_isTranslateOdd 2 below · cited by 1 · depth 35
• Representability of the formal upper half plane over a Noetherian base CerednikDrinfeld.FormalOmega.Omega.exists_scheme_equiv_nilpPoints_and_isOpenImmersion_of_isNoetherianRing 23 below · cited by 1 · depth 35
• Separatedness of a scheme representing the Deligne datum functor CerednikDrinfeld.FormalOmega.Omega.isSeparated_of_equiv_nilpPoints 1 below · cited by 1 · depth 35
• Reducedness and finite type of the edge chart ring over W(k)/p CerednikDrinfeld.FormalOmega.finiteType_and_isReduced_tensorProduct_chartERing_of_isAlgClosed 4 below · cited by 1 · depth 35
• Determinant index of a lattice cut out by an integral matrix CerednikDrinfeld.FormalOmega.hasDetIndex_of_forall_mem_iff_exists_mulVec_eq_pow_smul 0 below · cited by 2 · depth 35
• Lifting edge-chart coordinates along a nilpotent thickening CerednikDrinfeld.FormalOmega.DeligneDatum.exists_algHom_chartERing_comp_eq_of_isBaseChange_of_surjective 6 below · cited by 1 · depth 36
• Coincidence of two Deligne data is cut out by an ideal CerednikDrinfeld.FormalOmega.DeligneDatum.exists_ideal_forall_map_eq_iff_le_ker 0 below · cited by 1 · depth 36
• Uniqueness of the Pi-pair of a Drinfeld quadruple up to (u,u⁻¹) CerednikDrinfeld.FormalOmega.DrinfeldDatum.IsQuadrupleOf.exists_unit_eq_mul_chartERing_eta_of_line_eq 21 below · cited by 1 · depth 36
• Uniqueness of isomorphisms between Drinfeld data CerednikDrinfeld.FormalOmega.DrinfeldDatum.Iso.subsingleton 0 below · cited by 1 · depth 36
• Existence of the even translate when det(cg⁻¹) is a unit CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isTranslateEven_of_det_eq_algebraMap 0 below · cited by 1 · depth 36
• Existence of the odd translate of a Drinfeld datum CerednikDrinfeld.FormalOmega.DrinfeldDatum.exists_isTranslateOdd_of_det_mul_eq_algebraMap 0 below · cited by 1 · depth 36
• Explicit Drinfeld quadruple over the standard edge chart CerednikDrinfeld.FormalOmega.DeligneDatum.exists_isQuadrupleOf_and_pi_eq_smul_chartERing_of_line_eq 8 below · cited by 1 · depth 37
• Reversing the orientation of an edge chart CerednikDrinfeld.FormalOmega.DeligneDatum.inEdgeChart_act_scalarGL_inv_of_inEdgeChart 2 below · cited by 1 · depth 38
• Deligne data over a reduced ring are determined geometrically CerednikDrinfeld.FormalOmega.DeligneDatum.eq_of_forall_map_algClosed_eq_of_isReduced 0 below · cited by 2 · depth 39
• Recognition of the quadruple at an odd vertex CerednikDrinfeld.FormalOmega.DrinfeldDatum.isQuadrupleOf_iff_of_line_act_eq_span_of_pow_ne 8 below · cited by 1 · depth 43
• Recognising the Drinfeld quadruple of a vertex-interior chart point CerednikDrinfeld.FormalOmega.DrinfeldDatum.isQuadrupleOf_iff_of_line_stdFullLattice_eq_span_of_pow_ne 8 below · cited by 1 · depth 43
• Deligne data over a characteristic-p field with prescribed edge lines CerednikDrinfeld.FormalOmega.exists_deligneDatum_line_eq_span_of_mul_eq_zero_of_charP 3 below · cited by 3 · depth 43
• Recognising the node quadruple of the standard edge chart CerednikDrinfeld.FormalOmega.DrinfeldDatum.isQuadrupleOf_iff_of_line_eq_span_node 8 below · cited by 1 · depth 44

CerednikDrinfeld.FormalQuotientDatum 2

• Levelwise maps from a formal quotient datum to X CerednikDrinfeld.FormalQuotientDatum.exists_hom_pullback_comp_eq_theta_of_cerednikDrinfeld_quotient 0 below · cited by 3 · depth 28
• Formal quotient datum comparison maps are isomorphisms CerednikDrinfeld.FormalQuotientDatum.isIso_of_isPullback_of_cerednikDrinfeld_quotient 1 below · cited by 2 · depth 28

CerednikDrinfeld.GradedCartierModuleData 49

• Uniqueness of the canonical L-map on a special Cartier module CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.eq_of_isNilpotent 68 below · cited by 4 · depth 32
• Canonical L-maps transport along isomorphisms of graded Cartier data CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.exists_comp_eq_nMap_comp_of_bijective 0 below · cited by 2 · depth 32
• Base change of graded Cartier data composed with an isomorphism CerednikDrinfeld.GradedCartierModuleData.IsBaseChangeAlong.comp_of_bijective 0 below · cited by 1 · depth 33
• Base change compatibility of Cartier L-maps, p-torsion-free target CerednikDrinfeld.GradedCartierModuleData.IsCartierLMap.apply_comp_eq_nMap_apply_of_torsionFree 1 below · cited by 3 · depth 33
• Relative Cartier L-map determined by its values on a V-basis CerednikDrinfeld.GradedCartierModuleData.IsCartierLMap.comp_eq_nMap_comp_of_forall_apply_basis_eq 0 below · cited by 3 · depth 33
• Unique descent of an L-map along a surjective base change CerednikDrinfeld.GradedCartierModuleData.existsUnique_comp_eq_nMap_comp_and_isCartierLMap_of_surjective_of_isSpecialCartierModule 3 below · cited by 3 · depth 33
• Existence and uniqueness of L_M over p-torsion-free bases CerednikDrinfeld.GradedCartierModuleData.existsUnique_isCartierLMap_of_isSpecialCartierModule_of_torsionFree 2 below · cited by 5 · depth 33
• Universal property of base change for special graded Cartier modules CerednikDrinfeld.GradedCartierModuleData.exists_baseChange_comp_eq_and_unique 4 below · cited by 4 · depth 33
• Matched lifts of a special Cartier module admit a common domination CerednikDrinfeld.GradedCartierModuleData.exists_dominating_of_apply_basis_eq 1 below · cited by 2 · depth 33
• Lifting homogeneous V-bases along a unit-detecting base change CerednikDrinfeld.GradedCartierModuleData.exists_isHomogeneousVBasis_apply_eq_of_forall_isUnit 0 below · cited by 2 · depth 33
• Base change of a special graded Cartier module over a p-torsion-free ring CerednikDrinfeld.GradedCartierModuleData.exists_isSpecialCartierModule_and_baseChange_of_torsionFree 53 below · cited by 4 · depth 33
• η is functorial after multiplication by p CerednikDrinfeld.GradedCartierModuleData.nsmul_nMap_mem_eta_of_mem_eta_of_cast_eq_zero 1 below · cited by 5 · depth 33
• Naturality of the canonical L-map under base change CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.comp_eq_nMap_comp_of_comp_eq 71 below · cited by 6 · depth 34
• Naturality of canonical L-maps under base change CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.comp_eq_nMap_comp_of_isNilpotent 70 below · cited by 31 · depth 34
• Canonical L-maps persist under base change, label-free form CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.exists_of_isBaseChangeAlong_of_comp_eq 68 below · cited by 2 · depth 34
• Canonical L-maps commute with Pi CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.map_varpi 2 below · cited by 5 · depth 34
• The L-map on a V-basis vector over a p-torsion-free base CerednikDrinfeld.GradedCartierModuleData.IsCartierLMap.exists_smul_apply_eq_nMk_of_torsionFree 1 below · cited by 2 · depth 34
• An L-map carries ker f into ker N(f) CerednikDrinfeld.GradedCartierModuleData.IsCartierLMap.nMap_apply_eq_zero_of_apply_eq_zero 1 below · cited by 1 · depth 34
• A recognition criterion for base changes of special graded Cartier modules CerednikDrinfeld.GradedCartierModuleData.baseChange_of_map_smul_of_map_verschiebung_of_isHomogeneousVBasis 1 below · cited by 1 · depth 34
• Uniqueness of V-compatible semilinear maps on a homogeneous V-basis CerednikDrinfeld.GradedCartierModuleData.eq_of_map_smul_of_map_verschiebung_of_forall_apply_basis_eq 0 below · cited by 2 · depth 34
• Every special graded Cartier datum comes from a formal 𝒪_D-module CerednikDrinfeld.GradedCartierModuleData.exists_formalODModule_bijective_of_isSpecialCartierModule_of_torsionFree 51 below · cited by 1 · depth 34
• Existence of the base-change map on special graded Cartier modules CerednikDrinfeld.GradedCartierModuleData.exists_map_smul_map_verschiebung_apply_basis_eq_of_baseChange 0 below · cited by 1 · depth 34
• Frobenius lands in Pi M + VM for special Cartier modules CerednikDrinfeld.GradedCartierModuleData.frobenius_mem_range_lambda_of_isSpecialCartierModule 0 below · cited by 1 · depth 34
• Base change carries homogeneous V-bases to homogeneous V-bases CerednikDrinfeld.GradedCartierModuleData.isHomogeneousVBasis_map_of_baseChange 0 below · cited by 4 · depth 34
• Injectivity of λ given a homogeneous V-basis CerednikDrinfeld.GradedCartierModuleData.lambda_injective_of_isHomogeneousVBasis_of_torsionFree 0 below · cited by 12 · depth 34
• Kernel of λ on N(M) is killed by p CerednikDrinfeld.GradedCartierModuleData.nsmul_eq_zero_of_lambda_eq_zero_of_cast_eq_zero 0 below · cited by 1 · depth 34
• On η(L), N(V) agrees with Pi up to p CerednikDrinfeld.GradedCartierModuleData.nsmul_iterate_nMap_verschiebung_eq_nsmul_iterate_nVarpi_of_mem_eta 0 below · cited by 3 · depth 34
• Base change of special Cartier modules along a surjection is onto CerednikDrinfeld.GradedCartierModuleData.surjective_of_isBaseChangeAlong_of_surjective 0 below · cited by 4 · depth 34
• Base change compatibility of u on η(L) CerednikDrinfeld.GradedCartierModuleData.u_nMap_of_comp_eq 0 below · cited by 2 · depth 34
• Kernel of a graded Cartier base change to first V-order CerednikDrinfeld.GradedCartierModuleData.IsBaseChangeAlong.exists_eq_sum_teichmuller_smul_add_verschiebung_of_apply_eq_zero 0 below · cited by 2 · depth 35
• Canonical L-maps agree along every p-torsion-free lift CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.apply_comp_eq_nMap_apply_of_torsionFree 68 below · cited by 1 · depth 35
• Graded splitting of the φ_L-fixed subgroup η(L) CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.exists_mem_etaPiece_add_eq 3 below · cited by 13 · depth 35
• Canonical L-maps persist under base change CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.exists_of_isBaseChangeAlong 67 below · cited by 1 · depth 35
• Special Cartier module data with equal structure constants are isomorphic CerednikDrinfeld.GradedCartierModuleData.exists_bijective_map_eq_of_hasStructureConstants_of_torsionFree 2 below · cited by 1 · depth 35
• Structure constants for a homogeneous V-basis, with a_{0,0}a_{0,1}=p CerednikDrinfeld.GradedCartierModuleData.exists_hasStructureConstants_mul_eq_of_isHomogeneousVBasis 0 below · cited by 1 · depth 35
• Gradedness of the modified Cartier module N(M) CerednikDrinfeld.GradedCartierModuleData.isCompl_nPiece_zero_one 0 below · cited by 15 · depth 35
• Canonical L-maps preserve the ℤ/2-grading CerednikDrinfeld.GradedCartierModuleData.IsCanonicalLMap.apply_mem_nPiece 2 below · cited by 1 · depth 36
• η is invariant under square-zero thickenings killed by p CerednikDrinfeld.GradedCartierModuleData.bijOn_nMap_eta_of_sq_eq_zero_of_mul_eq_zero 2 below · cited by 3 · depth 36
• Nilpotence of φ_L on the kernel of N(f) CerednikDrinfeld.GradedCartierModuleData.phi_iterate_three_eq_zero_of_nMap_eq_zero_of_sq_eq_zero 0 below · cited by 1 · depth 37
• Equal colengths in both graded pieces of a V-commuting injection CerednikDrinfeld.GradedCartierModuleData.length_piece_quotient_eq_of_isHomogeneousVBasis_of_comm_verschiebung 0 below · cited by 1 · depth 38
• Bijectivity of N(f) on η under nilpotent thickenings CerednikDrinfeld.GradedCartierModuleData.bijective_eta_map_of_surjective_of_isNilpotent 7 below · cited by 4 · depth 39
• Structure constants of varpi on a homogeneous V-basis CerednikDrinfeld.GradedCartierModuleData.exists_varpi_eq_teichmuller_smul_add_verschiebung_mul_eq 0 below · cited by 3 · depth 39
• Uniqueness of V-compatible semilinear maps on a homogeneous V-basis CerednikDrinfeld.GradedCartierModuleData.eq_of_map_smul_of_map_verschiebung_of_forall_apply_basis_eq_of_isVAdicallyComplete 0 below · cited by 1 · depth 40
• Level-two digit recursion for varpi = V invariants CerednikDrinfeld.GradedCartierModuleData.exists_digits_eq_of_varpi_eq_verschiebung_level_two 0 below · cited by 1 · depth 40
• Base change of special graded Cartier modules along surjections CerednikDrinfeld.GradedCartierModuleData.exists_isSpecialCartierModule_and_isBaseChangeAlong_of_surjective 0 below · cited by 1 · depth 40
• Two V-digits of a Pi=V vector satisfy I⊆ I² CerednikDrinfeld.GradedCartierModuleData.exists_digits_mem_sq_of_varpi_eq_verschiebung 0 below · cited by 2 · depth 41
• Divisibility criterion for [c]γᵢ + Vn ∈ Pi Mᵢ₊₁ + VM CerednikDrinfeld.GradedCartierModuleData.exists_eq_varpi_add_verschiebung_iff_dvd_of_varpi_eq_teichmuller_smul_add 0 below · cited by 2 · depth 41
• Gluing η(L) along a Milnor square of Cartier data CerednikDrinfeld.GradedCartierModuleData.exists_mem_eta_nMap_eq_of_nMap_eq_nMap 0 below · cited by 1 · depth 41
• Artin–Schreier relation for the leading digit at a critical index CerednikDrinfeld.GradedCartierModuleData.exists_digit_eq_mul_pow_add_mul_of_varpi_eq_verschiebung 0 below · cited by 1 · depth 42

CerednikDrinfeld.HeckeTower 5

• Commuting Atkin–Lehner involutions from Čerednik descent data CerednikDrinfeld.HeckeTower.atkinLehner_involutive_comm_galois_of_descentIntertwining_one_zero 39 below · cited by 1 · depth 20
• Degeneracy maps intertwine Galois and Atkin–Lehner actions CerednikDrinfeld.HeckeTower.smul_phi_eq_phi_smul_of_descentIntertwining_one_zero 39 below · cited by 1 · depth 20
• Equal supports and degrees force equal Hecke correspondences CerednikDrinfeld.HeckeTower.correspondence_eq_of_support_eq_of_finrankAlong_eq 66 below · cited by 2 · depth 23
• Rigidity of Hecke tower data with equal divisor correspondences CerednikDrinfeld.HeckeTower.exists_algEquiv_forall_comp_phi_eq_of_correspondence_eq 67 below · cited by 2 · depth 23
• Primes other than q are units in a DVR with residue field of order q CerednikDrinfeld.HeckeTower.AwayPrime.isUnit_natCast_of_card_residue 0 below · cited by 9 · depth 24

CerednikDrinfeld.JPrimeTorsionDatum 2

• Counting the toric part of the 𝔪-torsion CerednikDrinfeld.JPrimeTorsionDatum.natCard_toric_inf_W_eq_pow_finrank_quotient 0 below · cited by 3 · depth 15
• Ribet exchange inequality for a purely toric torsion datum CerednikDrinfeld.JPrimeTorsionDatum.natCard_toric_inf_W_mul_natCard_smul_mem_toric_le_of_gal_eq_hecke_of_not_mem 0 below · cited by 1 · depth 16

CerednikDrinfeld.LevelU 3

• Lattice non-inclusion forces membership in the U_ℓ-set CerednikDrinfeld.LevelU.mem_levelHeckeUSet_of_not_le 39 below · cited by 1 · depth 18
• A U_ℓ-step lattice never contains the level lattice CerednikDrinfeld.LevelU.not_le_of_mem_levelHeckeUSet 33 below · cited by 1 · depth 18
• Level-lattice intersection along a U_ℓ-step CerednikDrinfeld.LevelU.ofFiniteIdele_mul_inf_ofFiniteIdele_mul_eq_of_mem_levelHeckeUSet 39 below · cited by 1 · depth 18

CerednikDrinfeld.Mumford 50

• Period uniformisation yields a toric uniformisation at each p ≠ r CerednikDrinfeld.Mumford.nonempty_toricUniformization_of_periodUniformization 29 below · cited by 1 · depth 17
• Restriction of an equivariant Mumford uniformisation to the torsion CerednikDrinfeld.Mumford.EquivariantUniformization.eFull_restrict_U_torsion_and_equiv 0 below · cited by 1 · depth 18
• The group U of a period datum is divisible CerednikDrinfeld.Mumford.PeriodDatum.U_divisible 0 below · cited by 1 · depth 18
• Ribbon kernel realised as ker π via periods CerednikDrinfeld.Mumford.PeriodDatum.exists_periodEquiv 0 below · cited by 1 · depth 18
• Gram adjointness of a Hecke map from its period identity CerednikDrinfeld.Mumford.PeriodDatum.period_adjoint_of_ord_Q 0 below · cited by 1 · depth 18
• Torsion points of a period Jacobian lie in imπ CerednikDrinfeld.Mumford.PeriodDatum.pi_surj_torsion 0 below · cited by 1 · depth 18
• p-torsion character and tame Kummer law for a Mumford period uniformisation CerednikDrinfeld.Mumford.PeriodUniformization.exists_torsionEquiv_tameCharacter_kummerLaw 18 below · cited by 1 · depth 18
• Frobenius on p-torsion of a Mumford period uniformisation CerednikDrinfeld.Mumford.PeriodUniformization.frobenius_toric_and_frobenius_quot_of_torsionEquiv 1 below · cited by 1 · depth 18
• p-torsion of a period datum's uniformising group CerednikDrinfeld.Mumford.PeriodDatum.exists_torsionEquiv 0 below · cited by 1 · depth 19
• Kummer law for p-th roots of Mumford periods CerednikDrinfeld.Mumford.PeriodDatum.kummer_of_ord_Q 0 below · cited by 1 · depth 19
• Cycles of a type-preserving tree action give a map on Gᵃᵇ CerednikDrinfeld.Mumford.exists_addMonoidHom_abelianization_ribbonKernel_apply_eq_pathCycle 0 below · cited by 8 · depth 20
• Harmonic morphism of oriented quotient data from g⁻¹Xg≤ Y CerednikDrinfeld.Mumford.exists_finiteHom_orientedQuotient_of_conj_le 2 below · cited by 4 · depth 20
• Splitting quotients of a two-coloured graph by a type-exchanging subgroup CerednikDrinfeld.Mumford.exists_quotVert_prod_equiv_and_quotEdge_equiv_oriented_of_exchanger 1 below · cited by 5 · depth 20
• Twisted D× C₂× C₂ symmetry of an invariant function field CerednikDrinfeld.Mumford.exists_symmetryGroup_semilinearAut_invariantFieldOf 0 below · cited by 1 · depth 20
• Pull-back of quotient-graph cycles equals transfer CerednikDrinfeld.Mumford.finiteHom_pullback_apply_eq_apply_transfer_of_forall_apply_eq_pathCycle_of_card_stabilizer 0 below · cited by 3 · depth 20
• Push-forward of path cycles along a subgroup inclusion CerednikDrinfeld.Mumford.finiteHom_pushforward_apply_eq_of_forall_addMonoidHom_apply_eq_pathCycle 0 below · cited by 3 · depth 20
• Cycle map Gᵃᵇ→ H₁(GbackslashT): surjectivity and kernel CerednikDrinfeld.Mumford.surjective_and_apply_eq_zero_iff_mem_closure_stabilizer_of_apply_eq_pathCycle 0 below · cited by 10 · depth 20
• Vertex types on a connected bipartite graph with automorphisms CerednikDrinfeld.Mumford.vertexType_add_one_of_adj_and_vertexType_smul_and_exists_typeCharacter 0 below · cited by 25 · depth 20
• Inertia-invariant points lift to inertia-invariant torus points CerednikDrinfeld.Mumford.EquivariantUniformization.exists_coeffMap_eq_and_eFull_eq_of_forall_inertia_gal_eq 2 below · cited by 1 · depth 21
• Naturality of period lattices under pullback along μ CerednikDrinfeld.Mumford.PeriodDatum.comp_pullback_mem_periodLattice_of_forall_eq 0 below · cited by 1 · depth 21
• Period lattices are natural along a finite harmonic morphism CerednikDrinfeld.Mumford.PeriodDatum.comp_pushforward_mem_periodLattice_of_forall_eq 0 below · cited by 1 · depth 21
• Finite-index subgroups induce finite morphisms of quotient degeneracy data CerednikDrinfeld.Mumford.exists_finiteHom_quotientDegeneracyData_of_subgroup 0 below · cited by 1 · depth 21
• Transporting a quotient-tree presentation along conjugation by p CerednikDrinfeld.Mumford.exists_presentation_conj_apply_eq_of_apply_eq_pathCycle 0 below · cited by 4 · depth 21
• Finite vertex stabilisers from finite dart stabilisers CerednikDrinfeld.Mumford.finite_stabilizer_of_finite_stabilizer_dart 0 below · cited by 1 · depth 21
• Inertia fixes every torus lift of an inertia-invariant point CerednikDrinfeld.Mumford.EquivariantUniformization.coeffMap_eq_of_mem_inertiaSubgroupIn_of_gal_eFull_eq 1 below · cited by 1 · depth 22
• Graph actions preserve the graph distance CerednikDrinfeld.Mumford.GraphAction.dist_smul_smul 0 below · cited by 2 · depth 22
• Equality of additive maps from periods and differences of places CerednikDrinfeld.Mumford.addMonoidHom_eq_of_apply_QL_eq_of_apply_eq_of_eq_pic0Mk_single_sub_single 6 below · cited by 1 · depth 22
• Equivariance of the cycle map under a normalising element CerednikDrinfeld.Mumford.apply_conj_eq_actZ_apply_of_apply_eq_pathCycle 5 below · cited by 2 · depth 22
• Stabiliser-trivial characters factor uniquely through the cycle lattice CerednikDrinfeld.Mumford.existsUnique_ribbonKernel_hom_comp_eq_of_forall_mem_stabilizer 4 below · cited by 4 · depth 22
• Galois transport of a theta-pinned Mumford torus point CerednikDrinfeld.Mumford.exists_monoidHom_theta_coeffMap_precomp_apply_eq_of_apply_eq 8 below · cited by 1 · depth 22
• Descending a symmetric pairing to a period datum CerednikDrinfeld.Mumford.exists_periodDatum_apply_eq_of_surjective_of_forall_ker 0 below · cited by 1 · depth 22
• Inertia-fixed subfield of a completion of ℚ̄ as a period field CerednikDrinfeld.Mumford.exists_periodField_completion 11 below · cited by 1 · depth 22
• Pinned theta multipliers exist for every pair of points CerednikDrinfeld.Mumford.exists_theta_multiplier_and_torusPoint_apply_eq_of_mumfordQuotient 49 below · cited by 1 · depth 22
• Total degree equals index of the conjugated level CerednikDrinfeld.Mumford.finiteHom_degTotal_eq_index_of_mumfordQuotient_conj 11 below · cited by 1 · depth 22
• Invariant field of a conjugate subgroup is its translate CerednikDrinfeld.Mumford.invariantFieldOf_map_conj_eq_map_fracAct 0 below · cited by 2 · depth 22
• Compositum of invariant fields of Γ and sΓ s⁻¹ CerednikDrinfeld.Mumford.invariantFieldOf_sup_map_conj_eq_inf_and_relfinrank_eq_relIndex 0 below · cited by 2 · depth 22
• Independence of the path cycle from the base vertex on a tree CerednikDrinfeld.Mumford.pathCycle_eq_pathCycle_of_isTree 0 below · cited by 4 · depth 22
• Coefficientwise invariance of torus points with invariant image CerednikDrinfeld.Mumford.EquivariantUniformization.coeffMap_eq_of_actZ_eq_one_of_gal_eFull_eq 0 below · cited by 1 · depth 23
• Quotient of two u-eigenvectors lies in the Γ-invariant field CerednikDrinfeld.Mumford.div_mem_invariantFieldOf_of_smul_eq_algebraMap_mul 0 below · cited by 2 · depth 23
• Transport of the quotient graph datum along an equivariant isomorphism CerednikDrinfeld.Mumford.exists_quotVert_equiv_quotEdge_equiv_of_iso_of_smul_eq 0 below · cited by 1 · depth 23
• Naturality of the cycle map under a normalising tree automorphism CerednikDrinfeld.Mumford.pathCycle_mulEquiv_eq_of_iso 1 below · cited by 1 · depth 23
• Uniqueness of weighted dart-orbit cochains on the Bruhat–Tits tree CerednikDrinfeld.Mumford.eq_zero_of_forall_sum_mul_mul_walkCycle_eq_zero 2 below · cited by 1 · depth 24
• Type-preserving subgroup is the kernel of a ℤ/2-character CerednikDrinfeld.Mumford.exists_monoidHom_ker_eq_typePreserving_and_index_dvd_two 0 below · cited by 2 · depth 24
• Finite vertex stabilisers give finitely many walk-overlapping translations CerednikDrinfeld.Mumford.finite_setOf_exists_mem_darts_smul_mem_darts 0 below · cited by 2 · depth 24
• Stabiliser-weighted overlap sum over G-translates of a walk CerednikDrinfeld.Mumford.finsum_walkOverlap_map_smulHom_eq_sum_stabWidth_mul_walkCycle_mul_walkCycle 0 below · cited by 2 · depth 24
• Four-point identity on a tree via signed dart overlap CerednikDrinfeld.Mumford.dist_add_dist_sub_dist_sub_dist_eq_two_mul_walkOverlap 0 below · cited by 2 · depth 25
• Vanishing on stabilisers: c as a product of cycle-vector powers CerednikDrinfeld.Mumford.exists_eq_prod_pow_of_forall_mem_stabilizer_of_forall_exists_pow_eq 1 below · cited by 2 · depth 25
• Harmonic quasi-invariant potentials give stabiliser-weighted cycles on the quotient graph CerednikDrinfeld.Mumford.exists_mem_ribbonKernel_and_sub_eq_sum_stabWidth_mul_walkCycle_of_dvd 0 below · cited by 1 · depth 25
• Finite generation from a cofinite action on a connected graph CerednikDrinfeld.Mumford.fg_of_finite_stabilizer_of_finite_quotVert_of_finite_quotEdge 0 below · cited by 1 · depth 28
• Torsion-free finite-index subgroups of inversion-free tree lattices are Schottky CerednikDrinfeld.Mumford.isSchottky_of_relIndex_ne_zero_of_forall_isOfFinOrder_imp_eq_one 0 below · cited by 1 · depth 30

CerednikDrinfeld.Omega 178

• Every pseudo-uniformiser of ℚ^{cl} exhausts the upper half plane CerednikDrinfeld.Omega.isExhausted_of_liesOverPrime 6 below · cited by 7 · depth 19
• Restriction of the theta character to a finite-index subgroup CerednikDrinfeld.Omega.comp_subtype_eq_prod_of_forall_eq_theta 15 below · cited by 2 · depth 20
• Theta multiplier of G equals the transfer from Γ' CerednikDrinfeld.Omega.eq_transfer_of_forall_eq_theta_of_forall_eq_theta_comp_subtype 26 below · cited by 3 · depth 20
• Points of the Mumford curve: orbits, surjectivity, evaluation CerednikDrinfeld.Omega.exists_place_invariantFieldOf_eq_iff_mem_orbit_and_evalAt_eq_of_map_le_typePreserving_of_isCurveOver_of_exists_v_le_of_v_card_stabilizer_eq_one 156 below · cited by 1 · depth 20
• Locally compact constants yield an exhausting pseudo-uniformiser CerednikDrinfeld.Omega.exists_pseudoUniformizer_isExhausted_of_isCompact 0 below · cited by 3 · depth 20
• Pseudo-uniformiser varpi=r for the closure of ℚ CerednikDrinfeld.Omega.exists_pseudoUniformizer_ratClosure_eq_natCast_of_liesOverPrime 6 below · cited by 1 · depth 20
• Coset norm of a χ-automorphic function has multiplier Ver(χ) CerednikDrinfeld.Omega.fracAct_prod_fracAct_eq_transfer_inv_mul 0 below · cited by 2 · depth 20
• Invariant field of the type-preserving part is a curve field CerednikDrinfeld.Omega.isCurveOver_invariantFieldOf_inf_typePreserving_of_exists_relIndex_ne_zero_of_exists_not_mem_range 84 below · cited by 3 · depth 20
• The ring of holomorphic functions on Ω over C_A is a domain CerednikDrinfeld.Omega.isDomain_holRing_of_liesOverPrime 15 below · cited by 1 · depth 20
• Units on Ω are pinned between two adjacent vertices CerednikDrinfeld.Omega.exists_adj_min_le_v_apply_le_max_of_isUnit 0 below · cited by 4 · depth 21
• thetaMer is represented by a theta pair CerednikDrinfeld.Omega.exists_isThetaPair_thetaMer_eq_mk 8 below · cited by 4 · depth 21
• Automorphy of the meromorphic theta function on Ω CerednikDrinfeld.Omega.exists_monoidHom_fracAct_thetaMer_eq 22 below · cited by 13 · depth 21
• Places of the invariant field at points of Ω CerednikDrinfeld.Omega.exists_place_invariantFieldOf_mem_iff_and_evalAt_eq_div_of_map_le_typePreserving 10 below · cited by 2 · depth 21
• A transcendental element with finite extension: Mumford field of Γ₊ CerednikDrinfeld.Omega.exists_transcendental_finiteDimensional_adjoin_invariantFieldOf_of_exists_relIndex_ne_zero_of_exists_not_mem_range 39 below · cited by 1 · depth 21
• Finiteness of {γ:ρ(γ)b∈Ωₙ} for discrete ρ CerednikDrinfeld.Omega.finite_setOf_pmoebius_mem_affinoid 1 below · cited by 21 · depth 21
• Finite stabiliser of the standard vertex implies discreteness CerednikDrinfeld.Omega.isDiscrete_of_finite_stabilizer_stdVertex 3 below · cited by 25 · depth 21
• The ring of holomorphic functions on Ω is a domain CerednikDrinfeld.Omega.isDomain_holRing 6 below · cited by 6 · depth 21
• Periods transport along a conjugating group isomorphism CerednikDrinfeld.Omega.period_pmoebius_pmoebius_of_mulEquiv_of_apply_eq_conj 1 below · cited by 1 · depth 21
• Fibres of the point-to-place map are ρ(Γ)-orbits CerednikDrinfeld.Omega.place_invariantFieldOf_eq_iff_exists_eq_smul_of_map_le_typePreserving_of_exists_v_le_of_v_card_stabilizer_eq_one 100 below · cited by 2 · depth 21
• Coset product of subgroup theta equals constant times theta CerednikDrinfeld.Omega.prod_theta_comp_subtype_pmoebius_eq_mul_theta_and_prod_fracAct_thetaMer_eq 16 below · cited by 1 · depth 21
• Every place of the invariant field comes from Ω CerednikDrinfeld.Omega.surjective_place_invariantFieldOf_of_mem_iff_of_map_le_typePreserving_of_isCurveOver_of_exists_v_le 60 below · cited by 2 · depth 21
• Defining relation thetaMer· H = F for a theta pair CerednikDrinfeld.Omega.thetaMer_mul_algebraMap_eq_of_isThetaPair 0 below · cited by 4 · depth 21
• Theta as a product over cosets of a finite-index subgroup CerednikDrinfeld.Omega.theta_eq_prod_theta_comp_subtype_and_thetaMer_eq_prod 14 below · cited by 1 · depth 21
• Nonemptiness of finitely punctured affinoids over algebraically closed K CerednikDrinfeld.Omega.affinoid_nonempty_of_exists_finset_cover 0 below · cited by 6 · depth 22
• Möbius maps agreeing pointwise with a countable group lie in it CerednikDrinfeld.Omega.apply_mem_map_of_forall_exists_pmoebius_eq_pmoebius 0 below · cited by 2 · depth 22
• Discreteness in the valuation sense forces G countable CerednikDrinfeld.Omega.countable_of_isDiscrete 0 below · cited by 4 · depth 22
• Invariance of the cross ratio under PGL₂(K₀) CerednikDrinfeld.Omega.crossRatio_pmoebius 0 below · cited by 12 · depth 22
• Divisibility in 𝒪(Ω) is decided by orders of vanishing CerednikDrinfeld.Omega.dvd_of_forall_ordAt_le 18 below · cited by 3 · depth 22
• Identity principle on a Drinfeld affinoid CerednikDrinfeld.Omega.eq_zero_of_mem_holOn_affinoid_of_forall_v_sub_lt_imp_eq_zero 3 below · cited by 7 · depth 22
• Liouville theorem for Λ-invariant functions on Ω CerednikDrinfeld.Omega.exists_eq_algebraMap_of_forall_smul_eq_of_forall_exists_smul_mem_affinoid 4 below · cited by 4 · depth 22
• Factorisation at a point: F=(w-z)^{ord_z F}G with G(z)≠ 0 CerednikDrinfeld.Omega.exists_eq_coordSub_pow_ordAt_mul_and_apply_ne_zero 7 below · cited by 9 · depth 22
• Characters as finite products of theta multipliers CerednikDrinfeld.Omega.exists_eq_prod_theta_of_forall_isOfFinOrder_of_colouring 92 below · cited by 1 · depth 22
• Balls in K₀ are covered by finitely many small discs CerednikDrinfeld.Omega.exists_finset_forall_v_sub_lt_of_finite_residueField 0 below · cited by 5 · depth 22
• Finitely many vertex orbits force a single absorbing affinoid CerednikDrinfeld.Omega.exists_forall_exists_smul_mem_affinoid_of_fintype_quotVert_map 12 below · cited by 2 · depth 22
• Holomorphic functions on a valued field do not vanish near a non-zero CerednikDrinfeld.Omega.exists_forall_v_sub_lt_imp_ne_zero_of_mem_holOn 0 below · cited by 6 · depth 22
• Theta product as a ratio of rigid-holomorphic functions CerednikDrinfeld.Omega.exists_holRing_div_eq_theta 7 below · cited by 3 · depth 22
• Invariant meromorphic functions separating two distinct ρ(Γ)-orbits CerednikDrinfeld.Omega.exists_mem_invariantFieldOf_apply_eq_zero_and_apply_ne_zero_of_forall_ne_smul_of_map_le_typePreserving_of_exists_v_le_of_v_card_stabilizer_eq_one 99 below · cited by 3 · depth 22
• A point of Ω avoiding two countable orbits CerednikDrinfeld.Omega.exists_mem_upperHalfPlane_forall_pmoebius_ne_and_of_countable_of_archimedean 0 below · cited by 2 · depth 22
• Automorphy of the cross-ratio theta product with multiplier CerednikDrinfeld.Omega.exists_monoidHom_isAutomorphicWithMultiplier_theta 8 below · cited by 4 · depth 22
• Stabiliser-weighted period pairing for tree lattices with torsion CerednikDrinfeld.Omega.exists_monoidHom_monoidHom_symm_mul_period_eq_one_v_eq_zpow_stabWidth 43 below · cited by 1 · depth 22
• A proper family of K-rational valuations on Δ-invariant meromorphic functions CerednikDrinfeld.Omega.exists_valuations_invariantFieldOf_of_finite_quotVert 30 below · cited by 1 · depth 22
• Additivity of the order of vanishing on Ω CerednikDrinfeld.Omega.ordAt_mul 9 below · cited by 8 · depth 22
• Order at a place times stabiliser order equals order of vanishing CerednikDrinfeld.Omega.ord_place_invariantFieldOf_mul_card_stabilizer_eq_ordAt_sub_ordAt_of_cast_card_ne_zero_of_map_le_typePreserving_of_exists_v_le_of_v_card_eq_one 110 below · cited by 3 · depth 22
• Vertex-fixing group elements have trivial Drinfeld period CerednikDrinfeld.Omega.period_eq_one_of_smul_vertex_eq 42 below · cited by 1 · depth 22
• Periods are independent of the auxiliary point a CerednikDrinfeld.Omega.period_eq_period_of_mem_upperHalfPlane 9 below · cited by 2 · depth 22
• Invariance of periods under the normaliser CerednikDrinfeld.Omega.period_pmoebius_pmoebius_mulEquiv 1 below · cited by 1 · depth 22
• Lower bound for z(cb+d)-(ab+β) on the affinoid Ωₙ CerednikDrinfeld.Omega.pow_le_v_phi_of_mem_affinoid_of_v_det_lt 0 below · cited by 3 · depth 22
• Pull-back of a point divisor under change of level CerednikDrinfeld.Omega.pullbackAlong_single_place_eq_sum_of_forall_ord_mul_card_stabilizer_eq_of_algEquiv 2 below · cited by 1 · depth 22
• Restriction of point places along a Mumford conjugation map CerednikDrinfeld.Omega.restrictAlong_place_eq_smul_inv_and_inertiaDegAlong_eq_one_of_forall_mem_iff 1 below · cited by 3 · depth 22
• Semilinear automorphism realised by (n,t) transports places accordingly CerednikDrinfeld.Omega.semilinearAut_smul_pt_eq_pt_smul_of_mem_toValuationSubring_iff 1 below · cited by 1 · depth 22
• Points of the level-zero affinoid determine their vertex CerednikDrinfeld.Omega.smul_stdVertex_eq_of_mem_affinoid_zero 10 below · cited by 7 · depth 22
• Theta product converges on all of Ω for discrete ρ CerednikDrinfeld.Omega.thetaMultipliable_of_isDiscrete_of_isExhausted 5 below · cited by 11 · depth 22
• Trivial theta multiplier at torsion elements CerednikDrinfeld.Omega.theta_apply_pmoebius_basePoint_eq_one_of_isOfFinOrder 25 below · cited by 11 · depth 22
• Equivariance of the theta product under isometric automorphisms CerednikDrinfeld.Omega.theta_isometricAut 0 below · cited by 3 · depth 22
• Base-point cocycle for the theta product CerednikDrinfeld.Omega.theta_mul_theta_basePoint 1 below · cited by 3 · depth 22
• Base-point independence of the theta automorphy multiplier CerednikDrinfeld.Omega.theta_pmoebius_basePoint_eq_theta_pmoebius_basePoint 5 below · cited by 3 · depth 22
• Conjugation invariance of theta on Drinfeld's upper half plane CerednikDrinfeld.Omega.theta_pmoebius_eq_theta_of_mulEquiv_of_apply_eq_conj 1 below · cited by 2 · depth 22
• Identity principle on an annulus with deleted residue classes CerednikDrinfeld.Omega.RatPair.identityPrinciple_annulus 2 below · cited by 1 · depth 23
• Identity principle on a disc minus residue classes CerednikDrinfeld.Omega.RatPair.identityPrinciple_disc 0 below · cited by 4 · depth 23
• Finite order of vanishing on Drinfeld's upper half plane CerednikDrinfeld.Omega.bddAbove_setOf_coordSub_pow_dvd 4 below · cited by 5 · depth 23
• Divisibility by w-z at a zero of a holomorphic function on Ω CerednikDrinfeld.Omega.coordSub_dvd_of_apply_eq_zero 1 below · cited by 6 · depth 23
• Cocycle identity for the cross ratio CerednikDrinfeld.Omega.crossRatio_mul_crossRatio 0 below · cited by 3 · depth 23
• Cross ratio is invariant under exchanging the two pairs CerednikDrinfeld.Omega.crossRatio_swap 0 below · cited by 1 · depth 23
• Schottky groups act freely on the affinoid g·Ω₀ CerednikDrinfeld.Omega.eq_one_of_pmoebius_eq_of_mem_affinoid_zero 11 below · cited by 1 · depth 23
• Theta factors tend to 1 uniformly on affinoids CerednikDrinfeld.Omega.eventually_cofinite_forall_mem_affinoid_v_thetaFactor_sub_one_lt 2 below · cited by 4 · depth 23
• Blaschke-type denominators for points escaping every affinoid CerednikDrinfeld.Omega.exists_blaschke_denominators 0 below · cited by 3 · depth 23
• Liouville theorem for Drinfeld's upper half plane CerednikDrinfeld.Omega.exists_eq_algebraMap_of_forall_v_apply_le 3 below · cited by 2 · depth 23
• Jacobi inversion with multipliers, divisor avoiding prescribed orbits CerednikDrinfeld.Omega.exists_eq_prod_theta_forall_ne_pmoebius_of_forall_isOfFinOrder_of_colouring 90 below · cited by 2 · depth 23
• Automorphic units on Ω have period multipliers: tame torsion case CerednikDrinfeld.Omega.exists_forall_eq_period_of_isUnit_of_apply_smul_eq_mul_of_forall_isOfFinOrder 75 below · cited by 1 · depth 23
• One affinoid meets every Γ''-orbit on Ω CerednikDrinfeld.Omega.exists_forall_exists_smul_mem_affinoid_of_relIndex_ne_zero 12 below · cited by 1 · depth 23
• Orders of vanishing of a theta pair on Ω CerednikDrinfeld.Omega.exists_isThetaPair_ordAt_eq_card 13 below · cited by 4 · depth 23
• The theta function Theta(a,α a;z₀;·) is a holomorphic unit CerednikDrinfeld.Omega.exists_isUnit_coe_eq_thetaMer_apply_smul_eq_period_mul 18 below · cited by 8 · depth 23
• Convergent infinite products of rational functions are holomorphic CerednikDrinfeld.Omega.exists_mem_holOn_hasProd_evalAt 0 below · cited by 3 · depth 23
• Zero-free holomorphic functions on the affinoid Ωₙ are invertible CerednikDrinfeld.Omega.exists_mem_holOn_mul_eq_one_of_forall_apply_ne_zero 3 below · cited by 2 · depth 23
• Invariant function vanishing to the stabiliser order at a point CerednikDrinfeld.Omega.exists_mk_mem_invariantFieldOf_apply_ne_zero_ordAt_eq_card_stabilizer_of_map_le_typePreserving_of_v_card_eq_one 108 below · cited by 1 · depth 23
• Bimultiplicativity of the Manin–Drinfeld period pairing CerednikDrinfeld.Omega.exists_monoidHom_monoidHom_eq_period 14 below · cited by 1 · depth 23
• Unit-residue layer of Jacobi inversion for theta multipliers CerednikDrinfeld.Omega.exists_pairs_v_prod_theta_div_sub_one_lt 68 below · cited by 1 · depth 23
• Valuations of characters as valuations of theta products CerednikDrinfeld.Omega.exists_pairs_v_prod_theta_eq 69 below · cited by 1 · depth 23
• Principal-unit characters as finite products of theta multipliers CerednikDrinfeld.Omega.exists_points_prod_theta_eq_of_v_sub_one_lt 79 below · cited by 1 · depth 23
• Finiteness of the zero set on the affinoid Ωₙ CerednikDrinfeld.Omega.finite_setOf_apply_eq_zero_of_mem_holOn_affinoid 8 below · cited by 7 · depth 23
• Balls of K₀ are covered by finitely many small balls CerednikDrinfeld.Omega.forall_exists_finset_v_sub_lt_pow_of_finite_quotient 0 below · cited by 10 · depth 23
• Torsion-free finite-index subgroups of tree lattices are Schottky CerednikDrinfeld.Omega.isSchottky_map_of_relIndex_ne_zero_of_forall_isOfFinOrder 6 below · cited by 2 · depth 23
• Stabiliser order divides vanishing orders of Γ-invariant functions CerednikDrinfeld.Omega.natCast_card_stabilizer_dvd_ordAt_sub_ordAt_of_mk_mem_invariantFieldOf_of_map_le_typePreserving 8 below · cited by 1 · depth 23
• Symmetry of the Manin–Drinfeld period pairing CerednikDrinfeld.Omega.period_symm 12 below · cited by 2 · depth 23
• Any theta pair presents `thetaMer` CerednikDrinfeld.Omega.thetaMer_eq_mk_of_isThetaPair 0 below · cited by 4 · depth 23
• Multipliability of the cross-ratio theta product on an affinoid CerednikDrinfeld.Omega.thetaMultipliable_of_isDiscrete_of_mem_affinoid 4 below · cited by 5 · depth 23
• Multiplicativity of the theta product in its divisor CerednikDrinfeld.Omega.theta_mul_theta_eq_theta 0 below · cited by 4 · depth 23
• The theta multiplier is a unit: c(β)c(β⁻¹)=1 CerednikDrinfeld.Omega.theta_pmoebius_basePoint_mul_inv 7 below · cited by 2 · depth 23
• Automorphy of the cross-ratio theta product under ρ(β) CerednikDrinfeld.Omega.theta_pmoebius_eq_mul 2 below · cited by 2 · depth 23
• Multiplicativity of the theta multiplier c(β) CerednikDrinfeld.Omega.theta_pmoebius_mul_basePoint 4 below · cited by 2 · depth 23
• Invariance of the theta product under ρ(β) CerednikDrinfeld.Omega.theta_pmoebius_pmoebius 1 below · cited by 3 · depth 23
• Cross-ratio theta product equals 1 at its base point CerednikDrinfeld.Omega.theta_self_eq_one 1 below · cited by 4 · depth 23
• Theta function attached to a torsion element is 1 CerednikDrinfeld.Omega.theta_smul_eq_one_of_isOfFinOrder 23 below · cited by 1 · depth 23
• Stabiliser-weighted Manin–Drinfeld period formula for tree lattices CerednikDrinfeld.Omega.v_period_eq_zpow_neg_sum_stabWidth_mul_pathCycle_mul_pathCycle 24 below · cited by 2 · depth 23
• Valuation exactly one on the level-0 affinoid under non-unit determinant CerednikDrinfeld.Omega.v_phi_eq_one_of_mem_affinoid_zero_of_v_det_lt_one 1 below · cited by 2 · depth 23
• Oscillation bound for rational functions bounded on a Drinfeld affinoid CerednikDrinfeld.Omega.RatPair.v_evalAt_sub_evalAt_le_mul_of_isPoleFreeOn_affinoid 2 below · cited by 1 · depth 24
• Continuity of functions holomorphic by rational approximation CerednikDrinfeld.Omega.continuous_of_mem_holOn 0 below · cited by 2 · depth 24
• Cross-ratio of Möbius images: the defect from 1 CerednikDrinfeld.Omega.crossRatio_pmoebius_pmoebius_sub_one_eq 0 below · cited by 1 · depth 24
• Cross ratio equals 1 at a coincident pair CerednikDrinfeld.Omega.crossRatio_self 0 below · cited by 2 · depth 24
• Constant-valuation holomorphic functions on Ω are constant CerednikDrinfeld.Omega.exists_eq_algebraMap_of_isUnit_of_v_apply_eq 4 below · cited by 1 · depth 24
• Constant valuation of a holomorphic function on a circle CerednikDrinfeld.Omega.exists_forall_v_apply_eq_on_circle_of_mem_holOn 0 below · cited by 3 · depth 24
• Valuation-one characters factor through the cycle map CerednikDrinfeld.Omega.exists_forall_v_eq_one_apply_eq_prod_zpow_pathCycle 4 below · cited by 2 · depth 24
• Unimodular cycle basis realised by group elements CerednikDrinfeld.Omega.exists_isUnit_det_pathCycle_and_span_pathCycle 6 below · cited by 2 · depth 24
• Valuation current of an automorphic unit as a weighted cycle CerednikDrinfeld.Omega.exists_mem_ribbonKernel_and_v_apply_smul_eq_mul_zpow_stabWidth_of_isUnit_of_forall_isOfFinOrder 16 below · cited by 1 · depth 24
• Avoiding countably many orbits in Drinfeld's upper half plane CerednikDrinfeld.Omega.exists_mem_upperHalfPlane_forall_pmoebius_ne_of_countable_of_archimedean 0 below · cited by 2 · depth 24
• Theta multiplier with prescribed unit power along one edge orbit CerednikDrinfeld.Omega.exists_pair_v_theta_eq_one_and_v_theta_mul_zpow_sub_one_lt 63 below · cited by 1 · depth 24
• Unit layer of Jacobi inversion for theta products, general position CerednikDrinfeld.Omega.exists_pairs_v_prod_theta_div_sub_one_lt_forall_ne_pmoebius 67 below · cited by 1 · depth 24
• Valuations of theta products realise any character, general position CerednikDrinfeld.Omega.exists_pairs_v_prod_theta_eq_forall_ne_pmoebius 68 below · cited by 2 · depth 24
• Principal-unit characters as theta products avoiding prescribed orbits CerednikDrinfeld.Omega.exists_points_prod_theta_eq_forall_ne_pmoebius_of_v_sub_one_lt 79 below · cited by 1 · depth 24
• Unimodular Jacobian of theta units on residue discs CerednikDrinfeld.Omega.exists_v_det_eq_one_of_isUnit_det_pathCycle_of_finite 62 below · cited by 2 · depth 24
• First-order Taylor estimate on the standard vertex affinoid CerednikDrinfeld.Omega.exists_v_sub_sub_mul_le_mul_sq_of_mem_affinoid_zero 0 below · cited by 3 · depth 24
• Finiteness of zeros in a disc minus finitely many discs CerednikDrinfeld.Omega.finite_setOf_apply_eq_zero_openDisc_sdiff_of_mem_holOn 3 below · cited by 1 · depth 24
• Gauss norm of a sum with separated poles CerednikDrinfeld.Omega.gaussNorm_add_eq_max_of_separated_poles 0 below · cited by 4 · depth 24
• Torsion-killing characters determined by a spanning family of cycles CerednikDrinfeld.Omega.monoidHom_eq_of_forall_isOfFinOrder_of_forall_apply_eq_of_span_pathCycle 3 below · cited by 2 · depth 24
• Swap symmetry of the Manin–Drinfeld period CerednikDrinfeld.Omega.period_swap 1 below · cited by 1 · depth 24
• Theta multiplier as a ratio of theta units CerednikDrinfeld.Omega.theta_pmoebius_mul_theta_eq_theta 0 below · cited by 4 · depth 24
• Units of holOn on the level-0 affinoid have constant valuation CerednikDrinfeld.Omega.v_apply_eq_of_mem_holOn_affinoid_zero_of_mul_eq_one 1 below · cited by 5 · depth 24
• Valuation of a cross ratio as v(varpi)^{geodesic overlap} CerednikDrinfeld.Omega.v_crossRatio_pmoebius_eq_zpow_walkOverlap 12 below · cited by 2 · depth 24
• Cross ratio of Möbius images via 2× 2 determinant pairings CerednikDrinfeld.Omega.crossRatio_pmoebius_eq_div_symp 0 below · cited by 1 · depth 25
• Normal form for units of 𝒪(Ω) on the standard star CerednikDrinfeld.Omega.exists_eq_mul_prod_zpow_mul_one_add_of_isUnit 0 below · cited by 2 · depth 25
• Integral jumps of v∘ F at neighbours of the standard vertex CerednikDrinfeld.Omega.exists_int_neighbours_sum_eq_zero_v_apply_smul_eq 5 below · cited by 2 · depth 25
• Division by a linear factor for rationally approximable functions CerednikDrinfeld.Omega.exists_mem_holOn_eq_sub_mul_of_apply_eq_zero 0 below · cited by 3 · depth 25
• Theta multipliers realising a prescribed unit, avoiding given orbits CerednikDrinfeld.Omega.exists_pair_v_theta_eq_one_and_v_theta_mul_zpow_sub_one_lt_forall_ne_pmoebius 62 below · cited by 2 · depth 25
• Theta multipliers realising a prescribed valuation on one dart orbit CerednikDrinfeld.Omega.exists_pairs_v_prod_theta_eq_v_zpow_stabWidth_mul_pathCycle_forall_ne 63 below · cited by 1 · depth 25
• Normalised derivative on a residue disc of a product normal form CerednikDrinfeld.Omega.exists_v_sub_sum_div_sub_le_of_forall_eq_mul_prod_zpow_mul_one_add 1 below · cited by 1 · depth 25
• Independence of 𝒪(Ω) and Ω₀ of the pseudo-uniformiser CerednikDrinfeld.Omega.holRing_eq_and_affinoid_zero_eq 0 below · cited by 4 · depth 25
• Tame fixed edge stabiliser order divides the fibre jump CerednikDrinfeld.Omega.natCard_dvd_of_v_apply_smul_eq_mul_zpow_of_forall_smul_eq 8 below · cited by 1 · depth 25
• Stabiliser of the standard vertex preserves the level-zero affinoid CerednikDrinfeld.Omega.pmoebius_mem_affinoid_zero_of_smul_stdVertex_eq 0 below · cited by 5 · depth 25
• Period law for theta units at arbitrary affinoid points CerednikDrinfeld.Omega.v_apply_smul_mul_zpow_sum_stabWidth_mul_pathCycle_mul_walkCycle_eq_of_isUnit_of_eq_theta 56 below · cited by 3 · depth 25
• Valuation of z-a for z in the level-zero affinoid CerednikDrinfeld.Omega.v_sub_eq_max_of_mem_affinoid_zero_of_not_mem 0 below · cited by 1 · depth 25
• Units are monomials modulo principal units on the edge tube CerednikDrinfeld.Omega.exists_v_apply_div_sub_one_lt_of_isUnit_of_mem_stdEdgeTube 0 below · cited by 1 · depth 26
• Units of 𝒪(Ω) are monomial on the standard edge tube CerednikDrinfeld.Omega.exists_v_apply_eq_mul_zpow_of_isUnit_of_mem_stdEdgeTube 0 below · cited by 3 · depth 26
• Tame tube stabiliser order divides the edge exponent CerednikDrinfeld.Omega.natCard_dvd_of_isUnit_of_forall_apply_smul_eq_of_v_apply_eq_mul_zpow 0 below · cited by 1 · depth 26
• Edge degrees of a unit of 𝒪(Ω) sum to zero CerednikDrinfeld.Omega.sum_add_eq_zero_of_isUnit_of_forall_v_apply_eq_mul_zpow 0 below · cited by 2 · depth 26
• Edge-tube monomial law extends to the two adjacent vertex fibres CerednikDrinfeld.Omega.v_apply_eq_and_v_apply_eq_mul_zpow_of_isUnit_of_forall_mem_stdEdgeTube 1 below · cited by 2 · depth 26
• Star formulas on overlapping stars differ by a power of v(varpi) CerednikDrinfeld.Omega.v_apply_mul_inv_eq_mul_zpow_neg_sum_of_star 0 below · cited by 2 · depth 26
• Single-dart period law for the theta unit CerednikDrinfeld.Omega.v_theta_pmoebius_mul_zpow_sum_stabWidth_mul_pathCycle_mul_walkCycle_eq 55 below · cited by 1 · depth 26
• Value of A/B at a point of local holomorphic quotient CerednikDrinfeld.Omega.exists_holRing_mul_eq_mul_and_apply_eq_of_holOn_disc 9 below · cited by 1 · depth 27
• Restriction of holomorphic functions to a subset CerednikDrinfeld.Omega.restrict_mem_holOn_of_subset 0 below · cited by 6 · depth 27
• Valuation of a theta value as a weighted cycle pairing CerednikDrinfeld.Omega.v_theta_pmoebius_eq_zpow_neg_sum_stabWidth_mul_pathCycle_mul_walkCycle 24 below · cited by 1 · depth 27
• Vanishing off a finite set on every affinoid CerednikDrinfeld.Omega.eq_zero_of_forall_finite_forall_apply_eq_zero 10 below · cited by 1 · depth 28
• Every point of Ω lies in a translate of the standard affinoid or edge tube CerednikDrinfeld.Omega.exists_pmoebius_inv_mem_affinoid_zero_or_v_lt_lt_one_of_isExhausted 0 below · cited by 1 · depth 28
• Identity principle: global relation HF=Φ holds at a point with local presentation CerednikDrinfeld.Omega.mul_apply_eq_of_forall_finite_mul_eq_of_holOn_disc 5 below · cited by 1 · depth 28
• Pole-free rational functions are bounded on the affinoids Ωₙ CerednikDrinfeld.Omega.RatPair.exists_forall_valuation_evalAt_le_of_isPoleFreeOn_affinoid 0 below · cited by 1 · depth 29
• Admissible edge regions cover the affinoid Ωₙ CerednikDrinfeld.Omega.edgeRegion_subset_affinoid_and_exists_mem_edgeRegion 0 below · cited by 2 · depth 29
• Affine edge-region chain cover of the Drinfeld affinoid CerednikDrinfeld.Omega.exists_chain_affine_edgeRegion_cover_affinoid 4 below · cited by 1 · depth 29
• Edge region of an affine chart: disc minus finitely many discs CerednikDrinfeld.Omega.exists_finset_edgeRegion_eq_tube_and_pmoebius_inv_eq_of_coe_eq_affine 2 below · cited by 2 · depth 29
• Invariant chartwise meromorphic functions on Ω are quotients CerednikDrinfeld.Omega.exists_holRing_forall_finite_mul_eq_of_forall_exists_mem_holOn_affinoid_mul_eq_of_invariant 54 below · cited by 1 · depth 29
• Equality loci of |p| and |q| are finite unions of disc conditions CerednikDrinfeld.Omega.exists_linearPieces_iff_v_eval_eq_v_eval_of_isAlgClosed 0 below · cited by 1 · depth 29
• A polynomial clearing denominators simultaneously on finitely many affinoid pieces CerednikDrinfeld.Omega.exists_polynomial_ne_zero_forall_mul_mem_holOn_of_forall_exists_mem_holOn_mul_eq_of_polynomial 0 below · cited by 1 · depth 29
• Finiteness of zeros on a disc minus residue classes CerednikDrinfeld.Omega.finite_setOf_apply_eq_zero_disc_of_mem_holOn 3 below · cited by 3 · depth 29
• Gluing holomorphy along a chain of pieces CerednikDrinfeld.Omega.mem_holOn_of_forall_mem_holOn_of_chain 4 below · cited by 1 · depth 29
• Chain clauses for a breadth-first list of edge discs CerednikDrinfeld.Omega.chain_clauses_of_ballEdges 0 below · cited by 1 · depth 30
• Uniqueness of the exponent in a local factorisation (z-p)^e F=φ CerednikDrinfeld.Omega.eq_of_sub_pow_mul_eq_of_sub_pow_mul_eq_of_mem_holOn 1 below · cited by 1 · depth 30
• Transporting a local presentation of an invariant function along ρ(γ) CerednikDrinfeld.Omega.exists_disc_forall_sub_pow_mul_eq_of_forall_pmoebius_eq_of_disc 0 below · cited by 1 · depth 30
• Breadth-first enumeration of discs of levels 1-n through n CerednikDrinfeld.Omega.exists_fin_ballEdges 0 below · cited by 1 · depth 30
• Weierstrass factorisation on an affinoid of Ω CerednikDrinfeld.Omega.exists_finset_eq_prod_sub_pow_mul_of_mem_holOn_affinoid 11 below · cited by 1 · depth 30
• Finite residue system of a discrete valuation ring in an overfield CerednikDrinfeld.Omega.exists_finset_residueSystem_of_finite_quotient 0 below · cited by 1 · depth 30
• Fundamental affinoid for a group acting through ρ on the tree CerednikDrinfeld.Omega.exists_forall_exists_pmoebius_mem_affinoid_of_finite_quotVert 13 below · cited by 1 · depth 30
• Lower bound for zero-free holomorphic functions on Ωₙ CerednikDrinfeld.Omega.exists_forall_le_v_apply_of_mem_holOn_affinoid_of_forall_ne_zero 1 below · cited by 1 · depth 30
• Gluing generic agreement on affinoids into a global holomorphic function CerednikDrinfeld.Omega.exists_holRing_forall_finite_eq_of_forall_exists_mem_holOn_eq 4 below · cited by 1 · depth 30
• Holomorphic function vanishing to prescribed orders along G-orbits CerednikDrinfeld.Omega.exists_holRing_ne_zero_forall_le_ordAt_smul 20 below · cited by 1 · depth 30
• Polynomial clearing of a piecewise meromorphic function on a tube CerednikDrinfeld.Omega.exists_polynomial_ne_zero_mul_mem_holOn_of_forall_mem_holOn_mul_eq_linearPiece_cover 14 below · cited by 1 · depth 30
• Strassman finiteness of zeros in an open residue class CerednikDrinfeld.Omega.finite_setOf_apply_eq_zero_of_v_sub_lt_of_mem_holOn 0 below · cited by 1 · depth 30
• Holomorphic inverse of a function bounded away from zero CerednikDrinfeld.Omega.inv_mem_holOn_of_forall_le_v_apply 0 below · cited by 2 · depth 30
• Gluing holomorphic functions across a sphere CerednikDrinfeld.Omega.mem_holOn_union_of_mem_holOn_of_forall_le_of_forall_mem_or_lt 3 below · cited by 3 · depth 30
• Vertex and edge tubes of an affine chart, explicitly CerednikDrinfeld.Omega.vertexTube_eq_and_edgeTube_eq_of_coe_eq_affine 0 below · cited by 2 · depth 30
• Mittag-Leffler splitting of a rational function at a disc CerednikDrinfeld.Omega.RatPair.exists_isPoleFreeOn_le_and_degree_lt_and_evalAt_eq_add 0 below · cited by 1 · depth 31
• Orthogonality of inner and outer parts on a sphere CerednikDrinfeld.Omega.RatPair.v_evalAt_lt_of_forall_v_evalAt_add_evalAt_lt_on_sphere 1 below · cited by 1 · depth 31
• Finite vanishing order at a point of a Drinfeld affinoid CerednikDrinfeld.Omega.exists_eq_sub_pow_mul_and_apply_ne_zero_of_mem_holOn_affinoid 6 below · cited by 1 · depth 31
• Weierstrass factorisation over a finite zero set CerednikDrinfeld.Omega.exists_finset_eq_prod_sub_pow_mul_of_mem_holOn_of_finite_setOf_eq_zero 3 below · cited by 1 · depth 31
• Minimum modulus on a tube for zero-free rigid functions CerednikDrinfeld.Omega.exists_forall_le_v_apply_of_mem_holOn_tube_of_forall_ne_zero 1 below · cited by 1 · depth 31
• Lower bound for a zero-free holomorphic function on a residue class CerednikDrinfeld.Omega.exists_forall_le_v_apply_of_v_sub_lt_of_forall_ne_zero 0 below · cited by 2 · depth 31
• A linear piece of a tube is a tube or empty CerednikDrinfeld.Omega.linearPiece_eq_empty_or_exists_tube_and_closedDisc_subset 0 below · cited by 1 · depth 31
• Gluing rigid-holomorphic functions along finite linear covers CerednikDrinfeld.Omega.mem_holOn_of_forall_mem_holOn_linearPiece_of_cover 5 below · cited by 1 · depth 31
• Finite order of vanishing at a point of a closed disc CerednikDrinfeld.Omega.exists_eq_sub_pow_mul_and_apply_ne_zero_of_mem_holOn_of_finite_setOf_eq_zero 2 below · cited by 1 · depth 32
• Maximum principle: a generic rim point dominates the closed disc CerednikDrinfeld.Omega.exists_finset_forall_v_apply_le_v_apply_of_mem_holOn_of_closedDisc_subset 0 below · cited by 2 · depth 32
• Gluing holomorphic functions along half-disc covers of a tube CerednikDrinfeld.Omega.mem_holOn_of_forall_mem_holOn_halfDiscPiece_of_cover 4 below · cited by 1 · depth 32

CerednikDrinfeld.OmegaNr 1

• Twisted Γ-orbits coincide with Γ'-orbits on Deligne data CerednikDrinfeld.OmegaNr.exists_isTwistedAct_iff_exists_even_isPullback 1 below · cited by 1 · depth 32

CerednikDrinfeld.Onr 3

• Faithful flatness of O^{nr} over a discrete valuation ring CerednikDrinfeld.Onr.faithfullyFlat 0 below · cited by 3 · depth 28
• Rigidity of 𝒪-algebra maps out of O^{nr} CerednikDrinfeld.Onr.algHom_eq_of_comp_eq_of_sq_zero 0 below · cited by 1 · depth 29
• Fixed subalgebra of an automorphism is a discrete valuation ring CerednikDrinfeld.Onr.isDiscreteValuationRing_equalizer_and_irreducible 0 below · cited by 1 · depth 32

CerednikDrinfeld.QM 1063

• Hecke neighbours as quotients by an extra level ℓ CerednikDrinfeld.QM.FakeEllipticCurve.heckeNeighbour_iff_exists_isLevelIsogeny 769 below · cited by 3 · depth 23
• Atkin–Lehner quotients at a ramified prime are Hecke neighbours CerednikDrinfeld.QM.FakeEllipticCurve.heckeNeighbour_of_isAtkinLehnerQuotient 726 below · cited by 3 · depth 23
• Two degeneracy maps generate the tower field CerednikDrinfeld.QM.ModuliTowerWitnessD.closure_range_phi_eq_top_of_two_mul_dvd 5,712 below · cited by 5 · depth 23
• Tower places versus ℓ-isogenies of fake elliptic curves CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_place_restrictAlong_iff_of_two_mul_dvd 728 below · cited by 4 · depth 23
• Degrees of the degeneracy arrows of the QM tower CerednikDrinfeld.QM.ModuliTowerWitnessD.finrankAlong_eq_arrowDegree_of_pt_pullback_of_two_mul_dvd_of_squarefree 5,799 below · cited by 3 · depth 23
• Existence of a moduli tower witness for fake elliptic curves CerednikDrinfeld.QM.exists_moduliTowerWitness_of_two_mul_dvd_of_neZero_of_squarefree 5,661 below · cited by 3 · depth 23
• Enumeration of the extra levels at ℓ on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_enum 735 below · cited by 8 · depth 24
• Kernel of an ℓ-isogeny as an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_of_finrank_kernel_eq 1 below · cited by 6 · depth 24
• Rank-one generator for n-torsion of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_generator_torsionPoints_of_isMaximalOrder_of_prime 722 below · cited by 9 · depth 24
• Fake elliptic curves over ℚ̄ extend over valuation rings CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_valuationSubring_of_isUnit_with_numberField_model 1,291 below · cited by 2 · depth 24
• Trivial kernel on k-points forces an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.isIso_of_forall_mapPt_eq_one_imp_eq_one 728 below · cited by 4 · depth 24
• Reducedness of the level scheme when N is invertible CerednikDrinfeld.QM.FakeEllipticCurve.isReduced_C_of_natCast_ne_zero 1 below · cited by 8 · depth 24
• Kernel of an isogeny of fake elliptic curves is reduced and finite CerednikDrinfeld.QM.FakeEllipticCurve.isReduced_pullback_one_of_natCast_ne_zero 9 below · cited by 9 · depth 24
• Level-ℓ isogeny quotients of isomorphic pairs are isomorphic CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isLevelIsogeny_of_iso_of_isOrder 727 below · cited by 4 · depth 24
• Atkin–Lehner quotients and the involution wᵣ on coarse moduli CerednikDrinfeld.QM.IsCoarseModuli.exists_atkinLehner_involution 774 below · cited by 2 · depth 24
• Čerednik–Drinfeld uniformisation of the coarse fake elliptic curve tower CerednikDrinfeld.QM.IsCoarseModuli.exists_cerednikDrinfeld_uniformization_of_span_eq_of_geometricallyConnected_of_squarefree_of_isUnit_two_of_geometricallyConnected_tower_of_isUnit_three 7,014 below · cited by 2 · depth 24
• Uniqueness of the coarse moduli curve over ℚ̄ CerednikDrinfeld.QM.IsCoarseModuli.exists_iso_pullback_awayD 10 below · cited by 1 · depth 24
• Coarse moduli curve over ℚ̄ is X_ℚ̄ CerednikDrinfeld.QM.IsCoarseModuli.exists_iso_pullback_of_two_mul_dvd 10 below · cited by 7 · depth 24
• Geometric reducedness and connectedness of the generic fibre CerednikDrinfeld.QM.IsCoarseModuli.geometricallyReduced_and_geometricallyConnected_of_curveModel_of_two_mul_dvd_of_isUnit_two_of_isUnit_three 3,834 below · cited by 4 · depth 24
• Geometrically reduced connected generic fibre of quaternionic coarse moduli CerednikDrinfeld.QM.IsCoarseModuliT.geometricallyReduced_and_geometricallyConnected_of_curveModel_of_prime_of_isUnit_two_of_isUnit_three 3,875 below · cited by 4 · depth 24
• Finiteness of the two ℓ-degeneracy morphisms CerednikDrinfeld.QM.IsCoarseModuliT.isFinite_degeneracy 4,438 below · cited by 2 · depth 24
• Surjectivity of the two degeneracy maps at ℓ CerednikDrinfeld.QM.IsCoarseModuliT.surjective_degeneracy_of_ne 818 below · cited by 2 · depth 24
• Hecke q- and q'-neighbours describe W₀ and W₁ CerednikDrinfeld.QM.ModuliTowerWitness.eq_smul_iff_heckeNeighbour_of_two_mul_dvd 773 below · cited by 1 · depth 24
• Hecke correspondence support as ℓ-Hecke neighbours of fake elliptic curves CerednikDrinfeld.QM.ModuliTowerWitness.mem_support_correspondence_single_iff_heckeNeighbour_of_two_mul_dvd 778 below · cited by 1 · depth 24
• Support of the ℓ-th push–pull as ℓ-isogenies of fake elliptic curves CerednikDrinfeld.QM.ModuliTowerWitness.mem_support_correspondence_single_iff_isLevelIsogeny_of_two_mul_dvd 732 below · cited by 1 · depth 24
• Tower laws for the quaternionic moduli tower CerednikDrinfeld.QM.ModuliTowerWitness.tower_laws_of_two_mul_dvd 767 below · cited by 1 · depth 24
• Commutation of Hecke correspondences at two primes on divisors CerednikDrinfeld.QM.ModuliTowerWitnessD.correspondence_comm_of_two_mul_dvd_of_squarefree 5,790 below · cited by 1 · depth 24
• Push–pull of a single place over all extra levels at ℓ CerednikDrinfeld.QM.ModuliTowerWitnessD.correspondence_single_eq_sum_exhaustive_of_pt_pullback_of_two_mul_dvd_of_squarefree 5,785 below · cited by 4 · depth 24
• Push-pull of a place as a sum of ℓ+1 places CerednikDrinfeld.QM.ModuliTowerWitnessD.correspondence_single_eq_sum_of_pt_pullback_of_two_mul_dvd_of_squarefree 5,786 below · cited by 1 · depth 24
• A place bijection exchanging the two tower legs at ℓ CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_bijective_place_restrictAlong_eq_of_not_dvd_of_two_mul_dvd 766 below · cited by 1 · depth 24
• Curve model for the ℓ-level layer of the QM tower CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_curveModel_level_of_pt_pullback_of_two_mul_dvd_of_squarefree 5,759 below · cited by 5 · depth 24
• Degeneracy restrictions separate places off a finite set CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_finite_restrictAlong_phi_injOn_of_two_mul_dvd 5,684 below · cited by 1 · depth 24
• Degree ℓ of the (ℓ,1) leg when ℓ ∣ N CerednikDrinfeld.QM.ModuliTowerWitnessD.finrankAlong_phi_one_eq_of_dvd_of_two_mul_dvd 5,746 below · cited by 1 · depth 24
• Atkin–Lehner involutions on the tower's upper function fields CerednikDrinfeld.QM.exists_WT_of_coarse_of_two_mul_dvd 775 below · cited by 1 · depth 24
• ℚ̄-linear Atkin–Lehner involutions at q and q' CerednikDrinfeld.QM.exists_W_of_coarse_of_two_mul_dvd 777 below · cited by 1 · depth 24
• Coarse moduli scheme of fake elliptic curves over ℤ[1/D] CerednikDrinfeld.QM.exists_coarseModuliScheme_of_squarefree_of_six_mul_dvd_of_neZero 5,591 below · cited by 2 · depth 24
• Factoring through a closed subscheme, detected on k-points CerednikDrinfeld.QM.exists_comp_eq_of_forall_factorsThrough_of_isReduced 0 below · cited by 26 · depth 24
• Semilinear Galois actions on the upper function fields of the tower CerednikDrinfeld.QM.exists_galT_of_coarse_of_two_mul_dvd 39 below · cited by 1 · depth 24
• Integral coarse moduli of fake elliptic curves, with degeneracy maps CerednikDrinfeld.QM.exists_isCoarseModuli_integral_of_isUnit_six_of_not_dvd_of_squarefree_of_isUnit_two_of_isUnit_three_of_ne 5,585 below · cited by 18 · depth 24
• Galois moves place representatives by base change along σ CerednikDrinfeld.QM.exists_isPullback_rep_gal_smul_of_two_mul_dvd 21 below · cited by 1 · depth 24
• Complex uniformisation of the quaternionic moduli curve with correspondences CerednikDrinfeld.QM.exists_uniformizedHeckeCurve_bcPlace_corr_eq_correspondence_of_two_mul_dvd 5,820 below · cited by 1 · depth 24
• Tower legs: forgetting the level and dividing by it CerednikDrinfeld.QM.isLevelRestrict_and_isLevelIsogeny_rep_restrictAlong_of_coarse_of_two_mul_dvd 33 below · cited by 1 · depth 24
• Tower representatives meet every level structure exactly once CerednikDrinfeld.QM.repT_surjective_and_injective_of_coarse_of_two_mul_dvd 0 below · cited by 1 · depth 24
• k-points of an extra level form (ℤ/ℓ)² CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_equiv_points 0 below · cited by 1 · depth 25
• Transfer of ℓ'-extra levels along an ℓ-isogeny CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_transfer_forall_factorsThrough_iff 1 below · cited by 1 · depth 25
• Finitely many level-ℓ isogeny sources over a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_fin_forall_isLevelIsogeny_iso 738 below · cited by 1 · depth 25
• Exactly ℓ level-ℓ preimages of a generic fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_fin_isLevelIsogeny_iso_of_dvd 5,712 below · cited by 1 · depth 25
• An ℓ-isogeny flip on fake elliptic curves with ℓ-level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_flip_of_not_dvd 759 below · cited by 1 · depth 25
• Atkin–Lehner quotients exist for fake elliptic curves with extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_isAtkinLehnerQuotient 50 below · cited by 1 · depth 25
• Quotient by an extra ℓ-level exists over ̄ k CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_isLevelIsogeny 32 below · cited by 6 · depth 25
• Base change of a fake elliptic curve with extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_isPullback 20 below · cited by 15 · depth 25
• Base change of Atkin–Lehner quotients with extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.isAtkinLehnerQuotient_of_isPullback 2 below · cited by 2 · depth 25
• Atkin–Lehner quotient relation is invariant under isomorphism of pairs CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.isAtkinLehnerQuotient_of_iso_of_iso 0 below · cited by 3 · depth 25
• Atkin–Lehner quotients of pairs commute with base change CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.isPullback_of_isAtkinLehnerQuotient_of_isAtkinLehnerQuotient 720 below · cited by 1 · depth 25
• Atkin–Lehner quotients at q and q' commute with extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_isAtkinLehnerQuotient_comm 726 below · cited by 1 · depth 25
• Twice-iterated Atkin–Lehner quotient with extra level is trivial CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_isAtkinLehnerQuotient_comp 740 below · cited by 1 · depth 25
• Double Atkin–Lehner quotient at a ramified prime returns the pair CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_isAtkinLehnerQuotient_comp_of_commRing 727 below · cited by 1 · depth 25
• Uniqueness of the Atkin–Lehner quotient with extra level at ℓ CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_isAtkinLehnerQuotient_of_isAtkinLehnerQuotient 726 below · cited by 1 · depth 25
• Atkin–Lehner quotients are isomorphism-invariant in the source CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_iso_of_isAtkinLehnerQuotient_of_isAtkinLehnerQuotient 713 below · cited by 2 · depth 25
• Base change of an extra level and a level-ℓ isogeny CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_of_isPullback 0 below · cited by 2 · depth 25
• Transport of extra level ℓ structures along isomorphisms CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_iso_of_iso 0 below · cited by 5 · depth 25
• Λ-stable (ℤ/ℓ)² of k-points is an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_of_equiv_points 5 below · cited by 3 · depth 25
• Level-injective morphisms of fake elliptic curves are level-surjective CerednikDrinfeld.QM.FakeEllipticCurve.exists_factorsThrough_lev_mapPt_eq_of_forall_eq_one 1 below · cited by 4 · depth 25
• Finitely many extra levels at ℓ up to isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.exists_fin_extraLevel_forall_exists_iso 710 below · cited by 1 · depth 25
• Rigidity of extra level-ℓ structures off finitely many classes CerednikDrinfeld.QM.FakeEllipticCurve.exists_fin_forall_factorsThrough_iff_of_iso_of_isLevelIsogeny 5,682 below · cited by 1 · depth 25
• Descent of fake elliptic curves over ℚ̄ to a number field CerednikDrinfeld.QM.FakeEllipticCurve.exists_intermediateField_finiteDimensional_isPullback_algebraMap_of_fg 19 below · cited by 1 · depth 25
• Existence of Atkin–Lehner quotients of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAtkinLehnerQuotient 50 below · cited by 4 · depth 25
• Existence of the ℓ-isogeny quotient by an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_isLevelIsogeny_of_isUnit 737 below · cited by 4 · depth 25
• Base change of a fake elliptic curve, with cartesian square and level CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_levelIff 20 below · cited by 112 · depth 25
• Extension of fake elliptic curves over valuation subrings of ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_valuationSubring_inf_of_isPullback_algebraMap_of_isUnit 1,278 below · cited by 1 · depth 25
• A ramified prime ideal acts nontrivially on r-torsion CerednikDrinfeld.QM.FakeEllipticCurve.exists_pushPt_act_ne_one_of_dvd_nrd_of_eq_or_eq 708 below · cited by 1 · depth 25
• Unique factorisation of isogenies of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_unique_comp_eq_of_forall_mapPt_eq_one 722 below · cited by 12 · depth 25
• Every fake elliptic curve is an ℓ-level isogeny target CerednikDrinfeld.QM.FakeEllipticCurve.exists_withExtraLevel_isLevelIsogeny 817 below · cited by 3 · depth 25
• Isogeny of degree prime to N carries level structure onto CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_iff_exists_mapPt_eq_of_coprime 1 below · cited by 17 · depth 25
• Level structures match exactly across a cartesian square CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_of_exists_comp_eq_of_isPullback 1 below · cited by 30 · depth 25
• Atkin–Lehner quotient at r commutes with the ℓ-isogeny leg CerednikDrinfeld.QM.FakeEllipticCurve.isAtkinLehnerQuotient_of_isLevelIsogeny_of_isLevelIsogeny 726 below · cited by 1 · depth 25
• Atkin–Lehner quotients are preserved under base change CerednikDrinfeld.QM.FakeEllipticCurve.isAtkinLehnerQuotient_of_isPullback 2 below · cited by 5 · depth 25
• Atkin–Lehner quotients transport along isomorphisms of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.isAtkinLehnerQuotient_of_iso_of_iso 0 below · cited by 3 · depth 25
• Level-ℓ isogenies are stable under isomorphism of the target CerednikDrinfeld.QM.FakeEllipticCurve.isLevelIsogeny_of_isLevelIsogeny_of_iso 0 below · cited by 1 · depth 25
• Level isogeny relation is invariant under isomorphism of the source pair CerednikDrinfeld.QM.FakeEllipticCurve.isLevelIsogeny_of_iso_of_isLevelIsogeny 0 below · cited by 5 · depth 25
• Atkin–Lehner quotients commute with base change CerednikDrinfeld.QM.FakeEllipticCurve.isPullback_of_isAtkinLehnerQuotient_of_isAtkinLehnerQuotient 720 below · cited by 2 · depth 25
• Uniqueness of r-Hecke neighbours at a ramified prime CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_heckeNeighbour_of_heckeNeighbour_of_ramified 768 below · cited by 1 · depth 25
• Atkin–Lehner quotients at q and q' commute CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isAtkinLehnerQuotient_comm 726 below · cited by 1 · depth 25
• Composing two Atkin–Lehner quotients recovers the curve CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isAtkinLehnerQuotient_comp 727 below · cited by 5 · depth 25
• Extra levels with equal ℚ̄-points give isomorphic quotients CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isLevelIsogeny_of_forall_factorsThrough_iff_of_isOrder 731 below · cited by 3 · depth 25
• Two-prime switch: the two iterated quotients agree CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isLevelIsogeny_transfer_swap 732 below · cited by 1 · depth 25
• Uniqueness up to isomorphism of pull-backs of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isPullback_of_isPullback 2 below · cited by 5 · depth 25
• Atkin–Lehner quotients are unique up to isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_iso_of_isAtkinLehnerQuotient_of_isAtkinLehnerQuotient 713 below · cited by 3 · depth 25
• One proper Λ-line inside the level structure iff ℓ ∣ N CerednikDrinfeld.QM.FakeEllipticCurve.natCard_properLine_image_subset_lev 15 below · cited by 1 · depth 25
• Action of n∈Λ is n-fold addition on points CerednikDrinfeld.QM.FakeEllipticCurve.pushPt_act_natCast_eq_nsmulPt 0 below · cited by 65 · depth 25
• Smoothness of relative dimension two for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.smoothOfRelativeDimension_two 4 below · cited by 37 · depth 25
• Uniqueness of the coarse moduli scheme of fake elliptic curves CerednikDrinfeld.QM.IsCoarseModuli.existsUnique_comp_eq_and_isIso 0 below · cited by 7 · depth 25
• Atkin–Lehner involution at r on coarse moduli, r invertible CerednikDrinfeld.QM.IsCoarseModuli.exists_atkinLehner_involution_of_isUnit 775 below · cited by 1 · depth 25
• Comparison morphism from a coarse moduli scheme to a base change CerednikDrinfeld.QM.IsCoarseModuli.exists_hom_pullback_bijective_points 0 below · cited by 4 · depth 25
• Coarse moduli of fake elliptic curves under base change to a field CerednikDrinfeld.QM.IsCoarseModuli.exists_isCoarseModuli_pullback_of_injective_of_isUnit_of_isUnit_two_of_isUnit_three 3,826 below · cited by 6 · depth 25
• An integral characteristic-zero geometric fibre of a coarse Shimura model CerednikDrinfeld.QM.IsCoarseModuli.exists_isIntegral_pullback_of_isAlgClosed_charZero_of_not_dvd_of_three_le_of_squarefree_of_isUnit_two_of_isUnit_three 5,402 below · cited by 2 · depth 25
• Integrality of the quaternionic coarse moduli scheme over k CerednikDrinfeld.QM.IsCoarseModuli.isIntegral_of_isAlgClosed_of_not_dvd_of_squarefree 5,512 below · cited by 1 · depth 25
• Properness of coarse moduli schemes of fake elliptic curves CerednikDrinfeld.QM.IsCoarseModuli.isProper_of_three_le_of_isUnit_two_of_isUnit_three 4,414 below · cited by 10 · depth 25
• Coarse moduli of fake elliptic curves is smooth of relative dimension one CerednikDrinfeld.QM.IsCoarseModuli.smoothOfRelativeDimension_one_of_isUnit_of_not_dvd_of_isUnit_two_of_isUnit_three 3,856 below · cited by 1 · depth 25
• Base automorphisms act on a coarse moduli scheme CerednikDrinfeld.QM.IsCoarseModuliT.existsUnique_hom_comp_eq_specMap_ringEquiv 0 below · cited by 1 · depth 25
• Degeneracy quotient map between coarse moduli schemes CerednikDrinfeld.QM.IsCoarseModuliT.exists_degeneracy_quotient 748 below · cited by 1 · depth 25
• Comparison morphism to a base-changed coarse moduli scheme CerednikDrinfeld.QM.IsCoarseModuliT.exists_hom_pullback_bijective_points 0 below · cited by 1 · depth 25
• Base change of the coarse moduli of pairs to a field CerednikDrinfeld.QM.IsCoarseModuliT.exists_isCoarseModuliT_pullback_of_injective_of_isUnit_of_isUnit_two_of_isUnit_three 3,866 below · cited by 6 · depth 25
• Uniqueness of the coarse moduli scheme with extra level ℓ CerednikDrinfeld.QM.IsCoarseModuliT.exists_iso_comp_eq_of_isCoarseModuliT 0 below · cited by 8 · depth 25
• Integrality of coarse moduli of pairs with extra ℓ-level CerednikDrinfeld.QM.IsCoarseModuliT.isIntegral_of_isAlgClosed_of_not_dvd_of_squarefree_of_ne 5,561 below · cited by 1 · depth 25
• Čerednik–Drinfeld uniformisation at fine level, tower and Atkin–Lehner CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_uniformization_fine_level_atkinLehner_of_geometricallyConnected_of_squarefree_of_isUnit_two_of_geometricallyConnected_tower_of_isUnit_three 7,008 below · cited by 1 · depth 25
• Coarse moduli at raised level ℓ from a line-stabiliser quotient CerednikDrinfeld.QM.IsFineModuli.exists_isCoarseModuliT_of_quotient_of_isIndefiniteRamifiedExactlyAt_of_isUnit_mem_iff 816 below · cited by 5 · depth 25
• Quotient of a fine moduli scheme by level twisting is coarse CerednikDrinfeld.QM.IsFineModuli.exists_isCoarseModuli_of_quotient_of_isIndefiniteRamifiedExactlyAt 773 below · cited by 10 · depth 25
• Existence of the level-twisting action on a fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_isLevelTwistAction 23 below · cited by 14 · depth 25
• Finite quotient of the fine moduli scheme is flat, locally of finite type CerednikDrinfeld.QM.IsFineModuli.flat_and_locallyOfFiniteType_of_quotient_of_isUnit_two_of_isUnit_three 3,923 below · cited by 4 · depth 25
• Geometric points of the fine-to-coarse map: surjectivity and G-orbits CerednikDrinfeld.QM.IsFineModuli.nilpPoints_quotient_surjective_and_iff_of_isAlgClosed_of_isMaximalOrder 735 below · cited by 1 · depth 25
• Smoothness of a finite quotient of the fine moduli scheme when 6qq' is invertible CerednikDrinfeld.QM.IsFineModuli.smoothOfRelativeDimension_one_of_quotient_of_isUnit_six 3,844 below · cited by 2 · depth 25
• Stabiliser of a left ideal under a level-twisting labelling CerednikDrinfeld.QM.IsLevelTwistAction.exists_subgroup_mem_iff_forall_mul_mem 3 below · cited by 5 · depth 25
• Hecke correspondences at two primes away from qq' commute CerednikDrinfeld.QM.ModuliTowerWitnessD.correspondence_comm_of_exhaustive_of_swap_of_two_mul_dvd 2 below · cited by 1 · depth 25
• Level-(N;ℓ) tower field is the function field of the coarse moduli curve CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_algEquiv_level_comp_phi_eq_of_pt_pullback_of_two_mul_dvd 5,726 below · cited by 1 · depth 25
• Ramification along the level-forgetting leg counts extra levels CerednikDrinfeld.QM.ModuliTowerWitnessD.ramificationIndexAlong_eq_card_of_pt_pullback_of_two_mul_dvd_of_squarefree 5,712 below · cited by 1 · depth 25
• Fine moduli scheme for fake elliptic curves with full level-m structure CerednikDrinfeld.QM.exists_isFineModuli_of_isUnit_two_of_isUnit_three 3,799 below · cited by 22 · depth 25
• Hecke correspondences on the uniformised fake-elliptic moduli curve CerednikDrinfeld.QM.exists_uniformizedHeckeCurve_bcPlace_corr_single_eq_sum_of_two_mul_dvd 5,502 below · cited by 1 · depth 25
• Every extra level at ℓ has the form L₀·(m/ℓ)P CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_fullLevel_forall_factorsThrough_iff 745 below · cited by 1 · depth 26
• Extra level at invertible ℓ determined by geometric points CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.factorsThrough_iff_of_forall_geomPoint 15 below · cited by 3 · depth 26
• Extra level K is reduced when ℓ is invertible in k CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.isReduced_K_of_natCast_ne_zero 1 below · cited by 7 · depth 26
• Twisting a full level-m structure by a unit mod mΛ CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_eq_pushPt_act_and_isTwist_of_mul_sub_one_eq_smul 0 below · cited by 3 · depth 26
• An extra level at ℓ from a full level-m structure CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_extraLevel_forall_factorsThrough_iff 23 below · cited by 3 · depth 26
• Twisting a full level structure transports the extra-level line CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_forall_factorsThrough_iff_of_mul_mem 2 below · cited by 3 · depth 26
• Isomorphic ℓ-level lifts pull back level-N structures alike CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_fin_forall_factorsThrough_mapPt_iff_of_iso_of_dvd 5,686 below · cited by 1 · depth 26
• Flat-local full level m along a fixed line L₀ CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_flat_surjective_withFullLevel_forall_factorsThrough_iff 787 below · cited by 1 · depth 26
• Level-Nℓ extensions of the level structure come from isogeny lifts CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_isogenyData_forall_mem_iff_of_levelExt 835 below · cited by 1 · depth 26
• Lifts of D with the same pulled-back level structure agree CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_forall_factorsThrough_mapPt_iff_of_dvd 760 below · cited by 1 · depth 26
• Uniqueness of the source of an ℓ-isogeny onto a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_forall_mapPt_eq_one_iff_of_forall_factorsThrough_mapPt_iff 737 below · cited by 2 · depth 26
• ψ-preimage of the level structure is a level-Nℓ extension CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.levelExt_setOf_factorsThrough_mapPt_of_dvd 752 below · cited by 1 · depth 26
• Base change of a fake elliptic curve with full level structure CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback 21 below · cited by 20 · depth 26
• Twisting a full level-m structure by a unit mod mΛ CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isTwist_and_fst_eq 1 below · cited by 4 · depth 26
• Twists by labels congruent modulo mΛ are isomorphic CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.iso_of_isTwist_of_isTwist_of_sub_eq_smul 0 below · cited by 2 · depth 26
• Drinfeld's trace condition descends along an étale equivariant map CerednikDrinfeld.QM.FakeEllipticCurve.act_trace_of_etale 1 below · cited by 2 · depth 26
• Kernel ideal of a non-isomorphic Hecke isogeny is neither rΛ nor Λ CerednikDrinfeld.QM.FakeEllipticCurve.annihilator_ne_of_not_isIso_of_nsmulPt_eq_one 730 below · cited by 1 · depth 26
• Ordinary–supersingular dichotomy for the ℓ+1 extra levels CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_or_forall_reducesToZero_of_extraLevels 771 below · cited by 1 · depth 26
• Level-N points of a fake elliptic curve form (ℤ/N)² CerednikDrinfeld.QM.FakeEllipticCurve.exists_equiv_levPoints 0 below · cited by 2 · depth 26
• Transport of an extra level along an isomorphism of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_mapPt_iff 0 below · cited by 5 · depth 26
• Extending an extra level and ℓ-isogeny over a valuation subring CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_of_isPullback_valuationSubring_of_coprime_of_one_mem_of_isPullback_inf 766 below · cited by 1 · depth 26
• Symmetry of ℓ-level isogenies of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_symm_of_not_dvd 737 below · cited by 1 · depth 26
• Extra ℓ-level recognised from its k-points CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_of_isClosedImmersion_of_equiv_points 3 below · cited by 2 · depth 26
• Rigidity of Λ-linear quasi-inverses of [n] off finitely many classes CerednikDrinfeld.QM.FakeEllipticCurve.exists_finset_forall_not_iso_forall_quasiInverse_exists_eq_nsmulPt 5,681 below · cited by 2 · depth 26
• Full level-m structures exist flat-locally on the base CerednikDrinfeld.QM.FakeEllipticCurve.exists_flat_surjective_withFullLevel_isPullback 761 below · cited by 4 · depth 26
• Existence of Frobenius twist and Verschiebung for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_hasFrobeniusVerschiebung_of_prime_not_dvd 33 below · cited by 5 · depth 26
• Descent of a finitely generated quaternionic action to large finite levels CerednikDrinfeld.QM.FakeEllipticCurve.exists_intermediateField_forall_exists_act_of_isPullback_algebraMap_of_fg 3 below · cited by 1 · depth 26
• Atkin–Lehner quotients of fake elliptic curves over r-invertible bases CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAtkinLehnerQuotient_of_isUnit 51 below · cited by 1 · depth 26
• Good reduction of the surface extends a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_algebraMap_of_abelianSchemePropertyBundle_of_mem_maximalIdeal 107 below · cited by 1 · depth 26
• Base change of a level-ℓ isogeny quotient of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_and_isLevelIsogeny_of_withExtraLevel_isPullback 23 below · cited by 5 · depth 26
• Two full level-m structures differ by a twist CerednikDrinfeld.QM.FakeEllipticCurve.exists_isTwist_of_fullLevel 0 below · cited by 1 · depth 26
• Quotient of a fake elliptic curve by an n-torsion subgroup CerednikDrinfeld.QM.FakeEllipticCurve.exists_quotient_core_of_isAlgClosed 31 below · cited by 5 · depth 26
• Quotient of a fake elliptic curve by a finite flat subgroup CerednikDrinfeld.QM.FakeEllipticCurve.exists_quotient_of_finiteFlat_stable_subgroup 728 below · cited by 1 · depth 26
• Annihilator of a torsion point: a left ideal containing rΛ CerednikDrinfeld.QM.FakeEllipticCurve.exists_submodule_mem_iff_mapPt_pushPt_act_eq_one 1 below · cited by 1 · depth 26
• Fake elliptic curves as ℓ-level isogeny images when ℓ ∣ N CerednikDrinfeld.QM.FakeEllipticCurve.exists_withExtraLevel_isLevelIsogeny_of_dvd 760 below · cited by 1 · depth 26
• Geometric points of the kernel of a dual ℓ-isogeny CerednikDrinfeld.QM.FakeEllipticCurve.exists_zmod_prod_equiv_factorsThrough_kernel_dual_of_not_dvd 696 below · cited by 2 · depth 26
• m-torsion of a fake elliptic curve has m⁴ points CerednikDrinfeld.QM.FakeEllipticCurve.finite_and_natCard_torsion_eq_pow_four_of_isUnit 706 below · cited by 7 · depth 26
• Kernel containment on k-points implies containment on all points CerednikDrinfeld.QM.FakeEllipticCurve.forall_mapPt_eq_one_of_forall_rationalPoint 11 below · cited by 4 · depth 26
• Level-ℓ quotient with a nontrivial point: E is Frobenius twist of d CerednikDrinfeld.QM.FakeEllipticCurve.hasFrobeniusVerschiebung_of_isLevelIsogeny_of_exists_factorsThrough_ne_one 788 below · cited by 1 · depth 26
• Rank ℓ² for the kernel of the partner isogeny CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_and_finrank_kernel_eq_of_comp_eq_nsmulPt_of_finrank_eq 705 below · cited by 5 · depth 26
• Finite flatness of morphisms of fake elliptic curves with quasi-inverse CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_flat_surjective_of_mapPt_mapPt_eq_nsmulPt 708 below · cited by 19 · depth 26
• Descent of the pull-back relation along S₀ → S₁ → S₂ CerednikDrinfeld.QM.FakeEllipticCurve.isPullback_of_isPullback_comp_of_isPullback 2 below · cited by 4 · depth 26
• Double Frobenius twist of a supersingular fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.iso_frobeniusTwist_frobeniusTwist_of_forall_nsmulPt_eq_one 769 below · cited by 1 · depth 26
• Level-ℓ quotient by a pointless extra level is the Frobenius twist CerednikDrinfeld.QM.FakeEllipticCurve.iso_frobeniusTwist_of_isLevelIsogeny_of_forall_factorsThrough_eq_one 764 below · cited by 2 · depth 26
• Uniqueness of the level-ℓ isogeny quotient over a general base CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isLevelIsogeny_of_isLevelIsogeny 712 below · cited by 3 · depth 26
• Transport of a level structure along a finite homomorphism CerednikDrinfeld.QM.FakeEllipticCurve.levelStructure_lev_comp_of_disjoint 0 below · cited by 1 · depth 26
• Exactly ℓ level extensions at ℓ ∣ N CerednikDrinfeld.QM.FakeEllipticCurve.natCard_levelExt_eq_of_dvd 753 below · cited by 1 · depth 26
• ℓ-torsion of a fake elliptic curve has order ℓ⁴ CerednikDrinfeld.QM.FakeEllipticCurve.natCard_torsionPoints_eq_pow_four 691 below · cited by 2 · depth 26
• Existence of full level-m structures over algebraically closed fields CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_fullLevel_of_isMaximalOrder 730 below · cited by 12 · depth 26
• Base change of coarse moduli of fake elliptic curves to a field CerednikDrinfeld.QM.IsCoarseModuli.exists_isCoarseModuli_pullback_fst_eq_of_injective_of_isUnit_of_isUnit_two_of_isUnit_three 3,825 below · cited by 3 · depth 26
• Uniqueness of coarse moduli schemes for fake elliptic curves CerednikDrinfeld.QM.IsCoarseModuli.exists_iso_comp_eq_of_isCoarseModuli 1 below · cited by 5 · depth 26
• Integrality of the complex coarse quaternionic Shimura curve CerednikDrinfeld.QM.IsCoarseModuli.isIntegral_of_complex_of_squarefree 5,377 below · cited by 2 · depth 26
• Uniqueness of the coarse moduli scheme up to unique isomorphism CerednikDrinfeld.QM.IsCoarseModuliT.existsUnique_comp_eq_and_isIso 0 below · cited by 5 · depth 26
• Integral geometric fibre of a coarse pairs model over ℤ[1/M] CerednikDrinfeld.QM.IsCoarseModuliT.exists_isIntegral_pullback_of_isAlgClosed_charZero_of_not_dvd_of_dvd_of_squarefree_of_isUnit_two_of_isUnit_three_of_ne 5,488 below · cited by 1 · depth 26
• Unique factorisation of invariant families through Theta_f CerednikDrinfeld.QM.IsFineModuli.existsUnique_factor_of_cerednikDrinfeld_uniformization_fine 878 below · cited by 1 · depth 26
• Formal Čerednik–Drinfeld quotient property at tower level ℓ CerednikDrinfeld.QM.IsFineModuli.existsUnique_factor_of_cerednikDrinfeld_uniformization_tower_of_isUnit_two 44 below · cited by 1 · depth 26
• Universal property of the fine moduli scheme of fake elliptic curves CerednikDrinfeld.QM.IsFineModuli.existsUnique_hom_ptF_comp_eq 22 below · cited by 2 · depth 26
• Universal property of the disjoint-line locus for pairs CerednikDrinfeld.QM.IsFineModuli.existsUnique_hom_ptT_comp_eq 51 below · cited by 1 · depth 26
• Fine-level Čerednik–Drinfeld uniformisation with Atkin–Lehner and level lifts CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_uniformization_fine_level_atkinLehner_minusT_liftT_of_squarefree_of_isUnit_two_of_pow_smul_mem 5,261 below · cited by 1 · depth 26
• Level-ℓ fine moduli scheme and its quotient presentation CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuliT_quotient_presentation_of_isSeparated 998 below · cited by 1 · depth 26
• Level-N fine moduli is finite étale over level 1 CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuli_finite_etale_of_isUnit 863 below · cited by 3 · depth 26
• Uniqueness of the fine moduli scheme for full level m CerednikDrinfeld.QM.IsFineModuli.exists_iso_of_isFineModuli 22 below · cited by 10 · depth 26
• Quotients of the fine moduli scheme commute with flat base change CerednikDrinfeld.QM.IsFineModuli.exists_iso_quotient_pullback_of_flat 34 below · cited by 2 · depth 26
• Flat base change of quotients, compatibly with moduli points CerednikDrinfeld.QM.IsFineModuli.exists_iso_quotient_pullback_of_flat_pointCompat 34 below · cited by 2 · depth 26
• An open and closed disjointness locus on the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_opens_isClosed_range_subset_iff_forall_factorsThrough_lev_imp 52 below · cited by 1 · depth 26
• Flatness of the fine moduli scheme of fake elliptic curves CerednikDrinfeld.QM.IsFineModuli.flat_of_isUnit_of_isUnit_two_of_isUnit_three 3,907 below · cited by 1 · depth 26
• Surjectivity of the Čerednik–Drinfeld parametrisation on geometric points CerednikDrinfeld.QM.IsFineModuli.forall_exists_eq_of_cerednikDrinfeld_uniformization_fine_minus_of_geometricallyConnected_of_squarefree_of_isUnit_two_of_isUnit_three 5,820 below · cited by 1 · depth 26
• Surjectivity of the tower-level Čerednik–Drinfeld uniformisation on geometric points CerednikDrinfeld.QM.IsFineModuli.forall_exists_eq_of_cerednikDrinfeld_uniformization_tower_minus_of_geometricallyConnected_of_isUnit_two_of_isUnit_three 5,870 below · cited by 1 · depth 26
• Properness of the fine moduli scheme of fake elliptic curves CerednikDrinfeld.QM.IsFineModuli.isProper_of_isUnit_two_of_isUnit_three 4,243 below · cited by 5 · depth 26
• Smoothness of relative dimension one for a fine moduli scheme of fake elliptic curves, with 6 invertible CerednikDrinfeld.QM.IsFineModuli.smoothOfRelativeDimension_one_of_isUnit_two_of_isUnit_three 3,798 below · cited by 1 · depth 26
• Relative smoothness of quotients of fake elliptic moduli schemes CerednikDrinfeld.QM.IsFineModuli.smoothOfRelativeDimension_one_of_quotient_of_isUnit_two_of_isUnit_three 3,843 below · cited by 3 · depth 26
• Label-compatible injection between level-twisting groups CerednikDrinfeld.QM.IsLevelTwistAction.exists_monoidHom_injective_label_congr_of_isOrder 0 below · cited by 6 · depth 26
• Level structures cutting out the same L₀-line agree over X_H CerednikDrinfeld.QM.IsLevelTwistAction.ptF_comp_eq_ptF_comp_of_forall_factorsThrough_levK_iff 36 below · cited by 1 · depth 26
• Moduli point on a G-invariant quotient ignores the full level CerednikDrinfeld.QM.IsLevelTwistAction.ptF_comp_eq_ptF_comp_of_fullLevel 36 below · cited by 1 · depth 26
• Decomposition-group count along the level-forgetting leg of the tower CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_galoisFrame_natCard_stabilizer_eq_of_two_mul_dvd_of_squarefree 5,708 below · cited by 1 · depth 26
• Geometric fibres of an affine invariant quotient are G-orbits CerednikDrinfeld.QM.exists_eq_comp_autHom_of_comp_eq_of_isAlgClosed 1 below · cited by 3 · depth 26
• Potential good reduction of abelian surfaces with quaternionic multiplication CerednikDrinfeld.QM.exists_intermediateField_abelianSchemePropertyBundle_isPullback_of_quaternionOrder_action_of_forall_isUnit_tensorProduct_padic 1,237 below · cited by 1 · depth 26
• Fine moduli of fake elliptic curves with full level m CerednikDrinfeld.QM.exists_isFineModuli_one_of_isUnit_two_of_isUnit_three 3,724 below · cited by 3 · depth 26
• Complex uniformisation of the fake elliptic moduli curve CerednikDrinfeld.QM.exists_period_algEquiv_pt_iff_bcPlace_of_uniformizedHeckeCurve_of_two_mul_dvd 5,470 below · cited by 1 · depth 26
• Descent of the point rule to pairs with extra level CerednikDrinfeld.QM.exists_ptT_eq_ptF_comp_of_isFineModuli_of_forall_ptF_comp_eq 26 below · cited by 1 · depth 26
• Descent of the point map along the forgetful morphism CerednikDrinfeld.QM.exists_pt_eq_ptF_comp_of_isFineModuli_of_forall_ptF_comp_eq 25 below · cited by 1 · depth 26
• Isogeny criterion: surjective, finite and flat CerednikDrinfeld.QM.surjective_and_isFinite_and_flat_of_mapPt_mapPt_eq_nsmulPt 720 below · cited by 6 · depth 26
• Clopen locus where the extra level is L₀·(m/ℓ)P CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_isClosed_range_subset_iff_forall_factorsThrough_iff 732 below · cited by 1 · depth 27
• Lattice between ℓΛ and Λ of index ℓ² cutting out an extra level CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_submodule_relIndex_eq_forall_factorsThrough_iff 4 below · cited by 3 · depth 27
• Transport of extra ℓ-levels along a base-change square CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.forall_exists_factorsThrough_iff_comp_of_isPullback_of_isAlgClosed 735 below · cited by 4 · depth 27
• Twisting a full level structure by a unit mod mΛ CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_P_eq_pushPt_act_and_isTwist 1 below · cited by 6 · depth 27
• Clopen locus where L₀·(m/ℓ)P meets the level structure trivially CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_isClosed_range_subset_iff_forall_factorsThrough_lev_imp_eq_one 21 below · cited by 2 · depth 27
• Pointwise transport of the extra level along a unit twist CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.forall_factorsThrough_iff_of_mul_mem_of_sectionAt_eq 0 below · cited by 1 · depth 27
• Pull-backs of a fake elliptic curve with extra level are unique CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_isPullback_of_isPullback 3 below · cited by 2 · depth 27
• Dual kernel equals the ℓ-torsion of the level structure CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.mapPt_eq_one_iff_factorsThrough_lev_of_dvd 750 below · cited by 3 · depth 27
• Extra levels of order N represented by a finite étale scheme CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_finite_etale_represents_extraLevel 846 below · cited by 1 · depth 27
• Base change of full level-m and extra level-ℓ structures CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_extraLevel_forall_factorsThrough_iff 24 below · cited by 1 · depth 27
• Base change of full and extra level structures on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_extraLevel_forall_factorsThrough_iff_of_ringHom 23 below · cited by 2 · depth 27
• Base change of fake elliptic curves with full and extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_extraLevel_forall_geomPoint_iff 23 below · cited by 1 · depth 27
• Base change invariance of the level factorisation condition CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.forall_factorsThrough_lev_imp_eq_one_iff_of_isPullback 2 below · cited by 1 · depth 27
• Twist-invariance of the lev-factorisation condition CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.forall_factorsThrough_lev_imp_eq_one_iff_of_isTwist 0 below · cited by 1 · depth 27
• Isomorphism invariance of the lev-vanishing condition at a geometric point CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.forall_factorsThrough_lev_imp_eq_one_iff_of_iso 0 below · cited by 1 · depth 27
• Invariance of the C-factorisation condition under extension of k CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.forall_factorsThrough_lev_imp_eq_one_iff_of_ringHom 0 below · cited by 2 · depth 27
• Atkin–Lehner element ̄ w in the endomorphism dictionary CerednikDrinfeld.QM.FakeEllipticCurve.exists_atkinLehnerDictionary_of_endomorphismDictionary_endIsoFull 818 below · cited by 1 · depth 27
• Kernel of Verschiebung as an extra level at ℓ CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_of_frobeniusVerschiebungData 716 below · cited by 1 · depth 27
• Kernel of an extended ℓ-isogeny as extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_ker_of_isPullback_valuationSubring_of_comp_eq_act_of_one_mem 720 below · cited by 1 · depth 27
• Cofinite rigidity of fake elliptic curves over ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.exists_finset_forall_not_iso_forall_hom_eq_id_or_mapPt_eq_inv 5,683 below · cited by 2 · depth 27
• Finitely many fake elliptic curves with quadratic multiplication CerednikDrinfeld.QM.FakeEllipticCurve.exists_finset_forall_not_iso_not_exists_mapPt_mapPt_mul_zpow_eq_zpow 5,680 below · cited by 1 · depth 27
• Equivariance of the rigidified dictionary: Γ, Atkin–Lehner, Hecke CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_isActBy_rigidifiedToG_star_of_isTranslateBy_of_isLevelIsogeny_of_isAtkinLehnerQuotient_of_endIsoFull 209 below · cited by 1 · depth 27
• Quadratic endomorphism with non-negative discriminant is an integer CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_mapPt_eq_zpow_of_forall_act_comp_eq_of_four_mul_le_sq 737 below · cited by 2 · depth 27
• Endomorphisms commuting with Λ satisfy a quadratic integer relation CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_mapPt_mapPt_mul_zpow_eq_zpow_of_forall_act_comp_eq 751 below · cited by 2 · depth 27
• The n-torsion of a fake elliptic curve is Λ-cyclic CerednikDrinfeld.QM.FakeEllipticCurve.exists_generator_torsionPoints_of_isMaximalOrder_of_isIndefiniteRamifiedExactlyAt 723 below · cited by 1 · depth 27
• Homomorphisms of fake elliptic curves extend over O CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_extension_of_isPullback_valuationSubring_of_isPullback_inf 69 below · cited by 1 · depth 27
• Orbits of an n-torsion subscheme lie in affine opens CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAffineOpen_forall_action_mem_of_nsmulPt_eq_one 709 below · cited by 1 · depth 27
• Clopen locus of full level-m generators in A[m] CerednikDrinfeld.QM.FakeEllipticCurve.exists_isClopen_genLocus_schemeKer 728 below · cited by 2 · depth 27
• Finite flat kernel of a morphism of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernel_of_isFinite_of_flat 0 below · cited by 3 · depth 27
• Relative Frobenius into the Frobenius twist of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_relFrobenius_of_isPullback_frobenius 0 below · cited by 1 · depth 27
• Forgetful map from rigidified fake elliptic curves to G-points CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedToG_natural_isRigTransport 889 below · cited by 1 · depth 27
• Level-N points as a Λ-stable overlattice of index N² CerednikDrinfeld.QM.FakeEllipticCurve.exists_submodule_forall_mem_iff_factorsThrough_lev_of_pointEquiv 0 below · cited by 11 · depth 27
• Transverse level-Nℓ lift of the level structure, ℓ ∣ N CerednikDrinfeld.QM.FakeEllipticCurve.exists_transverseLevelLift_of_dvd 730 below · cited by 2 · depth 27
• Transverse level lift yields an ℓ-level isogeny onto E₀ CerednikDrinfeld.QM.FakeEllipticCurve.exists_withExtraLevel_isLevelIsogeny_of_levelLift 738 below · cited by 1 · depth 27
• Extra level of a pullback pair is a full preimage CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_levK_of_exists_comp_eq_of_isPullback 1 below · cited by 9 · depth 27
• Isogeny with extra-level kernel maps level structure onto level structure CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_iff_exists_mapPt_eq_of_extraLevel 710 below · cited by 6 · depth 27
• Extension of a level-compatible morphism stays level-compatible CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_mapPt_of_isPullback_valuationSubring 1 below · cited by 1 · depth 27
• At most ℓ² rational ℓ-torsion points in characteristic ℓ CerednikDrinfeld.QM.FakeEllipticCurve.finite_and_natCard_torsionPoints_le_sq_of_charP_of_not_dvd 728 below · cited by 3 · depth 27
• Extra level at ℓ has no k-points iff it is ker F CerednikDrinfeld.QM.FakeEllipticCurve.forall_factorsThrough_eq_one_iff_isFrobeniusKernel 52 below · cited by 2 · depth 27
• Finite étale reduced closed subgroup with (ℤ/n)² points satisfies level axioms CerednikDrinfeld.QM.FakeEllipticCurve.forall_factorsThrough_levelPackage_of_isClosedImmersion_of_equiv_points 3 below · cited by 1 · depth 27
• Partner of a level-ℓ isogeny with a rational kernel point CerednikDrinfeld.QM.FakeEllipticCurve.forall_mapPt_eq_one_of_isLevelIsogeny_of_exists_factorsThrough_ne_one 762 below · cited by 1 · depth 27
• Transport of Frobenius–Verschiebung data along an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.hasFrobeniusVerschiebung_of_iso 0 below · cited by 1 · depth 27
• Frobenius–Verschiebung datum from a finite flat relative Frobenius CerednikDrinfeld.QM.FakeEllipticCurve.hasFrobeniusVerschiebung_of_relFrobenius_of_flat 4 below · cited by 1 · depth 27
• m-torsion of a fake elliptic curve is finite étale CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_and_etale_schemeKerStr_of_isUnit 13 below · cited by 28 · depth 27
• Finite flatness of an ℓ-isogeny from its kernel rank CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_flat_finrank_of_isClosedImmersion_kernel 722 below · cited by 5 · depth 27
• Two-sided n-isogenies of fake elliptic curves are finite and flat CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_flat_surjective_of_mapPt_mapPt_eq_nsmulPt_valuationSubring 708 below · cited by 2 · depth 27
• Isomorphism of level-N fake elliptic curves as lattice homothety CerednikDrinfeld.QM.FakeEllipticCurve.iso_iff_exists_smul_latt_eq_and_smul_lattLev_eq_of_pointEquiv 3 below · cited by 4 · depth 27
• Quotient by the Frobenius kernel is the Frobenius twist CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_isLevelIsogeny_of_isFrobeniusKernel 725 below · cited by 1 · depth 27
• Atkin–Lehner lift fixes the generic point of the geometric fibre CerednikDrinfeld.QM.IsCoarseModuli.base_genericPoint_eq_of_comp_fst_eq_fst_comp_of_isAtkinLehnerQuotient_of_not_dvd 765 below · cited by 2 · depth 27
• Lifted degeneracy maps are dominant on geometric generic fibres CerednikDrinfeld.QM.IsCoarseModuliT.base_genericPoint_eq_of_comp_fst_eq_fst_comp_degeneracy 818 below · cited by 4 · depth 27
• Integrality of the complex coarse Shimura curve with extra level at ℓ ∣ N CerednikDrinfeld.QM.IsCoarseModuliT.isIntegral_of_complex_of_squarefree_of_dvd 5,416 below · cited by 1 · depth 27
• Integrality of the complex coarse moduli scheme with extra level at ℓ CerednikDrinfeld.QM.IsCoarseModuliT.isIntegral_of_complex_of_squarefree_of_not_dvd_of_ne 5,394 below · cited by 1 · depth 27
• Invariance of a natural family under the Γₜ-orbit relation CerednikDrinfeld.QM.IsFineModuli.apply_eq_apply_of_isPullback_of_frobTwist_eq_of_invariant 8 below · cited by 2 · depth 27
• Atkin–Lehner relations for the Čerednik–Drinfeld fine family CerednikDrinfeld.QM.IsFineModuli.cerednikDrinfeld_fineFamily_atkinLehner_of_rigidifiedToG_heightNormalised_oneLegC5 914 below · cited by 1 · depth 27
• A Čerednik–Drinfel'd family on the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_fineFamily_of_rigidifiedToG_heightNormalised_eq_oneLegC5_h2 3,615 below · cited by 1 · depth 27
• Čerednik–Drinfeld uniformisation along the Hecke tower, one-leg form CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_towerFamily_liftT_of_rigidifiedToG_heightNormalised_eq_oneLegC5 1,301 below · cited by 1 · depth 27
• Fibres of the fine Čerednik–Drinfeld uniformisation, flat-locally CerednikDrinfeld.QM.IsFineModuli.exists_flat_family_isPullback_of_cerednikDrinfeld_uniformization_fine_eq 16 below · cited by 1 · depth 27
• Fpqc-local lifting through the fine Čerednik–Drinfeld uniformisation CerednikDrinfeld.QM.IsFineModuli.exists_flat_family_lift_of_cerednikDrinfeld_uniformization_fine 860 below · cited by 1 · depth 27
• Fine moduli at level (N;n) with extra level ℓ, finite étale CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuliT_finite_etale_forget 864 below · cited by 2 · depth 27
• Gluing a finite étale level-N cover of a fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuli_finite_etale_of_forall_local_via 36 below · cited by 1 · depth 27
• Fine moduli of fake elliptic curves is stable under base change CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuli_of_isPullback 0 below · cited by 7 · depth 27
• Openness of the image of a formally étale uniformisation family CerednikDrinfeld.QM.IsFineModuli.exists_isOpen_inter_eq_image_of_formallyEtale 858 below · cited by 1 · depth 27
• Fine moduli of fake elliptic curves: equivariant base change CerednikDrinfeld.QM.IsFineModuli.exists_iso_pullback_equivariant_of_isLevelTwistAction 27 below · cited by 3 · depth 27
• Flatness of fine moduli over any base from a torsion-free model over ℤ[1/6Nm] CerednikDrinfeld.QM.IsFineModuli.flat_of_exists_isFineModuli_localizationAway_six_mul_of_forall_mul_eq_zero 24 below · cited by 1 · depth 27
• Flatness over ℤ_q of the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.flat_padicInt_of_isUnit_of_isUnit_two_of_isUnit_three 3,905 below · cited by 1 · depth 27
• Fake elliptic curves near a period lie in one algebraic family CerednikDrinfeld.QM.IsFineModuli.forall_exists_algebraicFamily_isPullback_smul_latt_eq_of_analytic 4,810 below · cited by 1 · depth 27
• Surjectivity of the fine-level Čerednik–Drinfeld family on geometric points CerednikDrinfeld.QM.IsFineModuli.forall_exists_eq_of_geometricallyConnected_of_isOpen_of_nonempty_of_isUnit_two_of_isUnit_three 4,651 below · cited by 1 · depth 27
• Surjectivity of Čerednik–Drinfeld uniformisation at tower level ℓ CerednikDrinfeld.QM.IsFineModuli.forall_exists_eq_tower_of_geometricallyConnected_of_isOpen_of_nonempty_of_isUnit_two_of_isUnit_three 4,683 below · cited by 1 · depth 27
• A ρ_ℓ-invariant affine open around every point of M_ℓ CerednikDrinfeld.QM.IsFineModuli.forall_exists_isAffineOpen_mem_forall_preimage_eq_of_isFinite 0 below · cited by 1 · depth 27
• Every τ in H is a quaternionic period CerednikDrinfeld.QM.IsFineModuli.forall_exists_smul_latt_eq_qmPeriodLattice_of_isProper_of_analytic 4,809 below · cited by 5 · depth 27
• Properness of the fake-elliptic fine moduli scheme over ℤ[1/6Nm] CerednikDrinfeld.QM.IsFineModuli.isProper_localizationAway_six_mul 4,236 below · cited by 1 · depth 27
• Stabiliser index count for extra levels at a fine moduli point CerednikDrinfeld.QM.IsFineModuli.natCard_iso_extraLevel_mul_natCard_stabilizer_inf_eq 7 below · cited by 1 · depth 27
• Smoothness of relative dimension one for a fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.smoothOfRelativeDimension_one_of_finiteType_int 1,248 below · cited by 1 · depth 27
• Universal property of the level-ℓ Čerednik–Drinfeld uniformisation family CerednikDrinfeld.QM.IsFineModuliT.existsUnique_factor_of_cerednikDrinfeld_uniformization_fine 36 below · cited by 1 · depth 27
• Forgetting the full level: M_ℓ → Y_ℓ CerednikDrinfeld.QM.IsFineModuliT.exists_forgetLevel_toCoarseT 23 below · cited by 1 · depth 27
• Quotient of the level-ℓ fine scheme is mathcal Y_ℓ CerednikDrinfeld.QM.IsFineModuliT.exists_isIso_quotient_to_coarseT 786 below · cited by 1 · depth 27
• Level-twisting action lifts to the fine scheme of triples CerednikDrinfeld.QM.IsFineModuliT.exists_levelTwistAction_lift 24 below · cited by 1 · depth 27
• Fine moduli of fake elliptic curves is locally of finite presentation CerednikDrinfeld.QM.IsFineModuliT.locallyOfFinitePresentation 945 below · cited by 3 · depth 27
• Finiteness of a group acting by level twists CerednikDrinfeld.QM.IsLevelTwistAction.finite 0 below · cited by 2 · depth 27
• Galois frame on the restriction leg: stabiliser counts agree CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_galoisFrame_natCard_stabilizer_mul_eq_of_two_mul_dvd_of_squarefree 5,706 below · cited by 1 · depth 27
• Quaternionic action on the generic fibre extends to the abelian scheme CerednikDrinfeld.QM.exists_action_comp_eq_comp_of_isPullback_of_abelianSchemePropertyBundle 37 below · cited by 3 · depth 27
• Function field comparison for the fake elliptic curve moduli curve CerednikDrinfeld.QM.exists_algEquiv_realize_eventuallyEq_mem_pt_iff_of_periodMap_of_meromorphic_of_two_mul_dvd 120 below · cited by 1 · depth 27
• Base change to ℂ of a curve model, compatibly with places CerednikDrinfeld.QM.exists_curveModel_complex_pointEquivPlace_bcPlace_of_constantFieldExtension 82 below · cited by 1 · depth 27
• Supersingular base point with endomorphism dictionary and endomorphism-ring export CerednikDrinfeld.QM.exists_fakeEllipticCurve_isFormalModuleVia_hasHeight_four_endomorphismDictionary_endIsoFull_of_isUnit_two 4,038 below · cited by 1 · depth 27
• Invertible-level subgroup of the generic fibre extends étale CerednikDrinfeld.QM.exists_isClosedImmersion_etale_factorsThrough_iff_of_isPullback_of_isUnit 43 below · cited by 2 · depth 27
• Fine moduli for fake elliptic curves with full level m CerednikDrinfeld.QM.exists_isFineModuli_one_of_isFineModuli_thetaTypeLocally_of_qmStructure_of_isUnit_two_three_of_finiteType 3,170 below · cited by 1 · depth 27
• Uniformised locus cut out by an open subset of M CerednikDrinfeld.QM.exists_isOpen_forall_mem_and_iff_exists_uniformization_of_locallyOfFiniteType 16 below · cited by 1 · depth 27
• Coordinates on a maximal order at a ramified prime CerednikDrinfeld.QM.exists_isOrderCoord_of_isMaximalOrder 23 below · cited by 5 · depth 27
• Analytic uniformisation of fake elliptic curves over ℂ CerednikDrinfeld.QM.exists_latticeMap_pointEquiv_hom_iff_smul_le_analytic 314 below · cited by 4 · depth 27
• Period map and meromorphic realisation on a Shimura curve CerednikDrinfeld.QM.exists_periodMap_meromorphicRealization_of_uniformizedHeckeCurve_of_two_mul_dvd 5,428 below · cited by 1 · depth 27
• Isogeny bridge to the special formal mathcal O_D-module CerednikDrinfeld.QM.exists_quotientBridge_isIsogenyOfHeight_four_mul_of_isFormalModuleVia 66 below · cited by 1 · depth 27
• Maximal orders contain μ with μ²=-qq' normalising conjugation CerednikDrinfeld.QM.exists_sq_eq_neg_disc_and_forall_conj_mem_of_isMaximalOrder 55 below · cited by 4 · depth 27
• Drinfeld trace condition passes from the generic fibre to a smooth model CerednikDrinfeld.QM.trace_eq_of_isPullback_of_smoothOfRelativeDimension_two_of_mem_maximalIdeal 19 below · cited by 1 · depth 27
• Extra level structure pulls back along a cartesian comparison CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_forall_factorsThrough_iff_of_isPullbackVia 1 below · cited by 22 · depth 28
• Extra level of order N as clopen part of A[N] CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_opens_schemeKer_finrank_eq_forall_factorsThrough_iff 16 below · cited by 2 · depth 28
• Extra level ℓ read as a transversal sublattice of index ℓ² CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_submodule_forall_mem_iff_factorsThrough_transversal_of_pointEquiv 0 below · cited by 2 · depth 28
• Transport of a full level-m structure along an equivariant isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_P_eq_mapPt 0 below · cited by 5 · depth 28
• Full level structures pull back along base change CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_comp_eq_specMap_comp_of_isPullbackVia 0 below · cited by 21 · depth 28
• Formal mathcal O_D-module of a fake elliptic curve is special CerednikDrinfeld.QM.FakeEllipticCurve.IsFormalModuleOf.isSpecial 3 below · cited by 6 · depth 28
• Transport of formal 𝒪_D-module coordinates along an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.IsFormalModuleVia.mapPt_of_iso 0 below · cited by 3 · depth 28
• Composition of isogeny pairs multiplies degrees CerednikDrinfeld.QM.FakeEllipticCurve.IsIsogenyPair.comp 0 below · cited by 3 · depth 28
• Pasting pull-backs of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.IsPullbackVia.exists_comp_eq_and_isPullbackVia_of_comp_eq 0 below · cited by 14 · depth 28
• Germ identity for a rigidification transported along an isogeny CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.comp_comp_eq_comp_of_forall_nilEval_mapPt_of_comp_act_eq_comp_act 3 below · cited by 2 · depth 28
• Comparison of rigidifications: an identity of r-power series CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.comp_nthSeries_eq_comp_comp_of_forall_nilEval_of_comp_act_comp_eq_of_constantCoeff_eq_zero 3 below · cited by 4 · depth 28
• Corresponding rigidifications transport to the same η-point CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.eta_eq_of_isoVia_of_corr_of_isRigTransport 34 below · cited by 6 · depth 28
• Exponent bookkeeping for two rigidification transports linked by an isogeny CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_add_eq_add_two_mul_and_add_add_eq_add_add_of_isRigTransport_mapPt_of_comp_act_eq_comp_act_of_germ_of_isUnit 38 below · cited by 1 · depth 28
• Exponent bookkeeping for two transports of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_add_eq_add_two_mul_and_add_add_eq_add_add_of_isRigTransport_of_comp_act_eq_comp_of_germ 36 below · cited by 1 · depth 28
• Transporting a rigidification across an isogeny pair, dual side CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_comp_act_eq_comp_act_of_corr_of_isIsogenyPair 0 below · cited by 1 · depth 28
• Transporting a rigidification along an isogeny pair, dual form CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_comp_act_eq_comp_act_of_corr_of_isIsogenyPair_explicit 0 below · cited by 1 · depth 28
• Translating a rigidification, read on the dual isogenies CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_comp_act_eq_comp_of_isTranslateBy 0 below · cited by 1 · depth 28
• Correspondence packages between rigidifications pull back along base change CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_corr_of_isPullbackVia_of_isPullbackVia 16 below · cited by 1 · depth 28
• Rigidifications pull back along maps of coefficient algebras CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_of_isPullbackVia 23 below · cited by 28 · depth 28
• Corresponding rigidified isomorphisms descend to base change CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isoVia_corr_of_isPullbackVia 2 below · cited by 7 · depth 28
• Translation of rigidifications is stable under base change CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslateBy_of_isPullbackVia_of_isPullbackVia 16 below · cited by 1 · depth 28
• Translate relation between transported rigidifications along an isogeny CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslate_of_isRigTransport_mapPt_of_comp_act_eq_comp_act_of_germ_of_isUnit 9 below · cited by 1 · depth 28
• Transports of comparable rigidifications are translates CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslate_of_isRigTransport_of_comp_act_eq_comp_of_germ 7 below · cited by 1 · depth 28
• Integer multiplication commutes with the series representing φ' CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.nthSeries_comp_eq_comp_nthSeries_of_forall_nilEval 3 below · cited by 5 · depth 28
• Pairs over ℂ are isomorphic iff their lattice triples are homothetic CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_iff_exists_smul_latt_eq_and_smul_lattLev_eq_and_smul_lattK_eq_of_pointEquiv 4 below · cited by 1 · depth 28
• Rigidity of full level-m structures, m ≥ 3 CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.eq_refl_of_mapPt_eq_of_three_le 819 below · cited by 9 · depth 28
• Uniqueness and base change of extra-level representing schemes CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.existsUnique_hom_isPullback_of_represents_extraLevel 28 below · cited by 1 · depth 28
• Extra level-N structures represented by a finite étale scheme CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_finite_etale_represents_extraLevel_of_level 849 below · cited by 2 · depth 28
• Extra level structures descend along directed colimits of rings CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullbackVia_extraLevel_of_directed_colimit_of_isUnit 930 below · cited by 2 · depth 28
• Full-level fake elliptic curves extend over complete discrete valuation rings CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_algebraMap_of_isDiscreteValuationRing_of_finite_of_isAdicComplete 1,289 below · cited by 1 · depth 28
• First-order deformations of a fake elliptic curve form a line CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_fstHom_forall_exists_unique_smul_of_charP_of_isAlgClosed_of_three_le 1,143 below · cited by 1 · depth 28
• Square-zero lifting of fake elliptic curves with full level CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_of_sq_eq_bot_of_isArtinianRing 1,098 below · cited by 1 · depth 28
• Descent of triple isomorphisms along a directed colimit CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isoTVia_of_isoTVia_of_directed_colimit_of_isUnit 940 below · cited by 1 · depth 28
• Germ of [n] is n in End(X₀.F) CerednikDrinfeld.QM.FakeEllipticCurve.apply_nilEval_natCast_eq_mapPt_act_of_isFormalModuleVia 0 below · cited by 18 · depth 28
• L^{⊗ 4} is very ample with h⁰=64 CerednikDrinfeld.QM.FakeEllipticCurve.closedImmersionBySections_tensor_four_of_isCanonicalPol_of_isNoetherianRing 1,190 below · cited by 2 · depth 28
• Non-zero Λ-equivariant endomorphisms of fake elliptic curves are isogenies CerednikDrinfeld.QM.FakeEllipticCurve.endDegree_ne_zero_of_forall_act_comp_eq_of_ne_one 727 below · cited by 3 · depth 28
• Scalar automorphism ±[k] of a fake elliptic curve forces k=1 CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_isIso_of_forall_mapPt_eq_nsmulPt 723 below · cited by 1 · depth 28
• Frobenius is bijective on level points when ℓ ∤ N CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_factorsThrough_mapPt_relFrobenius_eq_of_not_dvd_of_isAlgClosed 2 below · cited by 1 · depth 28
• Chain decomposition of a CM endomorphism of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_chain_isLevelIsogeny_or_isAtkinLehnerQuotient_of_mapPt_mapPt_mul_zpow_eq_zpow 828 below · cited by 1 · depth 28
• Descent of the exhaustion clause to idempotent-free bases CerednikDrinfeld.QM.FakeEllipticCurve.exists_comp_act_comp_eq_of_isPullbackVia_of_forall_isIdempotentElem_of_closedImmersionBySections_of_isAlgClosed 452 below · cited by 1 · depth 28
• Level-preserving r-power self-isogenies are covered by the dictionary CerednikDrinfeld.QM.FakeEllipticCurve.exists_comp_act_eq_comp_act_of_isIsogenyPair_of_isPullback_prod_of_forall_exists_eq 0 below · cited by 1 · depth 28
• Common connected formal-coordinate cover for two fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_cover_connected_isFormalModuleVia_pair 51 below · cited by 7 · depth 28
• Formal mathcal O_D-modules over a connected basic-open cover CerednikDrinfeld.QM.FakeEllipticCurve.exists_cover_connected_isPullbackVia_isFormalModuleVia 47 below · cited by 14 · depth 28
• Existence of a dictionary family of level-preserving isogenies CerednikDrinfeld.QM.FakeEllipticCurve.exists_dictionary_family_of_isPullback_prod_of_forall_mem_awayUnits_iff 711 below · cited by 1 · depth 28
• Formal germ of the dual isogeny on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_end_forall_nilEval_eq_mapPt_and_mul_eq_of_isIsogenyPair 4 below · cited by 2 · depth 28
• Endomorphism killing q-torsion factors through [q] CerednikDrinfeld.QM.FakeEllipticCurve.exists_eq_act_comp_of_forall_nsmulPt_eq_one_imp_mapPt_eq_one 720 below · cited by 3 · depth 28
• Λ-linear endomorphisms of A× A come from R CerednikDrinfeld.QM.FakeEllipticCurve.exists_eq_act_of_mapPt_mul_of_isPullback_prod_of_forall_exists_eq 0 below · cited by 1 · depth 28
• Zariski gluing of extra level structures on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_of_openCover 2 below · cited by 2 · depth 28
• Admissible clopen subset of A[N] gives an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_of_opens_schemeKer 14 below · cited by 1 · depth 28
• Extra levels at ℓ and admissible sublattices of the period lattice CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_sublattice_lattLev_of_pointEquiv 762 below · cited by 1 · depth 28
• Level N reduced to level one for (t,n)-endomorphisms CerednikDrinfeld.QM.FakeEllipticCurve.exists_fin_forall_not_iso_not_exists_mapPt_mapPt_mul_zpow_eq_zpow_of_level_one 697 below · cited by 1 · depth 28
• Square-discriminant endomorphisms of fake elliptic curves are integers CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_mapPt_eq_zpow_of_forall_act_comp_eq_of_isSquare 734 below · cited by 1 · depth 28
• Labelling a conjugated quaternion action on a full level structure CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_mapPt_fullLevel_eq_pushPt_act_of_isOrder_of_conj_of_pow_modEq_one 0 below · cited by 1 · depth 28
• Generic-fibre squares for a fake elliptic curve over O∩ K' CerednikDrinfeld.QM.FakeEllipticCurve.exists_genericFibre_squares_of_isPullback_inf 21 below · cited by 1 · depth 28
• Dual germ, transports and kernel degree for an isogeny pair CerednikDrinfeld.QM.FakeEllipticCurve.exists_germ_dual_and_transport_and_hasKernelOfDegree_of_isIsogenyPair 55 below · cited by 1 · depth 28
• Dual germ rescaled by v⁻¹, transport, kernel degree CerednikDrinfeld.QM.FakeEllipticCurve.exists_germ_dual_rescaled_and_transport_and_hasKernelOfDegree_of_isIsogenyPair_of_isUnit 55 below · cited by 1 · depth 28
• Extending homomorphisms of fake elliptic curves over a DVR CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_extension_of_isPullback_of_isDiscreteValuationRing 41 below · cited by 1 · depth 28
• Descent of a homomorphism of fake elliptic curves to a finite extension CerednikDrinfeld.QM.FakeEllipticCurve.exists_intermediateField_forall_hom_of_isPullback_algebraMap 1 below · cited by 1 · depth 28
• Factoring a norm r^ē r endomorphism through the Atkin–Lehner quotient CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAtkinLehnerQuotientVia_isIsogenyPair_comp_eq_of_comp_eq_act_of_not_isIsogenyPair 811 below · cited by 1 · depth 28
• Canonical polarisation datum on a fake elliptic curve over an algebraically closed field CerednikDrinfeld.QM.FakeEllipticCurve.exists_isCanonicalPol_of_isAlgClosed_of_two_ne_zero 2,865 below · cited by 1 · depth 28
• Eichler order of level N inside the centralising quaternion algebra CerednikDrinfeld.QM.FakeEllipticCurve.exists_isEichlerOrder_forall_mem_awayUnits_iff_forall_exists_smul_mem_preservesLevel_and_exists_isMaximalOrder_inf_eq_of_isPullback_prod 79 below · cited by 1 · depth 28
• Global formal mathcal O_D-module over a local base CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalModuleOf_of_isLocalRing 47 below · cited by 8 · depth 28
• Base change of degree-d isogeny pairs of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isIsogenyPair_comp_eq_of_isPullbackVia_of_isPullbackVia 0 below · cited by 2 · depth 28
• Centraliser order acting on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isOrder_and_act_comp_eq_of_isPullback_prod_of_algHom_comm 0 below · cited by 1 · depth 28
• Transport of a product structure along an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_prod_and_act_eq_of_iso_of_isPullback_prod 0 below · cited by 1 · depth 28
• Relevelling a fake elliptic curve over an algebraically closed field CerednikDrinfeld.QM.FakeEllipticCurve.exists_iso_of_isIndefiniteRamifiedExactlyAt_of_linearMap_matrix_zmod_of_natCast_ne_zero 733 below · cited by 1 · depth 28
• Endomorphism-ring export of the quaternionic formal-module dictionary CerednikDrinfeld.QM.FakeEllipticCurve.exists_le_isOrder_forall_exists_pow_smul_mem_and_act_and_forall_exists_generalLinearGroup_and_exists_isMaximalOrder_inf_eq_of_isOrder_act_of_conj_of_injective 16 below · cited by 1 · depth 28
• Frobenius parity of the leg of Xi over connected bases CerednikDrinfeld.QM.FakeEllipticCurve.exists_le_one_rigidifiedToG_leg_eq_frobTwist_neg_of_forall_isIdempotentElem 52 below · cited by 9 · depth 28
• Level-N data and extra levels at N correspond CerednikDrinfeld.QM.FakeEllipticCurve.exists_levelOne_extraLevel_and_exists_of_extraLevel 0 below · cited by 3 · depth 28
• Enlarging the base field in a fake elliptic curve square CerednikDrinfeld.QM.FakeEllipticCurve.exists_levelRaise_squares_of_isPullback_inf 21 below · cited by 1 · depth 28
• Translating an endomorphism: ψ=φ-[k] and its quadratic relation CerednikDrinfeld.QM.FakeEllipticCurve.exists_mapPt_mapPt_mul_zpow_eq_zpow_sub_of_mapPt_mapPt_mul_zpow_eq_zpow 0 below · cited by 1 · depth 28
• Only r-power reduced norms act by r-power isogenies CerednikDrinfeld.QM.FakeEllipticCurve.exists_nrd_mul_pow_eq_of_isIsogenyPair_pow_of_endomorphismDictionary 8 below · cited by 2 · depth 28
• Gluing Xi from local transports: existence, leg, naturality CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedToG_natural_isRigTransport_of_cover 887 below · cited by 1 · depth 28
• Period lattices of Atkin–Lehner quotients at level N CerednikDrinfeld.QM.FakeEllipticCurve.exists_smul_latt_lattLev_atkinLehnerQuotient_of_pointEquiv 15 below · cited by 1 · depth 28
• Geometric fibres of an ℓ-torsion subgroup scheme are (ℤ/ℓ)² CerednikDrinfeld.QM.FakeEllipticCurve.exists_zmod_prod_equiv_factorsThrough_of_isPullback_valuationSubring 10 below · cited by 1 · depth 28
• Level-one structure meets exactly the unit section CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_iff_eq_one_of_level_one 0 below · cited by 3 · depth 28
• Faithfulness of the centraliser order acting through E CerednikDrinfeld.QM.FakeEllipticCurve.forall_act_comp_eq_imp_eq_of_isPullback_prod_of_injective 0 below · cited by 1 · depth 28
• Formal mathcal O_D-module of a fake elliptic curve has height four CerednikDrinfeld.QM.FakeEllipticCurve.hasHeight_four_of_isFormalModuleVia_of_one_mem 778 below · cited by 10 · depth 28
• Kernel of Verschiebung is finite of rank ℓ² CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_and_finrank_kernel_verschiebung_eq 713 below · cited by 2 · depth 28
• Transport of formal module coordinates along an isogeny of unit degree CerednikDrinfeld.QM.FakeEllipticCurve.isFormalModuleVia_mapPt_of_isIsogenyPair_of_isUnit 2 below · cited by 1 · depth 28
• Discriminant of a Λ-equivariant endomorphism is square or negative CerednikDrinfeld.QM.FakeEllipticCurve.isSquare_or_sq_lt_four_mul_of_forall_act_comp_eq 730 below · cited by 1 · depth 28
• Extensions over a valuation ring are automatically homomorphisms CerednikDrinfeld.QM.FakeEllipticCurve.mapPt_mul_and_act_comp_of_comp_eq_of_isPullback_valuationSubring 4 below · cited by 1 · depth 28
• Multiplication by ℓ coequalises the kernel pair of Frobenius CerednikDrinfeld.QM.FakeEllipticCurve.nsmulPt_eq_of_mapPt_relFrobenius_eq 0 below · cited by 1 · depth 28
• Coarse moduli of (N,ℓ)-pairs gives level Nℓ CerednikDrinfeld.QM.IsCoarseModuliT.exists_isCoarseModuli_mul_of_coprime_of_isUnit 30 below · cited by 1 · depth 28
• Atkin–Lehner operators on the Čerednik–Drinfel'd uniformisation CerednikDrinfeld.QM.IsFineModuli.cerednikDrinfeld_fineFamily_atkinLehner_of_rigidifiedToG_of_isNoetherianRing_heightNormalised_oneLegC5 891 below · cited by 1 · depth 28
• Orbit relation spreads from a field point to a localisation CerednikDrinfeld.QM.IsFineModuli.exists_apply_ne_zero_forall_isPullback_of_cerednikDrinfeld_uniformization_fine_eq 13 below · cited by 1 · depth 28
• Čerednik–Drinfeld uniformising family on the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_fineFamily_of_rigidifiedToG_of_isNoetherianRing_heightNormalised_eq_oneLegC5_h2 3,611 below · cited by 1 · depth 28
• Čerednik–Drinfeld uniformisation of the away-from-r̄ r Hecke tower CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_towerFamily_liftT_of_rigidifiedToG_of_isNoetherianRing_heightNormalised_eq_oneLegC5 1,194 below · cited by 1 · depth 28
• Non-emptiness: some τ is a fake elliptic period CerednikDrinfeld.QM.IsFineModuli.exists_exists_smul_latt_eq_qmPeriodLattice_of_analytic 4,731 below · cited by 1 · depth 28
• Gluing a finite étale level-N cover of a fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_isFineModuliT_finite_etale_of_forall_local 34 below · cited by 1 · depth 28
• Local period chart near a point of the fine moduli curve CerednikDrinfeld.QM.IsFineModuli.exists_isOpen_injOn_periodChart_of_analytic_of_isEichlerOrder 166 below · cited by 4 · depth 28
• Field-valued fibres of the fine Čerednik–Drinfeld uniformisation CerednikDrinfeld.QM.IsFineModuli.exists_isPullback_field_of_cerednikDrinfeld_uniformization_fine_eq 1 below · cited by 1 · depth 28
• Flatness over ℤ_q of the fine moduli scheme at level (1;m) CerednikDrinfeld.QM.IsFineModuli.flat_padicInt_one_of_isUnit 1,608 below · cited by 1 · depth 28
• Local algebraic charts for the quaternionic period map CerednikDrinfeld.QM.IsFineModuli.forall_exists_algebraicChart_of_periodMap_of_analytic 4,811 below · cited by 1 · depth 28
• Local algebraic families of fake elliptic curves with extra level ℓ CerednikDrinfeld.QM.IsFineModuli.forall_exists_algebraicFamily_withExtraLevel_isPullback_smul_latt_eq_of_analytic 5,331 below · cited by 1 · depth 28
• Properness makes the algebraic period locus closed CerednikDrinfeld.QM.IsFineModuli.isClosed_setOf_exists_smul_latt_eq_qmPeriodLattice_of_isProper_of_analytic 268 below · cited by 1 · depth 28
• Finiteness and surjectivity of the fine-to-coarse moduli map CerednikDrinfeld.QM.IsFineModuli.isFinite_and_surjective_of_isCoarseModuli_of_isLevelTwistAction_of_isUnit_two_of_isUnit_three 3,817 below · cited by 1 · depth 28
• Openness of the uniformised locus in H CerednikDrinfeld.QM.IsFineModuli.isOpen_setOf_exists_smul_latt_eq_qmPeriodLattice_of_analytic 172 below · cited by 1 · depth 28
• Fine moduli: base change iff composition of moduli points CerednikDrinfeld.QM.IsFineModuli.isPullback_iff_ptF_eq_specMap_comp_ptF 22 below · cited by 4 · depth 28
• Fine moduli of fake elliptic curves is locally of finite presentation CerednikDrinfeld.QM.IsFineModuli.locallyOfFinitePresentation 841 below · cited by 6 · depth 28
• Invariance at raised level of a twisted uniformising family CerednikDrinfeld.QM.IsFineModuliT.apply_eq_apply_of_isPullback_of_frobTwist_eq_of_invariant 8 below · cited by 1 · depth 28
• Morphisms out of the fine moduli scheme of triples CerednikDrinfeld.QM.IsFineModuliT.existsUnique_hom_ptFT_comp_eq 23 below · cited by 3 · depth 28
• Geometric fibres of the full-level forgetful map are G-orbits CerednikDrinfeld.QM.IsFineModuliT.exists_eq_comp_hom_of_comp_eq_of_isAlgClosed_of_isOrder 0 below · cited by 1 · depth 28
• Flat-local description of fibres of the level-ℓ fine uniformisation CerednikDrinfeld.QM.IsFineModuliT.exists_flat_family_isPullback_of_cerednikDrinfeld_uniformization_fine_eq 8 below · cited by 1 · depth 28
• Quotient of the level-ℓ fine scheme is coarse moduli CerednikDrinfeld.QM.IsFineModuliT.exists_isCoarseModuliT_of_quotient 784 below · cited by 2 · depth 28
• Finiteness and surjectivity of the level-forgetting map p_ℓ CerednikDrinfeld.QM.IsFineModuliT.isFinite_and_surjective_of_isCoarseModuliT_of_isUnit_two_of_isUnit_three 3,830 below · cited by 1 · depth 28
• Geometric fibres of the fine-to-coarse map are G-orbits CerednikDrinfeld.QM.IsLevelTwistAction.exists_eq_comp_hom_of_comp_eq_of_isAlgClosed_of_isOrder 0 below · cited by 1 · depth 28
• Galois frame and stabiliser balance for the level-ℓ leg CerednikDrinfeld.QM.ModuliTowerWitnessD.exists_galoisFrame_natCard_stabilizer_mul_eq_of_isFineModuli_of_quotient_of_two_mul_dvd_of_squarefree 5,698 below · cited by 1 · depth 28
• Analyticity of place evaluation along the period map CerednikDrinfeld.QM.analyticAt_evalAt_of_periodMap_of_algebraicChart_of_two_mul_dvd 23 below · cited by 1 · depth 28
• Edge-chart morphisms of a formally étale uniformisation are étale CerednikDrinfeld.QM.etale_edgeChartMorphism_of_cerednikDrinfeld_uniformization_fine 7 below · cited by 3 · depth 28
• N-torsion as a module over a maximal quaternion order CerednikDrinfeld.QM.exists_equiv_torsion_and_forall_pushPt_eq_mulVec_of_forall_exists_eq_of_isMaximalOrder 717 below · cited by 1 · depth 28
• The square A×_k A as a fake elliptic curve CerednikDrinfeld.QM.exists_fakeEllipticCurve_one_isPullback_and_act_eq_of_act_of_algHom_matrix_of_trace 19 below · cited by 1 · depth 28
• fpqc-local lifting for a formally étale uniformisation CerednikDrinfeld.QM.exists_flat_family_lift_of_formallyEtale_of_locallyOfFiniteType 18 below · cited by 1 · depth 28
• Fine moduli for full-level fake elliptic curves from QM pairs CerednikDrinfeld.QM.exists_isFineModuli_one_of_represents_qmStructure_pairs_of_isUnit_two 2,936 below · cited by 1 · depth 28
• Openness of the uniformised locus after nilpotent base change CerednikDrinfeld.QM.exists_isOpen_forall_mem_iff_exists_uniformization_of_isPullback 15 below · cited by 2 · depth 28
• Meromorphic realisation of the function field along a period map CerednikDrinfeld.QM.exists_meromorphicRealization_of_periodMap_of_analyticAt_evalAt_of_two_mul_dvd 1 below · cited by 1 · depth 28
• Deuring: supersingular curve whose endomorphisms are a maximal order CerednikDrinfeld.QM.exists_relativeGroupLaw_isMaximalOrder_act_injective_and_forall_exists_eq_of_charP 962 below · cited by 1 · depth 28
• Fine moduli of quaternionic-multiplication pairs over a Q-submoduli problem CerednikDrinfeld.QM.exists_represents_qmStructure_pairs_of_satisfying_isFineModuli_of_qmStructure_of_isUnit_two_of_finiteType 1,562 below · cited by 1 · depth 28
• Complex uniformisation of the level-one fake elliptic moduli curve CerednikDrinfeld.QM.exists_uniformizedHeckeCurve_place_corr_single_eq_sum_levelOne_of_two_mul_dvd_neZero 1,027 below · cited by 1 · depth 28
• Tangent space at the origin has dimension n CerednikDrinfeld.QM.finrank_eq_of_range_iff_isTangentVector_of_smoothOfRelativeDimension 2 below · cited by 10 · depth 28
• Subschemes killed by an isogeny with n invertible are reduced CerednikDrinfeld.QM.isReduced_of_mapPt_mapPt_eq_nsmulPt_of_natCast_ne_zero 6 below · cited by 1 · depth 28
• QM structures force local theta type (6,6) CerednikDrinfeld.QM.thetaTypeLocally_six_six_of_qmStructure_of_isUnit_two_three 1,375 below · cited by 1 · depth 28
• Extra level at ℓ as the period lattice of Λ t CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.factorsThrough_levK_symm_iff_exists_qmPeriodMap_mul_eq_of_forall_geomPoint 15 below · cited by 1 · depth 29
• Unique extension of full level-m structures over a DVR CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.existsUnique_comp_eq_of_isPullback_of_isUnit 700 below · cited by 1 · depth 29
• Unique lifting of full level-m structures along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.existsUnique_comp_eq_specMap_comp_of_isNilpotent_ker 17 below · cited by 7 · depth 29
• Transversality of L₀ (m/ℓ)P against the level structure CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.forall_factorsThrough_lev_imp_eq_one_of_smul_eq_qmPeriodMap_of_nrd_eq 64 below · cited by 1 · depth 29
• Base change of the formal 𝒪_D-module of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.IsFormalModuleOf.map_of_isPullback 0 below · cited by 3 · depth 29
• Composition of base-change comparisons for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.IsPullbackVia.comp 0 below · cited by 24 · depth 29
• Base change of formal mathcal O_D-module coordinates along a pull-back CerednikDrinfeld.QM.FakeEllipticCurve.IsPullbackVia.exists_isFormalModuleVia_map_and_comp_eq 0 below · cited by 18 · depth 29
• Composition of pull-backs of rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.IsPullbackVia.comp 1 below · cited by 4 · depth 29
• Transport of a rigidification to an admissible rigidified module CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isRigTransport_isAdmissible_of_isFormalModuleVia 845 below · cited by 1 · depth 29
• A common bridge for two transports along q CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_rho_eq_comp_and_nthSeries_comp_eq_of_isRigTransport_of_isRigTransport_mapPt 5 below · cited by 3 · depth 29
• Two transports on pull-backs give the same G-point CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.gPoint_eq_gPoint_of_isPullbackVia_of_isPullbackVia_of_isRigTransport 40 below · cited by 1 · depth 29
• Base change of the G-point of an admissible transport CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.gPoint_eq_map_gPoint_of_isPullbackVia_of_isRigTransport 750 below · cited by 1 · depth 29
• Base change of a Drinfeld transport of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isRigTransport_map_of_isPullbackVia 717 below · cited by 3 · depth 29
• Merging level N and extra level ℓ structures CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_fakeEllipticCurve_mul_forall_factorsThrough_iff_of_isUnit 4 below · cited by 1 · depth 29
• Flat-local existence of full level structures, with extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_flat_surjective_withFullLevel_isPullback 766 below · cited by 1 · depth 29
• Level splitting at (N,ℓ) commutes with base change CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.isPullback_of_isPullback_of_forall_factorsThrough_iff_mul 0 below · cited by 1 · depth 29
• Isomorphisms transport splittings of level Nℓ structures CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.iso_of_iso_of_forall_factorsThrough_iff_mul 0 below · cited by 1 · depth 29
• Isomorphisms over k[ε] are compatible with the comparison maps CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.comp_hom_eq_of_isPullbackVia_fstHom_of_three_le 820 below · cited by 1 · depth 29
• Base change of a fake elliptic curve with full and extra level CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_extraLevel_isPullbackVia_forall_factorsThrough_iff 24 below · cited by 3 · depth 29
• Spreading out an isomorphism of levelled fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_forall_isoTVia_of_isoTVia_of_isPullbackVia 40 below · cited by 1 · depth 29
• Factoring a base change through a pullback of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullbackVia_comp_eq_of_isPullbackVia_comp 0 below · cited by 1 · depth 29
• Scaling ε↦ cε preserves reduction, compatibly with comparison maps CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullbackVia_fstHom_comp_eq_of_isPullbackVia_map_smul_of_levelIff 0 below · cited by 1 · depth 29
• Full level structure on a Λ-equivariant bare deformation CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullbackVia_fstHom_iso_of_bareDeformation_of_act 64 below · cited by 1 · depth 29
• Base change of fake elliptic curves with full level-m structure CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_levelIff 21 below · cited by 20 · depth 29
• Fake elliptic curves with full level descend along directed colimits CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_of_directed_colimit_of_isUnit 835 below · cited by 2 · depth 29
• Twisting a full level structure by a unit modulo m CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isTwistVia_refl 1 below · cited by 7 · depth 29
• Inertia fixes ℓ-power torsion of full-level fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_prime_dvd_forall_inertiaSubgroupIn_comp_eq_of_isTorsionPoint 1,256 below · cited by 1 · depth 29
• Transversality clause descends, and ascends when N is invertible CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.forall_factorsThrough_imp_eq_one_imp_and_imp_of_isPullback 1 below · cited by 1 · depth 29
• Base-change calculus for fake elliptic curves with full level CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.isPullback_refl_comp_cancel_iso_unique_nsmulPt 2 below · cited by 5 · depth 29
• Uniqueness, composition, transport and transitivity for base-changed triples CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.isoTVia_unique_comp_transport_trans 3 below · cited by 2 · depth 29
• Rigidity for first-order deformations with full level structure CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.iso_of_comp_hom_eq_of_isPullbackVia_fstHom 48 below · cited by 1 · depth 29
• Base change of the L₀-generated extra level at ℓ CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.withExtraLevel_isPullback_of_isPullback_of_forall_factorsThrough_iff 17 below · cited by 1 · depth 29
• Drinfeld trace condition for the formal module of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.apply_trace_linearPart_addVia_eq_of_isFormalModuleVia 1 below · cited by 1 · depth 29
• Integer multiplication on a fake elliptic curve is epi CerednikDrinfeld.QM.FakeEllipticCurve.epi_act_of_ne_zero 719 below · cited by 4 · depth 29
• Rigidity: automorphisms fixing the n-torsion are trivial CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_forall_nsmulPt_eq_one_imp_mapPt_eq_of_three_le 772 below · cited by 1 · depth 29
• Even rigidified pairs: existence, uniqueness, base change, lifting CerednikDrinfeld.QM.FakeEllipticCurve.evenRigidifiedPair_exists_unique_pullback_lift_of_rigidifiedToG_conn_h2 3,498 below · cited by 1 · depth 29
• Unique formal completion of a Λ-linear morphism of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_hom_isFormalCompletionAlong_of_isFormalModuleVia 4 below · cited by 6 · depth 29
• Λ-equivariant lifting of fake elliptic curves along square-zero thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_bareDeformation_act_of_ker_mul_ker_eq_bot_of_isArtinianRing 1,056 below · cited by 2 · depth 29
• Chain of fake elliptic curves along a filtration of kerφ CerednikDrinfeld.QM.FakeEllipticCurve.exists_chain_of_filtration_mapPt_eq_one 806 below · cited by 1 · depth 29
• First-order deformations of a fake elliptic curve over k[ε] CerednikDrinfeld.QM.FakeEllipticCurve.exists_class_bareDeformation_dualNumber_forall_isIso_iff_of_isAlgClosed_of_charP 986 below · cited by 2 · depth 29
• Unique descent of homomorphic endomorphisms along a connected base change CerednikDrinfeld.QM.FakeEllipticCurve.exists_comp_eq_comp_unique_of_isPullbackVia_of_forall_isIdempotentElem_of_isAlgClosed 437 below · cited by 1 · depth 29
• Zariski-local existence of the formal 𝒪_D-module CerednikDrinfeld.QM.FakeEllipticCurve.exists_cover_isFormalModuleOf 45 below · cited by 4 · depth 29
• Formal germ of a dual isogeny, arbitrary degree CerednikDrinfeld.QM.FakeEllipticCurve.exists_end_forall_nilEval_eq_mapPt_and_mul_eq_natCast_of_isIsogenyPair 4 below · cited by 1 · depth 29
• Factoring a homomorphism killing q-torsion through [q] CerednikDrinfeld.QM.FakeEllipticCurve.exists_eq_act_comp_of_forall_nsmulPt_eq_one_imp_mapPt_eq_one_of_hom 720 below · cited by 2 · depth 29
• Points of a fake elliptic curve transport along a cartesian base change CerednikDrinfeld.QM.FakeEllipticCurve.exists_equiv_schemeHomOver_of_isPullback 0 below · cited by 5 · depth 29
• Twisted and Pi-translates carry even rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.exists_even_rigidification_of_isActBy_of_isPiTranslate 154 below · cited by 1 · depth 29
• Constant extra level from a homomorphism (ℤ/ℓ)² → A(S) CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_exists_sectionAt_eq 0 below · cited by 4 · depth 29
• Closed-open subgroup of N-torsion is an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_of_opens_schemeKer_of_level 14 below · cited by 1 · depth 29
• Extra level-ℓ structure from a transversal sublattice CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_smul_mem_of_transversal_of_pointEquiv 1 below · cited by 2 · depth 29
• Transport of extra level structures along a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_transport_and_iff_of_rigidification_normLevelTransport_oneLegC5 24 below · cited by 1 · depth 29
• Finitely many level-N structures on a fixed fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_fin_forall_iso_of_iso_hom_act 695 below · cited by 1 · depth 29
• Existence of a rigid fake elliptic curve over ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_hom_eq_id_or_mapPt_eq_inv_of_not_dvd_of_squarefree 5,686 below · cited by 1 · depth 29
• Norm level transport: existence and uniqueness of Pₙ CerednikDrinfeld.QM.FakeEllipticCurve.exists_fullLevel_transport_and_eq_of_rigidification_normLevelTransport 81 below · cited by 6 · depth 29
• Formal completions of q-power quasi-endomorphisms in matrix coordinates CerednikDrinfeld.QM.FakeEllipticCurve.exists_generalLinearGroup_forall_exists_centralizer_isFormalCompletionAlong_and_apply_eq_zpow_smul_conj 13 below · cited by 1 · depth 29
• Endomorphism dictionary extended to the Hecke element s CerednikDrinfeld.QM.FakeEllipticCurve.exists_heckeDictionary_star_and_comp_eq_of_endomorphismDictionary_endIsoFull 871 below · cited by 1 · depth 29
• Existence of the Atkin–Lehner quotient at ̄ r with universal property CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAtkinLehnerQuotientVia_epi_and_forall_existsUnique_comp_eq 770 below · cited by 1 · depth 29
• Existence and local uniqueness of the canonical polarisation datum CerednikDrinfeld.QM.FakeEllipticCurve.exists_isCanonicalPol_of_isUnit_two 2,864 below · cited by 4 · depth 29
• q-power torsion points of fake elliptic curves are infinitesimal CerednikDrinfeld.QM.FakeEllipticCurve.exists_isNilpotent_isInfinitesimal_of_nsmulPt_pow_eq_one_of_one_mem 765 below · cited by 8 · depth 29
• Comparison of two pull-backs of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_id_comp_eq_of_isPullbackVia_of_isPullbackVia_of_isUnit 15 below · cited by 13 · depth 29
• Fake elliptic curve structure on a Λ-linear bare deformation CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_of_bareDeformation_of_act_of_isArtinianRing 60 below · cited by 2 · depth 29
• Extending a fake elliptic curve from K to a DVR R CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_algebraMap_of_abelianSchemePropertyBundle_of_isPullback 99 below · cited by 1 · depth 29
• Forgetting the level structure of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_level_one_iso_hom_act 0 below · cited by 1 · depth 29
• Full level-m structures give units of Λ/mΛ CerednikDrinfeld.QM.FakeEllipticCurve.exists_mul_sub_one_eq_smul_of_fullLevel_of_smul_eq_qmPeriodMap 0 below · cited by 1 · depth 29
• Atkin–Lehner quotient at r as a Pi-translate, up to the centre CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_frobTwist_isPiTranslate_of_isAtkinLehnerQuotient 151 below · cited by 1 · depth 29
• Transporting rigidifications along the Atkin–Lehner quotient at ̄ r CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_of_isAtkinLehnerQuotient 717 below · cited by 1 · depth 29
• Lattice of a level-ℓ isogeny quotient, with level lattice CerednikDrinfeld.QM.FakeEllipticCurve.exists_smul_latt_lattLev_eq_of_isLevelIsogeny_of_pointEquiv 0 below · cited by 1 · depth 29
• Level preservation detected on a coordinate submodule W CerednikDrinfeld.QM.FakeEllipticCurve.exists_submodule_forall_preservesLevel_iff_forall_mem_of_isPullback_prod 9 below · cited by 1 · depth 29
• Splitting a level-Nℓ fake elliptic curve into level N and extra level ℓ CerednikDrinfeld.QM.FakeEllipticCurve.exists_withExtraLevel_forall_factorsThrough_iff_of_mul_of_isUnit 20 below · cited by 1 · depth 29
• Level subscheme of a pullback is the full pullback CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_lev_of_exists_comp_eq_comp_of_isPullbackVia 23 below · cited by 11 · depth 29
• Kernel of a quadratic endomorphism has n² points CerednikDrinfeld.QM.FakeEllipticCurve.finite_and_natCard_mapPt_eq_one_of_mapPt_mapPt_mul_zpow_eq_zpow 745 below · cited by 1 · depth 29
• Finiteness and base change for L^{⊗ 4} on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.finite_projective_sections_and_exists_linearEquiv_tensorProduct_pullback_tensor_four_of_isCanonicalPol_of_isNoetherianRing 1,074 below · cited by 1 · depth 29
• Two-dimensionality of the tangent space of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.finrank_eq_two_of_range_iff_isTangentVector 28 below · cited by 3 · depth 29
• Transversality spreads from one geometric point over a domain CerednikDrinfeld.QM.FakeEllipticCurve.forall_factorsThrough_lev_imp_eq_one_smul_fullLevel_of_isDomain 28 below · cited by 1 · depth 29
• The ̄ r-part of ker f is exactly A₀[mathfrak P_{̄ r}] CerednikDrinfeld.QM.FakeEllipticCurve.forall_mapPt_eq_one_iff_torsionPrime_of_comp_eq_act_of_not_isIsogenyPair 805 below · cited by 1 · depth 29
• Extra level at ℓ is preserved exactly on Γ∩ sΓ s⁻¹ CerednikDrinfeld.QM.FakeEllipticCurve.forall_preservesExtraLevel_iff_mem_inf_map_conj_of_heckeDictionary_star_of_comp_eq 45 below · cited by 1 · depth 29
• n-torsion of a fake elliptic curve is finite flat of rank n⁴ CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_flat_schemeKerStr_and_finrank_eq_pow_four 730 below · cited by 2 · depth 29
• Descent of an isogeny pair along a pullback CerednikDrinfeld.QM.FakeEllipticCurve.isIsogenyPair_of_isIsogenyPair_of_comp_eq_comp_of_isPullbackVia 0 below · cited by 1 · depth 29
• Reflexivity: a fake elliptic curve is its own base change along id_S CerednikDrinfeld.QM.FakeEllipticCurve.isPullbackVia_id 0 below · cited by 19 · depth 29
• Transport of pullbacks from (N,ℓ)-pairs to level Nℓ CerednikDrinfeld.QM.FakeEllipticCurve.isPullback_of_withExtraLevel_isPullback_of_forall_factorsThrough_iff_mul 0 below · cited by 1 · depth 29
• Isomorphic (N,ℓ)-pairs yield isomorphic level-Nℓ curves CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_withExtraLevel_iso_of_forall_factorsThrough_iff_mul 0 below · cited by 1 · depth 29
• Base-changed Atkin–Lehner map on lifted K-points of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.lift_pt_comp_atkinLehnerBaseChange_eq_of_isAtkinLehnerQuotient 24 below · cited by 2 · depth 29
• Existence of a fake elliptic curve over ℂ CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_complex 4,709 below · cited by 1 · depth 29
• Second factor through an Atkin–Lehner quotient preserves level CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_of_isAtkinLehnerQuotientVia_comp_eq_of_preservesLevel 2 below · cited by 1 · depth 29
• The quasi-inverse of a level-preserving isogeny preserves the level CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_of_isIsogenyPair_of_preservesLevel_of_coprime 710 below · cited by 1 · depth 29
• Level preservation descends along pull-backs of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_of_preservesLevel_of_comp_eq_comp_of_isPullbackVia 26 below · cited by 1 · depth 29
• Canonical polarisation datum is a symmetric square on geometric fibres CerednikDrinfeld.QM.IsCanonicalPolData.exists_relativeGroupLaw_geomFibre_exists_kernelTrivial_isSymmetric_iso_tensor_self_finrank_pos 1,061 below · cited by 5 · depth 29
• Geometric fibres of a canonical polarisation datum: h⁰ = 4 CerednikDrinfeld.QM.IsCanonicalPolData.exists_relativeGroupLaw_geomFibre_finite_kernelPts_and_geomFibreH0Finrank_eq_four_u0 1,125 below · cited by 3 · depth 29
• Fine Čerednik–Drinfeld family depends on ψ only through Frobenius invariants CerednikDrinfeld.QM.IsFineModuli.cerednikDrinfeld_uniformization_fine_eq_of_forall_frobFixed_eq 9 below · cited by 1 · depth 29
• Lifting the uniformisation map to the level-ℓ tower CerednikDrinfeld.QM.IsFineModuli.exists_fineFamilyT_lift_of_towerFamily_of_isNoetherianRing_heightNormalised_eq_oneLegC5 1,084 below · cited by 1 · depth 29
• Formally étale Ω̂× G-family of fine moduli points CerednikDrinfeld.QM.IsFineModuli.exists_fineFamily_of_evenRigidifiedPair_of_isNoetherianRing_heightNormalised_conn 989 below · cited by 1 · depth 29
• Holomorphic lattice frame over an analytic chart of fine moduli CerednikDrinfeld.QM.IsFineModuli.exists_holomorphic_latticeFrame_of_analytic_of_smooth_algebraicChart 68 below · cited by 1 · depth 29
• Local period chart on the fine moduli curve with full level CerednikDrinfeld.QM.IsFineModuli.exists_isOpen_injOn_periodChart_fullLevel_of_analytic_of_isEichlerOrder 165 below · cited by 1 · depth 29
• Regular prorepresentation of the stalk at a closed special point CerednikDrinfeld.QM.IsFineModuli.exists_isRegularLocalRing_prorepresents_stalk_of_isClosed 1,525 below · cited by 2 · depth 29
• A level homomorphism describing the Čerednik–Drinfeld fibres CerednikDrinfeld.QM.IsFineModuli.exists_levelHom_translate_fibre_of_fineFamily_of_isNoetherianRing_heightNormalised_conn_eq_oneLegC5 1,022 below · cited by 1 · depth 29
• Čerednik–Drinfeld uniformisation family on the Hecke tower CerednikDrinfeld.QM.IsFineModuli.exists_towerFamily_of_evenRigidifiedPair_of_heckeDictionary_of_isNoetherianRing_heightNormalised_oneLegC5 59 below · cited by 1 · depth 29
• Injectivity of τ on an analytic lattice-frame chart CerednikDrinfeld.QM.IsFineModuli.injOn_periodFunction_of_latticeFrame_of_analytic 5 below · cited by 1 · depth 29
• Stabiliser of a rigid fine moduli point lies in H CerednikDrinfeld.QM.IsFineModuli.mem_of_ptF_comp_eq_of_rigid 2 below · cited by 1 · depth 29
• Hecke translate by s_ℓ matches the d₁ degeneracy leg CerednikDrinfeld.QM.IsFineModuli.towerFamily_heckeTranslate_of_evenRigidifiedPair_of_isNoetherianRing_heightNormalised_oneLegC5 759 below · cited by 1 · depth 29
• Spreading of the Γ̃-orbit relation at level ℓ CerednikDrinfeld.QM.IsFineModuliT.exists_apply_ne_zero_forall_isPullback_of_cerednikDrinfeld_uniformization_fine_eq 6 below · cited by 1 · depth 29
• Uniqueness of the fine moduli scheme of triples CerednikDrinfeld.QM.IsFineModuliT.exists_iso_of_isFineModuliT 23 below · cited by 1 · depth 29
• Invariance of πcircpt_{F,ℓ} under change of full level CerednikDrinfeld.QM.IsFineModuliT.ptFT_comp_eq_ptFT_comp_of_fullLevel 38 below · cited by 1 · depth 29
• Functoriality of formal completion along the unit section CerednikDrinfeld.QM.IsFormalCompletionAlong.id_and_comp 1 below · cited by 7 · depth 29
• Pullback along a σ-twist acts as gal σ CerednikDrinfeld.QM.ModuliTowerWitnessD.germ_app_eq_ffEquiv_gal_smul_of_comp_fst_eq_of_comp_toBase_eq 15 below · cited by 2 · depth 29
• A linear presentation of tangent vectors at the origin CerednikDrinfeld.QM.exists_injective_range_iff_isTangentVector 3 below · cited by 7 · depth 29
• Lattice action on the tangent space at the origin CerednikDrinfeld.QM.exists_moduleEnd_apply_eq_pushPt_of_isTangentVector 0 below · cited by 5 · depth 29
• Level-one period map for fake elliptic curves over ℂ CerednikDrinfeld.QM.exists_periodMap_fakeEllipticCurve_complex_iso_iff_hecke_atkinLehner_levelOne 940 below · cited by 1 · depth 29
• Descent of the moduli rule from triples to pairs along π CerednikDrinfeld.QM.exists_ptT_eq_ptFT_comp_of_isFineModuliT_of_forall_ptFT_comp_eq 26 below · cited by 1 · depth 29
• Representability of quaternionic order actions, with degree strata CerednikDrinfeld.QM.exists_representsLatticeActions_of_closedImmersionBySections_of_topologicalKrullDim 1,279 below · cited by 1 · depth 29
• Gluing chart-wise QM point maps over a Q-fine moduli scheme CerednikDrinfeld.QM.exists_schemeHomOver_forall_comp_eq_ptZ_comp_openImmersion_of_affineCharts_satisfying 45 below · cited by 1 · depth 29
• Cartesian transition maps for QM-structure schemes over affine charts CerednikDrinfeld.QM.exists_transition_isPullback_of_represents_qmStructure_affineOpens_satisfying 868 below · cited by 1 · depth 29
• Mod N commutant of j(Λ) is blk(τ R) CerednikDrinfeld.QM.forall_commute_and_forall_exists_eq_blk_and_blk_eq_zero_iff_of_centraliser_of_isMaximalOrder 19 below · cited by 1 · depth 29
• Functoriality of the glued point map for QM pairs CerednikDrinfeld.QM.ptQ_eq_of_iso_and_ptQ_eq_comp_of_isPullback_of_affineCharts_satisfying 45 below · cited by 1 · depth 29
• Representability of QM structures from the affine charts CerednikDrinfeld.QM.ptQ_surjective_and_iso_of_ptQ_eq_of_affineCharts_of_isMaximalOrder_of_isUnit_two_satisfying 1,545 below · cited by 1 · depth 29
• Trace of a quaternionic matrix endomorphism on tangent vectors CerednikDrinfeld.QM.trace_eq_intCast_of_isTangentVector_prod_of_smoothOfRelativeDimension_one 3 below · cited by 2 · depth 29
• Étale extra level at invertible ℓ and its geometric points CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.etale_and_forall_factorsThrough_iff_of_isUnit 14 below · cited by 10 · depth 30
• Extra levels lift along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_forall_exists_comp_levK_eq_comp_of_isNilpotent_ker 19 below · cited by 3 · depth 30
• Extra level at ℓ transports uniquely along an isogeny CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_forall_factorsThrough_imp_of_isIsogenyPair_pow_of_ne_of_intCast_mem 1 below · cited by 6 · depth 30
• Extra level at invertible ℓ propagates along a base-change square CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.forall_factorsThrough_imp_exists_comp_eq_of_forall_geomPoint 16 below · cited by 2 · depth 30
• Uniqueness of the normalised level transport CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.eq_of_isNormLevelTransport_of_isNormLevelTransport 75 below · cited by 4 · depth 30
• Unique lifting of full level-m structures along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.existsUnique_lift_of_isNilpotent_ker 16 below · cited by 2 · depth 30
• Transport of full level-m structures along a degree-rᵈ isogeny CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_P_eq_mapPt_of_isIsogenyPair_pow_of_coprime_of_intCast_mem 0 below · cited by 2 · depth 30
• Lifting full level-m structures along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_comp_eq_specMap_comp_of_isNilpotent_ker 15 below · cited by 3 · depth 30
• Rigidifying an Atkin–Lehner quotient over the Frobenius-twisted leg CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_frobTwist_one_corr_relFrobenius_of_isAtkinLehnerQuotientVia_of_not_dvd 27 below · cited by 1 · depth 30
• Kernel degree of the quasi-inverse leg is an even power of r CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_hasKernelOfDegree_pow_two_mul_of_isODHom_of_represents 842 below · cited by 1 · depth 30
• Transport datum from the quasi-inverse leg of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isODHom_isRigTransport_of_isFormalModuleVia 6 below · cited by 4 · depth 30
• Isomorphisms of rigidified curves over one leg come from Γ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_mem_isPullback_of_isoVia_levelHom_of_translate_of_isAlgClosed_heightNormalised_eq_of_oneLeg_levelHomLaw 859 below · cited by 1 · depth 30
• Γ̃-translation of an even rigidification, with exact level transport CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_translate_level_eq_of_levelHom_of_character_of_isTwistedAct_heightNormalised_eq 207 below · cited by 1 · depth 30
• Admissible transports define the same G-point CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.gPoint_eq_of_isRigTransport_of_isRigTransport 36 below · cited by 1 · depth 30
• Closed subscheme [ℓ]⁻¹C∩[N]⁻¹K of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_isClosedImmersion_factorsThrough_iff_nsmulPt 0 below · cited by 1 · depth 30
• Geometric fibre of the combined level structure is (ℤ/Nℓ)² CerednikDrinfeld.QM.FakeEllipticCurve.WithExtraLevel.exists_zmod_mul_prod_equiv_factorsThrough_of_coprime 0 below · cited by 1 · depth 30
• Locally constant frame coordinates for action, level and generator CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_const_frameCoords_of_uniformization_family_of_smooth 8 below · cited by 2 · depth 30
• Descent of an isomorphism of level-ℓ triples to a finitely generated subalgebra CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_extraLevel_isPullbackVia_isoTVia_of_isoTVia 35 below · cited by 1 · depth 30
• Noetherian descent of full-level fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_isPullback_levelIff_of_isUnit_of_isIndefiniteRamifiedExactlyAt 821 below · cited by 2 · depth 30
• Descent of full-level isomorphisms along a directed colimit CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_iso_of_iso_of_directed_colimit_of_isUnit 833 below · cited by 1 · depth 30
• Cancelling a base change of fake elliptic curves with full level CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.isPullback_of_isPullback_comp_of_levelIff 0 below · cited by 1 · depth 30
• Universal bijectivity of T→Γ(A_T,𝒪) for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.bijective_algebraMap_sections_pullback 49 below · cited by 5 · depth 30
• Fibre dimension and trace condition pass to a bare deformation CerednikDrinfeld.QM.FakeEllipticCurve.dim_fibre_and_act_trace_of_bareDeformation 1 below · cited by 4 · depth 30
• Degree n² for quaternionic endomorphisms with t²<4n CerednikDrinfeld.QM.FakeEllipticCurve.endDegree_eq_natAbs_sq_of_mapPt_mapPt_mul_zpow_eq_zpow 741 below · cited by 1 · depth 30
• Unramifiedness of m-torsion: two m-torsion points agreeing on a thickening coincide CerednikDrinfeld.QM.FakeEllipticCurve.eq_of_nsmulPt_eq_one_of_comp_eq_of_isNilpotent_ker 2 below · cited by 1 · depth 30
• Field-valued q-power torsion of a fake elliptic curve is trivial CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_nsmulPt_pow_eq_one_of_field_of_one_mem 759 below · cited by 1 · depth 30
• The formal group of a fake elliptic curve has dimension 2 CerednikDrinfeld.QM.FakeEllipticCurve.eq_two_of_isFormalGroupAlong 30 below · cited by 1 · depth 30
• Level subscheme is étale, and membership is geometric CerednikDrinfeld.QM.FakeEllipticCurve.etale_lev_and_forall_factorsThrough_iff_of_isUnit 14 below · cited by 5 · depth 30
• Existence of an even rigidified pair with prescribed Ω̂-image CerednikDrinfeld.QM.FakeEllipticCurve.evenRigidifiedPair_exists_of_rigidifiedToG_of_isUnit_two 3,489 below · cited by 1 · depth 30
• Lifting even rigidifications along square-zero thickenings CerednikDrinfeld.QM.FakeEllipticCurve.evenRigidifiedPair_lift_of_rigidifiedToG 130 below · cited by 2 · depth 30
• Pull-back of even rigidified pairs with transported Deligne datum CerednikDrinfeld.QM.FakeEllipticCurve.evenRigidifiedPair_pullback_of_rigidifiedToG 24 below · cited by 5 · depth 30
• Uniqueness of rigidified pairs over a connected base CerednikDrinfeld.QM.FakeEllipticCurve.evenRigidifiedPair_unique_of_rigidifiedToG_of_forall_isIdempotentElem_of_isUnit_two 3,491 below · cited by 1 · depth 30
• Unique mathcal O_D-linear formal completion of a Λ-equivariant endomorphism CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_centralizer_isFormalCompletionAlong_of_isFormalModuleVia 5 below · cited by 2 · depth 30
• Lifting the Λ-action to a bare deformation CerednikDrinfeld.QM.FakeEllipticCurve.exists_act_of_forall_exists_comp_eq_comp_of_bareDeformation_of_isArtinianRing 25 below · cited by 1 · depth 30
• Jointly holomorphic uniformisation of a family of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_differentiableOn_eval_comp_uniformization_family_of_smooth_of_analytic 45 below · cited by 2 · depth 30
• Holomorphic ℤ-basis for the lattices of a family CerednikDrinfeld.QM.FakeEllipticCurve.exists_differentiableOn_latticeBasis_of_uniformization_family_of_smooth 1 below · cited by 2 · depth 30
• Factoring a level-preserving endomorphism through an ℓ-level structure CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogenyVia_isIsogenyPair_comp_eq_of_comp_eq_act_of_not_isIsogenyPair 830 below · cited by 1 · depth 30
• Level-ℓ isogeny between quotients of a Λ-stable filtration CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_of_quotients_of_card_eq_mul_sq 22 below · cited by 1 · depth 30
• Canonical polarisation datum exists faithfully flat locally when 2 is invertible CerednikDrinfeld.QM.FakeEllipticCurve.exists_faithfullyFlat_isCanonicalPolData_pullback_of_isUnit_two 2,857 below · cited by 1 · depth 30
• Vanishing of higher Čech cohomology of L^{⊗ 4} on field fibres CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_subsingleton_HSucc_pullback_tensor_four_of_isCanonicalPol 1,068 below · cited by 1 · depth 30
• Endomorphism dictionary matches any splitting up to q^{c₀} and conjugation CerednikDrinfeld.QM.FakeEllipticCurve.exists_generalLinearGroup_forall_apply_eq_smul_conj_of_isFormalCompletionAlong 12 below · cited by 1 · depth 30
• Descent of a canonical polarisation on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isCanonicalPol_of_faithfullyFlat_of_forall_locIsoOnBase 210 below · cited by 1 · depth 30
• n-torsion in the level structure: finite flat of rank n² CerednikDrinfeld.QM.FakeEllipticCurve.exists_isClosedImmersion_finrank_eq_sq_forall_factorsThrough_iff_nsmulPt_of_dvd_of_isUnit 19 below · cited by 1 · depth 30
• Formal mathcal O_D-module structure from a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalModuleVia_of_isFormalCoordinates 9 below · cited by 1 · depth 30
• Frobenius–Verschiebung datum along the reduced Frobenius lift CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_residueLeg_frobeniusLift_relFrobenius_verschiebung_of_not_dvd 35 below · cited by 3 · depth 30
• Extending a fake elliptic curve's level-N structure over a DVR CerednikDrinfeld.QM.FakeEllipticCurve.exists_level_of_isPullback_algebraMap_of_isUnit 49 below · cited by 1 · depth 30
• The r-power exponents of the endomorphism dictionary form a character mod n CerednikDrinfeld.QM.FakeEllipticCurve.exists_monoidHom_units_zmod_eq_pow_of_endomorphismDictionary_slack_of_comm 3 below · cited by 1 · depth 30
• Inertia acts with finite order on T_ℓ of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_pos_pow_rep_tateModule_eq_one_of_mem_inertiaSubgroupIn 1,255 below · cited by 1 · depth 30
• Quotient of a level-one fake elliptic curve by H CerednikDrinfeld.QM.FakeEllipticCurve.exists_quotient_forall_factorsThrough_iff_mem 35 below · cited by 1 · depth 30
• Iterated Frobenius rebase of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_frobTwist_zpow_isActBy_scalar_extraLevel_of_rigidifiedToG 153 below · cited by 1 · depth 30
• Transport of rigidifications along level-ℓ isogenies CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_of_isLevelIsogeny 729 below · cited by 1 · depth 30
• Infinitely many pairwise non-isomorphic fake elliptic curves over ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.exists_seq_forall_ne_not_iso_of_not_dvd_of_squarefree 5,587 below · cited by 1 · depth 30
• Free rank-one Tate module of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_tateModule_algPoints_forall_generator_torsion_of_isMaximalOrder_of_prime 731 below · cited by 3 · depth 30
• n-torsion of the level structure on a geometric fibre CerednikDrinfeld.QM.FakeEllipticCurve.exists_zmod_prod_equiv_factorsThrough_lev_and_nsmulPt_eq_one_of_dvd 0 below · cited by 2 · depth 30
• Independence of the geometric point over 𝔭 CerednikDrinfeld.QM.FakeEllipticCurve.forall_factorsThrough_lev_imp_eq_one_smul_fullLevel_of_ker_eq_ker 1 below · cited by 1 · depth 30
• Level sections killed by ℓ and by ̂ e(r^m̄ s) vanish CerednikDrinfeld.QM.FakeEllipticCurve.forall_factorsThrough_lev_nsmulPt_eq_one_mapPt_eq_one_imp_eq_one_of_levelHeckeUSet_of_endIsoFull 770 below · cited by 1 · depth 30
• Kernel of f on ̄ r-torsion is the P-torsion CerednikDrinfeld.QM.FakeEllipticCurve.forall_section_mapPt_eq_one_iff_torsionPrime_of_comp_eq_act_of_not_forall_mapPt_eq_one 758 below · cited by 1 · depth 30
• Rigidity of homomorphisms of fake elliptic curves over dual numbers CerednikDrinfeld.QM.FakeEllipticCurve.hom_eq_of_comp_eq_of_isPullback_fstHom 17 below · cited by 2 · depth 30
• Ramified step of a kernel filtration is an Atkin–Lehner quotient CerednikDrinfeld.QM.FakeEllipticCurve.isAtkinLehnerQuotient_of_quotients_of_card_eq_mul_sq 745 below · cited by 1 · depth 30
• Closedness of the L₀·(m/ℓ)P trivial-intersection locus CerednikDrinfeld.QM.FakeEllipticCurve.isClosed_setOf_forall_factorsThrough_lev_imp_eq_one_smul_fullLevel 22 below · cited by 1 · depth 30
• Automorphisms commuting with the quaternionic action have finite order CerednikDrinfeld.QM.FakeEllipticCurve.isOfFinOrder_of_forall_act_comp_eq 760 below · cited by 1 · depth 30
• Openness of the transversality locus of a full level structure CerednikDrinfeld.QM.FakeEllipticCurve.isOpen_setOf_forall_factorsThrough_lev_imp_eq_one_smul_fullLevel 20 below · cited by 1 · depth 30
• Atkin–Lehner quotient at r goes to a Pi-translate CerednikDrinfeld.QM.FakeEllipticCurve.isPiTranslate_rigidifiedToG_of_corr_relFrobenius_of_isAtkinLehnerQuotientVia 131 below · cited by 1 · depth 30
• Quotient by the kernel of an endomorphism is isomorphic to E CerednikDrinfeld.QM.FakeEllipticCurve.iso_of_quotient_ker_of_mapPt_mapPt_mul_zpow_eq_zpow 731 below · cited by 1 · depth 30
• Uniqueness of the canonical polarisation datum after base change CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPolData_pullback_of_isUnit_two 1,443 below · cited by 3 · depth 30
• Uniqueness of canonical polarisation data over any affine base CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_isUnit_two 1,442 below · cited by 4 · depth 30
• Endomorphism conditions on n-torsion descend from k₀-points CerednikDrinfeld.QM.FakeEllipticCurve.mapPt_eq_one_of_forall_section_of_nsmulPt_eq_one_of_isAlgClosed 25 below · cited by 2 · depth 30
• Existence of fake elliptic curves over ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_algebraicClosure_rat 4,708 below · cited by 1 · depth 30
• Tate module of a fake elliptic curve: rank four, equivariant, continuous CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_basis_tateModule_and_smul_pushPt_act_and_rep_sub_mem_of_prime 702 below · cited by 3 · depth 30
• Level preservation is detected on geometric points CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_iff_forall_factorsThrough_geomPoint_of_isAlgClosed 8 below · cited by 1 · depth 30
• Trace condition for an integral model at injective geometric points CerednikDrinfeld.QM.FakeEllipticCurve.trace_eq_of_isPullback_algebraMap_of_injective 0 below · cited by 1 · depth 30
• Canonical polarisation data are stable under base change CerednikDrinfeld.QM.IsCanonicalPolData.pullback_of_isPullback 5 below · cited by 15 · depth 30
• Tower family of coarse points over connected Noetherian bases CerednikDrinfeld.QM.IsCoarseModuliT.exists_towerFamily_connected_of_evenRigidifiedPair_of_heckeDictionary_of_isNoetherianRing_heightNormalised_oneLegC5 4 below · cited by 1 · depth 30
• Extension of the Čerednik–Drinfeld tower family to Noetherian bases CerednikDrinfeld.QM.IsCoarseModuliT.exists_towerFamily_of_towerFamily_connected_of_isNoetherianRing_heightNormalised_oneLegC5 50 below · cited by 1 · depth 30
• Tower and fine uniformisations agree through the degeneracy map d₀ CerednikDrinfeld.QM.IsCoarseModuliT.towerFamily_comp_dZero_eq_fineFamily_comp_of_isNoetherianRing_heightNormalised_oneLegC5 2 below · cited by 1 · depth 30
• Geometric fibres of the tower uniformisation maps Theta_T CerednikDrinfeld.QM.IsCoarseModuliT.towerFamily_eq_iff_exists_isTwistedAct_of_isAlgClosed_heightNormalised_oneLegC5 0 below · cited by 1 · depth 30
• Invariance of the tower parametrisation under Γ̃_ℓ CerednikDrinfeld.QM.IsCoarseModuliT.towerFamily_eq_of_isTwistedAct_of_mem_of_isNoetherianRing_heightNormalised_oneLegC5 3 below · cited by 1 · depth 30
• Equivariant Čerednik–Drinfeld family of fine moduli points CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_fineFamily_value_of_connected_of_isNoetherianRing_equivariant_heightNormalised_conn 798 below · cited by 1 · depth 30
• Čerednik–Drinfeld fine-level family on all Noetherian bases CerednikDrinfeld.QM.IsFineModuli.exists_cerednikDrinfeld_fineFamily_value_of_value_of_connected_equivariant_heightNormalised_conn 743 below · cited by 1 · depth 30
• Holomorphic lattice frame over a smooth analytic chart CerednikDrinfeld.QM.IsFineModuli.exists_holomorphic_latticeFrame_of_analytic_of_smooth_algebraicChart_noCoprime 68 below · cited by 1 · depth 30
• Pro-representing the stalk via the special formal module CerednikDrinfeld.QM.IsFineModuli.exists_ringHom_stalk_forall_existsUnique_algHom_of_prorepresents_deformations 1,324 below · cited by 1 · depth 30
• Fibres of the fine family over algebraically closed fields CerednikDrinfeld.QM.IsFineModuli.fineFamily_eq_iff_exists_mem_levelHom_of_isAlgClosed_heightNormalised_eq_hC5oneLeg 148 below · cited by 1 · depth 30
• Lifting Theta_f-values along square-zero surjections CerednikDrinfeld.QM.IsFineModuli.fineFamily_exists_lift_of_value_of_squareZero_heightNormalised_conn 959 below · cited by 1 · depth 30
• Uniqueness of the Ω-coordinate of lifts across square-zero thickenings CerednikDrinfeld.QM.IsFineModuli.fineFamily_lift_unique_of_value_of_squareZero_heightNormalised_conn 921 below · cited by 2 · depth 30
• Γₜ-equivariance of the fine family Theta_f CerednikDrinfeld.QM.IsFineModuli.fineFamily_twistedAct_levelHom_mul_eq_of_translate_heightNormalised_eq 51 below · cited by 3 · depth 30
• Injectivity of τ on an analytic period chart CerednikDrinfeld.QM.IsFineModuli.injOn_periodFunction_of_latticeFrame_of_analytic_noCoprime 5 below · cited by 1 · depth 30
• Level-ℓ fine uniformisation depends only on Frobenius-fixed coefficients CerednikDrinfeld.QM.IsFineModuliT.cerednikDrinfeld_uniformization_fine_eq_of_forall_frobFixed_eq 2 below · cited by 1 · depth 30
• Level-ℓ fine uniformisation family: existence, value, compatibilities CerednikDrinfeld.QM.IsFineModuliT.exists_fineFamilyT_value_compat_of_fineFamily_of_towerFamily_of_isNoetherianRing_oneLegC5 898 below · cited by 1 · depth 30
• Infinitesimal injectivity of the forgetful map π_ℓ on points CerednikDrinfeld.QM.IsFineModuliT.ext_of_comp_forget_eq_of_specMap_comp_eq_of_isNilpotent_ker 835 below · cited by 1 · depth 30
• Translation and fibres of the level-ℓ Čerednik–Drinfeld family CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_translate_fibre_of_value_of_isNoetherianRing_eq_oneLegC5 196 below · cited by 1 · depth 30
• Level homomorphism normalised by a character κ CerednikDrinfeld.QM.IsLevelTwistAction.exists_monoidHom_forall_pushPt_act_mapPt_eq_of_character 0 below · cited by 1 · depth 30
• Isomorphism of full-level fake elliptic curves from a homothety CerednikDrinfeld.QM.WithFullLevel.iso_of_smul_latt_subset_of_level_iff_of_generator 3 below · cited by 2 · depth 30
• Formal coordinates parametrise the tangent space at the origin CerednikDrinfeld.QM.exists_injective_range_isTangentVector_of_isFormalCoordinates 0 below · cited by 6 · depth 30
• Canonical cube polarisation datum extends over a discrete valuation ring CerednikDrinfeld.QM.exists_isCanonicalPolData_and_locIsoOnBase_of_isPullback_of_isDiscreteValuationRing 1,399 below · cited by 1 · depth 30
• Period lattices of level-one fake elliptic curves over ℂ CerednikDrinfeld.QM.exists_latticeMap_fakeEllipticCurve_complex_iso_iff_smul_eq_levelOne 897 below · cited by 1 · depth 30
• Point derivations at the unit are V⊗_κ M CerednikDrinfeld.QM.exists_pointDerivations_linearEquiv_tensor_of_isTangentVector 3 below · cited by 2 · depth 30
• Tangent presentations transport along a cartesian square over Spec e CerednikDrinfeld.QM.exists_presentation_comp_eq_iff_of_isPullback_of_ringEquiv 0 below · cited by 1 · depth 30
• Lattice-action scheme from a representing Hom-scheme CerednikDrinfeld.QM.exists_representsLatticeActions_of_represents_homScheme 12 below · cited by 1 · depth 30
• Homothety of period lattices from equal quaternionic periods CerednikDrinfeld.QM.smul_latt_subset_and_level_iff_and_generator_of_periodFunction_eq_of_latticeFrame 0 below · cited by 1 · depth 30
• Trace condition spreads from generic to all geometric points CerednikDrinfeld.QM.trace_eq_of_smooth_of_isCommutative_of_forall_injective_trace_eq 6 below · cited by 2 · depth 30
• Translation of a full level-m structure into 2g torsion sections CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.nsmul_pushPt_act_eq_one_and_finComb_injective_and_exists_finComb_eq 1 below · cited by 1 · depth 31
• Re-basing a rigidification along a twisted coefficient leg CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_comp_of_isPullbackVia_residueLeg_of_isogenyPair 21 below · cited by 3 · depth 31
• Locally norm-transported full level structure on a rigidified curve CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_fullLevel_locally_isNormLevelTransport_of_connected_conn 795 below · cited by 2 · depth 31
• Lifting rigidifications along a square-zero thickening CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_corr_of_squareZero_of_isNoetherianRing 108 below · cited by 2 · depth 31
• Translating a rigidification by a self-isogeny of A₀ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isTranslateBy_of_isIsogenyPair 3 below · cited by 2 · depth 31
• Isomorphic rigidified fake elliptic curves differ by Γₜ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_mem_isTwistedAct_isTranslateBy_corr_of_isoVia_of_isAlgClosed_heightNormalised_eq_oneLeg 731 below · cited by 1 · depth 31
• Quasi-inverse leg of a rigidification in formal coordinates CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_series_comp_eq_act_pow_and_comp_eq_act_pow_of_isODHom_of_represents 10 below · cited by 1 · depth 31
• Extra-level transport along an e_γ-translate, and stability detection CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_and_stable_of_isTranslateBy_corr_of_isAlgClosed_heightNormalised_eq_oneLeg 25 below · cited by 1 · depth 31
• Extra levels match along i iff e_γ stabilises K CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_iff_of_translate_corr_of_isAlgClosed_heightNormalised_eq_oneLeg 0 below · cited by 1 · depth 31
• Locally height-normalised full level structures coincide CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.fullLevel_eq_of_locally_isNormLevelTransport_conn 52 below · cited by 3 · depth 31
• Normalised level transport is stable under base change CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isNormLevelTransport_of_isPullbackVia_of_isNoetherianRing_frame 718 below · cited by 9 · depth 31
• Atkin–Lehner quotient at r induces a Pi-translate of rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isPiTranslate_of_isRigTransport_of_corr_relFrobenius_of_isAtkinLehnerQuotientVia 99 below · cited by 1 · depth 31
• Local normalised level transport passes along an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.locally_isNormLevelTransport_of_isoVia_of_corr_conn 69 below · cited by 3 · depth 31
• Transported level of a Γ-translate equals its χ-twist CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.mapPt_pushPt_eq_of_translate_corr_of_isNormLevelTransport_of_isAlgClosed_heightNormalised_eq_oneLeg_levelHomLaw 150 below · cited by 1 · depth 31
• Height-normalised level of a rigidification translate CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.normLevel_translate_eq_rpow_character_of_isTranslateBy_of_isActBy_heightNormalised_eq 89 below · cited by 2 · depth 31
• Constant frame coordinates for the full level-m generator CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_const_generator_frameCoords_of_uniformization_family_of_smooth 2 below · cited by 1 · depth 31
• Constant integer matrices for the Λ-action in a holomorphic frame CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_const_lambdaAction_frameCoords_of_uniformization_family_of_smooth 2 below · cited by 1 · depth 31
• Level-N points have locally constant frame coordinates CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_const_level_frameCoords_of_uniformization_family_of_smooth 1 below · cited by 1 · depth 31
• Extra level structures transport along an isomorphism of pairs CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_extraLevel_isoTVia_of_isoVia 2 below · cited by 1 · depth 31
• Descent of full-level isomorphisms to a finitely generated subalgebra CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_forall_iso_of_iso_of_isPullback 34 below · cited by 1 · depth 31
• Descent of extra-level isomorphism packages to finitely generated subalgebras CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_stdIsoPackage_extraLevel_of_stdIsoPackage 5 below · cited by 1 · depth 31
• Unpacking an isomorphism package with extra level after base change CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullbackVia_isoTVia_of_stdIsoPackage_extraLevel 24 below · cited by 1 · depth 31
• Iso package over L with extra level from a pull-back isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_stdIsoPackage_extraLevel_of_isoTVia_of_isPullbackVia 26 below · cited by 1 · depth 31
• Cube of a canonical polarisation: closed immersion with h⁰=36 CerednikDrinfeld.QM.FakeEllipticCurve.closedImmersionBySections_tensor_three_of_isCanonicalPol_of_isNoetherianRing 1,190 below · cited by 1 · depth 31
• Descent of level and torsion conditions to T-points CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_forall_section_of_factorsThrough_lev_of_nsmulPt_eq_one_of_isAlgClosed 25 below · cited by 1 · depth 31
• Fake elliptic curves have no q-power torsion in characteristic q CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_nsmulPt_pow_eq_one_of_isOrderCoord_of_charP 757 below · cited by 1 · depth 31
• Rigidity of m-torsion points on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_sectionAt_eq_of_nsmulPt_eq_one 11 below · cited by 3 · depth 31
• Descending m through two point-quotients of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_comp_eq_act_comp_of_quotients 3 below · cited by 2 · depth 31
• Cancellation of cartesian base changes along a composite ring map CerednikDrinfeld.QM.FakeEllipticCurve.exists_comp_eq_isPullback_levelIff_of_comp 0 below · cited by 1 · depth 31
• An integer prime to q acting as [n]_F CerednikDrinfeld.QM.FakeEllipticCurve.exists_coprime_natCast_mem_isFormalCompletionAlong_act_nthSeries 7 below · cited by 1 · depth 31
• Hecke and Pi translates of even rigidifications preserve levels CerednikDrinfeld.QM.FakeEllipticCurve.exists_even_rigidification_of_isActBy_of_isPiTranslate_normLevel_rpow 176 below · cited by 1 · depth 31
• Extra level at ℓ from the kernel of f on A₀[ℓ] CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_forall_factorsThrough_iff_of_comp_eq_act_of_not_isIsogenyPair 819 below · cited by 1 · depth 31
• Extra levels match ι(Λ)-stable sublattices of the period lattice CerednikDrinfeld.QM.FakeEllipticCurve.exists_extraLevel_isLevelIsogeny_sublattice_of_pointEquiv 763 below · cited by 1 · depth 31
• Spreading canonical polarisation data over a finite-type base CerednikDrinfeld.QM.FakeEllipticCurve.exists_faithfullyFlat_isCanonicalPolData_of_forall_isAdicComplete_charP_symmetricSqrt_of_finiteType 1,686 below · cited by 1 · depth 31
• Local existence of canonical polarisation data descends along base change CerednikDrinfeld.QM.FakeEllipticCurve.exists_faithfullyFlat_isCanonicalPolData_pullback_of_isPullback 6 below · cited by 1 · depth 31
• Spreading out a fake elliptic curve with finitely many sections CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_isPullback_levelIff_sections 174 below · cited by 2 · depth 31
• Spreading a full level-m structure to a finitely generated stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_le_generates_annihilator_of_isIndefiniteRamifiedExactlyAt 733 below · cited by 1 · depth 31
• Equality of two sections spreads out to a finitely generated stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_le_section_comp_eq 2 below · cited by 1 · depth 31
• Fake elliptic curves descend to finitely generated ℤ-algebras CerednikDrinfeld.QM.FakeEllipticCurve.exists_finiteType_int_isPullback 175 below · cited by 3 · depth 31
• Torsion basis induced by Λ gives full level-m structure CerednikDrinfeld.QM.FakeEllipticCurve.exists_fullLevel_of_torsionBasis_eq_pushPt_act 0 below · cited by 1 · depth 31
• Transport of the base full level along a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.exists_fullLevel_transport_and_eq_of_rigidification 20 below · cited by 1 · depth 31
• Finite field of definition for n-torsion of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_intermediateField_finiteDimensional_forall_smul_eq_of_mem_torsionBy 700 below · cited by 3 · depth 31
• Inertia fixes ℓ-power torsion over a finite extension CerednikDrinfeld.QM.FakeEllipticCurve.exists_intermediateField_forall_specMap_comp_eq_self_of_forall_exists_pos_pow_rep_eq_one 739 below · cited by 1 · depth 31
• Descent of canonical polarisation data along faithfully flat base change CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_locIsoOnBase_pullback_of_isCanonicalPolData_of_faithfullyFlat 62 below · cited by 1 · depth 31
• Quotient of a fake elliptic curve by an extra level CerednikDrinfeld.QM.FakeEllipticCurve.exists_isLevelIsogenyVia_epi_and_forall_existsUnique_comp_eq 32 below · cited by 1 · depth 31
• Rescaled fibrewise uniformisation, jointly holomorphic near the identity CerednikDrinfeld.QM.FakeEllipticCurve.exists_isLocalHom_differentiableOn_uniformization_family_near_zero 42 below · cited by 1 · depth 31
• Symmetric principal root on a fake elliptic curve over complete local R CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelTrivial_isSymmetric_isCanonicalPolData_tensor_pullback_negMor_of_isAdicComplete_of_isUnit_two_of_charP 2,606 below · cited by 1 · depth 31
• A locally constant GtimesFin 2 label on rigidified points CerednikDrinfeld.QM.FakeEllipticCurve.exists_label_natural_iff_exists_ptR_eq_of_rigidifiedToG_connInj_pr 825 below · cited by 2 · depth 31
• Lifting level-N structures along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_lev_finrank_eq_sq_forall_factorsThrough_iff_of_isPullback_of_isNilpotent_ker 15 below · cited by 1 · depth 31
• Lifting m-torsion points across nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_nsmulPt_eq_one_and_specMap_comp_eq_of_isNilpotent_ker 14 below · cited by 1 · depth 31
• Naturality of the Ω̂-family on rigidified points (connected-injectivity edition) CerednikDrinfeld.QM.FakeEllipticCurve.exists_omega_family_natural_apply_ptR_eq_of_rigidifiedToG_connInj_pr 29 below · cited by 2 · depth 31
• Local constancy of n-torsion sections of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_opens_mem_and_comp_eq_comp_of_nsmulPt_eq_one_of_isAlgClosed 24 below · cited by 2 · depth 31
• Finite-order inertia on every ℓ-adic Tate module CerednikDrinfeld.QM.FakeEllipticCurve.exists_pos_pow_rep_tateModule_eq_one_of_forall_specMap_comp_eq_self_of_forall_mem 1,192 below · cited by 1 · depth 31
• Finite inertia order on T_ℓ of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_pos_pow_rep_tateModule_eq_one_of_mem_inertiaSubgroupIn_of_forall_isUnit_tensorProduct_padic 725 below · cited by 1 · depth 31
• Representability of the rigidified-pair functor on each edge chart CerednikDrinfeld.QM.FakeEllipticCurve.exists_represents_inEdgeChart_of_rigidifiedToG_of_bdd 1,087 below · cited by 2 · depth 31
• Frobenius re-basing of rigidifications, up to a central scalar CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_frobTwist_isActBy_scalar_levelCompat_of_rigidifiedToG 147 below · cited by 1 · depth 31
• Serre–Tate dictionary for deformations of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_serreTateDictionary_of_prorepresents_deformations 1,258 below · cited by 1 · depth 31
• Period lattice of an Atkin–Lehner quotient at a ramified prime CerednikDrinfeld.QM.FakeEllipticCurve.exists_smul_latt_atkinLehnerQuotient_of_pointEquiv 15 below · cited by 1 · depth 31
• Compatible Λ-generators of the ℓ-adic Tate module CerednikDrinfeld.QM.FakeEllipticCurve.exists_tateModule_forall_generator_torsion_of_isMaximalOrder_of_prime 724 below · cited by 1 · depth 31
• Level points over injective geometric points form (ℤ/N)² CerednikDrinfeld.QM.FakeEllipticCurve.exists_zmod_prod_equiv_factorsThrough_of_isPullback_algebraMap_of_injective 0 below · cited by 1 · depth 31
• Kernel of ̂ e(r^m̄ s) misses the level ℓ-line CerednikDrinfeld.QM.FakeEllipticCurve.forall_section_factorsThrough_lev_nsmulPt_eq_one_mapPt_eq_one_imp_eq_one_of_forall_pow_smul_star_mul_mul_ne_smul_of_endIsoFull 742 below · cited by 1 · depth 31
• Canonical polarisation data descend along faithfully flat base change CerednikDrinfeld.QM.FakeEllipticCurve.isCanonicalPol_of_locIsoOnBase_pullback_of_faithfullyFlat 164 below · cited by 1 · depth 31
• Complex fake elliptic curves: isomorphism iff homothetic lattices CerednikDrinfeld.QM.FakeEllipticCurve.iso_iff_exists_smul_latt_eq_of_pointEquiv 0 below · cited by 1 · depth 31
• Uniqueness of canonical polarisation data, local-to-global reduction CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_forall_isLocalRing_of_isUnit_two 43 below · cited by 1 · depth 31
• Uniqueness of the canonical polarisation over a local base CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_isLocalRing_of_isUnit_two 1,438 below · cited by 1 · depth 31
• Order r² for mathfrak Pᵣ-torsion of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.natCard_setOf_forall_pushPt_act_eq_one_of_eq_or_eq 723 below · cited by 1 · depth 31
• Geometric n-torsion of a fake elliptic curve has order n⁴ CerednikDrinfeld.QM.FakeEllipticCurve.natCard_torsionBy_algPoints_eq_pow_four 698 below · cited by 2 · depth 31
• Fake elliptic curves exist over ℚ̄ when q'≥ 5 CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_algebraicClosure_rat_of_five_le 4,700 below · cited by 1 · depth 31
• Fake elliptic curves of discriminant 6 over ℚ̄ CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_algebraicClosure_rat_of_isIndefiniteRamifiedExactlyAt_two_three 890 below · cited by 1 · depth 31
• Full level-m structure yields a torsion basis CerednikDrinfeld.QM.FakeEllipticCurve.nsmulPt_eq_one_and_torsionBasis_of_nsmul_eq_one_of_finComb 0 below · cited by 1 · depth 31
• Bijectivity of θ on label-(g,j) points at algebraically closed fields CerednikDrinfeld.QM.FakeEllipticCurve.omega_family_bijective_label_of_isAlgClosed_of_rigidifiedToG_connInj_pr 1,447 below · cited by 2 · depth 31
• Unique infinitesimal lifting of θ-points of label (g,j) CerednikDrinfeld.QM.FakeEllipticCurve.omega_family_existsUnique_lift_label_of_isArtinianRing_of_rigidifiedToG_connInj_pr 1,417 below · cited by 2 · depth 31
• Level preservation descends to the second factor of an ℓ-isogeny CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_of_isLevelIsogenyVia_comp_eq_of_preservesLevel 711 below · cited by 1 · depth 31
• Dual Atkin–Lehner map preserves level structure for r ∤ N CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_symm_of_isAtkinLehnerQuotientVia_of_not_dvd 3 below · cited by 1 · depth 31
• Zariski gluing of canonical polarisation data over the base CerednikDrinfeld.QM.IsCanonicalPolData.exists_of_forall_away_of_locIsoOnBase 52 below · cited by 1 · depth 31
• Canonical polarisation data transport along a compatible isomorphism CerednikDrinfeld.QM.IsCanonicalPolData.pullback_inv_of_iso 9 below · cited by 3 · depth 31
• Rigidified-pair functor over the fine moduli scheme, with unramified strata CerednikDrinfeld.QM.IsFineModuli.exists_algFunctor_represents_rigidifiedCurve_strata_unramified 3,182 below · cited by 2 · depth 31
• Full level and moduli point lift to an Artinian deformation CerednikDrinfeld.QM.IsFineModuli.exists_fullLevel_ringHom_stalk_ptF_eq_of_isPullbackVia 42 below · cited by 1 · depth 31
• Rigidity of level-m moduli points over an Artinian base CerednikDrinfeld.QM.IsFineModuli.exists_iso_comp_eq_of_ptF_eq_of_three_le 820 below · cited by 1 · depth 31
• Base change of a full-level fake elliptic curve along A → A' CerednikDrinfeld.QM.IsFineModuli.exists_withFullLevel_isPullbackVia_comp_eq_ptF_eq_of_algHom 22 below · cited by 1 · depth 31
• Curves over Artin local rings from points of the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_withFullLevel_isPullbackVia_ptF_eq_of_ringHom_stalk 42 below · cited by 1 · depth 31
• Invariance of the height-normalised fine family over Noetherian bases CerednikDrinfeld.QM.IsFineModuli.fineFamily_apply_eq_apply_of_isPullback_of_frobTwist_eq_of_isNoetherianRing_heightNormalised_eq 86 below · cited by 1 · depth 31
• Fibres of the fine family over geometric points CerednikDrinfeld.QM.IsFineModuli.fineFamily_eq_iff_exists_mem_levelHom_isTwistedAct_of_isAlgClosed_heightNormalised_eq 56 below · cited by 1 · depth 31
• Cross-leg fibre description of the fine Čerednik–Drinfeld family CerednikDrinfeld.QM.IsFineModuli.fineFamily_eq_iff_exists_mem_levelHom_of_oneLeg_of_legBlind_of_twistedAct_heightNormalised_eq 51 below · cited by 1 · depth 31
• Local infinitesimal lifting for the fine family CerednikDrinfeld.QM.IsFineModuli.fineFamily_exists_lift_locally_of_value_of_squareZero 874 below · cited by 1 · depth 31
• Local uniqueness of the Ω-coordinate of a square-zero lift CerednikDrinfeld.QM.IsFineModuli.fineFamily_lift_unique_locally_of_value_of_squareZero 919 below · cited by 1 · depth 31
• Γₜ-equivariance of the fine Čerednik–Drinfeld family CerednikDrinfeld.QM.IsFineModuli.fineFamily_twistedAct_levelHom_mul_eq_of_translate_of_isFormalModuleVia_of_forall_isIdempotentElem_heightNormalised_eq 0 below · cited by 1 · depth 31
• Isomorphic level-m data over A give the same moduli point CerednikDrinfeld.QM.IsFineModuli.ptF_eq_of_iso_comp_eq 19 below · cited by 1 · depth 31
• Level-ℓ fine family on connected Noetherian bases CerednikDrinfeld.QM.IsFineModuliT.exists_fineFamilyT_connected_value_of_fineFamily_of_towerFamily_of_isNoetherianRing_oneLegC5 871 below · cited by 1 · depth 31
• Extending the level-ℓ fine uniformisation family to Noetherian bases CerednikDrinfeld.QM.IsFineModuliT.exists_fineFamilyT_value_of_fineFamilyT_connected_of_isNoetherianRing_oneLegC5 752 below · cited by 1 · depth 31
• Compatibility of the level-ℓ formal family with π_ℓ CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_comp_forgetExtraLevel_eq_fineFamily_of_natural_equivariant_value_oneLegC5 50 below · cited by 1 · depth 31
• Level-ℓ formal family composed with p_ℓ equals tower family CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_comp_forgetFullLevel_eq_towerFamily_of_natural_value_oneLegC5 50 below · cited by 1 · depth 31
• Geometric fibres of the level-(n;ℓ) uniformisation CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_fibre_of_translate_of_value_of_isAlgClosed_eq_oneLegC5 190 below · cited by 1 · depth 31
• Γ̃_ℓ-invariance of the level-(n;ℓ) parametrisation CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_translate_of_value_of_isNoetherianRing_eq_oneLegC5 53 below · cited by 1 · depth 31
• Additivity of formal completion along the coordinates CerednikDrinfeld.QM.IsFormalCompletionAlong.of_forall_mapPt_eq_mul_of_isFormalCoordinates 1 below · cited by 3 · depth 31
• Lattice actions as β-tuples satisfying the multiplication table CerednikDrinfeld.QM.LatticeAction.table_and_existsUnique_of_table 0 below · cited by 1 · depth 31
• Unique formal completion of a homomorphism along unit sections CerednikDrinfeld.QM.existsUnique_hom_isFormalCompletionAlong_of_isFormalCoordinates 2 below · cited by 3 · depth 31
• Period lattices of level-one fake elliptic curves over ℂ CerednikDrinfeld.QM.exists_latticeMap_pointEquiv_quotient_hom_iff_smul_le_levelOne 315 below · cited by 1 · depth 31
• Propagating (ℤ/N)² points to all geometric points CerednikDrinfeld.QM.exists_zmod_prod_equiv_factorsThrough_of_etale_of_forall_injective 3 below · cited by 1 · depth 31
• Finite étale d-torsion of a closed N-torsion subscheme CerednikDrinfeld.QM.isFinite_etale_pullback_schemeKer_and_factorsThrough_iff_of_dvd_of_isUnit 14 below · cited by 2 · depth 31
• Closed N-torsion subscheme is étale and open in A[N] CerednikDrinfeld.QM.isFinite_etale_schemeKerStr_and_etale_isOpenImmersion_of_forall_nsmulPt_eq_one_of_isUnit 13 below · cited by 4 · depth 31
• Tangent vectors at the unit are point derivations on an affine chart CerednikDrinfeld.QM.isTangentVector_specMap_fromSpec_iff_pointDerivations 0 below · cited by 1 · depth 31
• The group law adds point derivations at the unit CerednikDrinfeld.QM.mul_eq_specMap_fromSpec_of_pointDerivations_add 0 below · cited by 1 · depth 31
• Homothety of period lattices at equal τ CerednikDrinfeld.QM.smul_latt_subset_and_level_iff_and_generator_of_periodFunction_eq_of_latticeFrame_noCoprime 0 below · cited by 1 · depth 31
• Period lattice attached to an extra level at ℓ CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.exists_submodule_forall_mem_iff_factorsThrough_of_pointEquiv 5 below · cited by 1 · depth 32
• Extra level at ℓ equals ℓ-torsion killed by f CerednikDrinfeld.QM.FakeEllipticCurve.ExtraLevel.factorsThrough_levK_iff_nsmulPt_eq_one_and_mapPt_eq_one_of_forall_point 752 below · cited by 1 · depth 32
• Full level-m structure gives a fibrewise torsion basis CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.torsionBasis_pushPt_act_of_neZero 0 below · cited by 1 · depth 32
• Relative Frobenius correspondence of rigidifications in formal coordinates CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.act_pow_comp_map_comp_eq_act_pow_comp_comp_frob_of_corr_relFrobenius_of_represents 3 below · cited by 1 · depth 32
• Base change stability of the rigidification correspondence data CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_corr_of_isPullbackVia_of_isPullbackVia_of_comp_eq_of_comp_eq 1 below · cited by 1 · depth 32
• Square-zero descent of two rigidifications to a common correspondence CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_corr_of_rigidifiedToG_of_isNormLevelTransport_of_squareZero_conn 883 below · cited by 1 · depth 32
• Bounded rigidification exponent on a fixed edge chart CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_le_equiv_of_inEdgeChart_of_isArtinianRing 900 below · cited by 2 · depth 32
• Transported ℓ-levels agree only if e_γ stabilises K₀ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_and_stable_of_isTranslateBy_corr_of_isAlgClosed_heightNormalised_eq_oneLeg_det 23 below · cited by 1 · depth 32
• Transported ℓ-level agrees under an e_γ-translate of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_and_stable_of_isTranslateBy_corr_of_isAlgClosed_heightNormalised_eq_oneLeg_ell 19 below · cited by 1 · depth 32
• Prime-to-r extra levels agree for compatible rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_levK_iff_of_comp_eq_comp_of_forall_geomPoint_iff 19 below · cited by 2 · depth 32
• Norm level transport is preserved by isomorphisms of rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isNormLevelTransport_of_isoVia_of_corr_of_isFormalModuleVia 43 below · cited by 10 · depth 32
• Exponent identity for a Γ̃-translate of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.n_eq_n_add_sub_slackExponent_of_isRigTransport_of_isRigTransport_translate_of_isActBy 69 below · cited by 1 · depth 32
• Rigidity of the Ω-datum along a square-zero thickening CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.omega_eq_of_rigidifiedToG_of_isPullbackVia_corr_of_squareZero 69 below · cited by 1 · depth 32
• Point identity for a Γ-translate of a rigidified section CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.pushPt_slack_nsmulPt_translate_section_eq_nsmulPt_pushPt_character_section_of_isTranslateBy 0 below · cited by 1 · depth 32
• Descent of a full-level isomorphism to a finitely generated subalgebra CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_isPullback_iso_of_iso_of_isPullback 33 below · cited by 1 · depth 32
• Descent of the trace condition along a cartesian square CerednikDrinfeld.QM.FakeEllipticCurve.act_trace_descend_of_isPullback_of_fg_of_isCommutative 8 below · cited by 1 · depth 32
• Rigidity of m-torsion sections modulo a nilpotent ideal CerednikDrinfeld.QM.FakeEllipticCurve.eq_of_specMap_comp_eq_of_nsmulPt_eq_one_of_isNilpotent_ker 16 below · cited by 4 · depth 32
• Unique lift of an isomorphism of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_iso_lift_of_isFormalModuleVia_of_one_mem 824 below · cited by 2 · depth 32
• Spreading out equality of rigidified-pair points to a basic open CerednikDrinfeld.QM.FakeEllipticCurve.exists_away_map_eq_of_atPrime_map_eq_of_rigidifiedToG 28 below · cited by 1 · depth 32
• Labelled presentation on a connected basic open at each prime CerednikDrinfeld.QM.FakeEllipticCurve.exists_away_presentationLabel_of_isPrime_pr 113 below · cited by 1 · depth 32
• Spreading canonical polarisation data from the closed points CerednikDrinfeld.QM.FakeEllipticCurve.exists_faithfullyFlat_isCanonicalPolData_of_forall_isMaximal_faithfullyFlat_symmetricSqrt_of_isNoetherianRing 1,682 below · cited by 1 · depth 32
• Enlarging the finitely generated subalgebra: inverting m and s CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_le_isUnit_isPullback_section_of_isPullback 21 below · cited by 1 · depth 32
• Descent of fake elliptic curve data and sections to a finitely generated base CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_rawData_isPullback_sections 124 below · cited by 1 · depth 32
• Compatible tower of symmetric principal roots on thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_kernelTrivial_isSymmetric_isCanonicalPolData_thickening_of_isUnit_two_of_charP 2,547 below · cited by 1 · depth 32
• Full level-m structures are Zariski-local on the base CerednikDrinfeld.QM.FakeEllipticCurve.exists_fullLevel_eq_of_forall_isPullbackVia_isLocalizationAway_of_span_eq_top 0 below · cited by 1 · depth 32
• Lifting r^mφ₀ across a square-zero thickening CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_comp_eq_nsmulPt_pow_comp_of_squareZero_of_isNoetherianRing 44 below · cited by 1 · depth 32
• Uniqueness of the formal 𝒪_D-module presentation of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_isIso_forall_apply_eq_apply_nilEval_of_isFormalModuleVia 4 below · cited by 1 · depth 32
• Frobenius re-basing moves the dictionary by a central scalar CerednikDrinfeld.QM.FakeEllipticCurve.exists_isActBy_scalar_zpow_rigidifiedToG_frobTwist_neg_one_of_comp_verschiebung 101 below · cited by 2 · depth 32
• Frobenius re-basing shifts the dictionary by a central scalar rᶜ CerednikDrinfeld.QM.FakeEllipticCurve.exists_isActBy_scalar_zpow_rigidifiedToG_frobTwist_one_of_comp_relFrobenius 99 below · cited by 2 · depth 32
• Base change of Atkin–Lehner quotients of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isAtkinLehnerQuotientVia_comp_eq_of_isPullbackVia 24 below · cited by 1 · depth 32
• Transporting formal coordinates along an isomorphism of formal mathcal O_D-modules CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalModuleVia_and_apply_nilEval_eq_of_isFormalModuleVia_of_isIso 1 below · cited by 3 · depth 32
• Formal 𝒪_D-coordinates exist over a field where q is nilpotent CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalModuleVia_of_field 53 below · cited by 5 · depth 32
• Algebraisation of compatible invertible modules on adic thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_of_forall_thickening 231 below · cited by 2 · depth 32
• Level transport, G-label and Frobenius parity over a local base CerednikDrinfeld.QM.FakeEllipticCurve.exists_isNormLevelTransport_and_eq_pushPt_act_and_leg_eq_frobTwist_of_isLocalRing_of_rigidifiedToG_pr 115 below · cited by 1 · depth 32
• Formal germ of a Λ-linear map is an 𝒪_D-homomorphism CerednikDrinfeld.QM.FakeEllipticCurve.exists_isODHom_forall_comp_eq_apply_nilEval_of_isFormalModuleVia 5 below · cited by 6 · depth 32
• Open window in a rigidified stratum over an edge chart CerednikDrinfeld.QM.FakeEllipticCurve.exists_isOpenImmersion_stratum_window_of_rigidifiedToG_of_bdd_local 1,057 below · cited by 1 · depth 32
• Inverse Frobenius twist with its Verschiebung and relative Frobenius CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_residueLeg_frobeniusLift_symm_verschiebung_relFrobenius_of_not_dvd 37 below · cited by 2 · depth 32
• Serre–Tate lifting of fake elliptic curves, Artinian local base CerednikDrinfeld.QM.FakeEllipticCurve.exists_lift_of_isFormalModuleVia_of_isArtinianRing_of_isAlgClosed_of_one_mem 1,232 below · cited by 3 · depth 32
• Pull-back of a presentation along a C-algebra map CerednikDrinfeld.QM.FakeEllipticCurve.exists_pullback_presentation_ptR_eq_map_pr 26 below · cited by 1 · depth 32
• Frobenius re-basing of rigidifications acts by a central scalar CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_frobTwist_isActBy_scalar_levelCompat_heightNormalised_of_rigidifiedToG 158 below · cited by 2 · depth 32
• Frobenius rebasing of a rigidification: Xi moves by Zᶜ CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidification_frobTwist_zpow_isActBy_scalar_extraLevel_normLevel_of_rigidifiedToG 175 below · cited by 1 · depth 32
• Uniformising lattice of a level-ℓ isogeny quotient CerednikDrinfeld.QM.FakeEllipticCurve.exists_smul_latt_eq_of_isLevelIsogeny_of_pointEquiv 0 below · cited by 1 · depth 32
• Kernel of f on ℓ-torsion is (ℤ/ℓ)² CerednikDrinfeld.QM.FakeEllipticCurve.exists_zmod_prod_injective_mul_iff_nsmulPt_eq_one_and_mapPt_eq_one_of_not_isIsogenyPair 774 below · cited by 1 · depth 32
• Finite projectivity and base change for L^{⊗ 3} CerednikDrinfeld.QM.FakeEllipticCurve.finite_projective_sections_and_exists_linearEquiv_tensorProduct_pullback_tensor_three_of_isCanonicalPol_of_isNoetherianRing 1,074 below · cited by 1 · depth 32
• Full m-level conditions descend along an injective base change CerednikDrinfeld.QM.FakeEllipticCurve.generates_annihilator_of_isPullback_of_injective_of_isUnit 731 below · cited by 1 · depth 32
• Isogeny pairs lift along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.isIsogenyPair_pow_add_of_comp_eq_nsmulPt_pow_comp_of_isNilpotent_ker 51 below · cited by 3 · depth 32
• Canonical polarisation datum passes from thickenings to complete local base CerednikDrinfeld.QM.FakeEllipticCurve.kernelTrivial_isSymmetric_isCanonicalPolData_of_forall_thickening 1,202 below · cited by 1 · depth 32
• Geometric fibres of the descended level subgroup are (ℤ/N)² CerednikDrinfeld.QM.FakeEllipticCurve.lev_fibre_descend_of_isPullback_of_fg_of_isCommutative 38 below · cited by 1 · depth 32
• Uniqueness of canonical polarisation data over a local base CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_locIsoOnBase_sqrt_of_isLocalRing_of_isUnit_two 1,436 below · cited by 1 · depth 32
• d-torsion in the level subgroup has d² geometric points CerednikDrinfeld.QM.FakeEllipticCurve.natCard_factorsThrough_lev_nsmulPt_eq_one_eq_sq 0 below · cited by 2 · depth 32
• Existence of level-N fake elliptic curves over algebraically closed fields CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_of_isIndefiniteRamifiedExactlyAt_of_linearMap_matrix_zmod_of_natCast_ne_zero 733 below · cited by 3 · depth 32
• θ separates equally labelled points over Artinian local bases CerednikDrinfeld.QM.FakeEllipticCurve.omega_family_eq_of_label_of_theta_eq_of_isArtinianRing_of_rigidifiedToG_connInj_pr 1,403 below · cited by 1 · depth 32
• Existence of labelled lifts along square-zero surjections for θ CerednikDrinfeld.QM.FakeEllipticCurve.omega_family_exists_lift_label_of_isArtinianRing_of_rigidifiedToG_connInj_pr 1,415 below · cited by 1 · depth 32
• Labelled presentations of R-points are stable under base change CerednikDrinfeld.QM.FakeEllipticCurve.presentationLabel_map_of_presentationLabel_pr 728 below · cited by 1 · depth 32
• Twisting the full level by χ(h) shifts the label from g to hg CerednikDrinfeld.QM.FakeEllipticCurve.presentationLabel_twist_pr 2 below · cited by 1 · depth 32
• Uniqueness of the presentation label and of the Frobenius leg CerednikDrinfeld.QM.FakeEllipticCurve.presentationLabel_unique_and_leg_eq_of_ptR_eq_pr 93 below · cited by 1 · depth 32
• Level preservation lifts along a nilpotent thickening CerednikDrinfeld.QM.FakeEllipticCurve.preservesLevel_of_comp_eq_comp_of_preservesLevel_of_isNilpotent_ker_of_isUnit 39 below · cited by 3 · depth 32
• Bijectivity of the dictionary Xi at Artinian local points CerednikDrinfeld.QM.FakeEllipticCurve.rigidifiedToG_surjective_injective_of_isArtinianRing_of_isAlgClosed_residueField 1,388 below · cited by 3 · depth 32
• Atkin–Lehner quotient series generate the ideal of varpi CerednikDrinfeld.QM.FakeEllipticCurve.span_range_eq_span_range_varpi_of_isAtkinLehnerQuotientVia_of_forall_comp_eq_apply_nilEval 11 below · cited by 1 · depth 32
• Canonical polarisation data descend along a basic open cover CerednikDrinfeld.QM.IsCanonicalPolData.of_forall_away 22 below · cited by 1 · depth 32
• Canonical polarisation data transfer along base-local isomorphisms CerednikDrinfeld.QM.IsCanonicalPolData.of_locIsoOnBase 6 below · cited by 5 · depth 32
• Invariance upgrade for natural Γ̃-equivariant families on nilpotent algebras CerednikDrinfeld.QM.IsFineModuli.apply_eq_apply_of_isPullback_of_frobTwist_eq_of_invariant_lite 8 below · cited by 1 · depth 32
• Formal étaleness of the rigidified-curve functor over the fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.existsUnique_lift_of_sheaf_of_represents_rigidifiedCurve 867 below · cited by 1 · depth 32
• Gluing isogeny-pair representing schemes over a fine moduli scheme CerednikDrinfeld.QM.IsFineModuli.exists_locallyOfFinitePresentation_forall_representsOn_hom_isPullback_of_forall_withFullLevel 28 below · cited by 1 · depth 32
• Representability of level-preserving rᵈ-isogeny pairs over arbitrary bases CerednikDrinfeld.QM.IsFineModuli.exists_representsOn_isogenyPair_of_withFullLevel_of_isPullbackVia 3,105 below · cited by 1 · depth 32
• Formal unramifiedness of the glued isogeny-pair stratum CerednikDrinfeld.QM.IsFineModuli.formallyUnramified_of_forall_representsOn_hom_isPullback 66 below · cited by 1 · depth 32
• Invariance of the level-(n;ℓ) formal family under twisted translation CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_apply_eq_apply_of_isPullback_of_frobTwist_eq_of_translate_of_isNoetherianRing_eq 36 below · cited by 1 · depth 32
• One-leg fibre criterion for Theta_{f,ℓ} over algebraically closed fields CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_eq_iff_exists_isTwistedAct_of_translate_of_value_of_isAlgClosed_eq 56 below · cited by 1 · depth 32
• Cross-leg description of the geometric fibres of Theta_{f,ℓ} CerednikDrinfeld.QM.IsFineModuliT.fineFamilyT_fibre_of_oneLeg_of_legBlind_of_translate_of_value_of_isAlgClosed_eq 96 below · cited by 1 · depth 32
• A level-twist action is by a finite group CerednikDrinfeld.QM.IsLevelTwistAction.finite_of_isOrder 0 below · cited by 2 · depth 32
• Saturation, full rank and products for an order coordinate CerednikDrinfeld.QM.IsOrderCoord.exists_eq_pow_smul_and_exists_nsmul_mem_and_exists_coprime 0 below · cited by 3 · depth 32
• Order elements of reduced norm divisible by r approximating Pi CerednikDrinfeld.QM.IsOrderCoord.exists_mul_star_eq_and_fst_mem_span_pow_and_snd_sub_one_mem_span_pow 1 below · cited by 2 · depth 32
• First r-adic coordinate divides r when rmidnrd CerednikDrinfeld.QM.IsOrderCoord.fst_mem_span_natCast_of_mul_star_eq_intCast_mul 1 below · cited by 2 · depth 32
• Canonical polarisation datum for a quaternionic action on an abelian surface CerednikDrinfeld.QM.LatticeAction.exists_isCanonicalPolData_and_forall_locIsoOnBase_of_isUnit_two 2,865 below · cited by 1 · depth 32
• Zariski gluing for the rigidified pair class functor PR CerednikDrinfeld.QM.RigidifiedPairClass.PR.existsUnique_map_eq_of_span_eq_top 759 below · cited by 1 · depth 32
• Equal `ptR`-class implies isomorphism up to an r-power shift CerednikDrinfeld.QM.RigidifiedPairClass.exists_isoVia_corr_of_ptR_eq_of_forall_isIdempotentElem 861 below · cited by 1 · depth 32
• Surjectivity of the rigidified-pair point map `ptR` CerednikDrinfeld.QM.RigidifiedPairClass.exists_ptR_eq 27 below · cited by 1 · depth 32
• Strata points of rigidified fake elliptic curves CerednikDrinfeld.QM.RigidifiedPairClass.exists_stratumPoint_of_forall_representsOn_of_isPullback 29 below · cited by 1 · depth 32
• Push-forward preserves the relation on rigidified presented points CerednikDrinfeld.QM.RigidifiedPairClass.mapCompat_of_ptX_natural 26 below · cited by 1 · depth 32
• Naturality of the rigidified-pair class under base change CerednikDrinfeld.QM.RigidifiedPairClass.map_ptR_eq_ptR_of_isPullbackVia 25 below · cited by 2 · depth 32
• Isomorphism invariance of the rigidified-pair class ptR CerednikDrinfeld.QM.RigidifiedPairClass.ptR_eq_of_isoVia_of_corr 26 below · cited by 2 · depth 32
• A supersingular fake elliptic curve over O^{nr}/π CerednikDrinfeld.QM.exists_fakeEllipticCurve_isFormalModuleOf_hasHeight_four 1,744 below · cited by 1 · depth 32
• Elliptic curve over k with an action of ℤ[ω] CerednikDrinfeld.QM.exists_relativeGroupLaw_smoothOfRelativeDimension_one_act_span_one_omega_of_isAlgClosed_of_charZero 176 below · cited by 1 · depth 32
• Level-one fake elliptic curves from matrix actions in characteristic zero CerednikDrinfeld.QM.nonempty_fakeEllipticCurve_one_of_act_of_algHom_matrix_of_trace_of_charZero 20 below · cited by 1 · depth 32
• Two full level-n structures over a connected base differ by a label CerednikDrinfeld.QM.FakeEllipticCurve.FullLevel.exists_P_eq_pushPt_act_of_forall_isIdempotentElem 38 below · cited by 2 · depth 33
• Base change of a scheme representing isogeny pairs CerednikDrinfeld.QM.FakeEllipticCurve.IsogenyPair.existsUnique_hom_isPullback_of_representsOn 26 below · cited by 2 · depth 33
• Complementary parities and exponent shift for Frobenius-twisted transports CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.add_eq_one_and_two_mul_n_add_eq_of_isRigTransport_of_comp_frobSeries_eq_act_comp 25 below · cited by 2 · depth 33
• Rigidity of rigidification correspondences along square-zero thickenings CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_corr_of_isPullbackVia_of_isPullbackVia_of_squareZero 67 below · cited by 1 · depth 33
• Transport of a rigidification along an isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_corr_of_isoVia 0 below · cited by 2 · depth 33
• Uniform degree relation for correspondences of rigidified fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_forall_corr_mul_pow_eq_of_forall_isIdempotentElem 738 below · cited by 1 · depth 33
• Height of a transported rigidified isogeny under a correspondence CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isIsogenyOfHeight_comp_of_act_pow_comp_eq_of_isAdmissible_of_nontrivial 27 below · cited by 1 · depth 33
• Lifting rigidifications along a surjection with kernel in (π) CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_corr_of_ker_le_span 25 below · cited by 1 · depth 33
• Re-basing data for rigidifications pull back along coefficient maps CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_rebase_of_isPullbackVia 23 below · cited by 2 · depth 33
• Transport of a Frobenius-rebased rigidification and multiplication by r CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isRigTransport_comp_frobSeries_of_isRigTransport_frobTwist_one_of_X_eq 13 below · cited by 2 · depth 33
• Equal Xi-image over an Artinian base: compatible isomorphisms CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isoVia_corr_and_formalIso_of_rigidifiedToG_eq_of_isArtinianRing 854 below · cited by 1 · depth 33
• Lifting a rigidified isomorphism along a nilpotent thickening CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isoVia_corr_of_isoVia_corr_of_formalIso_of_isNilpotent_ker 827 below · cited by 1 · depth 33
• Bounded rigidification depth on an edge chart, local bases CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_le_equiv_of_inEdgeChart_of_isLocalRing 900 below · cited by 1 · depth 33
• Bounded rigidification degree from bounded transport exponent CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_le_equiv_of_isAdmissible_of_n_le_of_isArtinianRing 898 below · cited by 1 · depth 33
• Rigidifications lift along square-zero thickenings, up to r-power scalars CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_lift_corr_and_equiv_of_isPullbackVia_corr_of_ker_sq_eq_bot 103 below · cited by 1 · depth 33
• Level equation transports along a correspondence of rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_section_nsmulPt_pow_eq_of_corr_of_nsmulPt_pow_eq 1 below · cited by 1 · depth 33
• Dual-isogeny series for the transported rigidification and the correspondence identity CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_series_isODHom_represents_of_isoVia_of_corr_of_isRigTransport 12 below · cited by 1 · depth 33
• Descent of extra-level e_γ-stability from ̄ k to k₀ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_and_stable_of_isTranslateBy_corr_of_isAlgClosed_heightNormalised_eq_oneLeg_det_descent 16 below · cited by 1 · depth 33
• Equal transported extra levels force e_γ-stability over the residue field CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.forall_factorsThrough_iff_and_stable_of_isTranslateBy_corr_of_isAlgClosed_heightNormalised_eq_oneLeg_det_kbar 19 below · cited by 1 · depth 33
• Full level transports across inverse-Frobenius re-basing differ by rᶜ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.fullLevel_eq_of_isNormLevelTransport_of_isNormLevelTransport_frobTwist_neg_one_of_isActBy 68 below · cited by 1 · depth 33
• Normalised full level transports across Frobenius re-basing CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.fullLevel_eq_of_isNormLevelTransport_of_isNormLevelTransport_frobTwist_one_of_isActBy 71 below · cited by 1 · depth 33
• Inverse Frobenius re-basing raises the transport exponent by one CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isRigTransport_succ_of_isRigTransport_frobTwist_neg_one 10 below · cited by 2 · depth 33
• Parity-one transport as a ([r],0,2)-translate CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslate_act_zero_two_of_isRigTransport_one_of_comp_frobSeries_eq_act_comp 2 below · cited by 1 · depth 33
• Parity-zero transport: t' is an identity translate of t CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslate_id_zero_zero_of_isRigTransport_zero_of_comp_frobSeries_eq_act_comp 3 below · cited by 1 · depth 33
• Transports of parities j and j+2 differ by Frobenius square CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isTranslate_one_of_isRigTransport_add_two 1 below · cited by 1 · depth 33
• Uniqueness of the Drinfeld transport of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.parity_eq_and_n_eq_and_eta_eq_of_isRigTransport_of_isRigTransport 36 below · cited by 6 · depth 33
• Parity and exponent of admissible transports of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.parity_mod_two_eq_of_isRigTransport_of_isAdmissible 25 below · cited by 2 · depth 33
• Descent of a full-level isomorphism package to a f.g. subalgebra CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_fg_subalgebra_stdIsoPackage_of_stdIsoPackage 5 below · cited by 1 · depth 33
• Isomorphism of standard base changes yields isomorphic base changes CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_isPullback_iso_of_stdIsoPackage 22 below · cited by 1 · depth 33
• Isomorphism of base changes yields standard pullback isomorphism package CerednikDrinfeld.QM.FakeEllipticCurve.WithFullLevel.exists_stdIsoPackage_of_iso_of_isPullback 25 below · cited by 1 · depth 33
• Serre–Tate lifting of homomorphisms of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_hom_lift_of_isFormalModuleVia_of_one_mem 811 below · cited by 1 · depth 33
• Unique lifting of level structures along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.existsUnique_lev_lift 17 below · cited by 1 · depth 33
• Canonical polarisation data on basic opens give a faithfully flat cover CerednikDrinfeld.QM.FakeEllipticCurve.exists_faithfullyFlat_isCanonicalPolData_of_forall_isMaximal_away 25 below · cited by 1 · depth 33
• Descent of fake elliptic curve data to a finitely generated subalgebra CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_abelianScheme_act_level_isPullback 123 below · cited by 1 · depth 33
• Descent of a fake elliptic curve to a finitely generated subring CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_isPullback_hom 179 below · cited by 2 · depth 33
• Vanishing of higher Čech cohomology of L^{⊗ 3} on field fibres CerednikDrinfeld.QM.FakeEllipticCurve.exists_forall_subsingleton_HSucc_pullback_tensor_three_of_isCanonicalPol 1,068 below · cited by 1 · depth 33
• Algebraising invertible systems on thickenings of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_of_forall_thickening_of_forall_exists_isCoherent 31 below · cited by 1 · depth 33
• Descent of canonical polarisation data to a local Noetherian stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_isNoetherianRing_injective_isLocalHom_isPullback_isCanonicalPol_nonempty_pullback_iso_of_locIsoOnBase_sqrt_of_isLocalRing 314 below · cited by 1 · depth 33
• Symmetric invertible root of the canonical polarisation over the residue field CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelTrivial_isSymmetric_isCanonicalPolData_pullback_residue_pow_one_of_isUnit_two 2,474 below · cited by 1 · depth 33
• Upgrading a bare deformation to a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_lift_of_isFormalModuleVia_of_bareDeformation 856 below · cited by 1 · depth 33
• Representability of level-preserving degree rᵈ isogeny pairs CerednikDrinfeld.QM.FakeEllipticCurve.exists_locallyOfFinitePresentation_represents_isIsogenyPair_preservesLevel_of_closedImmersionBySections_of_intCast_mem 1,005 below · cited by 1 · depth 33
• Spreading a canonical polarisation datum out to a basic open CerednikDrinfeld.QM.FakeEllipticCurve.exists_not_mem_isCanonicalPolData_away_of_faithfullyFlat_symmetricSqrt_atPrime 1,667 below · cited by 1 · depth 33
• Ratio-window open and degree laws on the exponent-D stratum CerednikDrinfeld.QM.FakeEllipticCurve.exists_opens_ratioWindow_degree_laws_of_rigidifiedToG_of_bdd 744 below · cited by 1 · depth 33
• Lifting a symmetric canonical polarisation datum along a small surjection CerednikDrinfeld.QM.FakeEllipticCurve.exists_pullback_iso_kernelTrivial_isSymmetric_isCanonicalPolData_of_surjective_of_ker_mul_maximalIdeal_of_isUnit_two_of_isArtinianRing_of_charP_residueField 1,533 below · cited by 1 · depth 33
• Relative lifting of rigidified curves over Artinian local algebras CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedCurve_lift_of_isArtinianRing 1,304 below · cited by 2 · depth 33
• Rigidified lift over an Artinian base with prescribed period CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedCurve_lift_pullback_isoVia_of_label_of_isArtinianRing_of_rigidifiedToG_connInj_pr 1,389 below · cited by 1 · depth 33
• Strata points for two classes agreeing at a prime CerednikDrinfeld.QM.FakeEllipticCurve.exists_strata_point_specMap_comp_eq_of_atPrime_map_eq_of_rigidifiedToG 26 below · cited by 1 · depth 33
• Height 4 of the formal group of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.finrank_quotient_span_nthSeries_map_eq_pow_four_of_isFormalModuleVia 779 below · cited by 1 · depth 33
• Formal unramifiedness of a scheme representing degree-rᵈ isogeny pairs CerednikDrinfeld.QM.FakeEllipticCurve.formallyUnramified_of_represents_isIsogenyPair_preservesLevel 65 below · cited by 1 · depth 33
• Base change invariance of the level-m generator conditions CerednikDrinfeld.QM.FakeEllipticCurve.generates_annihilator_iff_of_isPullback 0 below · cited by 1 · depth 33
• Positivity of h⁰ on all geometric fibres from the closed fibre CerednikDrinfeld.QM.FakeEllipticCurve.geomFibreH0Finrank_pos_of_isCanonicalPolData_thickening_zero 1,142 below · cited by 1 · depth 33
• Kernel conditions pass from thickenings to complete local base CerednikDrinfeld.QM.FakeEllipticCurve.kernelTrivial_and_kernelIsTwoTorsion_of_forall_thickening 807 below · cited by 1 · depth 33
• Uniqueness of canonical polarisation data over local noetherian bases CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_isLocalRing_of_isNoetherianRing_of_isUnit_two 1,327 below · cited by 1 · depth 33
• d-torsion of the descended level subgroup has d² points CerednikDrinfeld.QM.FakeEllipticCurve.natCard_factorsThrough_nsmulPt_eq_one_eq_sq_of_isPullback_of_fg_of_isCommutative 35 below · cited by 1 · depth 33
• Invertible modules isomorphic on all thickenings are isomorphic CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_iso_of_forall_nonempty_pullback_thickening_iso 139 below · cited by 2 · depth 33
• Surjectivity of Xi on points over an algebraically closed field CerednikDrinfeld.QM.FakeEllipticCurve.rigidifiedToG_surjective_of_isAlgClosed 144 below · cited by 1 · depth 33
• Rosati compatibility descends from all infinitesimal thickenings CerednikDrinfeld.QM.FakeEllipticCurve.rosatiCompatible_of_forall_thickening 150 below · cited by 1 · depth 33
• Trace condition descends along a cartesian square CerednikDrinfeld.QM.FakeEllipticCurve.trace_eq_of_isPullback_of_comp_eq 0 below · cited by 3 · depth 33
• Principal square root clause is local on the base CerednikDrinfeld.QM.IsCanonicalPolData.exists_faithfullyFlat_sqrt_of_forall_away_of_isInvertible 16 below · cited by 1 · depth 33
• Canonical polarisation data transport along module isomorphisms CerednikDrinfeld.QM.IsCanonicalPolData.of_iso 7 below · cited by 3 · depth 33
• Zariski gluing of presented stratum points up to r-power correspondence CerednikDrinfeld.QM.IsFineModuli.exists_strata_point_locally_corr_of_span_eq_top 744 below · cited by 1 · depth 33
• Invariance of a natural family under twisted Γ̃_ℓ-action CerednikDrinfeld.QM.IsFineModuliT.apply_eq_apply_of_isPullback_of_frobTwist_eq_of_invariant_lite 8 below · cited by 1 · depth 33
• Trace condition transports along a group pullback CerednikDrinfeld.QM.LatticeAction.forall_trace_eq_iff_of_isGroupPullback 0 below · cited by 1 · depth 33
• Base change of rigidified pairs; naturality of `ptX` CerednikDrinfeld.QM.RigidifiedPairClass.exists_pullback_ptX_eq_specMap_comp 25 below · cited by 1 · depth 33
• The rigidified-pair relation is an equivalence relation CerednikDrinfeld.QM.RigidifiedPairClass.rel_equivalence 42 below · cited by 2 · depth 33
• Zariski-locality of the rigidified-pair relation over the base CerednikDrinfeld.QM.RigidifiedPairClass.rel_of_forall_rel_map 2 below · cited by 1 · depth 33
• Differential at the origin of a homomorphism of points CerednikDrinfeld.QM.existsUnique_linearMap_forall_eq_pushPt 0 below · cited by 4 · depth 33
• Matrix action of a lattice on A×_R A through j CerednikDrinfeld.QM.exists_act_prod_of_algHom_matrix_of_one_mem_of_mul_mem 0 below · cited by 1 · depth 33
• Zariski-local trace function for a quaternionic action CerednikDrinfeld.QM.exists_cover_forall_trace_eq_algebraMap_of_smooth_of_isCommutative 6 below · cited by 2 · depth 33
• Ring action on a relative group law differentiates to Λ → End_κ(W) CerednikDrinfeld.QM.exists_ringHom_moduleEnd_forall_eq_pushPt 1 below · cited by 1 · depth 33
• Fake elliptic curves with level N exist in characteristic q CerednikDrinfeld.QM.nonempty_fakeEllipticCurve_of_isAlgClosed_of_charP 1,644 below · cited by 1 · depth 33
• Tangent trace of a matrix quaternion action equals reduced trace CerednikDrinfeld.QM.trace_eq_intCast_of_isTangentVector_prod_of_smoothOfRelativeDimension_one_of_charZero 3 below · cited by 1 · depth 33
• Drinfeld's trace condition descends from generising geometric points CerednikDrinfeld.QM.trace_eq_of_smooth_of_isCommutative_of_forall_exists_ker_le_trace_eq 6 below · cited by 1 · depth 33
• Frobenius and Verschiebung on sections of a re-based rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.comp_eq_and_comp_eq_nsmulPt_of_frobTwist_one_sections 0 below · cited by 1 · depth 34
• Verschiebung compares the rebased sections of a rigidification CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.comp_eq_of_frobTwist_neg_one_sections 0 below · cited by 1 · depth 34
• Cancelling [r^k] in a comparison of rigidifications CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.comp_phi_comp_eq_phi_of_comp_act_pow_eq 710 below · cited by 1 · depth 34
• Uniqueness of the series representing a rigidification's isogeny CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.eq_of_represents_of_represents_of_constantCoeff_eq_zero 0 below · cited by 6 · depth 34
• Equivalence of rigidifications descends along square-zero surjections CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.equiv_of_equiv_of_isPullbackVia_of_ker_sq_eq_bot 64 below · cited by 1 · depth 34
• Lifting rigidifications along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_corr_of_isNilpotent_ker_of_isNoetherianRing 98 below · cited by 1 · depth 34
• Lifting rigidifications along square-zero thickenings, up to r-power scalars CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isPullbackVia_corr_of_squareZero 98 below · cited by 1 · depth 34
• Bounded rigidification exponent over Noetherian local bases CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_le_equiv_of_isAdmissible_of_n_le_of_isLocalRing 898 below · cited by 1 · depth 34
• Formal divisibility by [r^k] implies divisibility of φ' CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_schemeNsmul_comp_eq_of_represents_comp_of_isInfinitesimalTorsion_of_constantCoeff_eq_zero 770 below · cited by 2 · depth 34
• Pointwise rank shift along an r-power correspondence CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.finrank_mul_finrank_act_pow_eq_of_corr 711 below · cited by 3 · depth 34
• Exponent shift under the inverse Frobenius re-basing: n_{t'}=nₜ+c CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.n_eq_n_add_of_isRigTransport_of_isRigTransport_frobTwist_neg_one_of_isActBy 49 below · cited by 1 · depth 34
• Transport exponent shifts by 1+c under Frobenius re-basing CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.n_eq_of_isRigTransport_of_isRigTransport_frobTwist_one_of_isActBy 52 below · cited by 1 · depth 34
• Kernel of the divided leg is killed by r^h CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.nsmulPt_pow_eq_one_of_comp_eq_one_of_represents_of_comp_eq_act_pow 770 below · cited by 2 · depth 34
• Rigidity of Drinfeld transport along a nilpotent thickening CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.parity_eq_and_isIsomorphic_of_isRigTransport_of_isPullbackVia_corr_of_rigidified 758 below · cited by 1 · depth 34
• Divided leg of a rigidification is represented by σ₁ CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.represents_of_schemeNsmul_comp_eq_of_represents_comp 814 below · cited by 2 · depth 34
• Rigidification transports at levels j and j+1 differ by Frobenius CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.rho_eq_comp_frobSeries_of_isRigTransport_succ 1 below · cited by 2 · depth 34
• Rigidity of isogeny pairs along a nilpotent thickening CerednikDrinfeld.QM.FakeEllipticCurve.comp_act_eq_comp_act_of_isPullbackVia_of_isIsogenyPair_of_ker_pow_eq_bot 51 below · cited by 2 · depth 34
• Lifting the Λ-action to a bare deformation CerednikDrinfeld.QM.FakeEllipticCurve.exists_act_of_bareDeformation_of_isFormalCoordinates 827 below · cited by 1 · depth 34
• Descent of fake elliptic curve data and level subscheme CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_abelianScheme_act_levelData_isPullback 122 below · cited by 1 · depth 34
• Spreading out a fake elliptic curve with four bundles CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_isPullback_kernelIsTwoTorsion_kernelTrivial_nonempty_pullback_iso 274 below · cited by 1 · depth 34
• Lifting q^{nμ}φ₀ to a homomorphism of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_lift_nsmul_pow 8 below · cited by 1 · depth 34
• Descent of canonical polarisation data to S_𝔭 CerednikDrinfeld.QM.FakeEllipticCurve.exists_isCanonicalPolData_localizationAtPrime_of_faithfullyFlat_of_isUnit_two 1,478 below · cited by 1 · depth 34
• Level-preserving locus is a finitely presented closed subscheme CerednikDrinfeld.QM.FakeEllipticCurve.exists_isClosedImmersion_locallyOfFinitePresentation_preservesLevel_iff_of_represents_isIsogenyPair 32 below · cited by 1 · depth 34
• Rosati-compatible invertible sheaf lifts along nilpotent thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_rosatiCompatible_of_isPullbackVia_of_isArtinianRing_of_isAlgClosed_residueField_of_isUnit_two_of_charP_residueField 1,260 below · cited by 2 · depth 34
• Dividing an isogeny by [r^{d-k}] on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isIsogenyPair_preservesLevel_comp_act_eq_of_schemeNsmul_comp_eq_of_forall_ker 721 below · cited by 2 · depth 34
• Localising a noetherian base at the contraction of mathfrak m_R CerednikDrinfeld.QM.FakeEllipticCurve.exists_isLocalRing_injective_isLocalHom_isPullback_comp_eq_of_injective 21 below · cited by 1 · depth 34
• Serre–Tate lifting with prescribed rigidified formal module CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_rigidified_of_hom_isIso 1,234 below · cited by 1 · depth 34
• Symmetric lift of an invertible sheaf across a small thickening CerednikDrinfeld.QM.FakeEllipticCurve.exists_isSymmetric_pullback_iso_of_isSymmetric_of_pullback_iso_of_isUnit_two 148 below · cited by 1 · depth 34
• Formal isomorphism of transported rigidifications descends to the curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isoVia_corr_inducing_of_formalIso_of_isRigTransport_of_isAlgClosed 846 below · cited by 1 · depth 34
• Rosati-compatible principal bundle on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelTrivial_rosatiCompatible_of_isAlgClosed_of_isUnit_two 2,457 below · cited by 1 · depth 34
• Representability of degree rᵈ isogeny pairs of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_locallyOfFinitePresentation_isSeparated_represents_isIsogenyPair_of_closedImmersionBySections 436 below · cited by 1 · depth 34
• Spreading canonical polarisation data from a prime to a basic open CerednikDrinfeld.QM.FakeEllipticCurve.exists_not_mem_isCanonicalPolData_away_of_isCanonicalPolData_atPrime_of_symmetricSqrt 1,652 below · cited by 1 · depth 34
• Admissible rigidified modules arise from rigidified fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedCurve_isRigTransport_zero_isIsomorphic_of_isAdmissible_of_isAlgClosed 132 below · cited by 1 · depth 34
• Fibrewise h⁰>0 from positivity for the symmetrisation CerednikDrinfeld.QM.FakeEllipticCurve.geomFibreH0Finrank_pos_of_kernelTrivial_of_geomFibreH0Finrank_tensor_pullback_negMor_pos 972 below · cited by 1 · depth 34
• Intertwining of forward isogenies forces intertwining of backwards CerednikDrinfeld.QM.FakeEllipticCurve.hom_comp_eq_comp_hom_of_isIsogenyPair_of_isoVia 710 below · cited by 1 · depth 34
• Canonical polarisation data over a finite product of base algebras CerednikDrinfeld.QM.FakeEllipticCurve.isCanonicalPolData_pi_of_forall 22 below · cited by 1 · depth 34
• Canonicity of a polarisation datum descends along an injective local homomorphism CerednikDrinfeld.QM.FakeEllipticCurve.isCanonicalPol_of_isCanonicalPol_pullback_of_injective_of_isLocalHom 232 below · cited by 1 · depth 34
• Isogeny pairs of fake elliptic curves: finite, flat, surjective CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_flat_surjective_of_isIsogenyPair 709 below · cited by 4 · depth 34
• Rank of a finite flat morphism: fibrewise and locally constant CerednikDrinfeld.QM.FakeEllipticCurve.isLocallyConstant_finrank_one 0 below · cited by 4 · depth 34
• Trivial kernel descends along a small surjection CerednikDrinfeld.QM.FakeEllipticCurve.kernelTrivial_of_kernelTrivial_of_pullback_iso_of_surjective_of_ker_mul_maximalIdeal_of_isNoetherianRing 158 below · cited by 1 · depth 34
• Uniqueness of canonical polarisation data over a complete local base CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_iso_of_isCanonicalPol_of_isAdicComplete_of_isAlgClosed_of_isUnit_two 1,316 below · cited by 1 · depth 34
• Triviality of the Mumford bundle over a small thickening CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_mumfordBundle_iso_unit_of_pullback_iso_unit_of_ker_mul_maximalIdeal 138 below · cited by 1 · depth 34
• Rank of [r^k] exceeds 1 on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.one_lt_finrank_act_pow_of_isFinite_of_flat 717 below · cited by 3 · depth 34
• The G-point Xi(x) is pinned by any admissible transport CerednikDrinfeld.QM.FakeEllipticCurve.rigidifiedToG_eq_of_isRigTransport_of_isIsomorphic 37 below · cited by 1 · depth 34
• Rosati compatibility descends from the symmetrisation of L CerednikDrinfeld.QM.FakeEllipticCurve.rosatiCompatible_of_rosatiCompatible_tensor_pullback_negMor_of_kernelTrivial_of_isArtinianRing 867 below · cited by 1 · depth 34
• Symmetrisation yields canonical polarisation data over an algebraically closed field CerednikDrinfeld.QM.isCanonicalPolData_tensor_pullback_negMor_of_kernelTrivial 606 below · cited by 1 · depth 34
• Symmetrisation L⊗[-1]^*L is a canonical polarisation datum CerednikDrinfeld.QM.isCanonicalPolData_tensor_pullback_negMor_of_kernelTrivial_of_commRing 386 below · cited by 1 · depth 34
• Kernel of quaternionic multiplication has rank n² CerednikDrinfeld.QM.isFinite_endKerStr_act_and_finrank_eq_natAbs_sq 720 below · cited by 1 · depth 34
• Level-one fake elliptic curves exist in characteristic q CerednikDrinfeld.QM.nonempty_fakeEllipticCurve_one_of_isAlgClosed_of_charP 1,015 below · cited by 1 · depth 34
• Action of c = 6(1 + b₀^⋆ b₀) on points CerednikDrinfeld.QM.pushPt_act_eq_nsmul_mul_pushPt_act_star_pushPt_act_of_eq_smul_one_add_star_mul 0 below · cited by 1 · depth 34
• Exactly intertwined germs of padded isogenies on nilpotent points CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_germs_represents_comp_eq_of_represents_of_act_pow_comp_eq_of_constantCoeff_eq_zero 3 below · cited by 1 · depth 35
• Lifting an isomorphism of reductions of rigidified fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_isoVia_corr_inducing_of_iso_reduction_of_act_pow_comp_eq_of_isAlgClosed 5 below · cited by 1 · depth 35
• Cancelling the Frobenius leg in an intertwining of transports CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.exists_represents_comp_act_pow_eq_of_isRigTransport_of_act_comp_hom_comp_eq 37 below · cited by 1 · depth 35
• Transport of rigidifications along an isomorphism of formal 𝒪_D-modules CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.isRigTransport_comp_nilEval_of_isRigTransport_of_isODHom_of_constantCoeff_eq_zero 1 below · cited by 1 · depth 35
• Corresponding rigidifications: equal parity, exponents and transported series CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.parity_eq_and_n_eq_and_act_pow_comp_eq_of_isoVia_of_corr_of_isRigTransport_of_isFormalCompletionAlong 34 below · cited by 1 · depth 35
• Kernel subgroup scheme of a formal mathcal O_D-isogeny CerednikDrinfeld.QM.FakeEllipticCurve.exists_closedSubgroup_factorsThrough_iff_nilEval_eq_zero_of_isIsogenyOfHeight_of_isAlgClosed 43 below · cited by 1 · depth 35
• Descent of a fake elliptic curve's abelian scheme with Λ-action CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_abelianScheme_act_isPullback 112 below · cited by 1 · depth 35
• Invertible module descends to an enlarged finitely generated stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_extension_isInvertible_nonempty_pullback_iso 47 below · cited by 1 · depth 35
• Spreading K(M)=E[2] to a finitely generated stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_extension_kernelIsTwoTorsion_pullback 169 below · cited by 1 · depth 35
• Triviality of the Mumford kernel over a finitely generated stage CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_extension_kernelTrivial_pullback 169 below · cited by 1 · depth 35
• Descent of a Rosati-compatible principal sheaf to a finite extension CerednikDrinfeld.QM.FakeEllipticCurve.exists_finiteDimensional_isPullback_kernelTrivial_rosatiCompatible_of_isAlgClosure 111 below · cited by 1 · depth 35
• Lifting r^mφ₀ along a nilpotent thickening CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_comp_eq_nsmulPt_pow_comp_of_isNilpotent_ker 66 below · cited by 1 · depth 35
• Lifting r^m-multiples of homomorphisms along square-zero thickenings CerednikDrinfeld.QM.FakeEllipticCurve.exists_hom_comp_eq_nsmulPt_pow_comp_of_squareZero 66 below · cited by 1 · depth 35
• Level preservation cut out by a finitely generated ideal CerednikDrinfeld.QM.FakeEllipticCurve.exists_ideal_fg_forall_preservesLevel_iff_map_eq_bot_of_isPullbackVia 4 below · cited by 1 · depth 35
• Formal module of the quotient by a formal isogeny kernel CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalCoordinates_quotient_comp_eq_nilEval_of_factorsThrough_iff_nilEval_eq_zero_of_isAlgClosed 57 below · cited by 1 · depth 35
• Descent of the canonical polarisation module to S_𝔭 CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_localizationAtPrime_locIsoOnBase_of_isCanonicalPolData_of_faithfullyFlat_of_isUnit_two 1,466 below · cited by 1 · depth 35
• Lifting a Rosati-compatible invertible module along a small extension CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_of_rosatiCompatible_of_ker_mul_maximalIdeal_of_isArtinianRing_of_isAlgClosed 1,223 below · cited by 2 · depth 35
• Descent of invertible modules along small extensions away from 2qq' CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_of_rosatiCompatible_of_ker_mul_maximalIdeal_of_isUnit_of_isUnit_two 1,056 below · cited by 1 · depth 35
• Quotient by a finite Λ-stable n-torsion subgroup is again fake elliptic CerednikDrinfeld.QM.FakeEllipticCurve.exists_isIsogenyPair_preservesLevel_isFormalModuleVia_of_quotient_groupCore_of_coprime_germ 1 below · cited by 1 · depth 35
• Fake elliptic curve pull-backs factor through intermediate quotients CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_quotient_comp_eq_of_isPullbackVia_of_le_ker 21 below · cited by 3 · depth 35
• Lifting level-one fake elliptic curves to characteristic-zero complete DVRs CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_of_isDiscreteValuationRing_charZero_of_isAlgClosed_one_of_isUnit_two 2,174 below · cited by 1 · depth 35
• Formal mathcal O_D-isomorphism of germs induces isomorphism of targets CerednikDrinfeld.QM.FakeEllipticCurve.exists_iso_comp_eq_of_formalIso_comp_germ_eq_of_isIsogenyPair_of_hasHeight_four_of_isAlgClosed_of_constantCoeff_eq_zero 807 below · cited by 1 · depth 35
• Principal Rosati-compatible bundle on a fake elliptic curve, char 0 CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelTrivial_rosatiCompatible_of_isAlgClosed_of_charZero 1,270 below · cited by 1 · depth 35
• Specialisation of a Rosati-compatible principal sheaf over a DVR CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelTrivial_rosatiCompatible_of_isPullback_of_isDiscreteValuationRing 1,089 below · cited by 1 · depth 35
• Canonical polarisation datum spreads from S_𝔭 to S_g CerednikDrinfeld.QM.FakeEllipticCurve.exists_not_mem_isCanonicalPolData_away_of_four_spread_clauses 1,454 below · cited by 1 · depth 35
• Quotient of a fake elliptic curve by a finite Λ-stable subgroup CerednikDrinfeld.QM.FakeEllipticCurve.exists_quotient_core_of_isAlgClosed_of_nsmulPt_eq_one 24 below · cited by 1 · depth 35
• Representability of Λ-linear isogeny pairs between fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_represents_homPair_act_comp_eq_of_closedImmersionBySections 435 below · cited by 1 · depth 35
• Rigidified curve realising a given admissible rigidified formal module CerednikDrinfeld.QM.FakeEllipticCurve.exists_rigidifiedCurve_isRigTransport_zero_isIsomorphic_of_quotient_of_isFormalModuleVia 47 below · cited by 1 · depth 35
• Halving an invertible module trivial along a small extension CerednikDrinfeld.QM.FakeEllipticCurve.exists_tensor_self_iso_of_pullback_iso_unit_of_isUnit_two 48 below · cited by 1 · depth 35
• Descent of a canonical polarisation datum to the local ring CerednikDrinfeld.QM.FakeEllipticCurve.isCanonicalPolData_localizationAtPrime_of_locIsoOnBase_of_isCanonicalPolData_of_faithfullyFlat 162 below · cited by 1 · depth 35
• Uniqueness of the canonical polarisation over an algebraically closed field CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_isAlgClosed_of_two_ne_zero 1,222 below · cited by 1 · depth 35
• Canonical polarisation data agreeing on the closed fibre, 2 invertible CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_residue_of_isUnit_two 137 below · cited by 1 · depth 35
• Triviality of t^*(mathcal L₁'^∨⊗[-1]^*mathcal L₁') over a symmetric base CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_pullback_dual_tensor_pullback_negMor_iso_unit_of_isSymmetric_of_pullback_iso 3 below · cited by 1 · depth 35
• Triviality of d⊗[-1]^*d along a small thickening CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_tensor_pullback_negMor_iso_unit_of_pullback_iso_unit_of_isUnit_two 131 below · cited by 1 · depth 35
• Rosati compatibility lifts along nilpotent thickenings of Artinian local base CerednikDrinfeld.QM.FakeEllipticCurve.rosatiCompatible_of_pullback_iso_of_rosatiCompatible_of_isPullbackVia_of_isArtinianRing 150 below · cited by 4 · depth 35
• Quaternion order action read in the endomorphism group CerednikDrinfeld.QM.act_add_mul_zsmul_neg_pointCommGroup 0 below · cited by 1 · depth 35
• Maximal quaternion order acting on a supersingular elliptic curve CerednikDrinfeld.QM.exists_relativeGroupLaw_smoothOfRelativeDimension_one_isMaximalOrder_act_of_charP 962 below · cited by 1 · depth 35
• Drinfeld's trace condition for special formal mathcal O_D-modules CerednikDrinfeld.QM.forall_trace_eq_intCast_of_isFormalCoordinates_of_isSpecial 2 below · cited by 1 · depth 35
• Level-one fake elliptic curve from a quaternionic matrix representation CerednikDrinfeld.QM.nonempty_fakeEllipticCurve_one_of_act_of_algHom_matrix_of_trace 20 below · cited by 1 · depth 35
• Descending a cancelled germ identity from k/(r) to k/(π) CerednikDrinfeld.QM.FakeEllipticCurve.Rigidification.map_hom_comp_germ_eq_germ_of_act_pow_comp_map_comp_act_pow_eq 0 below · cited by 1 · depth 36
• Transport of the Λ-action along an 𝒪_D-linear formal quotient CerednikDrinfeld.QM.FakeEllipticCurve.apply_nilEval_addVia_act_eq_pushPt_of_isODHom_of_comp_eq_nilEval 10 below · cited by 1 · depth 36
• Vanishing of ⋆-balanced alternating tensors at a ramified prime CerednikDrinfeld.QM.FakeEllipticCurve.eq_zero_of_eq_smul_tmul_sub_tmul_of_forall_map_eq_map_star_of_isRamified 1,050 below · cited by 1 · depth 36
• Vanishing of star-balanced alternating tensors for non-scalar Φ(μ) CerednikDrinfeld.QM.FakeEllipticCurve.eq_zero_of_eq_smul_tmul_sub_tmul_of_forall_map_eq_map_star_of_isSplit_of_charP 868 below · cited by 1 · depth 36
• Descent of a fake elliptic curve's abelian scheme with endomorphisms CerednikDrinfeld.QM.FakeEllipticCurve.exists_fg_subalgebra_abelianScheme_endomorphisms_isPullback 110 below · cited by 1 · depth 36
• Lifting level-one fake elliptic curves in residue characteristic q or q' CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_of_isDiscreteValuationRing_charZero_of_isAlgClosed_one_of_charP_of_eq_or_eq 2,009 below · cited by 1 · depth 36
• Lifting a level-one fake elliptic curve to a complete DVR CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullback_of_isDiscreteValuationRing_charZero_of_isAlgClosed_one_of_charP_of_ne_of_ne_two 1,828 below · cited by 1 · depth 36
• Existence of a ⋆-compatible polarisation on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_kernelPts_finite_geomFibreH0Finrank_pos_rosatiCompatible_of_isAlgClosed 930 below · cited by 2 · depth 36
• Canonical polarisation datum over an open neighbourhood of 𝔭 CerednikDrinfeld.QM.FakeEllipticCurve.exists_not_mem_isCanonicalPolData_away_of_stage_datum 1,447 below · cited by 1 · depth 36
• Spreading a canonical polarisation datum from S_𝔭 to S_{g_0} CerednikDrinfeld.QM.FakeEllipticCurve.exists_not_mem_stage_kernelIsTwoTorsion_isSymmetric_rosatiCompatible_of_isCanonicalPolData_atPrime 50 below · cited by 1 · depth 36
• Rosati-compatible sheaves are powers of a ⋆-primitive one CerednikDrinfeld.QM.FakeEllipticCurve.exists_rosatiCompatible_iso_tensorPow_tensor_forall_le_one_of_kernelPts_finite_of_charZero 726 below · cited by 1 · depth 36
• Factoring through the kernel-algebra point iff γ(s)=0 CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_kerAlgebra_iff_nilEval_eq_zero_of_isIsogenyOfHeight_of_isAlgClosed 1 below · cited by 2 · depth 36
• Kernel of a formal mathcal O_D-isogeny: subgroup, Λ-stable, r-power torsion CerednikDrinfeld.QM.FakeEllipticCurve.factorsThrough_kerAlgebra_one_mul_inv_act_nsmulPt_of_isIsogenyOfHeight_of_isAlgClosed 35 below · cited by 1 · depth 36
• Rank two and reduced traces on A₁ for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.finrank_eq_two_and_trace_restrict_eq_of_charP_of_isIndefiniteRamifiedExactlyAt_of_isUnit 866 below · cited by 2 · depth 36
• Formal isogeny kernel embeds as finite flat closed subscheme CerednikDrinfeld.QM.FakeEllipticCurve.isClosedImmersion_formalCoordinates_kerAlgebra_of_isIsogenyOfHeight_of_isAlgClosed 6 below · cited by 1 · depth 36
• Trivial kernel for ⋆-primitive Rosati-compatible bundles on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.kernelTrivial_of_forall_iso_tensorPow_tensor_le_one_of_rosatiCompatible_of_charZero 1,004 below · cited by 1 · depth 36
• Uniqueness of the canonical polarisation over ̄ k CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_of_isCanonicalPol_of_isAlgClosed 1,221 below · cited by 1 · depth 36
• Elements of Λ congruent mod mΛ agree on m-torsion CerednikDrinfeld.QM.FakeEllipticCurve.pushPt_act_eq_pushPt_act_of_sub_eq_smul_of_nsmulPt_eq_one 1 below · cited by 1 · depth 36
• Rosati compatibility lifts along small thickenings of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.rosatiCompatible_of_pullback_iso_of_rosatiCompatible_of_isPullbackVia_of_ker_mul_maximalIdeal_of_isArtinianRing 140 below · cited by 1 · depth 36
• Specialisation of Rosati compatibility for fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.rosatiCompatible_pullback_special_of_rosatiCompatible_generic_of_isDiscreteValuationRing 53 below · cited by 1 · depth 36
• Rosati compatibility: two pullbacks of the obstruction cocycle are cohomologous CerednikDrinfeld.QM.FakeEllipticCurve.unitPullback_sub_unitPullback_mem_range_d_of_rosatiCompatible_of_isPicObstructionCocycle_mumfordBundle 17 below · cited by 2 · depth 36
• Matrix representation Λ → M₂(𝒪) acting on A ×_R A CerednikDrinfeld.QM.exists_act_prod_of_algHom_matrix_of_isOrder 0 below · cited by 1 · depth 36
• Trace of an endomorphism equals the trace of its linear part CerednikDrinfeld.QM.forall_trace_eq_apply_trace_linearPart_of_isFormalCoordinates 1 below · cited by 1 · depth 36
• Tate module of a fake elliptic curve: Λ-generated ℤ_ℓ-basis CerednikDrinfeld.QM.FakeEllipticCurve.exists_basis_tateModule_eq_apply_of_isMaximalOrder_of_prime 735 below · cited by 2 · depth 37
• Algebraisation of an adic tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_fakeEllipticCurve_forall_isPullbackVia_of_tower_of_finiteBySections 445 below · cited by 2 · depth 37
• Rosati-compatible finite-by-sections invertible sheaf on a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_finiteBySections_rosatiCompatible_of_isAlgClosed 729 below · cited by 2 · depth 37
• Compatible invertible sheaves along a versal tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_compatible_finiteBySections_of_tower_versal_of_ne_of_isUnit_two 1,279 below · cited by 1 · depth 37
• Versal deformation tower for fake elliptic curves away from qq' CerednikDrinfeld.QM.FakeEllipticCurve.exists_isRegularRing_ringKrullDim_le_two_tower_isPullbackVia_versal_one_of_isAlgClosed_of_ne 1,196 below · cited by 1 · depth 37
• Λ acts ℤ_ℓ-linearly on the Tate module CerednikDrinfeld.QM.FakeEllipticCurve.exists_tateModule_linearMap_forall_apply_eq_pushPt_act 1 below · cited by 2 · depth 37
• Compatible tower of Rosati-polarised fake elliptic curves over O'/𝔪ⁿ⁺¹ CerednikDrinfeld.QM.FakeEllipticCurve.exists_tower_isPullbackVia_isInvertible_rosatiCompatible_of_isFormalModuleVia_of_forall_map_eq 1,514 below · cited by 1 · depth 37
• Rank two and quaternionic trace on Čech H¹ in characteristic p CerednikDrinfeld.QM.FakeEllipticCurve.finrank_eq_two_and_trace_restrict_eq_of_charP_of_isIndefiniteRamifiedExactlyAt 1,012 below · cited by 1 · depth 37
• Pullback relations of fake elliptic curves compose CerednikDrinfeld.QM.FakeEllipticCurve.isPullback_comp_of_isPullback_of_isPullback 0 below · cited by 1 · depth 37
• Fake elliptic curve base change along the inverse isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.isPullback_symm_of_ringEquiv_of_levelIff 0 below · cited by 1 · depth 37
• Uniqueness of symmetrised principal Rosati-compatible bundles on fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.locIsoOnBase_tensor_pullback_negMor_of_kernelTrivial_of_rosatiCompatible_of_isAlgClosed 1,092 below · cited by 1 · depth 37
• Trace on primitives of a pinned endomorphism when p∤ qq' CerednikDrinfeld.QM.FakeEllipticCurve.trace_restrict_primitives_eq_intCast_of_charP_of_not_dvd 860 below · cited by 2 · depth 37
• Simultaneous principal square roots over an algebraically closed extension CerednikDrinfeld.QM.IsCanonicalPolData.exists_isAlgClosed_forall_isPullback_exists_kernelTrivial_locIsoOnBase_pair 6 below · cited by 1 · depth 37
• A pinned Λ-action on the Hopf algebra of E[p] CerednikDrinfeld.QM.FakeEllipticCurve.exists_algHom_pinned_forall_primitives 1 below · cited by 1 · depth 38
• Algebraisation of a finite-over-P^r tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_fakeEllipticCurve_forall_isPullbackVia_of_tower_of_isFinite_proj 369 below · cited by 1 · depth 38
• Finite maps to one P^r_R along a formal tower CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFinite_proj_tower_of_finiteBySections 89 below · cited by 1 · depth 38
• Line bundles on an algebraised tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_forall_pullback_iso_of_tower_of_isPullbackVia 232 below · cited by 1 · depth 38
• Invertible sheaf finite by sections descends along a base isomorphism CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_finiteBySections_of_isPullbackVia_of_bijective 5 below · cited by 1 · depth 38
• Versal deformation tower of a level-one fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isRegularRing_tower_isPullbackVia_versal_algebra_of_isDiscreteValuationRing_of_isAlgClosed_residueField_one_of_ne 1,193 below · cited by 1 · depth 38
• Rosati-compatible Serre–Tate lifting of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_lift_isInvertible_pullback_iso_rosatiCompatible_of_isFormalModuleVia_of_isArtinianRing_of_isAlgClosed 1,513 below · cited by 1 · depth 38
• Primitives of the p-torsion Hopf algebra have rank 2 CerednikDrinfeld.QM.FakeEllipticCurve.finrank_primitives_eq_two 837 below · cited by 2 · depth 38
• Non-zero quaternionic endomorphisms of a fake elliptic curve are finite CerednikDrinfeld.QM.FakeEllipticCurve.isFinite_act_of_ne_zero 711 below · cited by 1 · depth 38
• Biadditivity of the Mumford bundle under the Λ-action CerednikDrinfeld.QM.FakeEllipticCurve.nonempty_pullback_map_act_add_mumfordBundle_iso_tensor 579 below · cited by 1 · depth 38
• Trace on primitives of E[p] equals the reduced trace CerednikDrinfeld.QM.FakeEllipticCurve.trace_restrict_primitives_eq_intCast_of_charP 1,006 below · cited by 1 · depth 38
• Algebraisation of a tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_fakeEllipticCurve_forall_isPullbackVia_of_tower_of_forall_isPullback 366 below · cited by 1 · depth 39
• Scheme-level algebraisation of a tower finite over P^r_R CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFinite_proj_forall_isPullback_of_tower_of_isFinite_proj 111 below · cited by 1 · depth 39
• Rosati-compatible invertible modules lift along deformations of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isInvertible_pullback_iso_rosatiCompatible_of_isPullbackVia_of_isArtinianRing_of_isAlgClosed 1,256 below · cited by 1 · depth 39
• Compatible P^r_R-presentations along a tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_proj_tower_of_forall_projPresentation_sigma_eq 5 below · cited by 1 · depth 39
• Non-trivial formal tower of fake elliptic curves over O[[t]] CerednikDrinfeld.QM.FakeEllipticCurve.exists_tower_powerSeries_isPullbackVia_nontrivial_of_isDiscreteValuationRing_of_isAlgClosed_residueField_one_of_ne 1,152 below · cited by 1 · depth 39
• Versality of a non-trivial tower over O[[t]] CerednikDrinfeld.QM.FakeEllipticCurve.forall_existsUnique_isPullbackVia_powerSeries_of_tower_nontrivial_of_isAlgClosed_residueField_one_of_ne 1,099 below · cited by 1 · depth 39
• Trace on primitives of E[p]: ramified case p ∣ qq' CerednikDrinfeld.QM.FakeEllipticCurve.trace_restrict_primitives_eq_intCast_of_charP_of_dvd 990 below · cited by 1 · depth 39
• Existence of a non-trivial n-torsion point, n≥ 2 CerednikDrinfeld.QM.exists_torsion_point_ne_one_of_two_le 714 below · cited by 1 · depth 39
• Rigidity: pull-back realisations over an Artinian local base coincide CerednikDrinfeld.QM.FakeEllipticCurve.eq_of_isPullbackVia_of_isPullbackVia_of_comp_eq_comp 17 below · cited by 2 · depth 40
• Points through the level subscheme are the unit when N=1 CerednikDrinfeld.QM.FakeEllipticCurve.eq_one_of_factorsThrough_lev_of_level_one 0 below · cited by 3 · depth 40
• First-order deformations of a fake elliptic curve form a line CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_fstHom_forall_existsUnique_smul_of_level_one_of_isAlgClosed_of_charP 1,062 below · cited by 2 · depth 40
• Level-one fake elliptic curves have one deformation over k CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_id_comp_eq_of_isPullbackVia_of_bijective_of_level_one 1 below · cited by 1 · depth 40
• Lifting level-one fake elliptic curves along small surjections CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_of_surjective_of_ker_mul_maximalIdeal_eq_bot_of_ne_of_isAlgClosed 1,095 below · cited by 1 · depth 40
• Fibre-product exactness for deformations of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_pullbackRing_of_isPullbackVia_of_isArtinianRing 45 below · cited by 1 · depth 40
• Algebraisation of group law and Λ-action over the tower CerednikDrinfeld.QM.FakeEllipticCurve.exists_mul_unit_inv_act_hom_forall_comp_eq_of_tower_of_forall_isPullback 225 below · cited by 1 · depth 40
• Primitives of E[p] via the special formal mathcal O_D-module CerednikDrinfeld.QM.FakeEllipticCurve.exists_specialFormalODModule_linearMap_primitives_of_ramified 834 below · cited by 1 · depth 40
• Generators of the deformation tangent line are non-trivial CerednikDrinfeld.QM.FakeEllipticCurve.not_exists_isPullbackVia_algebraMap_dualNumber_of_forall_existsUnique_smul 1 below · cited by 1 · depth 40
• Smooth proper fibres of dimension two for an algebraised tower CerednikDrinfeld.QM.FakeEllipticCurve.smooth_and_isConnected_fibre_and_topologicalKrullDim_fibre_of_tower_of_forall_isPullback_of_flat 156 below · cited by 1 · depth 40
• Drinfeld's trace condition at every geometric point of Z CerednikDrinfeld.QM.FakeEllipticCurve.trace_eq_of_tower_of_forall_isPullback_of_isCommutative_of_smoothOfRelativeDimension 23 below · cited by 1 · depth 40
• Algebraising inversion and Λ-action along a fake elliptic tower CerednikDrinfeld.QM.FakeEllipticCurve.exists_inv_act_forall_comp_eq_of_tower_of_forall_isPullback 218 below · cited by 1 · depth 41
• E[p] as truncation of the formal mathcal O_D-module, ramified case CerednikDrinfeld.QM.FakeEllipticCurve.exists_isFormalModuleVia_algEquiv_kerAlgebra_forall_apply_eq_nilEval_of_ramified 826 below · cited by 1 · depth 41
• First-order bare deformations with Λ-action are fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_fstHom_iso_of_bareDeformation_of_act 60 below · cited by 1 · depth 41
• Rigidifying an isomorphism of first-order deformations of a fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_id_of_iso_comp_eq 48 below · cited by 1 · depth 41
• Fake elliptic curves glue along B'×_B B'' CerednikDrinfeld.QM.FakeEllipticCurve.exists_isPullbackVia_pullbackFst_pullbackSnd_isPushout_of_surjective_of_isNilpotent 23 below · cited by 1 · depth 41
• Uniqueness of the glued fake elliptic curve over B'×_B B'' CerednikDrinfeld.QM.FakeEllipticCurve.exists_iso_of_isPushout_of_isPullbackVia_pullbackFst_pullbackSnd 33 below · cited by 1 · depth 41
• Algebraising the group law along a tower of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_mul_forall_coe_mul_comp_eq_lift_comp_of_tower_of_forall_isPullback 220 below · cited by 1 · depth 41
• Algebraising the unit sections of a formal fake elliptic curve CerednikDrinfeld.QM.FakeEllipticCurve.exists_section_forall_coe_one_comp_eq_of_tower_of_forall_isPullback 219 below · cited by 1 · depth 41
• Group law and Λ-action identities on the algebraised model CerednikDrinfeld.QM.FakeEllipticCurve.mul_assoc_comm_and_act_identities_of_forall_comp_eq_of_tower_of_forall_isPullback 220 below · cited by 1 · depth 41
• Trace condition for a lattice action via a trace function CerednikDrinfeld.QM.LatticeAction.forall_trace_eq_iff_forall_apply_eq_of_smoothOfRelativeDimension 6 below · cited by 1 · depth 41
• Trace of a lattice action over a local base CerednikDrinfeld.QM.exists_forall_trace_eq_apply_of_isLocalRing_of_smooth_of_isCommutative 7 below · cited by 1 · depth 41
• Drinfeld's trace condition holds automatically in characteristic zero CerednikDrinfeld.QM.trace_eq_intCast_of_charZero_of_smoothOfRelativeDimension_two 6 below · cited by 1 · depth 41
• A cubic relation for the trace of a quaternionic differential CerednikDrinfeld.QM.trace_sub_mul_sq_sub_eq_zero_of_smoothOfRelativeDimension_two 4 below · cited by 1 · depth 41
• Glued group law and Λ-action on a pushout of fake elliptic curves CerednikDrinfeld.QM.FakeEllipticCurve.exists_mul_unit_inv_act_forall_comp_eq_of_isPushout_of_isPullbackVia 17 below · cited by 1 · depth 42
• Comparison morphism of two gluings is a homomorphism CerednikDrinfeld.QM.FakeEllipticCurve.mapPt_mul_and_factorsThrough_iff_of_isPushout_of_comp_eq_of_comp_eq 13 below · cited by 1 · depth 42
• Gluing the multiplication morphism over a pushout of total spaces CerednikDrinfeld.QM.FakeEllipticCurve.exists_mul_forall_coe_mul_comp_eq_lift_comp_of_isPushout_of_isPullbackVia 13 below · cited by 1 · depth 43
• Gluing unit, inversion and Λ-action over a pushout CerednikDrinfeld.QM.FakeEllipticCurve.exists_unit_inv_act_forall_comp_eq_of_isPushout_of_isPullbackVia 1 below · cited by 1 · depth 43
• Group law and Λ-action axioms over the fibre ring CerednikDrinfeld.QM.FakeEllipticCurve.mul_assoc_comm_and_act_identities_of_forall_comp_eq_of_isPushout_of_isPullbackVia 13 below · cited by 1 · depth 43