Namespace LanglandsTunnell 1,543 theorems
Landmarks here: Weight-one form attached to a surjective mod-3 representation
— 215 · ArchBessel 11 · ArchPlace 7 · Artin 11 · Converse 173 · CubicInduction 656 · CubicLambda 9 · ExplicitLift 4 · HeckeTate 5 · P2 21 · RankinSelberg 339 · RealArchParam 1 · TateLocal 91
directly in LanglandsTunnell 215
- Langlands–Tunnell with controlled level for tamely ramified ρ̄
LanglandsTunnell.exists_isWeightOneChiNegThreeRealized_not_nine_dvd_not_cube_dvd_of_natCard_inertia_eq_two_of_coprime6,900 below · cited by 1 · depth 8 - Determinant of the explicit lift equals χ₋₃ at Frobenius
LanglandsTunnell.det_lift_eq_chiNegThree_of_isFrobeniusAt1 below · cited by 2 · depth 9 - landmark Weight-one form attached to a surjective mod-3 representation
LanglandsTunnell.exists_isWeightOneChiNegThreeRealized_eq_trace_lift6,804 below · cited by 2 · depth 9 - Langlands–Tunnell with cube-free level away from 3
LanglandsTunnell.exists_isWeightOneChiNegThreeRealized_three_dvd_not_cube_dvd_of_coprime6,896 below · cited by 1 · depth 9 - Trivial inertia above all but finitely many rational primes
LanglandsTunnell.exists_finset_forall_inertia_eq_bot0 below · cited by 11 · depth 10 - Continuous surjective mod-3 representations come from finite Galois towers
LanglandsTunnell.exists_galoisTower_of_continuous_surjective0 below · cited by 1 · depth 10 - Langlands–Tunnell: cusp form attached to a mod-3 representation
LanglandsTunnell.exists_isWeightOneChiNegThreeRealized_eq_trace_lift_cuspForm6,806 below · cited by 1 · depth 10 - Descent of Deligne–Serre output to a ℤ[√-2]-valued eigensystem
LanglandsTunnell.exists_isWeightOneChiNegThreeRealized_of_deligneSerre_output8 below · cited by 1 · depth 10 - Lift-valued Hecke system and its χ₋₃-twist are both cuspidal
LanglandsTunnell.exists_liftValued_isCusp_pair_of_detDictionaryRow6,801 below · cited by 1 · depth 10 - Determinant of the lifted mod 3 representation at Frobenius
LanglandsTunnell.det_map_comp_lift_eq_chiNegThree_of_isFrobeniusAt0 below · cited by 1 · depth 11 - Lift-valued cuspidal Hecke system for octahedral GL₂(𝔽₃)-extensions
LanglandsTunnell.exists_liftValued_isCusp_pair_of_detDictionaryRow_of_coversModCentre6,800 below · cited by 1 · depth 11 - Frobenius trace on inertia invariants lies in ι(ℤ[√-2])
LanglandsTunnell.trace_restrict_invariants_mem_range_of_lift0 below · cited by 1 · depth 11 - Octahedral Langlands–Tunnell over ℚ at cubic-resolvent grain
LanglandsTunnell.exists_agreesLiftTraceSeed_isCusp_pair_of_detDictionaryRow_of_coversModCentre5,218 below · cited by 1 · depth 12 - Lift-valued Frobenius table from cubic base-change agreement
LanglandsTunnell.exists_liftValued_of_agreesLiftTraceSeed_isCusp_pair2,325 below · cited by 1 · depth 12 - Quadratic base-change fibre over the cubic resolvent
LanglandsTunnell.agreesAwayFromFinite_or_twist_bcWeight_of_formalBaseChange_agree_sylowH898 below · cited by 1 · depth 13 - Automorphic induction of a ray class character of E/F
LanglandsTunnell.exists_agreesAwayFromFinite_isGenuineCusp_of_raySymbol_eq_one355 below · cited by 1 · depth 13 - Cubic base change to the Sylow fixed field of GL₂(𝔽₃)
LanglandsTunnell.exists_agreesFormalBaseChange_arithGenuineCuspRealizable_sylowH_of_quatH_of_unitary_resolvent2,556 below · cited by 1 · depth 13 - Boundedness of aₚ from agreement with the lift-trace seed
LanglandsTunnell.exists_forall_norm_a_le_of_formalBaseChange_agrees_liftTraceSeed1 below · cited by 1 · depth 13 - Weight-one holomorphic descent along a non-Galois cubic base change
LanglandsTunnell.exists_genuineCuspRealization_weightOne_of_formalBaseChange_cubic_of_not_isGalois_of_not_agreesAwayFromFinite_twist2,926 below · cited by 1 · depth 13 - Quadratic descent to ℚ of a cusp-realizable Hecke eigensystem
LanglandsTunnell.exists_isArithBoundedGenuineCuspRealizable_pair_agrees_liftTraceSeed_quatH3,136 below · cited by 1 · depth 13 - Cubic descent of the lift-trace seed to the determinant-kernel field
LanglandsTunnell.exists_isConstantOnFibers_b_formalBaseChange_arithBoundedGenuineCuspRealizable_detKer_of_quatH3,359 below · cited by 1 · depth 13 - Ray class character realising the Q₈ seed table
LanglandsTunnell.exists_quadratic_rayClassChar_table_liftTraceSeed_quatH_of_detDictionaryRow96 below · cited by 1 · depth 13 - Seed table over the cubic resolvent as a theta table
LanglandsTunnell.exists_quadratic_rayClassChar_table_liftTraceSeed_sylowH_of_detDictionaryRow93 below · cited by 1 · depth 13 - Resolvent sign character and non-self-twist guard for GL₂(𝔽₃) towers
LanglandsTunnell.exists_resolventSign_not_agreesAwayFromFinite_twist_sylowH_of_liftTraceSeed_quatH27 below · cited by 1 · depth 13 - Frobenius read-off at an unramified prime, up to the cubic partner
LanglandsTunnell.face_liftValuedUpToPartner_of_b_agreesAt_liftTraceSeed_detKer_sylowH0 below · cited by 2 · depth 13 - Determinant table of the lift-trace seed equals χ₋₃ of the norm
LanglandsTunnell.liftTraceSeed_b_eq_chiNegThree_of_detDictionaryRow2 below · cited by 3 · depth 13 - The fixed field of `sylowH` is not Galois over ℚ
LanglandsTunnell.not_isGalois_fixFld_sylowH0 below · cited by 1 · depth 13 - Holomorphic weight-one descent along a cubic base change
LanglandsTunnell.exists_genuineRealization_archWeightOne_holomorphic_of_formalBaseChange_cubic_of_not_isGalois_of_not_agreesAwayFromFinite_twist2,925 below · cited by 1 · depth 14 - Spherical Hecke coset system away from the level
LanglandsTunnell.exists_heckeCosetSystem_productionPinsCompact_of_not_dvd0 below · cited by 2 · depth 14 - Hecke coset system at a place prime to the level
LanglandsTunnell.exists_heckeCosetSystem_productionPinsGeneral_of_not_dvd0 below · cited by 4 · depth 14 - Twisting a bounded genuine cuspidal eigensystem by a Hecke character
LanglandsTunnell.exists_isArithBoundedGenuineCuspRealizable_twist_centreCut19 below · cited by 3 · depth 14 - Determinant eigenvalues come from a finite-order Hecke character
LanglandsTunnell.exists_isFiniteOrderHeckeChar_det_heckeGen_eq_b_of_isArithGenuineCuspRealizable2 below · cited by 1 · depth 14 - Automorphic induction of a ray class symbol to weight one
LanglandsTunnell.exists_isGenuineCusp_archWeightOne_a_eq_of_raySymbol_eq_prod_of_finrank_eq_two353 below · cited by 1 · depth 14 - Niceness of generic twisted base-change data over a cubic field
LanglandsTunnell.exists_isNicePinned_twistedDatum_formalBaseChange_superset_generic_of_norm_eq_one_of_summable2,507 below · cited by 1 · depth 14 - Sign character of a GL₂(𝔽₃)-tower over ℚ
LanglandsTunnell.exists_resolventSignChar_sylowH1 below · cited by 1 · depth 14 - Split primes with distinct order-four Artin values
LanglandsTunnell.exists_split_place_artinValue4_ne17 below · cited by 1 · depth 14 - Split places with distinct Artin values above them
LanglandsTunnell.exists_split_place_artinValue_ne17 below · cited by 1 · depth 14 - The fixed field of `sylowH` is cubic over ℚ
LanglandsTunnell.finrank_fixFld_sylowH0 below · cited by 1 · depth 14 - Induced trace identities for the quaternionic lift seed
LanglandsTunnell.liftTraceSeed_quatH_table_eq_artinValue41 below · cited by 1 · depth 14 - Induced-character table for the lift-trace seed over the cubic field
LanglandsTunnell.liftTraceSeed_sylowH_table_eq_artinValue1 below · cited by 1 · depth 14 - Base change to the `sylowH` fixed field is not Eisenstein
LanglandsTunnell.not_agreesAwayFromFinite_formalBaseChange_sylowH_eisensteinTableOf_of_quatH220 below · cited by 1 · depth 14 - No self-twist by the determinant sign character
LanglandsTunnell.not_agreesAwayFromFinite_twist_resolventSign_of_liftTraceSeed_quatH24 below · cited by 1 · depth 14 - Cubic base change: archimedean types and weight-one holomorphy
LanglandsTunnell.archOccursInClassOf_formalBaseChange_iff_of_finrank_eq_three_of_not_agreesAwayFromFinite_twist2,919 below · cited by 1 · depth 15 - Per-place Artin value transfer across the C₄ ⊂ C₈ step
LanglandsTunnell.artinValue4_eq_artinValue_under_pow0 below · cited by 1 · depth 15 - Non-vanishing Gauss-sum twist at some admitted modulus
LanglandsTunnell.exists_admitsModulus_gaussSumFn_ne_zero14 below · cited by 2 · depth 15 - Lift-trace seed vanishes above primes with e(σ) of order eight
LanglandsTunnell.exists_finset_liftTraceSeed_quatH_a_eq_zero_of_orderOf_eq_eight2 below · cited by 1 · depth 15 - Unramified prime with order-eight Frobenius in a GL₂(𝔽₃)-tower
LanglandsTunnell.exists_inertia_eq_bot_isArithFrobAt_orderOf_eq_eight17 below · cited by 1 · depth 15 - Automorphic induction of a quadratic Hecke character, weight one
LanglandsTunnell.exists_isGenuineCusp_archWeightOne_a_eq_of_isFiniteOrderHeckeChar_of_finrank_eq_two352 below · cited by 1 · depth 15 - Gauss-sum twist of a cuspidal function on GL₂
LanglandsTunnell.fnTwist_gaussSumFn_isSmoothCuspData0 below · cited by 2 · depth 15 - Twisted Gauss-sum combination is invariant at level Nf²
LanglandsTunnell.fnTwist_gaussSumFn_level_invariant0 below · cited by 4 · depth 15 - Quaternion-layer formal base change at Frobenius of order eight
LanglandsTunnell.formalBaseChange_quatH_a_eq_of_orderOf_eq_eight2 below · cited by 1 · depth 15 - Hecke eigenvalue of a twisted Gauss-sum combination
LanglandsTunnell.isHeckeCosetEigenfunctionAt_fnTwist_gaussSumFn2 below · cited by 3 · depth 15 - Cubic base change: archimedean class-level ascent, non-self-twist case
LanglandsTunnell.archOccursInClassOf_formalBaseChange_of_archOccursInClassOf_of_finrank_eq_three_of_not_agreesAwayFromFinite_twist2,912 below · cited by 1 · depth 16 - Entire twisted L-functions of an induced Hecke eigensystem
LanglandsTunnell.exists_differentiable_hasProd_eulerProduct_induced_twist_of_isFiniteOrderHeckeChar_of_finrank_eq_two70 below · cited by 1 · depth 16 - Determinant character of a Hecke character induced from a quadratic extension
LanglandsTunnell.exists_isAdmissibleTwist_apply_uniformizerIdele_eq_det_induced_of_isFiniteOrderHeckeChar_of_finrank_eq_two131 below · cited by 1 · depth 16 - Induced datum from a finite-order Hecke character is nicely pinned
LanglandsTunnell.exists_isNicePinned_twistedDatum_induced_of_isFiniteOrderHeckeChar_of_finrank_eq_two128 below · cited by 1 · depth 16 - Archimedean parameters and Whittaker factorisation of a cusp realisation over ℚ
LanglandsTunnell.exists_realArchParam_whittaker_factorization_apply_one_ne_zero_localSpaceAt_of_continuous_realization450 below · cited by 2 · depth 16 - Rigidity of holomorphy for weight-one realisations at a real place
LanglandsTunnell.isArchHolomorphicAt_of_agreesAwayFromFinite_of_weightOne_of_coversModCentre509 below · cited by 2 · depth 16 - Frobenius density for a division of an order-8 class
LanglandsTunnell.towerDirichletDensity_add_of_orderOf_eq_eight15 below · cited by 1 · depth 16 - Archimedean transfer of cubic base change at a real place
LanglandsTunnell.archOccursInClassOf_formalBaseChange_archCasimirAt_of_archOccursInClassOf_of_finrank_eq_three_of_not_agreesAwayFromFinite_twist2,877 below · cited by 1 · depth 17 - Central character of formal base change at a real place
LanglandsTunnell.centralChar_archCentralUnit_eq_of_agreesAwayFromFinite_formalBaseChange_of_isReal14 below · cited by 1 · depth 17 - Value of χ_A·|·|_A at det of Hecke generators
LanglandsTunnell.dirichletIdeleChar_mul_modulus_det_gen2 below · cited by 2 · depth 17 - Minimal-weight Casimir eigenvector inside one cuspidal constituent
LanglandsTunnell.exists_archCasimir_eigenvector_minimalWeight_mem_isCuspConstituent_whittaker_diagOne_ne_zero_of_continuous_realization408 below · cited by 1 · depth 17 - Inductivity of conductor and root number for a quadratic extension
LanglandsTunnell.exists_heckeRootNumber_eq_mul_pinnedRootNumber_and_heckeConductor_eq_induced_of_finrank_eq_two49 below · cited by 1 · depth 17 - Non-triviality of ξ·(χ∘ N) on the norm-one ideles
LanglandsTunnell.exists_mem_normOneIdeles_mul_comp_idelicNorm_ne_one_of_finrank_eq_two61 below · cited by 1 · depth 17 - Weight family and Whittaker factorisation with C(1,1)≠ 0
LanglandsTunnell.exists_whittaker_factorization_apply_one_ne_zero_localSpaceAt_of_archCasimir_eigenvector_minimalWeight373 below · cited by 2 · depth 17 - Whittaker factorisation at an odd principal archimedean parameter over ℚ
LanglandsTunnell.exists_whittaker_factorization_apply_one_ne_zero_localSpaceAt_of_archCasimir_eigenvector_weightOne_of_ne370 below · cited by 1 · depth 17 - Weight-one adelic lift is bounded on Siegel windows
LanglandsTunnell.isBoundedOnSiegelWindows_weightOneLift0 below · cited by 2 · depth 17 - Cuspidality of the adelic weight-one lift
LanglandsTunnell.isCuspidalFn_weightOneLift12 below · cited by 2 · depth 17 - Adelic weight-one lift is a Hecke coset eigenfunction
LanglandsTunnell.isHeckeCosetEigenfunctionAt_weightOneLift7 below · cited by 2 · depth 17 - Smoothness of the adelic weight-one lift
LanglandsTunnell.isKfSmooth_weightOneLift6 below · cited by 2 · depth 17 - Central law of the weight-one lift at Hecke generators
LanglandsTunnell.weightOneLift_centralScalar_det_gen_mul12 below · cited by 2 · depth 17 - Central character of the adelic weight-one lift
LanglandsTunnell.weightOneLift_centralScalar_mul11 below · cited by 3 · depth 17 - Artin induction of L- and Γ-factors in a quadratic extension
LanglandsTunnell.wellFormed_converges_twistedDatum_and_archFactor_lFun_heckeDatum_eq_induced_of_finrank_eq_two12 below · cited by 1 · depth 17 - Archimedean type profile of a class from its minimal type
LanglandsTunnell.archOccursInClassOf_archCasimirAt_iff_of_archOccursInClassOf_minimalType_laplaceEigenvalue_of_coversModCentre394 below · cited by 3 · depth 18 - Converse theorem at the base change of a real archimedean parameter
LanglandsTunnell.archOccursInClassOf_whittakerCoefficient_fibre_eq_archW_archOfParam_of_forall_isNicePinned120 below · cited by 1 · depth 18 - Fibre Whittaker factorisation transports along a norm twist
LanglandsTunnell.archOccursInClassOf_whittakerCoefficient_fibre_eq_archW_twist_of_archOccursInClassOf_rat11 below · cited by 2 · depth 18 - Real archimedean agreement of central characters under base change
LanglandsTunnell.centralChar_archCentralUnit_eq_of_centralChar_uniformizer_pow_inertiaDeg_productionPinsOf_of_isReal13 below · cited by 1 · depth 18 - Minimal-weight Casimir eigenvector for a continuous cuspidal realization over ℚ
LanglandsTunnell.exists_archCasimir_eigenvector_minimalWeight_of_continuous_realization344 below · cited by 1 · depth 18 - Casimir eigenvalue from a Whittaker factorisation on one finite fibre
LanglandsTunnell.exists_archOccursInClassOf_archCasimirAt_laplaceEigenvalue_of_whittakerCoefficient_fibre_eq359 below · cited by 5 · depth 18 - Pinned niceness of twisted L-data of a cubic formal base change
LanglandsTunnell.exists_forall_isNicePinned_twistedDatum_formalBaseChange_archOfParam_of_whittakerCoefficient_fibre_eq_archW_of_not_agreesAwayFromFinite_twist_of_isCasimirEigen2,832 below · cited by 1 · depth 18 - Twist-stability of arithmetic genuine cusp-realizability on a covering window
LanglandsTunnell.exists_isArithGenuineCuspRealizable_twist_of_coversModCentre_centreCut94 below · cited by 1 · depth 18 - Isotypic cusp form replaced inside one cuspidal constituent
LanglandsTunnell.exists_isCuspConstituent_mem_isIsotypicCuspFormAt_of_isIsotypicCuspFormAt_of_rightConv_eq340 below · cited by 2 · depth 18 - Whittaker non-vanishing on the torus, with J-rigidity transfer
LanglandsTunnell.exists_mem_isCuspConstituent_isIsotypicCuspFormAt_whittakerCoefficient_diagOne_ne_zero_J_rigid_of_hasArchCharacterAt224 below · cited by 3 · depth 18 - Archimedean parameter and Whittaker datum of a cuspidal class over ℚ
LanglandsTunnell.exists_realArchParam_archDatumR_whittakerCoefficient_fibre_eq_isCasimirEigen_of_archOccursInClassOf_rat464 below · cited by 1 · depth 18 - Twisting a bounded genuine cusp realisation by a finite-order Hecke character
LanglandsTunnell.exists_smoothCuspRealizationAt_fnTwist_gaussSumFn_centreCut19 below · cited by 1 · depth 18 - Pure-tensor factorisation of a Whittaker function over ℚ
LanglandsTunnell.exists_whittakerCoefficient_eq_archWhittaker_mul_finWhittaker_of_isIsotypicCuspFormAt3 below · cited by 3 · depth 18 - Whittaker factorization for reflected-lowering eigencombinations at weight one
LanglandsTunnell.exists_whittaker_factorization_add_smul_reflect_lower_of_archCasimir_eigenvector_weightOne_of_ne363 below · cited by 2 · depth 18 - Whittaker factorisation of a minimal-weight Casimir eigenvector over ℚ
LanglandsTunnell.exists_whittaker_factorization_eq_or_eq_smul_raise_of_archCasimir_eigenvector_minimalWeight367 below · cited by 1 · depth 18 - Local Whittaker relations at a good place over ℚ
LanglandsTunnell.finWhittaker_unipotent_levelOne_hecke_centre_of_isIsotypicCuspFormAt1 below · cited by 3 · depth 18 - Formal base change commutes with norm twists
LanglandsTunnell.formalBaseChange_twist_rpow_absNorm_agreesAwayFromFinite1 below · cited by 1 · depth 18 - Archimedean Γ-factors for a quadratic extension of number fields
LanglandsTunnell.prod_gammaR_mul_prod_gammaC_infinitePlace_induced_eq_of_finrank_eq_two0 below · cited by 1 · depth 18 - Strict Satake bound at good places with unitary determinant
LanglandsTunnell.satake_norm_lt_sqrt_absNorm_of_norm_b_eq_one22 below · cited by 3 · depth 18 - Formal base change commutes with twisting
LanglandsTunnell.agrees_formalBaseChange_twist0 below · cited by 1 · depth 19 - Weight, lowering and raising relations in torus coordinates
LanglandsTunnell.archDerivAt_E_sub_Fm_eq_and_splitTorus_lowering_raising_relations_of_hasArchCharacterAt0 below · cited by 2 · depth 19 - Whittaker coefficients match a model datum up to sign twist
LanglandsTunnell.archOccursInClassOf_whittakerCoefficient_fibre_eq_archW_or_twist_sign_of_archOccursInClassOf_rat420 below · cited by 1 · depth 19 - General-pins Whittaker link from a Casimir-eigen minimal-weight datum
LanglandsTunnell.exists_agreesAwayFromFinite_isArithGenuineCuspRealizable_twist_whittaker_link_localSpaceAt_of_whittakerCoefficient_fibre_eq_archW_of_isCasimirEigen550 below · cited by 1 · depth 19 - Admissible twist matching the unitary formal base change of Φ
LanglandsTunnell.exists_isAdmissibleTwist_eq_twist_formalBaseChange_b_isArchCompAt_archOfParam_of_whittakerCoefficient_fibre_eq_archW40 below · cited by 1 · depth 19 - Pinned niceness of twisted base-change L-data over cubic fields
LanglandsTunnell.exists_isNicePinned_twistedDatum_formalBaseChange_archOfParam_superset_generic_of_whittaker_factorization_of_norm_eq_one_of_summable_of_localSpaceAt2,507 below · cited by 1 · depth 19 - Mellin transform of a weight-one Whittaker profile
LanglandsTunnell.exists_mellin_whittakerProfile_eq_archFactor_of_whittaker_ode_weightOne6 below · cited by 5 · depth 19 - Some cuspidal constituent meets the isotypic archimedean-type space
LanglandsTunnell.exists_mem_finset_inf_isotypicCuspSubmodule_inf_archCutSubmodule_ne_bot_of_mem_sup7 below · cited by 1 · depth 19 - Existence of a real archimedean parameter for a cuspidal class
LanglandsTunnell.exists_realArchParam_archOccursInClassOf_minimalType_laplaceEigenvalue_of_coversModCentre394 below · cited by 3 · depth 19 - Bi-finite isotypic smoothing of a continuous cuspidal realization
LanglandsTunnell.exists_rightConv_ne_zero_mem_isotypicCuspSubmodule_mem_archCutSubmodule80 below · cited by 2 · depth 19 - Nonvanishing first Whittaker coefficient at a real torus point
LanglandsTunnell.exists_whittakerCoefficient_diagOne_archUnitHom_mul_ne_zero_of_isIsotypicCuspFormAt24 below · cited by 2 · depth 19 - Whittaker factorisation for a weight-zero cusp form and its raising
LanglandsTunnell.exists_whittaker_factorization_self_and_smul_raise_of_archCasimir_eigenvector_weightZero364 below · cited by 1 · depth 19 - Formal base change commutes with norm twists, primewise
LanglandsTunnell.formalBaseChange_twist_rpow_absNorm_a_eq_and_b_eq0 below · cited by 1 · depth 19 - Archimedean derivatives and Casimir commute with Whittaker integration
LanglandsTunnell.isArchSmoothAt_whittakerCoefficient_and_archDerivAt_comm0 below · cited by 12 · depth 19 - Raising operator: isotypy, weight k+2, Whittaker coefficients
LanglandsTunnell.isIsotypicCuspFormAt_smul_archRaise_and_whittakerCoefficient_archRaise_archLower340 below · cited by 2 · depth 19 - Mellin transform of a lowering-annihilated Whittaker profile is Γ_ℂ
LanglandsTunnell.mellin_whittakerProfile_eq_GammaC_of_lowering_eq_zero0 below · cited by 4 · depth 19 - Strict bound on Satake parameters away from level and exceptional set
LanglandsTunnell.satake_norm_lt_sqrt_absNorm_of_not_dvd_level_of_not_mem_exceptionalSet20 below · cited by 2 · depth 19 - Lowering operator forces Whittaker vanishing on the negative torus
LanglandsTunnell.whittakerCoefficient_diagOne_neg_eq_zero_of_isIsotypicCuspFormAt_of_lowering_eq_zero102 below · cited by 1 · depth 19 - Torus structure of the first Whittaker coefficient over ℚ
LanglandsTunnell.whittakerCoefficient_splitTorus_structure_of_isIsotypicCuspFormAt_of_archCasimirAt_eq102 below · cited by 3 · depth 19 - Reflection J acts by (-1)^{a₁} on the weight-zero class
LanglandsTunnell.archOccursInClassOf_archWeightChar_zero_archCasimirAt_apply_mul_J_eq_neg_one_pow_of_whittakerCoefficient_fibre_eq_archW_of_isCasimirEigen386 below · cited by 3 · depth 20 - Selection of a minimal-weight cuspidal constituent with nonvanishing Whittaker vector
LanglandsTunnell.exists_agreesAwayFromFinite_twist_archCasimir_eigenvector_minimalWeight_mem_isCuspConstituent_whittaker_diagOne_ne_zero_of_whittakerCoefficient_fibre_eq_archW_of_isCasimirEigen484 below · cited by 1 · depth 20 - Selecting a weight-one cusp form: odd principal case
LanglandsTunnell.exists_agreesAwayFromFinite_twist_archCasimir_eigenvector_weightOne_whittakerCoefficient_torus_eq_archW_mem_isCuspConstituent_whittaker_diagOne_ne_zero_of_whittakerCoefficient_fibre_eq_archW_of_ne_of_ne535 below · cited by 1 · depth 20 - Unitary archimedean datum forces ‖bₚ‖ = Np almost everywhere
LanglandsTunnell.exists_finset_norm_b_eq_absNorm_of_whittakerCoefficient_fibre_eq_archW_of_re_centralExponent_eq_zero10 below · cited by 4 · depth 20 - Weight-one Whittaker factorisation over the torus fibre
LanglandsTunnell.exists_whittaker_factorization_apply_one_ne_zero_localSpaceAt_of_archCasimir_eigenvector_weightOne_of_ne_of_torus_profile_eigen370 below · cited by 1 · depth 20 - Moderate-growth solutions of the Whittaker equation are dependent
LanglandsTunnell.linearDependent_of_whittaker_ode_of_moderateGrowth_complexParam0 below · cited by 14 · depth 20 - Weight-one lowering operator on shifted Gaussian convolutions
LanglandsTunnell.lowering_principal_profile_sum_eq0 below · cited by 1 · depth 20 - Mellin transform of a Gaussian convolution equals Γ_ℝΓ_ℝ
LanglandsTunnell.mellin_mulConvGaussian_eq_archFactor_principal0 below · cited by 6 · depth 20 - Mellin transforms of the weight-zero and weight-two torus profiles
LanglandsTunnell.mellin_whittakerProfile_eq_archFactor_of_whittaker_ode_weightZero5 below · cited by 4 · depth 20 - Exponential decay of the Gaussian multiplicative convolution
LanglandsTunnell.norm_mulConvGaussian_le_rpow_max_mul_exp0 below · cited by 2 · depth 20 - Non-vanishing of the principal-series archimedean profile
LanglandsTunnell.principal_profile_exists_ne_zero1 below · cited by 2 · depth 20 - Weight-one Whittaker equation for the summed Gaussian convolutions
LanglandsTunnell.principal_profile_sum_solves_whittaker_ode_weightOne0 below · cited by 1 · depth 20 - Whittaker's ODE on the split torus at a real place
LanglandsTunnell.whittaker_ode_splitTorus_of_isArchSmoothAt_of_archCasimirAt_eq0 below · cited by 6 · depth 20 - Transfer of a cleared functional equation to an equal rational factor
LanglandsTunnell.clearedFE_of_clearedFE_of_forall_eval_mul_eval_mul_cpow_eq0 below · cited by 4 · depth 21 - Polynomially bounded solutions of f'=(α/t+β)f vanish when Reβ>0
LanglandsTunnell.eq_zero_of_deriv_eq_div_add_mul_of_re_pos_of_isBigO_pow0 below · cited by 1 · depth 21 - Descent to a cuspidal constituent keeping a Whittaker non-vanishing
LanglandsTunnell.exists_isCuspConstituent_mem_isIsotypicCuspFormAt_of_isIsotypicCuspFormAt_of_rightConv_eq_whittakerCoefficient_add_smul_reflect_lower_ne_zero340 below · cited by 1 · depth 21 - Whittaker fibre of a weight-one cusp form is a multiple of W_∞
LanglandsTunnell.exists_whittakerCoefficient_fibre_eq_archW_mul_of_apply_mul_archRealGLAt_J_eq_mul_lower_of_mem_isCuspConstituent_weightOne_of_ne_bot424 below · cited by 1 · depth 21 - Whittaker factorisation for a minimal-weight Casimir eigenvector
LanglandsTunnell.exists_whittaker_factorization_of_archCasimir_eigenvector_minimalWeight361 below · cited by 1 · depth 21 - Weight-one Whittaker factorisation with pinned fibre eigenvalue
LanglandsTunnell.exists_whittaker_factorization_of_archCasimir_eigenvector_weightOne_of_ne_of_fibre_profile_eigen361 below · cited by 1 · depth 21 - Uniqueness of the factor in a cleared functional equation
LanglandsTunnell.forall_eval_mul_eval_mul_cpow_eq_of_clearedFE_of_clearedFE_of_ne_zero0 below · cited by 2 · depth 21 - Even principal profile solves the Whittaker equation
LanglandsTunnell.principal_profile_solves_whittaker_ode0 below · cited by 1 · depth 21 - Vanishing of moderate-growth Whittaker solutions of wrong sign
LanglandsTunnell.whittaker_ode_neg_weight_eq_zero_of_moderateGrowth_of_mellin_eq_GammaC_mul6 below · cited by 2 · depth 21 - Gamma-factor rigidity forces vanishing of the leading term at 0
LanglandsTunnell.eq_zero_of_mellin_eq_GammaC_mul_of_sub_rpow_bound_near_zero0 below · cited by 1 · depth 22 - Power bound near zero for solutions of the Whittaker equation
LanglandsTunnell.exists_rpow_bound_near_zero_of_whittaker_ode_of_abs_re_lt_half0 below · cited by 2 · depth 22 - Near-zero power bound for moderate-growth Whittaker solutions
LanglandsTunnell.exists_rpow_bound_near_zero_of_whittaker_ode_of_discrete_tower_of_moderateGrowth1 below · cited by 2 · depth 22 - Trace-and-norm counting in a cubic extension of a finite field
LanglandsTunnell.ncard_charpoly_coeff_pair_eq_ncard_symm_pair_of_finrank_eq_three0 below · cited by 1 · depth 22 - Hasse–Davenport lifting relation in degree three
LanglandsTunnell.sum_mulChar_norm_mul_addChar_trace_eq_gaussSum_pow_three_of_finrank_eq_three0 below · cited by 1 · depth 22 - Logarithmic two-term behaviour at 0 for the Whittaker equation
LanglandsTunnell.whittaker_ode_exists_sub_log_mul_sqrt_bound_near_zero_of_zero0 below · cited by 1 · depth 22 - Behaviour at 0 of solutions of Whittaker's equation, half-integral ν
LanglandsTunnell.whittaker_ode_exists_sub_mul_rpow_bound_near_zero_of_half_integer1 below · cited by 1 · depth 22 - Vanishing of decaying L¹-Mellin solutions of the negative-weight Whittaker equation
LanglandsTunnell.whittaker_ode_neg_weight_eq_zero_of_tendsto_zero_of_mellinConvergent0 below · cited by 1 · depth 22 - Vanishing of ν=0 wrong-sign Whittaker solutions
LanglandsTunnell.whittaker_ode_neg_weight_zero_param_eq_zero_of_tendsto_of_mellinConvergent0 below · cited by 1 · depth 22 - Whittaker's equation on the split torus from the Casimir eigen-equation
LanglandsTunnell.whittaker_ode_splitTorus_of_casimir_of_archWeightChar_of_unipotent0 below · cited by 1 · depth 22 - Bargmann-type constraint on λ from a weight set
LanglandsTunnell.casimir_pos_or_discrete_or_zero_of_weightSet0 below · cited by 2 · depth 23 - Uniform decay of a bounded SU(2)-string of torus Whittaker functions
LanglandsTunnell.exists_norm_le_mul_rpow_of_torus_system_even_casimir_real_nonpos_of_bounded2 below · cited by 1 · depth 23 - Uniform decay near 0 for an SU(2)-string Whittaker system
LanglandsTunnell.exists_pos_norm_le_mul_rpow_of_torus_system_casimir_eq_real_pos2 below · cited by 1 · depth 23 - Gaussian moments as derivatives of the Gaussian
LanglandsTunnell.integral_ofReal_pow_mul_exp_neg_pi_mul_sq_mul_cexp_eq_iteratedDeriv0 below · cited by 31 · depth 23 - Moderate-growth solutions of the Whittaker equation are dependent
LanglandsTunnell.linearDependent_of_whittaker_ode_of_moderateGrowth0 below · cited by 2 · depth 23 - Mellin transform of w^N e^-b(w²+w⁻²): convergence and positivity
LanglandsTunnell.mellinConvergent_and_mellin_ofReal_pos_rpow_mul_exp_neg_mul_sq_add_inv_sq0 below · cited by 7 · depth 23 - Near-zero bounds for solutions of Whittaker's equation
LanglandsTunnell.norm_le_mul_rpow_half_sub_abs_re_near_zero_of_whittaker_ode0 below · cited by 2 · depth 23 - Near-zero bounds for the forced K-Bessel equation
LanglandsTunnell.norm_le_mul_rpow_near_zero_of_bessel_ode_of_forcing_of_apriori0 below · cited by 2 · depth 23 - Decay O(y^{1-ε}) for a complex-place torus Whittaker system
LanglandsTunnell.norm_le_mul_rpow_of_torus_system_of_casimir_ne_of_re_eq1 below · cited by 1 · depth 23 - Gaussian convolution profile from principal archimedean Mellin data
LanglandsTunnell.add_pow_mul_apply_neg_eq_mul_mulConvGaussian_of_mellin_eq_archFactor3 below · cited by 10 · depth 24 - Discrete Whittaker profile solves the weight-(n+1) Whittaker equation
LanglandsTunnell.discrete_profile_solves_whittaker_ode0 below · cited by 2 · depth 24 - Uniqueness of coefficients of a finite sum sum N^{-iu}aᵢ
LanglandsTunnell.eq_of_forall_finsum_cpow_neg_mul_eq0 below · cited by 2 · depth 24 - Archimedean triple integral as tfrac12Γ_ℝ times a Mellin transform
LanglandsTunnell.exists_forall_integrable_and_mellinConvergent_and_setIntegral_cpow_mul_torusKernel_eq_half_GammaR_mul_mellin0 below · cited by 6 · depth 24 - Identity of exponential polynomials extends off a half-plane
LanglandsTunnell.forall_cpow_mul_eval_eq_of_forall_lt_re0 below · cited by 2 · depth 24 - Two-term contiguity identity for a Gaussian torus triple integral
LanglandsTunnell.integral_mulConvGaussian_torusGauss_two_term_eq_GammaR_prod_div4 below · cited by 6 · depth 24 - Sign-quadrant fold for integrable functions on ℝ²
LanglandsTunnell.integral_prod_eq_setIntegral_Ioi_setIntegral_Ioi_sum_reflections0 below · cited by 14 · depth 24 - Moderate-growth Whittaker solution strings on GL₂(ℂ) are proportional
LanglandsTunnell.linearDependent_string_of_gl2Complex_whittaker_system_of_moderateGrowth1 below · cited by 1 · depth 24 - Mellin transform of a multiplicative convolution
LanglandsTunnell.mellinConvergent_integral_mul_comp_mul_and_mellin_eq_mellin_mul_mellin0 below · cited by 2 · depth 24 - Near-zero decay for a forced first-order Euler equation
LanglandsTunnell.norm_le_mul_rpow_near_zero_of_first_order_euler_of_forcing0 below · cited by 1 · depth 24 - Decay at a regular singular point for a diagonal first-order system
LanglandsTunnell.norm_le_mul_rpow_near_zero_of_first_order_system_of_diag0 below · cited by 1 · depth 24 - Near-zero bound for forced Whittaker equation solutions
LanglandsTunnell.norm_le_mul_rpow_near_zero_of_whittaker_ode_of_forcing0 below · cited by 1 · depth 24 - Raising and lowering operators on the split torus
LanglandsTunnell.raising_lowering_splitTorus_of_archWeightChar_of_unipotent0 below · cited by 1 · depth 24 - One-parity Whittaker sheet is the Gaussian convolution profile
LanglandsTunnell.add_pow_mul_apply_neg_eq_mul_mulConvGaussian_of_mellin_sheet_eq_archFactor3 below · cited by 8 · depth 25 - Pointwise eventual constancy of entire families is uniform
LanglandsTunnell.exists_forall_le_eq_of_differentiable_of_forall_exists_forall_le_apply_eq0 below · cited by 1 · depth 25 - Torus–Gauss integral of two Gaussian convolutions in Euler form
LanglandsTunnell.integral_mulConvGaussian_torusGauss_eq_GammaR_mul_integral_Ioi_integral_Ioi_cpow1 below · cited by 1 · depth 25 - A torus--Gauss triple integral as a ratio of Γ_ℝ-products
LanglandsTunnell.integral_mulConvGaussian_torusGauss_eq_GammaR_prod_div_of_balance2 below · cited by 14 · depth 25 - Gaussian convolution with shifts (γ,γ+1) is elementary
LanglandsTunnell.mulConvGaussian_add_one_eq_two_mul_cpow_mul_exp_neg_two_pi_mul0 below · cited by 3 · depth 25 - Archimedean calibration of ξ and non-vanishing of the Whittaker coefficient
LanglandsTunnell.centralExponent_modulus_and_whittaker_ne_zero_of_mellin_archFactor_rat1 below · cited by 1 · depth 26 - Discrete-series Whittaker profile identified by Mellin inversion
LanglandsTunnell.eq_mul_cpow_mul_exp_of_mellin_eq_archFactor_discrete1 below · cited by 1 · depth 26 - Unitarity and polynomial bounds for the twisted Hecke table over ℚ
LanglandsTunnell.exists_finset_twistedTable_ne_zero_bound_unitarity_of_isArithGenuineCuspRealizable_rat22 below · cited by 1 · depth 26 - Non-vanishing of a finite Hermite sum on the real ray
LanglandsTunnell.exists_gt_and_hermiteSum_GammaR_mul_mellin_ne_zero_of_shift_ratio_real0 below · cited by 2 · depth 26 - Non-vanishing of a finite Hermite sum on the real ray
LanglandsTunnell.exists_gt_and_hermiteSum_GammaR_mul_mellin_ne_zero_of_shift_ratio_real_of_parity0 below · cited by 1 · depth 26 - Mellin growth for the Gauss–torus transform: non-vanishing, shift ratio, half-steps
LanglandsTunnell.mellin_gaussTorusTransform_ne_zero_and_shift_ratio_and_halfStep10 below · cited by 3 · depth 26 - Unitarity of the archimedean principal-series parameter over ℚ
LanglandsTunnell.re_sub_eq_zero_or_im_sub_eq_zero_of_isIsotypicCuspFormAt_of_mellin_eq_archFactor_principal_of_minimalWeight399 below · cited by 1 · depth 26 - Mellin uniqueness: a Γ_ℂ-transform forces the discrete-series shape
LanglandsTunnell.eq_mul_cpow_mul_exp_of_continuousOn_of_mellin_div_eq_mul_GammaC0 below · cited by 1 · depth 27 - Mixed parity excluded for real non-zero u₁-u₂ in minimal weight
LanglandsTunnell.eq_of_im_sub_eq_zero_of_re_sub_ne_zero_of_isIsotypicCuspFormAt_of_mellin_eq_archFactor_principal_of_minimalWeight132 below · cited by 1 · depth 27 - Asymptotic phase coherence of the tilted Gaussian double integral
LanglandsTunnell.exists_forall_mul_integral_norm_tiltKernel_le_norm_integral5 below · cited by 2 · depth 27 - Concentration of the tilt kernel in an explicit log-box
LanglandsTunnell.exists_forall_mul_setIntegral_le_setIntegral_logBox_tiltKernel2 below · cited by 3 · depth 27 - Isotypic cusp forms over ℚ are archimedean Casimir eigenfunctions
LanglandsTunnell.exists_isArchSmoothAt_and_archCasimirAt_eq_smul_of_isIsotypicCuspFormAt_of_rightConv_eq_of_ne_bot_rat357 below · cited by 1 · depth 27 - Reality of (u₁-u₂)² for a real Casimir eigenvalue
LanglandsTunnell.exists_sub_sq_eq_ofReal_of_archCasimirAt_eq_smul_of_mellin_eq_archFactor_principal115 below · cited by 1 · depth 27 - Fourfold reflection fold of a Hermite-type sum
LanglandsTunnell.hermite_fourfold_reflection_eq_sum_filter0 below · cited by 1 · depth 27 - Four-fold reflection fold of a Hermite sum with two sheet weights
LanglandsTunnell.hermite_fourfold_twoSheet_reflection_eq_sum_filter0 below · cited by 1 · depth 27 - Reality of the archimedean Casimir eigenvalue over ℚ
LanglandsTunnell.im_eq_zero_of_archCasimirAt_eq_smul_of_isIsotypicCuspFormAt_rat98 below · cited by 1 · depth 27 - Gaussian moment of (t+iu)^m as an explicit Hermite sum
LanglandsTunnell.integral_exp_neg_pi_sq_mul_ofReal_add_I_mul_pow_eq_hermite_sum0 below · cited by 2 · depth 27 - Gaussian torus transform: Mellin transform in J-form
LanglandsTunnell.mellinConvergent_and_mellin_gaussTorusTransform_eq_Gamma_mul_Jintegral0 below · cited by 2 · depth 27 - Half-step decay of the shifted Mellin transform
LanglandsTunnell.norm_mellin_gaussTorusTransform_halfStep_le9 below · cited by 1 · depth 27 - Laplace–Mellin transform of the Gaussian convolution profile
LanglandsTunnell.setIntegral_mulConvGaussian_mul_cpow_mul_exp_eq_betaIntegral_mul_GammaR2 below · cited by 4 · depth 27 - Longitudinal Laplace concentration for the sheared J-integrand
LanglandsTunnell.exists_forall_integrable_and_setIntegral_longitudinal_compl_window_le_mul_integral1 below · cited by 1 · depth 28 - Integrability of the one-sided torus-pair integrand
LanglandsTunnell.exists_forall_integrable_oneSided_torusPair_integrand_of_torusBound_of_polyBound0 below · cited by 3 · depth 28 - A crude bound for |Γ(y+d+iτ')/Γ(y+iτ)|
LanglandsTunnell.exists_forall_norm_Gamma_add_mul_I_le_mul_rpow_mul_norm_Gamma1 below · cited by 1 · depth 28 - Concentration of the tilted kernel on a log-box
LanglandsTunnell.exists_logBox_mul_setIntegral_le_setIntegral_tiltKernel3 below · cited by 1 · depth 28 - Phase coherence for v^{-S}w^Ar^B on a logarithmic box
LanglandsTunnell.exists_unit_forall_mem_logBox_cos_mul_norm_tiltKernel_le_re0 below · cited by 1 · depth 28 - Integrability of the Gaussian–Mellin factor against |u|^{w+1}K
LanglandsTunnell.integrable_cpow_mul_exp_mul_of_integrable_abs_cpow_mul0 below · cited by 1 · depth 28 - Integrability transport through the dual fibre maps
LanglandsTunnell.integrable_dualFibres_of_integrable_oneSided0 below · cited by 1 · depth 28 - Twisted Gaussian moment as a shifted Hermite integral
LanglandsTunnell.integral_exp_neg_pi_sq_div_sq_mul_affine_pow_mul_cexp_eq_mul_hermiteMoment0 below · cited by 2 · depth 28 - Mellin transform of Gaussian convolution times discrete profile
LanglandsTunnell.mellin_mulConvGaussian_mul_discreteProfile_eq_GammaC_mul_GammaC_div1 below · cited by 1 · depth 28 - Casimir eigenvalue equals the principal-series Laplace eigenvalue
LanglandsTunnell.ofReal_eq_laplaceEigenvalue_principal_of_archCasimirAt_eq_smul_of_mellin_eq_archFactor_principal114 below · cited by 2 · depth 28 - Outer Gaussian–Mellin integration producing the factor Γ_ℝ(w+1)
LanglandsTunnell.setIntegral_cpow_mul_exp_mul_eq_GammaReal_mul_setIntegral_of_integrable0 below · cited by 1 · depth 28 - Dual fibres as two one-sided torus integrals
LanglandsTunnell.setIntegral_dualFibres_eq_oneSided_torusPair0 below · cited by 1 · depth 28 - Dual torus-pair scaling identity for the inner integrals
LanglandsTunnell.setIntegral_dualTorusPair_scaling0 below · cited by 1 · depth 28 - Fibre collapse of the one-sided torus integral with flat bracket
LanglandsTunnell.setIntegral_oneSided_torusPair_flatBracket_eq_const_mul_laplaceMellin_and_mirror_eq_zero12 below · cited by 2 · depth 28 - |Γ(σ+iτ)| is close to Γ(σ) for large σ
LanglandsTunnell.exists_forall_sub_mul_Gamma_le_norm_Gamma_add_mul_I_of_abs_le0 below · cited by 1 · depth 29 - Torus sheets of a factorised Whittaker function over ℚ
LanglandsTunnell.exists_torusSheets_whittakerODE_of_isIsotypicCuspFormAt_of_archCasimirAt_eq_of_whittaker_factorisation_rat104 below · cited by 1 · depth 29 - Rodrigues formula for Gaussian moments of σ-iz
LanglandsTunnell.exp_neg_pi_mul_sq_mul_integral_sub_I_mul_pow_mul_exp_eq_iteratedDeriv0 below · cited by 4 · depth 29 - Second-kind beta integral on (0,∞) equals B(b,a)
LanglandsTunnell.integrableOn_and_integral_cpow_mul_one_add_cpow_neg_eq_betaIntegral0 below · cited by 1 · depth 29 - One-sided reduction of the unfolded torus integral
LanglandsTunnell.setIntegral_oneSided_torusPair_eq_setIntegral_fiber4 below · cited by 1 · depth 29 - Fibre integral of a Gaussian average over a hyperbolic region
LanglandsTunnell.setIntegral_setIntegral_cpow_mul_pow_mul_exp_mul_gaussianAverage_eq_Gamma_mul_exp_mul_eval_of_isHomogeneous5 below · cited by 1 · depth 29 - Cauchy–Schlömilch Gaussian integral int₀^∞ e^{-π(w^2+ρ^2/w^2)} dw=tfrac12 e^{-2πρ}
LanglandsTunnell.integrableOn_and_integral_Ioi_exp_neg_pi_mul_sq_add_sq_div_sq_eq_half_exp0 below · cited by 1 · depth 30 - Integrability of a Gaussian–hyperbolic integrand on σ w>v
LanglandsTunnell.integrableOn_cpow_mul_zpow_mul_exp_mul_gaussMoment_hyperbolicRegion1 below · cited by 1 · depth 30 - n-fold integration by parts against a Gaussian on a half-line
LanglandsTunnell.integral_Ioi_cpow_mul_iteratedDeriv_exp_neg_pi_mul_sq_eq_prod_mul_integral_cpow_sub_mul_exp0 below · cited by 1 · depth 30 - Fubini over the hyperbolic region σ,w>0, σ w>v
LanglandsTunnell.integral_Ioi_integral_Ioi_div_eq_setIntegral_and_swap_of_integrableOn_hyperbolicRegion0 below · cited by 1 · depth 30
LanglandsTunnell.ArchBessel 11
- Mellin uniqueness for a product of two Γ_ℝ-factors
LanglandsTunnell.ArchBessel.eq_mul_cpow_mul_besselKernel_of_continuousOn_of_mellin_eq_mul_GammaR_mul_GammaR1 below · cited by 2 · depth 27 - Mellin transform of the Bessel kernel k_ν
LanglandsTunnell.ArchBessel.mellin_besselKernel_eq_mul_Gamma_mul_Gamma0 below · cited by 2 · depth 28 - Mellin transform of a product of two Bessel kernels
LanglandsTunnell.ArchBessel.mellin_besselKernel_mul_besselKernel_eq3 below · cited by 1 · depth 28 - Second integral representation of the Bessel kernel
LanglandsTunnell.ArchBessel.besselKernel_eq_cpow_mul_integral_exp_neg_sub_sq_div0 below · cited by 1 · depth 29 - Bessel profiles of mixed-sign Whittaker solutions force (μ+tfrac12)²=ν²
LanglandsTunnell.ArchBessel.sq_eq_sq_of_whittakerODE_pair_of_add_eq_mul_besselKernel_of_sub_eq_mul_besselKernel6 below · cited by 1 · depth 29 - Whittaker pair with Bessel profile forces μ²=ν²
LanglandsTunnell.ArchBessel.sq_eq_sq_of_whittakerODE_pair_of_add_mul_eq_mul_cpow_mul_besselKernel6 below · cited by 1 · depth 29 - The Bessel kernel k_ν does not vanish identically
LanglandsTunnell.ArchBessel.exists_besselKernel_ne_zero1 below · cited by 2 · depth 30 - Modified Bessel equation for the kernel k_ν
LanglandsTunnell.ArchBessel.hasDerivAt_besselKernel_and_hasDerivAt_deriv_besselKernel3 below · cited by 2 · depth 30 - Derivative in x of the Bessel kernel k_ν
LanglandsTunnell.ArchBessel.hasDerivAt_besselKernel1 below · cited by 1 · depth 31 - Index recurrence for the Bessel kernel
LanglandsTunnell.ArchBessel.mul_besselKernel_eq_mul_sub1 below · cited by 1 · depth 31 - Integrability of the K-Bessel integrand on (0,∞)
LanglandsTunnell.ArchBessel.integrableOn_exp_neg_mul_add_inv_mul_cpow0 below · cited by 2 · depth 32
LanglandsTunnell.ArchPlace 7
- The complex Gaussian is self-dual for the self-dual measure
LanglandsTunnell.ArchPlace.tateFourier_complexTestFun_zero_self0 below · cited by 1 · depth 16 - Complex local zeta integral equals π Γ_ℂ(s+u+|k|/2)
LanglandsTunnell.ArchPlace.complexZeta_complexTestFun_complexCharFun_eq_pi_mul_GammaComplex0 below · cited by 3 · depth 18 - Fourier transform of an archimedean pure tensor
LanglandsTunnell.ArchPlace.fourierIntegral_mixedSpace_pureTensor3 below · cited by 3 · depth 18 - Real Tate integral of the Gaussian equals Γ_ℝ
LanglandsTunnell.ArchPlace.realZeta_realTestFun_realCharFun_eq_GammaReal0 below · cited by 4 · depth 18 - Fourier transform of the complex-place test functions
LanglandsTunnell.ArchPlace.tateFourier_psiComplex_complexTestFun0 below · cited by 1 · depth 19 - Gaussian eigenfunctions of the real Tate–Fourier transform
LanglandsTunnell.ArchPlace.tateFourier_psiReal_realTestFun0 below · cited by 2 · depth 19 - Continuous quasi-characters of ℝ^× and ℂ^×
LanglandsTunnell.ArchPlace.forall_continuous_exists_eq_realCharFun_and_forall_continuous_exists_eq_complexCharFun0 below · cited by 6 · depth 30
LanglandsTunnell.Artin 11
- Artin reciprocity at a modulus admissible for exponent n
LanglandsTunnell.Artin.artinSymbol_surjective_and_ker_eq_normRaySubgroup_of_pow_eq_one_of_isAdmissibleModulusOfDegree113 below · cited by 5 · depth 14 - Artin reciprocity at an admissible modulus of ℓ-power degree
LanglandsTunnell.Artin.artinSymbol_surjective_and_ker_eq_normRaySubgroup_of_isAdmissibleModulusOfDegree112 below · cited by 1 · depth 15 - Determinant one in GL₂(𝔽₃): identity or commutator
LanglandsTunnell.Artin.eq_one_or_eq_commutator_of_det_eq_one0 below · cited by 1 · depth 15 - Artin transfer data for cyclic cubic extensions
LanglandsTunnell.Artin.exists_transferData_of_finrank_eq_three109 below · cited by 1 · depth 15 - Trivial symbol forces the norm into the norm-ray subgroup
LanglandsTunnell.Artin.Ni_mem_normRaySubgroup_of_symbol_eq_one0 below · cited by 3 · depth 16 - Crossing two auxiliary data at a prescribed automorphism
LanglandsTunnell.Artin.exists_Ni_eq_Ni_and_symbol_eq_of_artinPairCore0 below · cited by 3 · depth 16 - Artin auxiliary field and pair data at degree ℓ^k
LanglandsTunnell.Artin.exists_artinFieldCore_exists_artinPairCore_of_isAdmissibleModulusOfDegree1 below · cited by 1 · depth 16 - Uniform existence of Artin field cores and pair cores
LanglandsTunnell.Artin.exists_artinFieldCore_nonempty_artinPairCore1 below · cited by 2 · depth 16 - Artin transfer data for a quadratic extension at an admissible modulus
LanglandsTunnell.Artin.exists_transferData_of_finrank_eq_two68 below · cited by 1 · depth 16 - Cyclotomic levels prime to a finite bad set are disjoint from F
LanglandsTunnell.Artin.exists_badPrimes_finrank_sup_adjoin_eq_mul_totient0 below · cited by 2 · depth 17 - Artin reciprocity at an admissible modulus, exponent dividing 24
LanglandsTunnell.Artin.artinSymbol_surjective_and_ker_eq_normRaySubgroup_of_dvd_twentyFour111 below · cited by 1 · depth 19
LanglandsTunnell.Converse 173
- Admissible twist matching base-changed central entries at unramified places
LanglandsTunnell.Converse.exists_isAdmissibleTwist_eq_formalBaseChange_b_of_isArithGenuineCuspRealizable12 below · cited by 1 · depth 14 - Converse theorem for GL(2) with pinned root number and central character
LanglandsTunnell.Converse.exists_isArithGenuineCuspRealizable_of_forall_isNicePinned_of_centralChar_of_generic126 below · cited by 1 · depth 14 - Existence of a non-zero complex archimedean Whittaker datum
LanglandsTunnell.Converse.exists_archDatumC_W_ne_zero4 below · cited by 3 · depth 15 - Existence of a non-zero archimedean Whittaker datum at every real parameter
LanglandsTunnell.Converse.exists_archDatumR_W_ne_zero4 below · cited by 1 · depth 15 - Existence of a non-vanishing finite Whittaker datum
LanglandsTunnell.Converse.exists_finWhittakerDatum_Wf_ne_zero18 below · cited by 3 · depth 15 - Genericity of the formal base change outside finitely many primes
LanglandsTunnell.Converse.exists_formalBaseChange_generic_of_isArithGenuineCuspRealizable26 below · cited by 2 · depth 15 - Converse theorem for GL₂: nice L-data give cuspidal realisations
LanglandsTunnell.Converse.exists_isArithGenuineCuspRealizable_of_isJLNice109 below · cited by 3 · depth 15 - Nice pinned twisted L-data yield Jacquet–Langlands S-data
LanglandsTunnell.Converse.exists_isJLNice_of_forall_isNicePinned7 below · cited by 2 · depth 15 - Holomorphy at real places of half-determinant twisted translate sums
LanglandsTunnell.Converse.CuspSynthesis.isArchHolomorphicAt_translateSum_halfDet27 below · cited by 1 · depth 16 - Invariance of the Jacquet–Langlands Whittaker series under GL₂(K)
LanglandsTunnell.Converse.CuspSynthesis.jlSeries_globalPoints_mul_eq_of_isJLNice74 below · cited by 1 · depth 16 - Square-integrability of translate sums on a Siegel window
LanglandsTunnell.Converse.CuspSynthesis.memLp_translateSum45 below · cited by 1 · depth 16 - Polynomial bound for finite Whittaker values on the rational torus
LanglandsTunnell.Converse.FinWhittakerDatum.exists_norm_Wf_globalPoints_diagOne_mul_le19 below · cited by 4 · depth 16 - Character-twisted Laplace uniqueness on a compact group times ℝ
LanglandsTunnell.Converse.MellinUniqueness.eq_of_forall_continuous_char_laplace_eq1 below · cited by 2 · depth 16 - Unipotent-invariant Hecke eigenfunctions are zero or Eisenstein
LanglandsTunnell.Converse.eq_zero_or_exists_agreesAwayFromFinite_eisensteinTableOf_of_unipotent_invariant16 below · cited by 1 · depth 16 - Non-vanishing archimedean Whittaker datum for real principal series
LanglandsTunnell.Converse.exists_archDatumR_principal_W_ne_zero3 below · cited by 1 · depth 16 - Archimedean components of a continuous idele class character
LanglandsTunnell.Converse.exists_archParams_of_continuous0 below · cited by 6 · depth 16 - Even idele class characters of ℚ with prescribed conductors
LanglandsTunnell.Converse.exists_even_isAdmissibleTwist_hasConductorExponentAt_of_three_le3 below · cited by 18 · depth 16 - Unitarity bounds for Hecke eigenvalues of genuine cusp realizations over ℚ
LanglandsTunnell.Converse.exists_finset_sq_eq_real_mul_b_and_norm_sq_lt_of_isArithGenuineCuspRealizable23 below · cited by 3 · depth 16 - Converse theorem with pinned constants and weight-one real components
LanglandsTunnell.Converse.exists_isArithGenuineCuspRealizable_archWeightOne_isArchHolomorphicAt_of_forall_isNicePinned_of_centralChar_of_generic121 below · cited by 1 · depth 16 - Existence of an admissible twist with prescribed unit characters on S
LanglandsTunnell.Converse.exists_isJLTwist0 below · cited by 4 · depth 16 - Elements of a number field with prescribed valuations
LanglandsTunnell.Converse.exists_ne_zero_valuation_eq_exp_neg0 below · cited by 5 · depth 16 - One-term dual S-part for products of standard root numbers
LanglandsTunnell.Converse.exists_sPartDual_eq_of_forall_cancel_units9 below · cited by 2 · depth 16 - Admissible twists pull back along the idelic norm
LanglandsTunnell.Converse.isAdmissibleTwist_comp_idelicNorm_genuineBaseChange2 below · cited by 6 · depth 16 - Existence of representatives for order vectors at finitely many places
LanglandsTunnell.Converse.nonempty_sOrderReps1 below · cited by 1 · depth 16 - Entire twisted Euler products exclude Eisenstein eigensystems
LanglandsTunnell.Converse.not_agreesAwayFromFinite_eisensteinTableOf_of_hasProd_eulerProduct_unitary_twist72 below · cited by 3 · depth 16 - Dual S-part series equals the S-part series of μ⁻¹
LanglandsTunnell.Converse.sPartDual_eq_sPart_inv0 below · cited by 3 · depth 16 - Coefficient translation multiplies the S-part series by a monomial
LanglandsTunnell.Converse.sPart_shift0 below · cited by 3 · depth 16 - Genericity of degree-f Satake power sums under a local bound
LanglandsTunnell.Converse.satakePow_sq_ne_of_sq_eq_real_mul_of_norm_sq_lt1 below · cited by 1 · depth 16 - Right-invariant derivatives of complex archimedean Whittaker data
LanglandsTunnell.Converse.ArchDatumC.exists_W_eq_fderivWithin_mul0 below · cited by 1 · depth 17 - Near-zero derivative bound for a complex archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumC.norm_iteratedFDerivWithin_diagOne_le0 below · cited by 1 · depth 17 - Left-invariant derivative of a real archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.exists_W_eq_fderivWithin_mul0 below · cited by 1 · depth 17 - Small-|y| derivative bounds for a real archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.norm_iteratedFDerivWithin_diagOne_le0 below · cited by 8 · depth 17 - Estimates for the Whittaker series of a nice JL datum
LanglandsTunnell.Converse.CuspSynthesis.exists_growth_exponent_and_local_majorant_and_bounded_on_siegel_of_isJLNice23 below · cited by 3 · depth 17 - Haar measure and a common entire torus transform
LanglandsTunnell.Converse.CuspSynthesis.exists_isHaarMeasure_torusTransform_eq_of_isJLNice69 below · cited by 1 · depth 17 - Character-twisted Laplace uniqueness on C × ℝ
LanglandsTunnell.Converse.MellinUniqueness.eq_of_forall_continuous_char_exists_laplace_eq1 below · cited by 1 · depth 17 - Uniqueness for two-sided Laplace transforms on opposite half-planes
LanglandsTunnell.Converse.MellinUniqueness.eq_of_laplace_eq_of_boundedOnStrips0 below · cited by 2 · depth 17 - Torus recursions for a unipotent-invariant Hecke eigenfunction
LanglandsTunnell.Converse.eq_zero_or_exists_continuous_torus_recursion_of_unipotent_invariant3 below · cited by 1 · depth 17 - Non-vanishing weight-one archimedean datum at the odd Artin parameter
LanglandsTunnell.Converse.exists_archDatumR_oddArtin_archWeightChar_one_mdifferentiable_W_ne_zero0 below · cited by 1 · depth 17 - Finite-order idele characters have trivial infinity type
LanglandsTunnell.Converse.exists_isArchCompAt_zero_of_isOfFinOrder2 below · cited by 3 · depth 17 - Archimedean components of a character composed with the idelic norm
LanglandsTunnell.Converse.isArchCompAt_comp_idelicNorm_genuineBaseChange2 below · cited by 7 · depth 17 - Pinned niceness from an entire pair satisfying a functional equation
LanglandsTunnell.Converse.isNicePinned_of_entire_pair0 below · cited by 3 · depth 17 - Dual S-part series with one-point support is a monomial
LanglandsTunnell.Converse.sPartDual_eq_single0 below · cited by 1 · depth 17 - No small subgroups in ℂ^×
LanglandsTunnell.Converse.units_eq_one_of_forall_zpow_norm_sub_one_lt0 below · cited by 1 · depth 17 - Scaling of the entire zeta function under diag(A,1)
LanglandsTunnell.Converse.ArchDatumC.zetaEntire_diagOne_mul0 below · cited by 1 · depth 18 - Determinant twist of a real archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.exists_twist_W_eq_abs_det_rpow_mul0 below · cited by 5 · depth 18 - Diagonal scaling law for the entire archimedean zeta function
LanglandsTunnell.Converse.ArchDatumR.zetaEntire_diagOne_mul0 below · cited by 2 · depth 18 - Unramified idele characters have local conductor exponent 0
LanglandsTunnell.Converse.conductorExponentAt_localChar_eq_zero_of_isUnramifiedCharAt0 below · cited by 5 · depth 18 - Assembled archimedean Whittaker function is a Casimir eigenfunction
LanglandsTunnell.Converse.continuous_archW_and_isArchSmoothAt_and_archCasimirAt_eq_of_isCasimirEigen0 below · cited by 6 · depth 18 - Assembling archimedean Rankin–Selberg integrals from diagonal torus identities
LanglandsTunnell.Converse.exists_const_sum_rsArchIntegral_eq_mul_of_torus_identities3 below · cited by 2 · depth 18 - Archimedean local component at a complex place is a quasi-character
LanglandsTunnell.Converse.exists_isArchCompAt_of_isComplex0 below · cited by 9 · depth 18 - Local component of an idele character at a real place
LanglandsTunnell.Converse.exists_isArchCompAt_of_isReal0 below · cited by 11 · depth 18 - Haar measure on GL₂(A_ℚ) splits as a product
LanglandsTunnell.Converse.exists_isHaarMeasure_map_adelicGLHaar_eq_prod_archMeasure3 below · cited by 5 · depth 18 - Haar splitting of the adelic unipotent group of GL₂/ℚ
LanglandsTunnell.Converse.exists_isHaarMeasure_map_unipotentHaar_eq_prod_map_val2 below · cited by 5 · depth 18 - Finiteness of the pinned exponent of a continuous idele character
LanglandsTunnell.Converse.finite_setOf_pinnedExp_ne_zero_of_continuous10 below · cited by 2 · depth 18 - Unramified idele characters have local conductor exponent zero
LanglandsTunnell.Converse.hasConductorExponentAt_localChar_zero_of_isUnramifiedCharAt0 below · cited by 11 · depth 18 - The measure |det g|⁻² dg is a bi-invariant Haar measure on GL₂(ℝ)
LanglandsTunnell.Converse.isHaarMeasure_and_isMulRightInvariant_archMeasure0 below · cited by 9 · depth 18 - Sign twist of an archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.exists_twist_sign_W_eq_sign_det_mul0 below · cited by 3 · depth 19 - Idelic determinant character factors over a finite set of primes
LanglandsTunnell.Converse.chiDetGL_eq_prod_localChar_det_componentAt3_of_isArchCompAt_zero_zero1 below · cited by 3 · depth 19 - Minimal-weight archimedean Whittaker datum for a real parameter
LanglandsTunnell.Converse.exists_archDatumR_archWeightChar_minimalType_isCasimirEigen_W_ne_zero15 below · cited by 2 · depth 19 - Iwasawa reduction of the archimedean Rankin–Selberg integral
LanglandsTunnell.Converse.exists_const_rsArchIntegral_eq_mul_integral_diagonal2 below · cited by 3 · depth 19 - Jacquet–Langlands data from pinned niceness of twisted L-data
LanglandsTunnell.Converse.exists_jlData_isJLNice_of_forall_isNicePinned4 below · cited by 1 · depth 19 - Iwasawa majorant for density-weighted integrals on GL₂(ℝ)
LanglandsTunnell.Converse.exists_lintegral_mul_density_archMeasure_le_lintegral_iwasawa3 below · cited by 2 · depth 19 - Finite Rankin–Selberg integrand integrable, or archimedean integral vanishes
LanglandsTunnell.Converse.integrable_rsFinIntegrand_or_rsArchIntegral_eq_zero_of_integrable4 below · cited by 2 · depth 19 - Archimedean–finite splitting of the unipotent-quotient Rankin–Selberg integral
LanglandsTunnell.Converse.integral_unipotentQuotient_eq_rsArchIntegral_mul_rsFinIntegral_of_integrable4 below · cited by 3 · depth 19 - Pinned niceness depends on the eigensystem only outside S
LanglandsTunnell.Converse.isNicePinned_twistedDatum_iff_of_forall_notMem_a_eq_b_eq0 below · cited by 1 · depth 19 - Vanishing of the discrete-series Whittaker datum on det<0
LanglandsTunnell.Converse.ArchDatumR.W_eq_zero_of_det_neg_of_discrete_of_archWeightChar_of_isCasimirEigen10 below · cited by 3 · depth 20 - Weight-one limit-of-discrete-series datum vanishes on negative determinants
LanglandsTunnell.Converse.ArchDatumR.W_eq_zero_of_det_neg_of_principal_of_ne_of_archWeightChar_one_of_isCasimirEigen10 below · cited by 3 · depth 20 - Reflection law for weight-zero real principal Whittaker data
LanglandsTunnell.Converse.ArchDatumR.W_mul_diag_eq_neg_one_pow_mul_of_principal_of_archWeightChar_zero_of_isCasimirEigen10 below · cited by 4 · depth 20 - Reflection by diag(-1,1) as a lowering derivative
LanglandsTunnell.Converse.ArchDatumR.exists_W_mul_diag_eq_mul_lower_of_principal_of_ne_of_ne_of_archWeightChar_one_of_isCasimirEigen11 below · cited by 2 · depth 20 - Complex linear combinations of archimedean data
LanglandsTunnell.Converse.ArchDatumR.exists_lincomb0 below · cited by 1 · depth 20 - Sign-of-determinant twist shifts both parities of a principal-series archimedean datum
LanglandsTunnell.Converse.ArchDatumR.exists_sgnTwist0 below · cited by 1 · depth 20 - Weight-one Whittaker datum: W(xJ)=κ (LW)(x) with κ²(u₁-u₂)²=1
LanglandsTunnell.Converse.ArchDatumR.exists_sq_mul_sq_eq_one_and_W_mul_diag_eq_mul_lower_of_principal_of_ne_of_ne_of_archWeightChar_one_of_isCasimirEigen11 below · cited by 4 · depth 20 - Transformation laws and torus ODE for an archimedean datum
LanglandsTunnell.Converse.ArchDatumR.laws_and_torus_ode_of_archWeightChar_of_isCasimirEigen2 below · cited by 8 · depth 20 - Torus rays determine a ψ-Whittaker function of weight k
LanglandsTunnell.Converse.ArchR.eq_mul_of_unip_law_of_central_law_of_archWeightChar_of_torus_eq_of_sign_det0 below · cited by 7 · depth 20 - Right rotation equivariance of the discrete-series Whittaker function
LanglandsTunnell.Converse.DiscreteFamily.W_archWeightChar0 below · cited by 1 · depth 20 - Nonvanishing of the discrete-series Whittaker function
LanglandsTunnell.Converse.DiscreteFamily.exists_W_ne_zero0 below · cited by 1 · depth 20 - Discrete-series Whittaker function as an archimedean datum
LanglandsTunnell.Converse.DiscreteFamily.exists_archDatumR_W_eq0 below · cited by 1 · depth 20 - Discrete-series Whittaker function is a Casimir eigenfunction
LanglandsTunnell.Converse.DiscreteFamily.matrixCasimir_W0 below · cited by 1 · depth 20 - Right rotation invariance of the even principal Whittaker function
LanglandsTunnell.Converse.PrincipalFamily.Wmem_zero_zero_archWeightChar0 below · cited by 1 · depth 20 - Non-vanishing of the real principal-series Whittaker function
LanglandsTunnell.Converse.PrincipalFamily.exists_Wmem_ne_zero3 below · cited by 1 · depth 20 - Principal-series Whittaker function comes from an archimedean datum
LanglandsTunnell.Converse.PrincipalFamily.exists_archDatumR_W_eq3 below · cited by 1 · depth 20 - Non-vanishing of the odd weight-one Whittaker combination
LanglandsTunnell.Converse.PrincipalFamily.exists_oddComb_ne_zero3 below · cited by 1 · depth 20 - Sign-twisted two-term Whittaker combinations are Casimir eigenfunctions
LanglandsTunnell.Converse.PrincipalFamily.isCasimirEigen_of_W_eq_comb3 below · cited by 1 · depth 20 - Weight-one equivariance of the odd principal-series Whittaker combination
LanglandsTunnell.Converse.PrincipalFamily.oddComb_archWeightChar3 below · cited by 1 · depth 20 - Admissible twist on K matching a formal base change
LanglandsTunnell.Converse.exists_isAdmissibleTwist_eq_formalBaseChange_b_isArchCompAt_archOfParam18 below · cited by 1 · depth 20 - Unitary local character at p extends to an idele class character
LanglandsTunnell.Converse.exists_isAdmissibleTwist_localChar_eq_of_hasConductorExponentAt_of_norm_eq_one13 below · cited by 1 · depth 20 - Deep-twist product law for priced local root numbers above p
LanglandsTunnell.Converse.finprod_stdRootNumberAt_twist_mul_twist_eq_sq_of_le_floor22 below · cited by 1 · depth 20 - Index shift of S-part coefficients preserves pinned niceness
LanglandsTunnell.Converse.isNicePinned_sPart_shift2 below · cited by 1 · depth 20 - Pinned conductor exponent unchanged by a shallow norm twist
LanglandsTunnell.Converse.pinnedExp_comp_idelicNorm_mul_eq_pinnedExp_of_hasConductorExponentAt_le_of_depth_floor3 below · cited by 3 · depth 20 - Uniform strip bounds and continuity in g of the entire zeta function
LanglandsTunnell.Converse.ArchDatumR.exists_norm_zetaEntire_le_mul_pow_mul_exp_and_continuousOn1 below · cited by 1 · depth 21 - Whittaker ODE, growth and Mellin shape on the negative sheet
LanglandsTunnell.Converse.ArchDatumR.negSheet_ode_and_growth_and_mellin_eq_of_archWeightChar_of_isCasimirEigen1 below · cited by 2 · depth 21 - Mellin uniqueness on the ideles of ℚ
LanglandsTunnell.Converse.MellinUniqueness.eq_smul_of_forall_isAdmissibleTwist_mellin_eq8 below · cited by 1 · depth 21 - Twisting an admissible character by the idelic norm
LanglandsTunnell.Converse.isAdmissibleTwist_mul_comp_idelicNorm_of_isFiniteOrderHeckeChar2 below · cited by 2 · depth 21 - Iwasawa bound for W_D(diag(at,1)e⁻¹)
LanglandsTunnell.Converse.ArchDatumR.norm_W_diagOne_mul_inv_le_of_iwasawa0 below · cited by 1 · depth 22 - Unitarity bounds at almost all primes for cusp-realizable eigensystems
LanglandsTunnell.Converse.exists_finset_sq_eq_real_mul_b_and_norm_sq_lt_of_isArithGenuineCuspRealizable_of_coversModCentre20 below · cited by 2 · depth 22 - Iwasawa factorisation of a weight-k archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.W_diagOne_mul_iwasawa_eq_psi_mul_centralChar_mul_archWeightChar_mul_W_diagOne1 below · cited by 7 · depth 23 - Torus profile of a weight-zero real archimedean Whittaker datum
LanglandsTunnell.Converse.ArchDatumR.continuousOn_and_exists_ne_zero_W_diagOne_of_weightZero0 below · cited by 8 · depth 23 - Continuity on (0,∞) of a Gaussian-damped torus transform of W
LanglandsTunnell.Converse.ArchDatumR.continuousOn_gaussian_mul_integral_W_diagOne_torusKernel0 below · cited by 5 · depth 23 - Weight ≥ 1 real Whittaker profiles: both parity sheets non-vanishing
LanglandsTunnell.Converse.ArchDatumR.exists_W_diagOne_add_mul_W_diagOne_neg_ne_zero_of_one_le_weight32 below · cited by 9 · depth 23 - Mellin non-vanishing of a Gaussian torus transform beyond any abscissa
LanglandsTunnell.Converse.ArchDatumR.exists_lt_re_mellin_gaussian_mul_integral_W_diagOne_torusKernel_ne_zero3 below · cited by 3 · depth 23 - Mellin non-vanishing far right for two-sheet torus-kernel transforms
LanglandsTunnell.Converse.ArchDatumR.exists_lt_re_mellin_gaussian_mul_integral_twoSheet_torusKernel_ne_zero2 below · cited by 2 · depth 23 - Existence of an odd admissible twist for ℚ
LanglandsTunnell.Converse.exists_odd_isAdmissibleTwist4 below · cited by 8 · depth 23 - Archimedean Γ-slots for a principal parameter, signature (1,1)
LanglandsTunnell.Converse.prod_map_GammaR_twistedGammaR_archOfParamR_principal_one_real_one_complex0 below · cited by 3 · depth 23 - Explicit Γ_ℝ-slot for a principal parameter, three real places
LanglandsTunnell.Converse.prod_map_GammaR_twistedGammaR_archOfParamR_principal_three_real0 below · cited by 6 · depth 23 - Dual Γ-factors and archimedean root number in signature (1,1)
LanglandsTunnell.Converse.prod_map_GammaR_twistedGammaR_dual_and_archRootNumber_one_real_one_complex0 below · cited by 5 · depth 23 - Dual Γ-products and ε_∞ at three real places
LanglandsTunnell.Converse.prod_map_GammaR_twistedGammaR_dual_and_archRootNumber_three_real0 below · cited by 7 · depth 23 - Non-vanishing of the torus Mellin transform beyond any point
LanglandsTunnell.Converse.ArchDatumR.exists_lt_mellin_W_diagOne_ne_zero_of_weightZero_of_parity0 below · cited by 1 · depth 24 - Dual torus-triple evaluation for the block-harmonic section
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_blockHarmonic_eq_mul_prod_GammaR5 below · cited by 1 · depth 24 - Dual torus-triple evaluation, conjugate block, discrete branch
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_conjBlock_eq_mul_prod_GammaR_of_discreteProfile8 below · cited by 1 · depth 24 - Dual torus triple evaluation: flat conjugate block, two sheets, n=0
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_conjBlock_eq_mul_prod_GammaR_of_twoSheetProfile7 below · cited by 1 · depth 24 - Dual torus pair equals tfrac12Γ_ℝ times torus triple
LanglandsTunnell.Converse.dualTorusPair_iwasawa_eq_const_mul_integral_torusTriple_blockHarmonic_of_re_gt3 below · cited by 1 · depth 24 - Dual torus pair equals Γ_ℝ times conjugate-block torus triple
LanglandsTunnell.Converse.dualTorusPair_iwasawa_eq_const_mul_integral_torusTriple_conjBlock_of_re_gt3 below · cited by 2 · depth 24 - Iwasawa-unfolded dual torus pair as a torus-triple integral
LanglandsTunnell.Converse.dualTorusPair_iwasawa_eq_const_mul_integral_torusTriple_minor_of_re_gt3 below · cited by 1 · depth 24 - Integrability of dual quadruple and torus-triple integrands
LanglandsTunnell.Converse.exists_forall_integrable_dualQuadruple_and_torusTriple_blockHarmonic_of_mulConvGaussian_sheets0 below · cited by 2 · depth 24 - Integrability of the flat dual quadruple and torus-triple integrands
LanglandsTunnell.Converse.exists_forall_integrable_dualQuadruple_and_torusTriple_conjBlock_of_mulConvGaussian_sheets0 below · cited by 4 · depth 24 - Integrability of dual four- and three-variable minor integrands
LanglandsTunnell.Converse.exists_forall_integrable_dualQuadruple_and_torusTriple_minor_of_mulConvGaussian_sheets0 below · cited by 1 · depth 24 - Integrability of the post-Gaussian conjugate-block torus integrand
LanglandsTunnell.Converse.exists_forall_integrable_postGaussian_torusTriple_conjBlock_of_mulConvGaussian_profile0 below · cited by 6 · depth 24 - Integrability of the post-Gaussian minor-section torus triple integrand
LanglandsTunnell.Converse.exists_forall_integrable_postGaussian_torusTriple_minor_of_mulConvGaussian_sheets0 below · cited by 1 · depth 24 - Integrability of the θ-free block-harmonic Iwasawa integrand
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_blockHarmonic_of_mulConvGaussian_sheets0 below · cited by 1 · depth 24 - Integrability of the θ-free conjugate-block Iwasawa integrand
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_conjBlock_of_mulConvGaussian_profile0 below · cited by 2 · depth 24 - Integrability of the θ-free Iwasawa integrand, two-sheet profile
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_minor_of_mulConvGaussian_sheets0 below · cited by 1 · depth 24 - Integrability of the unfolded (x,t)-integrand for Gaussian-convolution profiles
LanglandsTunnell.Converse.exists_forall_integrable_xAffineGaussian_psi_mul_torusPair_of_mulConvGaussian_profiles0 below · cited by 1 · depth 24 - Integrability of the block-harmonic dual Iwasawa integrand
LanglandsTunnell.Converse.integrable_dualConfig_iwasawaIntegrand_blockHarmonic0 below · cited by 1 · depth 24 - Integrability of the conjugate-block dual Iwasawa integrand
LanglandsTunnell.Converse.integrable_dualConfig_iwasawaIntegrand_conjBlock0 below · cited by 2 · depth 24 - Integrability of the minor-section dual Iwasawa integrand
LanglandsTunnell.Converse.integrable_dualConfig_iwasawaIntegrand_minor0 below · cited by 1 · depth 24 - Dual configuration integral equals 2π times Iwasawa integral
LanglandsTunnell.Converse.integral_dualConfig_blockHarmonic_eq_two_pi_mul_integral_iwasawa_of_weightZero4 below · cited by 1 · depth 24 - Dual Godement integral in Iwasawa coordinates, weight n+1
LanglandsTunnell.Converse.integral_dualConfig_conjBlock_eq_two_pi_mul_integral_iwasawa_of_archWeightChar4 below · cited by 2 · depth 24 - Dual Godement integral in Iwasawa coordinates, weight zero
LanglandsTunnell.Converse.integral_dualConfig_minor_eq_two_pi_mul_integral_iwasawa_of_weightZero4 below · cited by 1 · depth 24 - Evaluation of the post-Gaussian block-harmonic torus-triple integral
LanglandsTunnell.Converse.integral_postGaussian_torusTriple_blockHarmonic_eq_mul_prod_GammaR5 below · cited by 1 · depth 24 - Post-Gaussian conjugate-block torus triple for a discrete profile
LanglandsTunnell.Converse.integral_postGaussian_torusTriple_conjBlock_eq_mul_prod_GammaR_of_discreteProfile8 below · cited by 1 · depth 24 - Conjugate-block torus-triple integral: six Γ_ℝ factors
LanglandsTunnell.Converse.integral_postGaussian_torusTriple_conjBlock_eq_mul_prod_GammaR_of_twoSheetProfile8 below · cited by 1 · depth 24 - Gaussian x-moment step for the block-harmonic Iwasawa integrand
LanglandsTunnell.Converse.integral_thetaFree_iwasawaIntegrand_blockHarmonic_eq_integral_postGaussian_torusTriple2 below · cited by 1 · depth 24 - Gaussian x-integration of the conjugate-block Iwasawa integrand
LanglandsTunnell.Converse.integral_thetaFree_iwasawaIntegrand_conjBlock_eq_integral_postGaussian_torusTriple2 below · cited by 2 · depth 24 - Weight-zero Whittaker datum on the Iwasawa matrix
LanglandsTunnell.Converse.ArchDatumR.W_diagOne_mul_iwasawa_eq_psi_mul_centralChar_mul_W_diagOne_of_weightZero1 below · cited by 3 · depth 25 - Dual torus triple integral as six Γ_ℝ-factors
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_detPow_blockQuadratic_colHarmonicTwo_eq_mul_prod_GammaR_of_weightZeroProfile5 below · cited by 1 · depth 25 - Γ-evaluation of the dual torus triple: discrete branch
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_detPow_colHarmonic_eq_mul_prod_GammaR_of_evenPrincipal_of_discreteProfile5 below · cited by 1 · depth 25 - Γ-evaluation of the dual torus triple, even principal two-sheet profile
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_detPow_colHarmonic_eq_mul_prod_GammaR_of_evenPrincipal_of_twoSheetProfile4 below · cited by 1 · depth 25 - Γ-evaluation of the dual torus triple, even weight-zero case
LanglandsTunnell.Converse.GammaR_mul_integral_dualTorusTriple_detPow_colHarmonic_eq_mul_prod_GammaR_of_evenPrincipal_of_weightZeroProfile5 below · cited by 1 · depth 25 - Dual torus-pair identity for the block-quadratic section
LanglandsTunnell.Converse.dualTorusPair_iwasawa_eq_const_mul_integral_torusTriple_detPow_blockQuadratic_colHarmonic_of_re_gt5 below · cited by 1 · depth 25 - Iwasawa-unfolded dual torus pair as a torus-triple integral
LanglandsTunnell.Converse.dualTorusPair_iwasawa_eq_const_mul_integral_torusTriple_detPow_colHarmonic_of_re_gt3 below · cited by 3 · depth 25 - Fixed (a₁,a₂) fibre of the major dual torus pair
LanglandsTunnell.Converse.dualTorusPair_iwasawa_fibre_eq_const_mul_integral_torusQuadruple_blockHarmonic1 below · cited by 1 · depth 25 - Fibrewise Iwasawa identity for the conjugate-block flat section
LanglandsTunnell.Converse.dualTorusPair_iwasawa_fibre_eq_const_mul_integral_torusQuadruple_conjBlock1 below · cited by 1 · depth 25 - Fibre identity for the dual torus pair integral
LanglandsTunnell.Converse.dualTorusPair_iwasawa_fibre_eq_const_mul_integral_torusQuadruple_minor1 below · cited by 1 · depth 25 - Integrability of dual quadruple and torus-triple quadratic integrands
LanglandsTunnell.Converse.exists_forall_integrable_dualQuadruple_and_torusTriple_detPow_blockQuadratic_colHarmonic_of_evenSheet0 below · cited by 2 · depth 25 - Integrability of dual quadruple and torus-triple even-sheet integrands
LanglandsTunnell.Converse.exists_forall_integrable_dualQuadruple_and_torusTriple_detPow_colHarmonic_of_evenSheet0 below · cited by 6 · depth 25 - Integrability of the post-Gaussian torus-triple integrand, quadratic section
LanglandsTunnell.Converse.exists_forall_integrable_postGaussian_torusTriple_detPow_blockQuadratic_colHarmonicTwo_of_mulConvGaussian_sheet0 below · cited by 2 · depth 25 - Integrability of the post-Gaussian torus-triple integrand
LanglandsTunnell.Converse.exists_forall_integrable_postGaussian_torusTriple_detPow_colHarmonic_of_mulConvGaussian_sheet0 below · cited by 3 · depth 25 - Integrability of the θ-free Iwasawa integrand for a quadratic section
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_detPow_blockQuadratic_colHarmonicTwo_of_mulConvGaussian_sheet0 below · cited by 1 · depth 25 - Integrability of the θ-free Iwasawa integrand, one Gaussian sheet
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_detPow_colHarmonic_of_mulConvGaussian_sheet0 below · cited by 3 · depth 25 - Integrability of an affine Gaussian–Whittaker torus integrand
LanglandsTunnell.Converse.exists_forall_integrable_xAffineGaussian_psi_mul_torusPair_of_archDatumR0 below · cited by 2 · depth 25 - Integrability of the x-affine Gaussian against a torus pair
LanglandsTunnell.Converse.exists_forall_integrable_xAffineGaussian_psi_mul_torusPair_of_mulConvGaussian_sheet0 below · cited by 3 · depth 25 - Non-vanishing scalar in the weight-one torus profile identity
LanglandsTunnell.Converse.exists_ne_zero_and_W_diagOne_add_eq_mul_mulConvGaussian_of_weightOneLevi33 below · cited by 1 · depth 25 - Non-vanishing scalar in the discrete-series Whittaker profile
LanglandsTunnell.Converse.exists_ne_zero_and_W_diagOne_eq_mul_exp_and_eq_zero_of_discreteLevi33 below · cited by 1 · depth 25 - Non-vanishing scalar in the weight-zero torus profile
LanglandsTunnell.Converse.exists_ne_zero_and_W_diagOne_eq_mul_mulConvGaussian_of_weightZeroLevi17 below · cited by 1 · depth 25 - Integrability of the quadratic-section dual Iwasawa integrand
LanglandsTunnell.Converse.integrable_dualConfig_iwasawaIntegrand_detPow_blockQuadratic_colHarmonicTwo0 below · cited by 1 · depth 25 - Integrability of the dual Iwasawa integrand for the det^δ section
LanglandsTunnell.Converse.integrable_dualConfig_iwasawaIntegrand_detPow_colHarmonic0 below · cited by 3 · depth 25 - Dual Godement integral in Iwasawa coordinates at weight zero
LanglandsTunnell.Converse.integral_dualConfig_detPow_blockQuadratic_colHarmonicTwo_eq_two_pi_mul_integral_iwasawa_of_weightZero4 below · cited by 1 · depth 25 - Iwasawa form of the dual-configuration integral at weight k₀=n
LanglandsTunnell.Converse.integral_dualConfig_detPow_colHarmonic_eq_two_pi_mul_integral_iwasawa_of_archWeightChar4 below · cited by 3 · depth 25 - Post-Gaussian torus triple equals six Γ_ℝ-factors
LanglandsTunnell.Converse.integral_postGaussian_torusTriple_detPow_blockQuadratic_colHarmonicTwo_eq_mul_prod_GammaR_of_weightZeroProfile5 below · cited by 1 · depth 25 - x-integration of the quadratic θ-free Iwasawa integrand
LanglandsTunnell.Converse.integral_thetaFree_iwasawaIntegrand_detPow_blockQuadratic_colHarmonicTwo_eq_integral_postGaussian_torusTriple3 below · cited by 1 · depth 25 - Factorisation of the unipotent-quotient integral into archimedean and finite Rankin–Selberg factors
LanglandsTunnell.Converse.integral_unipotentQuotient_eq_rsArchIntegral_mul_rsFinIntegral4 below · cited by 1 · depth 25 - Archimedean Γ-factor splitting for a cubic field, principal case
LanglandsTunnell.Converse.prod_twistedGammaR_mul_prod_twistedGammaC_archOfParam_eq_archFactor_mul_of_principal0 below · cited by 2 · depth 25 - Fubini and the archimedean Gaussian integral for a dual torus pair
LanglandsTunnell.Converse.setIntegral_integral_dite_eq_const_mul_GammaR_mul_integral_triple_of_fibre0 below · cited by 5 · depth 25 - Fibrewise quadratic identity for the dual torus pair
LanglandsTunnell.Converse.dualTorusPair_iwasawa_fibre_eq_const_mul_integral_torusQuadruple_detPow_blockQuadratic_colHarmonic3 below · cited by 1 · depth 26 - Iwasawa fibre of a dual torus pair as quadruple integral
LanglandsTunnell.Converse.dualTorusPair_iwasawa_fibre_eq_const_mul_integral_torusQuadruple_detPow_colHarmonic1 below · cited by 1 · depth 26 - Joint integrability of a quadratic Gaussian against the torus pair
LanglandsTunnell.Converse.exists_forall_integrable_xQuadraticGaussian_psi_mul_torusPair_of_mulConvGaussian_sheet0 below · cited by 1 · depth 26 - Second Gaussian moment against the additive character
LanglandsTunnell.Converse.integral_ofReal_sq_mul_exp_neg_pi_mul_sq_div_sq_mul_psi1 below · cited by 2 · depth 26 - Discrete-series Γ-slots and root number over a (1,1) field
LanglandsTunnell.Converse.prod_map_Gamma_twistedGamma_and_dual_and_archRootNumber_discrete_one_real_one_complex0 below · cited by 2 · depth 26 - Discrete-series Γ-slots and root number, three real places
LanglandsTunnell.Converse.prod_map_Gamma_twistedGamma_and_dual_and_archRootNumber_discrete_three_real0 below · cited by 2 · depth 26 - Third Gaussian moment against the character ψ(cx)
LanglandsTunnell.Converse.integral_ofReal_pow_three_mul_exp_neg_pi_mul_sq_div_sq_mul_psi1 below · cited by 1 · depth 27 - Integrability of the θ-free Iwasawa integrand, one-sided profile
LanglandsTunnell.Converse.exists_forall_integrable_thetaFree_iwasawaIntegrand_conjBlockPow_colHarmonic_of_oneSided_profile0 below · cited by 1 · depth 29 - Integrability of a one-sided Gaussian torus integrand
LanglandsTunnell.Converse.exists_forall_integrable_xPowGaussian_psi_mul_torusPair_of_oneSided_profile0 below · cited by 1 · depth 29 - Integrability of the θ-free dual Iwasawa integrand
LanglandsTunnell.Converse.integrable_dualThetaFree_integrand0 below · cited by 1 · depth 29
LanglandsTunnell.CubicInduction 656
- Admissible character induced from a cubic field
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_eulerCoeff_eq_inducedE3_of_finrank_eq_three270 below · cited by 3 · depth 16 - Dual Whittaker function as reflected Whittaker function
LanglandsTunnell.CubicInduction.CubicInductionForm.dualWhittaker_eq_dualWhittakerFn30 below · cited by 3 · depth 18 - Factorisation of the dual Whittaker function over a finite set
LanglandsTunnell.CubicInduction.CubicInductionForm.dualWhittaker_eq_dualWhittakerFn3_whittakerArch_mul_prod0 below · cited by 1 · depth 18 - Twisting a cubic induction form by a character of the determinant
LanglandsTunnell.CubicInduction.CubicInductionForm.twist_det_package2 below · cited by 7 · depth 18 - Non-vanishing of every local Whittaker factor
LanglandsTunnell.CubicInduction.CubicInductionForm.whittakerLoc_ne_zero0 below · cited by 2 · depth 18 - Local Whittaker functions scale by the local central character
LanglandsTunnell.CubicInduction.CubicInductionForm.whittakerLoc_scalar_mul_eq_localChar_centralChar_mul0 below · cited by 7 · depth 18 - Level-one invariance and torus table for a dual GL₃ Whittaker function
LanglandsTunnell.CubicInduction.dualWhittakerFn3_localLevelOne_and_torusValues_const_sq_of_localRankinSelbergFE12 below · cited by 2 · depth 18 - Admissibility of the cyclic space of a spherical Whittaker function
LanglandsTunnell.CubicInduction.exists_finset_mem_span_of_mem_gl3CyclicSubspace_of_isInducedSphericalAt_of_isUnitaryChar18 below · cited by 3 · depth 18 - Local functional equation at one deeply twisted prime
LanglandsTunnell.CubicInduction.exists_forall_mem_gl3CyclicSubspace_localZeta31_fe_one_of_cubicInductionForm_twist_deepAt593 below · cited by 1 · depth 18 - Local constants of twisted cubic induction on the cyclic span
LanglandsTunnell.CubicInduction.exists_forall_mem_gl3CyclicSubspace_localZetaDual31_eq_mul_of_cubicInductionForm_twisted_badPlaces_noFE32_adm598 below · cited by 1 · depth 18 - Gauge majorant descends to the local Whittaker function at v
LanglandsTunnell.CubicInduction.exists_gauge_whittakerLoc_of_isGaugeMajorised3_of_form_ne_zero0 below · cited by 5 · depth 18 - Central character of cubic induction, with conductor bound
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_eulerCoeff_eq_inducedE3_of_finrank_eq_three_conductorBound272 below · cited by 1 · depth 18 - Explicit K₁(p^{3B+Δ})-invariant bump vector for twisted cubic induction
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_twist_whittakerLoc_congruenceK1_invariant_iotaGL_bump_of_conductor_le_ed3111 below · cited by 1 · depth 18 - Finitely many bad places for a continuous idele character
LanglandsTunnell.CubicInduction.finite_setOf_isBadPlace_of_continuous3 below · cited by 10 · depth 18 - Cyclicity of induced spherical Whittaker functions on GL₃(ℚᵥ)
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_of_isInducedSphericalAt_of_isUnitaryChar32 below · cited by 3 · depth 18 - Local GL₃timesGL₂ functional equation on the cyclic space
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_rsLocalIntegral_fe32_of_forall_localZeta31_fe_of_gauge87 below · cited by 4 · depth 18 - Finiteness of torus coefficients in the twisted local cyclic space
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_torusFinite_of_cubicInductionForm_twisted_noFE32_level19 below · cited by 1 · depth 18 - Cubic automorphic induction: existence of a cubic induction form
LanglandsTunnell.CubicInduction.hasCubicInductionForm_arch_torusValues_localPackage_bad1,687 below · cited by 1 · depth 18 - Unramified twist by χᵥ∘det preserves induced spherical data
LanglandsTunnell.CubicInduction.hasSphericalTorusValuesAt_twist_det_of_isUnramifiedCharAt6 below · cited by 14 · depth 18 - Unramified twist preserves induced level and K₁(vᶜ)-invariance
LanglandsTunnell.CubicInduction.inducedLevelAt_twist_eq_of_isUnramifiedCharAt4 below · cited by 8 · depth 18 - Right translation stability of cubic-induction Whittaker data
LanglandsTunnell.CubicInduction.isGaugeMajorised3_hasIotaMoments_hasWhittakerHalfPlane_comp_mul_right28 below · cited by 3 · depth 18 - Openness of the integral subgroup of GL₃(Kᵥ)
LanglandsTunnell.CubicInduction.isOpen_localMaximalCompact30 below · cited by 10 · depth 18 - Twisting by χᵥ∘det preserves the cyclic subspace
LanglandsTunnell.CubicInduction.mem_gl3CyclicSubspace_twist_det0 below · cited by 7 · depth 18 - Twisting a local GL₃ package by χᵥ ∘ det
LanglandsTunnell.CubicInduction.twist_det_localPackage1 below · cited by 6 · depth 18 - Contragredient duality for two-row GL₃ torus tables
LanglandsTunnell.CubicInduction.twoRowTable_contragredient_eq_inv_pow_mul0 below · cited by 2 · depth 18 - Archimedean value of an idele character through a section of the infinite part
LanglandsTunnell.CubicInduction.apply_of_infPart_eq_of_isArchCompAt0 below · cited by 11 · depth 19 - Archimedean root sizes of a GL₂ block image and its dual
LanglandsTunnell.CubicInduction.archRoot_iota_archRealGLAt_and_dual0 below · cited by 1 · depth 19 - Spherical Whittaker Hecke eigenfunctions on GL₃ come from principal series
LanglandsTunnell.CubicInduction.exists_eq_coefficientFn_principalSeries3_of_isCosetEigenfunction_of_norm_eq_one15 below · cited by 2 · depth 19 - Finite-dimensionality of U-invariant matrix coefficients of I(χ)
LanglandsTunnell.CubicInduction.exists_finset_coefficientFn_mem_span_of_isOpen5 below · cited by 4 · depth 19 - Finite support of the GL₃ Whittaker type integrals
LanglandsTunnell.CubicInduction.exists_finset_typeIntegral_eq_zero_of_eq_coefficientFn_of_le_conductorExponentAt23 below · cited by 1 · depth 19 - Vanishing of type integrals outside finitely many torus shells
LanglandsTunnell.CubicInduction.exists_finset_typeIntegral_eq_zero_of_forall_exists_finset_eq_zero_betaFinCS0 below · cited by 1 · depth 19 - Local GL₃timesGL₁ constants of a cubic induction at one bad place
LanglandsTunnell.CubicInduction.exists_forall_mem_gl3CyclicSubspace_localZetaDual31_eq_mul_of_isCubicInductionDataOn_deepAt539 below · cited by 1 · depth 19 - Span-wide local constants for deep cubic induction data
LanglandsTunnell.CubicInduction.exists_forall_mem_gl3CyclicSubspace_localZetaDual31_eq_mul_of_isCubicInductionDataOn_deep_badPlaces550 below · cited by 1 · depth 19 - Existence of a cubic-induction datum: archimedean and bad-place package
LanglandsTunnell.CubicInduction.exists_isCubicInductionDataOn_arch_torusValues_localPackage_bad1,686 below · cited by 1 · depth 19 - Re-choosing one local Whittaker factor within its cyclic span
LanglandsTunnell.CubicInduction.exists_isCubicInductionDataOn_whittakerLoc_eq_of_mem_gl3CyclicSubspace28 below · cited by 2 · depth 19 - Normalising cubic induction data at bad places
LanglandsTunnell.CubicInduction.exists_isCubicInductionDataOn_whittakerLoc_one_eq_one_of_isBadPlace31 below · cited by 2 · depth 19 - Central character of cubic induction data at unramified primes
LanglandsTunnell.CubicInduction.exists_localChar_centralChar_eq_finprod_mul_of_not_isRamifiedIn_of_isCubicInductionDataOn274 below · cited by 9 · depth 19 - Bump function on GL₂ from translates of a GL₃ Whittaker function
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_iotaGL_bump_of_isCompact_of_isOpen1 below · cited by 3 · depth 19 - Gauge majorisation passes to cyclic translates and duals on GL₃
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_exists_gauge_and_exists_gauge_dualWhittakerFn30 below · cited by 15 · depth 19 - Local GL₃ zeta data passes to the cyclic span
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_localZeta30_localZetaDual31_eulerData_of_forall2 below · cited by 7 · depth 19 - Cyclicity of a unitary spherical Whittaker function on GL₃
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_of_isCosetEigenfunction_of_norm_eq_one31 below · cited by 1 · depth 19 - Two-sided determinant moments from gauge-majorised mirabolic expansions
LanglandsTunnell.CubicInduction.hasIotaMoments_of_hasSum_mirabolicTranslate_of_isGaugeMajorised326 below · cited by 3 · depth 19 - Twisting cubic induction data by χ∘det
LanglandsTunnell.CubicInduction.isCubicInductionDataOn_twist_det16 below · cited by 2 · depth 19 - Gauge majorisation passes to spans of right translates
LanglandsTunnell.CubicInduction.isGaugeMajorised3_of_mem_gl3CyclicSubspace0 below · cited by 5 · depth 19 - Unramified twist preserves spherical Hecke and torus conditions
LanglandsTunnell.CubicInduction.isInducedSphericalAt_iff_and_hasSphericalTorusValuesAt_iff_localChar_mul0 below · cited by 2 · depth 19 - Archimedean zeta package for an explicit GL₃ Whittaker vector
LanglandsTunnell.CubicInduction.jacquetVector3_archZeta_package32 below · cited by 1 · depth 19 - Deep rational twists stay deep over a cubic field
LanglandsTunnell.CubicInduction.le_conductorExponentAt_localChar_mul_comp_idelicNorm_of_hasConductorExponentAt_of_forall_le11 below · cited by 1 · depth 19 - Local GL(3) functional-equation package from cleared denominators
LanglandsTunnell.CubicInduction.localZeta31_fe_one_of_forall_exists_mul_eval_eq_of_eval_mul_eq0 below · cited by 5 · depth 19 - Local component at v of a global additive character: non-triviality and level clauses
LanglandsTunnell.CubicInduction.psiLoc_ne_one_and_level_clauses_of_isGlobalAddChar17 below · cited by 13 · depth 19 - Unitary idele class characters have purely imaginary archimedean exponent
LanglandsTunnell.CubicInduction.re_eq_zero_of_isArchCompAt_of_isUnitaryChar0 below · cited by 11 · depth 19 - Multiplicativity of the local GL₃timesGL₂ functional equation, unramified partner
LanglandsTunnell.CubicInduction.rsLocalIntegral_fe32_of_forall_localZeta31_fe_of_gauge86 below · cited by 1 · depth 19 - Unfolding N(ℚ)-invariant integrals along NbackslashGL₂(ℚ)
LanglandsTunnell.CubicInduction.setLIntegral_eq_setLIntegral_tsum_mirabolicRep2 below · cited by 4 · depth 19 - Mirabolic series and integrals of a gauge-majorised function on GL₃
LanglandsTunnell.CubicInduction.summable_growth_continuous_halfPlane_integrable_of_isGaugeMajorised325 below · cited by 14 · depth 19 - Dual Whittaker function as the reflected Whittaker function
LanglandsTunnell.CubicInduction.CubicInductionData.dualWhittaker_eq_dualWhittakerFn30 below · cited by 2 · depth 20 - Archimedean zeta integral of `jacquetVector3`, unfolded
LanglandsTunnell.CubicInduction.archZeta30_jacquetVector3_eq_archFactor_mul3 below · cited by 1 · depth 20 - Archimedean functional equation for the induced GL₃ zeta integrals
LanglandsTunnell.CubicInduction.archZetaDual31_jacquetVector3_mul_archFactor_eq12 below · cited by 1 · depth 20 - Twisting a GL₃ cusp form by a character of the determinant
LanglandsTunnell.CubicInduction.continuous_and_isCuspidalAlong_and_whittaker3_fnTwist3_eq_chiDetGL_mul0 below · cited by 1 · depth 20 - Continuity and gauge majorisation of factorisable functions on adelic GL₃
LanglandsTunnell.CubicInduction.continuous_and_isGaugeMajorised3_of_eq_mul_prod0 below · cited by 1 · depth 20 - Transpose-inverse dual of a GL₃ cusp form
LanglandsTunnell.CubicInduction.continuous_isCuspidalAlong_isModerateGrowth3_dualForm0 below · cited by 2 · depth 20 - Spherical Whittaker eigenfunction on GL₃ vanishing at the identity
LanglandsTunnell.CubicInduction.eq_zero_of_isCosetEigenfunction_of_isGL3PsiWhittakerFn_of_apply_one_eq_zero6 below · cited by 2 · depth 20 - Explicit root number in the GL₃ functional equation at v
LanglandsTunnell.CubicInduction.eval_mul_eq_finprod_rootNumber_mul_eval_of_forall_localZeta31_fe_one_of_isCubicInductionDataOn_of_addCharLevel493 below · cited by 2 · depth 20 - Archimedean unfolding integral equals Γ_ℝ times an entire function
LanglandsTunnell.CubicInduction.exists_differentiable_unfoldingIntegral_eq_GammaR_mul4 below · cited by 1 · depth 20 - Entire twisted Euler product for a non-norm cubic induction
LanglandsTunnell.CubicInduction.exists_entire_eulerTwist_induced_of_not_exists_eq_pow_inertiaDeg71 below · cited by 1 · depth 20 - Levi-central units act trivially on a deeply contracted Whittaker function
LanglandsTunnell.CubicInduction.exists_forall_apply_diagonal3_mul_diagonal3_mul_eq_of_isCompact_of_valued_le_fst1 below · cited by 1 · depth 20 - Deep central units diag(t,t,s) act trivially on contracted Whittaker values
LanglandsTunnell.CubicInduction.exists_forall_apply_diagonal3_mul_diagonal3_mul_eq_of_isCompact_of_valued_le_snd1 below · cited by 1 · depth 20 - Whittaker functions vanish deep in the GL₂-torus
LanglandsTunnell.CubicInduction.exists_forall_apply_iotaGL_mul_eq_zero_of_lt_neg4 below · cited by 3 · depth 20 - Local rationality and functional equation at a bad place
LanglandsTunnell.CubicInduction.exists_forall_exists_mul_eval_eq_of_isCubicInductionDataOn_of_forall_mem_bad_of_addCharLevel514 below · cited by 4 · depth 20 - Principal series vectors of p-adic GL₃ are smooth
LanglandsTunnell.CubicInduction.exists_forall_gl3AmbientRightTranslate_eq_of_mem_principalSeries32 below · cited by 20 · depth 20 - Local Whittaker data at bad places of a saturated cubic induction
LanglandsTunnell.CubicInduction.exists_forall_le_exists_localWhittaker_saturated_and_laurent_fe_of_mem_bad65 below · cited by 1 · depth 20 - Uniform root-size bound for induced spherical Whittaker functions
LanglandsTunnell.CubicInduction.exists_forall_rootSize_bound_of_isInducedSphericalAt_of_isUnitaryChar3 below · cited by 1 · depth 20 - Type integrals of deep GL₃ Whittaker coefficients vanish eventually
LanglandsTunnell.CubicInduction.exists_forall_typeIntegral_eq_zero_of_le_fst7 below · cited by 1 · depth 20 - Vanishing of GL₃ type integrals for large n₂
LanglandsTunnell.CubicInduction.exists_forall_typeIntegral_eq_zero_of_le_snd11 below · cited by 1 · depth 20 - Conductor bound at every place for the induced central character
LanglandsTunnell.CubicInduction.exists_hasConductorExponentAt_le_finsum_pinnedExp_of_eulerCoeff_eq_inducedE3354 below · cited by 1 · depth 20 - Conductor exponent bound at places unramified in the cubic field
LanglandsTunnell.CubicInduction.exists_hasConductorExponentAt_le_inducedLevelAt_of_eulerCoeff_eq_inducedE3280 below · cited by 2 · depth 20 - Conductor bound for the local central character at unramified v
LanglandsTunnell.CubicInduction.exists_hasConductorExponentAt_localChar_centralChar_le_inducedLevelAt_of_isCubicInductionDataOn278 below · cited by 4 · depth 20 - Odd admissible twist with non-vanishing archimedean GL₃ × GL₁ zeta
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_ne_zero_odd_of_isCubicInductionDataOn6 below · cited by 2 · depth 20 - Archimedean zeta non-vanishing far right for a suitable translate
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_ne_zero_of_isCubicInductionDataOn1 below · cited by 6 · depth 20 - Re-choosing one local Whittaker factor within its cyclic span
LanglandsTunnell.CubicInduction.exists_isCubicInductionDataOn_whittakerLoc_eq_of_mem_gl3CyclicSubspace_of_isOpen29 below · cited by 1 · depth 20 - Good-place spherical Whittaker function for cubic induction
LanglandsTunnell.CubicInduction.exists_isGL3PsiWhittakerFn_isInducedSphericalAt_of_not_isBadPlace8 below · cited by 1 · depth 20 - Uniform smoothness of a GL₃ principal-series coefficient under right translation
LanglandsTunnell.CubicInduction.exists_isOpen_forall_apply_mul_iotaGL_mul_eq1 below · cited by 2 · depth 20 - Non-zero Whittaker functional on a GL₃ principal series
LanglandsTunnell.CubicInduction.exists_isWhittakerFunctional3_ne_zero0 below · cited by 1 · depth 20 - Local newvector of level K₁(ℓᵥ) at twist-ramified primes
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_congruenceK1_torusValues_of_isCubicInductionDataOn615 below · cited by 1 · depth 20 - Bump vector in the cyclic span of a GL₃ Whittaker function
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_iotaGL_bump0 below · cited by 5 · depth 20 - Congruence-invariant vector in the local cyclic space at a ramified bad place
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_principalLevel_le_of_isRamifiedIn_of_isCubicInductionDataOn_of_conductorBound615 below · cited by 1 · depth 20 - A twist-independent constant in the deep-place GL₃× GL₁ functional equation
LanglandsTunnell.CubicInduction.exists_ne_zero_forall_eval_mul_eq_mul_rootNumber_mul_eval_of_forall_localZeta31_fe_twist_of_isCubicInductionDataOn_of_deep_of_archPackage_of_inv_eq_psiQ_of_whittakerLoc_one502 below · cited by 2 · depth 20 - Root-size support and growth bound for local GL₃ Whittaker functions
LanglandsTunnell.CubicInduction.exists_rootSize_bound_of_isGL3PsiWhittakerFn1 below · cited by 7 · depth 20 - Convergence and rationality of local GL₃× GL₂ Rankin–Selberg integrals
LanglandsTunnell.CubicInduction.exists_rsLocalIntegral_and_dual_integrable_and_eq_rational_sphericalWhittaker_of_forall_localZeta31_fe_of_gauge13 below · cited by 2 · depth 20 - Spherical vector in an unramified principal series of GL₃
LanglandsTunnell.CubicInduction.exists_spherical_mem_principalSeries3_isCosetEigenfunction3 below · cited by 1 · depth 20 - Pure tensor Whittaker vector with local multiplicity one at S
LanglandsTunnell.CubicInduction.exists_whittaker3_eq_mul_prod_hasWhittakerMultOne_of_isCuspidalAlong354 below · cited by 1 · depth 20 - Converse-theorem input for the cubic induction from an archimedean Whittaker vector
LanglandsTunnell.CubicInduction.exists_whittaker_zeta_fe_of_forall_not_mem_isInducedSphericalAt_of_arch145 below · cited by 1 · depth 20 - Finite-dimensional level-n fixed vectors in the GL₃ principal series
LanglandsTunnell.CubicInduction.finiteDimensional_fixedPoints_principalSeries31 below · cited by 4 · depth 20 - Only finitely many rational primes ramify in a number field
LanglandsTunnell.CubicInduction.finite_setOf_isRamifiedIn0 below · cited by 4 · depth 20 - Product formula (prodᵥλᵥ²) λ_∞²=1 for a cubic induction
LanglandsTunnell.CubicInduction.finprod_sq_mul_lamSqArch_eq_one_of_forall_ne_zero_localZeta31_fe_rootNumber_of_isCubicInductionDataOn_of_archPackage_of_inv_eq_psiQ538 below · cited by 1 · depth 20 - Coefficients of unitary principal series of GL₃ generate irreducibly
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_of_mem_gl3CyclicSubspace_coefficientFn17 below · cited by 3 · depth 20 - Rapid vertical decay of the archimedean zeta integral `archZeta30`
LanglandsTunnell.CubicInduction.forall_pow_mul_norm_archZeta30_jacquetVector3_le3 below · cited by 1 · depth 20 - Polynomial decay of a dual archimedean zeta integral on strips
LanglandsTunnell.CubicInduction.forall_pow_mul_norm_archZetaDual31_jacquetVector3_le3 below · cited by 1 · depth 20 - Moments of all orders for Φ and its dual along GL₂
LanglandsTunnell.CubicInduction.hasIotaMoments_and_dualForm_of_hasSum_of_hasWhittakerHalfPlane2 below · cited by 1 · depth 20 - Mirabolic Whittaker expansion of a cuspidal GL₃ form over ℚ
LanglandsTunnell.CubicInduction.hasSum_whittaker3_mirabolicTranslate_mul_of_summable_of_isCuspidalAlong10 below · cited by 1 · depth 20 - Whittaker multiplicity one for a normalised spherical Whittaker function
LanglandsTunnell.CubicInduction.hasWhittakerMultOne_of_isCosetEigenfunction7 below · cited by 1 · depth 20 - Whittaker multiplicity one for cyclic spans on GL₃(ℚᵥ)
LanglandsTunnell.CubicInduction.hasWhittakerMultOne_of_ne_one_of_forall_mem_gl3CyclicSubspace5 below · cited by 9 · depth 20 - Dual archimedean factor of the twisted Hecke–Tate L-datum
LanglandsTunnell.CubicInduction.heckeDatum_archFactorDual_eq_archFactor_dual_twist_mul_GammaR0 below · cited by 2 · depth 20 - Splitting off the Γ_ℝ-factor at one real place
LanglandsTunnell.CubicInduction.heckeDatum_archFactor_eq_archFactor_twist_mul_GammaR0 below · cited by 2 · depth 20 - Twisting a cubic idelic character by χ ∘ N
LanglandsTunnell.CubicInduction.inducedCoeff_mul_comp_idelicNorm_and_isBadPlace_iff_of_conductorExponentAt_le24 below · cited by 1 · depth 20 - Cubing of E₃ under fibrewise scaling by χ(πᵥ)^f
LanglandsTunnell.CubicInduction.inducedE3_eq_pow_three_mul_of_fibre_eq_pow_inertiaDeg_mul0 below · cited by 1 · depth 20 - Compactness of the integral maximal compact subgroup of GL₃(ℚᵥ)
LanglandsTunnell.CubicInduction.isCompact_localMaximalCompact30 below · cited by 14 · depth 20 - Cuspidality of a GL₃ form from entire tempered Euler products
LanglandsTunnell.CubicInduction.isCuspidalAlong_form_of_tempered_of_entire_eulerTwists760 below · cited by 1 · depth 20 - Twisting a local GL₃ Whittaker datum by χ∘det
LanglandsTunnell.CubicInduction.isGL3PsiWhittakerFn_and_hasWhittakerMultOne_of_eq_mul_det0 below · cited by 1 · depth 20 - Whittaker law for the GL₃ Jacquet vector
LanglandsTunnell.CubicInduction.isGL3PsiWhittakerFn_jacquetVector30 below · cited by 2 · depth 20 - Whittaker transformation law for the GL₃ coefficient
LanglandsTunnell.CubicInduction.isGL3PsiWhittakerFn_whittaker3_of_forall_upperUnipotent3_mul_eq0 below · cited by 1 · depth 20 - K-finiteness of the polynomial-times-Gaussian Jacquet vector on GL₃
LanglandsTunnell.CubicInduction.isKFinite_jacquetVector32 below · cited by 1 · depth 20 - Integrability and continuity of the GL₃ Jacquet vector
LanglandsTunnell.CubicInduction.jacquetIntegrand3_integrable_and_jacquetVector3_continuous1 below · cited by 2 · depth 20 - Convergence half-planes for archimedean GL₃timesGL₁ zeta integrals
LanglandsTunnell.CubicInduction.jacquetVector3_isArchZetaConvergentAbove4 below · cited by 1 · depth 20 - Rapid decay of the GL₃ Jacquet–Whittaker vector
LanglandsTunnell.CubicInduction.jacquetVector3_norm_archComponent3_le6 below · cited by 1 · depth 20 - Central character of the explicit GL₃ Jacquet vector
LanglandsTunnell.CubicInduction.jacquetVector3_scalar_mul1 below · cited by 2 · depth 20 - Sign of the discriminant: Brill's rule for λ²_∞
LanglandsTunnell.CubicInduction.lamSqArch_eq_neg_one_pow_nrComplexPlaces0 below · cited by 2 · depth 20 - Local central character at good places equals e₃
LanglandsTunnell.CubicInduction.localChar_centralChar_eq_one_and_apply_uniformizerUnit_eq_inducedE3_of_not_isBadPlace_of_isCubicInductionDataOn0 below · cited by 1 · depth 20 - Identified local functional equation passes to the cyclic span
LanglandsTunnell.CubicInduction.localZeta31_identified_of_mem_gl3CyclicSubspace1 below · cited by 2 · depth 20 - Converse theorem on GL₃/ℚ with finite exceptional set
LanglandsTunnell.CubicInduction.nonempty_automorphyDatum31_of_zeta_fe22 below · cited by 1 · depth 20 - Non-norm condition is stable under twisting by base-changed characters
LanglandsTunnell.CubicInduction.not_exists_eq_pow_inertiaDeg_mul_comp_idelicNorm_of_not_exists12 below · cited by 1 · depth 20 - Local γ-factor identity for GL₃× GL₂ with gauge majorant
LanglandsTunnell.CubicInduction.rsLocalIntegral_fe32_of_eq_rational_of_forall_localZeta31_fe_of_gauge85 below · cited by 1 · depth 20 - S-part integrability of the GL₃ zeta and dual integrands
LanglandsTunnell.CubicInduction.sPart_integrable_and_dual_of_isCubicInductionDataOn_of_isGaugeMajorised353 below · cited by 4 · depth 20 - Central character law for the archimedean Whittaker function
LanglandsTunnell.CubicInduction.whittakerArch_scalar_mul_eq_centralChar_mul_of_isCubicInductionDataOn0 below · cited by 4 · depth 20 - Whittaker functionals do not vanish on the spherical vector
LanglandsTunnell.CubicInduction.whittakerFunctional3_spherical_ne_zero_of_norm_eq_one4 below · cited by 1 · depth 20 - Nonvanishing of every local Whittaker factor
LanglandsTunnell.CubicInduction.whittakerLoc_ne_zero_of_isCubicInductionDataOn0 below · cited by 7 · depth 20 - Smoothness, admissibility and inverse Whittaker law for the dual function
LanglandsTunnell.CubicInduction.admissible_gl3CyclicSubspace_dualWhittakerFn3_rightTranslate1 below · cited by 5 · depth 21 - Invariance of bi-Whittaker forms on GL₃ under g↦ w ^tg w
LanglandsTunnell.CubicInduction.apply_comp_longWeyl3_conj_transpose_eq_apply_of_ne_one0 below · cited by 1 · depth 21 - Dual archimedean (3,1) zeta integral of a unipotent average
LanglandsTunnell.CubicInduction.archZeta30_integral_dualWhittakerFn3_eq_archZetaDual31_of_eq_map_ringEquiv_mixedSpace1 below · cited by 2 · depth 21 - Radical Fourier coefficient of the dual mirabolic GL₃ series
LanglandsTunnell.CubicInduction.box_integral_dualSeries_radical_eq_ideleNorm_mul_tsum_integral_dual1 below · cited by 2 · depth 21 - Coset sums preserve right U-invariance and the Whittaker law
LanglandsTunnell.CubicInduction.cosetSum_isRightInvariant_and_isGL3PsiWhittakerFn0 below · cited by 1 · depth 21 - Archimedean constants for one real and one complex place, exponent zero
LanglandsTunnell.CubicInduction.dualConstants_eq_epsilonFactor_mul_signEpsilon_of_isComplex_of_eq_zero0 below · cited by 1 · depth 21 - Archimedean constants for one real and one complex place
LanglandsTunnell.CubicInduction.dualConstants_eq_epsilonFactor_mul_signEpsilon_of_isComplex_of_ne_zero0 below · cited by 1 · depth 21 - Archimedean epsilon constants at three real places
LanglandsTunnell.CubicInduction.dualConstants_eq_epsilonFactor_mul_signEpsilon_of_isReal_split0 below · cited by 1 · depth 21 - Dual Whittaker function of cubic induction data
LanglandsTunnell.CubicInduction.dualWhittaker_eq_dualWhittakerFn3_of_isCubicInductionDataOn0 below · cited by 6 · depth 21 - Factorisation of the dual Whittaker function over a finite set of places
LanglandsTunnell.CubicInduction.dualWhittaker_eq_dualWhittakerFn3_whittakerArch_mul_prod_of_isCubicInductionDataOn0 below · cited by 6 · depth 21 - Vanishing of Whittaker functions off the dominant cone
LanglandsTunnell.CubicInduction.eq_zero_of_coe_eq_diagonal_of_valued_lt_of_isGL3PsiWhittakerFn0 below · cited by 8 · depth 21 - Independence of the functions (a₁a₂)^j h_{d-2j}(a₁,a₂)
LanglandsTunnell.CubicInduction.eq_zero_of_forall_sum_mul_pow_mul_heckeRecursionSeq_eq_zero0 below · cited by 5 · depth 21 - Diagonal value as e₃ᶜ times a two-row torus value
LanglandsTunnell.CubicInduction.eq_zpow_mul_twoRowPointLocal_of_coe_eq_diagonal_of_isRightInvariant0 below · cited by 4 · depth 21 - Adelic GL₃ element supported at a single finite place
LanglandsTunnell.CubicInduction.exists_adelicGL3_archComponent3_eq_one_componentAt3_eq0 below · cited by 3 · depth 21 - Stable complements in the unitary principal series of GL₃
LanglandsTunnell.CubicInduction.exists_compl_of_forall_le_comap_gl3AmbientRightTranslate7 below · cited by 1 · depth 21 - Entire continuation and functional equation of GL₃ zeta integrals
LanglandsTunnell.CubicInduction.exists_entire_eq_globalZeta30_eq_mul_globalZetaDual31_of_isCubicInductionDataOn42 below · cited by 7 · depth 21 - Schur's lemma for the GL₃ principal series at equal characters
LanglandsTunnell.CubicInduction.exists_eq_smul_id_of_gl3AmbientRightTranslate_comm_of_apply_eq4 below · cited by 1 · depth 21 - Scalarity of translation-equivariant endomorphisms of GL₃ principal series
LanglandsTunnell.CubicInduction.exists_eq_smul_id_of_gl3AmbientRightTranslate_comm_of_injective6 below · cited by 1 · depth 21 - Schur's lemma for a GL₃ principal series with two equal characters
LanglandsTunnell.CubicInduction.exists_eq_smul_id_of_gl3AmbientRightTranslate_comm_of_not_injective5 below · cited by 1 · depth 21 - Uniqueness of the ψ-Whittaker functional on GL₃ principal series
LanglandsTunnell.CubicInduction.exists_eq_smul_of_isWhittakerFunctional33 below · cited by 2 · depth 21 - Iwasawa decomposition NTK for GL₃ over a completion of ℚ
LanglandsTunnell.CubicInduction.exists_eq_upperUnipotent3_mul_diagonal_mul_mem_localMaximalCompact30 below · cited by 18 · depth 21 - Stable local functional equation under highly ramified twists
LanglandsTunnell.CubicInduction.exists_exists_forall_localZetaDual31_eq_mul_localZeta30_and_exists_localZeta30_ne_zero20 below · cited by 6 · depth 21 - Non-vanishing of a local GL₃ Whittaker zeta integral
LanglandsTunnell.CubicInduction.exists_exists_localZeta30_selfDual_ne_zero_of_isGL3PsiWhittakerFn_of_ne_zero0 below · cited by 6 · depth 21 - Unramified characters of ℚᵥ^× are powers of the modulus
LanglandsTunnell.CubicInduction.exists_forall_apply_eq_modulus_cpow1 below · cited by 6 · depth 21 - Convergence of the local GL₃timesGL₂ Rankin–Selberg integral
LanglandsTunnell.CubicInduction.exists_forall_integrable_rsLocalIntegrand_of_gauge8 below · cited by 4 · depth 21 - Vanishing of GL₃ cell-section Whittaker functions outside a cone
LanglandsTunnell.CubicInduction.exists_forall_jacquetWhittaker3_eq_zero_of_rootSize_gt14 below · cited by 3 · depth 21 - Stabilisation of truncated Jacquet integrals of translated cell sections
LanglandsTunnell.CubicInduction.exists_forall_le_integrableOn_and_jacquetTruncated3_eq_of_cellSectionOf8 below · cited by 13 · depth 21 - Level vectors killed by N_{(2,1)}-invariant functionals on I(χ)
LanglandsTunnell.CubicInduction.exists_forall_linearMap_apply_eq_zero_of_radicalP21_of_mem_principalSeries37 below · cited by 2 · depth 21 - Vanishing along the opposite (2,1)-radical for ramified GL₃ principal series
LanglandsTunnell.CubicInduction.exists_forall_linearMap_apply_eq_zero_of_transposeInv3_radicalP21_of_mem_principalSeries38 below · cited by 1 · depth 21 - Non-vanishing of a GL₃ Whittaker zeta integral on a half-plane
LanglandsTunnell.CubicInduction.exists_forall_localZeta30_selfDual_ne_zero_of_isGL3PsiWhittakerFn_of_ne_zero0 below · cited by 9 · depth 21 - Stability of the GL₃timesGL₁ local functional equation under ramified twists
LanglandsTunnell.CubicInduction.exists_forall_localZetaDual31_eq_mul_localZeta30_of_isGL3PsiWhittakerFn_of_norm_eq_one20 below · cited by 6 · depth 21 - Local gauge bound for the cubic Whittaker function
LanglandsTunnell.CubicInduction.exists_gauge_whittakerLoc_of_isGaugeMajorised3_of_form_ne_zero_of_isCubicInductionDataOn0 below · cited by 2 · depth 21 - Conductor bound at ramified places, cofinite Euler data
LanglandsTunnell.CubicInduction.exists_hasConductorExponentAt_le_finsum_addCharLevel_of_eulerCoeff_eq_inducedE3_one_cofinite_of_isRamifiedIn327 below · cited by 1 · depth 21 - Euler factorisation of the GL₃timesGL₁ zeta integral outside S
LanglandsTunnell.CubicInduction.exists_hasProd_sphericalShellSums_and_globalZeta30_eq_mul_integral_sPart6 below · cited by 7 · depth 21 - Rationality in Nᵥ^{-s} of local GL₃× GL₂ integrals
LanglandsTunnell.CubicInduction.exists_integrable_and_rsLocalIntegral_mul_eval_eq_of_isGL3PsiWhittakerFn12 below · cited by 2 · depth 21 - Integrable majorant and measurability for the GL₃ Jacquet integrand
LanglandsTunnell.CubicInduction.exists_integrable_majorant_jacquetIntegrand3_and_aestronglyMeasurable_prod1 below · cited by 17 · depth 21 - Coset system for diag(varpi,1,1) and three-term Whittaker sum
LanglandsTunnell.CubicInduction.exists_isHeckeCosetSystem_heckeGen1_cosetSum_twoRowPointLocal0 below · cited by 2 · depth 21 - Coset system for diag(varpi,varpi,1) and Whittaker coset sums
LanglandsTunnell.CubicInduction.exists_isHeckeCosetSystem_heckeGen2_cosetSum_twoRowPointLocal0 below · cited by 2 · depth 21 - Non-vanishing of the local GL₃× GL₁ zeta integral
LanglandsTunnell.CubicInduction.exists_isLocalZeta30ConvergentAbove_and_forall_exists_localZeta30_ne_zero_of_admissible_of_ne_zero13 below · cited by 2 · depth 21 - Local functional equation for spherical GL₃ Whittaker zeta integrals
LanglandsTunnell.CubicInduction.exists_laurent_localZeta_eq_gl3LFactorPoly_of_sphericalData24 below · cited by 1 · depth 21 - Local Laurent form and functional equation of the GL₃ Whittaker zeta integrals
LanglandsTunnell.CubicInduction.exists_laurent_localZeta_fe_of_jacquetWhittaker3_mul_antidiagonal352 below · cited by 1 · depth 21 - Local zeta functional equation at a ramified place
LanglandsTunnell.CubicInduction.exists_localZeta31_fe_one_inducedEulerPoly_rational_of_isCubicInductionDataOn_of_isRamifiedIn527 below · cited by 2 · depth 21 - Local functional equation at a bad place unramified in K
LanglandsTunnell.CubicInduction.exists_localZeta31_fe_one_inducedEulerPoly_rational_of_isCubicInductionDataOn_of_not_isRamifiedIn527 below · cited by 2 · depth 21 - K-invariant vector in the cyclic span with unchanged local integrals
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_iotaGL_invariant_rsLocalIntegral_eq9 below · cited by 5 · depth 21 - Unfolded (3,2) functional equation for deformed spherical vectors
LanglandsTunnell.CubicInduction.exists_mvPolynomial_forall_dominant_rsLocalIntegral_deformedSpherical_eq_and_fe_of_forall_localZeta31_fe_of_gauge80 below · cited by 1 · depth 21 - Rationality of the local GL₃timesGL₂ Rankin–Selberg integral
LanglandsTunnell.CubicInduction.exists_mvPolynomial_forall_rsLocalIntegral_mul_eq_eval_of_iotaGL_invariant14 below · cited by 3 · depth 21 - Gauge bound for GL₃ Jacquet–Whittaker cell functions
LanglandsTunnell.CubicInduction.exists_norm_jacquetWhittaker3_le_of_rootSize_le14 below · cited by 3 · depth 21 - Normalised K₁(v^ℓ)-newvector with prescribed Rankin–Selberg integral
LanglandsTunnell.CubicInduction.exists_normalisedNewvector_of_isLocalWhittakerDatum_of_localFE32_of_inducedE3_eq_zero42 below · cited by 1 · depth 21 - Newvector in the cyclic span from local GL₃timesGL₂ functional equations
LanglandsTunnell.CubicInduction.exists_normalisedNewvector_of_isLocalWhittakerDatum_of_localFE32_of_ne_zero42 below · cited by 1 · depth 21 - Invariant separating pairing for irreducible smooth admissible GL₃ representations
LanglandsTunnell.CubicInduction.exists_pairing_transposeInv3_of_isIrreducibleRep1 below · cited by 1 · depth 21 - Rationality in Nᵥ^{-s} of two GL₃ local zeta integrals
LanglandsTunnell.CubicInduction.exists_polynomial_mul_localZeta30_eq_and_dual_of_isGL3PsiWhittakerFn5 below · cited by 3 · depth 21 - Three local characters at a bad prime of cubic induction
LanglandsTunnell.CubicInduction.exists_prod_eq_localChar_and_prod_stdRootNumberAt_eq_of_saturated21 below · cited by 1 · depth 21 - Rational representatives of polynomially bounded gauge on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.exists_rational_inv_mul_mem_converseCongruence_gauge3_le0 below · cited by 1 · depth 21 - Local translates at v of a cuspidal GL₃ Hecke eigenform
LanglandsTunnell.CubicInduction.exists_sum_eq_span_translates_of_isCuspidalAlong_of_isCosetEigenfunction336 below · cited by 1 · depth 21 - Whittaker-type function on GL₂(ℚᵥ) with prescribed torus values
LanglandsTunnell.CubicInduction.exists_unipotent_localLevelOne_scalarPi_diagZ_torusFactor_of_ne_zero0 below · cited by 6 · depth 21 - Valuation exponents of a dominant diagonal element of GL₃
LanglandsTunnell.CubicInduction.exists_valued_eq_exp_of_coe_eq_diagonal_of_not_valued_lt0 below · cited by 2 · depth 21 - Essential Whittaker vector at p with non-vanishing value at 1
LanglandsTunnell.CubicInduction.exists_whittaker_localLevelOne_centralChar_admissible_principalSeries2_apply_one_ne_zero_of_norm_eq_one_of_higherUnitsAt50 below · cited by 1 · depth 21 - Unramified primes in a cubic field: residue degrees sum to three
LanglandsTunnell.CubicInduction.finsum_primeFibre_inertiaDeg_eq_three_of_not_isRamifiedIn0 below · cited by 2 · depth 21 - Local GL₃timesGL₁ functional equation spreads to the cyclic space
LanglandsTunnell.CubicInduction.forall_mem_gl3CyclicSubspace_localZeta31_fe_of_forall_localZeta31_fe0 below · cited by 4 · depth 21 - Euler factorisation of the GL₃ zeta integral outside S
LanglandsTunnell.CubicInduction.globalZeta30_eq_sPart_mul_inducedL_of_isCubicInductionDataOn32 below · cited by 6 · depth 21 - Euler factorisation of the unipotent GL₃× GL₁ zeta integral
LanglandsTunnell.CubicInduction.globalZeta31_eq_mul_integral_sPart_mul_of_hasProd_localZeta31_of_integrable10 below · cited by 6 · depth 21 - Euler product of dual (3,1) zeta integrals at good primes
LanglandsTunnell.CubicInduction.hasProd_localZeta31_dualWhittakerFn3_of_isInducedSphericalAt_of_three_le12 below · cited by 6 · depth 21 - Spherical torus values from Rankin–Selberg local integrals
LanglandsTunnell.CubicInduction.hasSphericalTorusValuesAt_inducedCoeff_of_rsLocalIntegral_eq_cellVolume17 below · cited by 1 · depth 21 - Torus sum as radical coefficient of the mirabolic series
LanglandsTunnell.CubicInduction.hasSum_torus_radicalCoefficient_mirabolicSeries1 below · cited by 2 · depth 21 - Whittaker multiplicity one passes to elements of the cyclic span
LanglandsTunnell.CubicInduction.hasWhittakerMultOne_of_mem_gl3CyclicSubspace_of_isOpen0 below · cited by 1 · depth 21 - No cubic term at primes ramified in a cubic field
LanglandsTunnell.CubicInduction.inducedE3_eq_zero_of_isRamifiedIn_of_finrank_eq_three0 below · cited by 1 · depth 21 - Integrability of the dual GL₃ zeta integrand of a Jacquet vector
LanglandsTunnell.CubicInduction.integrable_dualWhittakerFn3_jacquetVector3_prod2 below · cited by 1 · depth 21 - Integrability of an idelic product forces factorwise integrability or vanishing
LanglandsTunnell.CubicInduction.integrable_factors_or_eq_zero_of_integrable_sPart3 below · cited by 6 · depth 21 - Joint integrability of the dilated Jacquet integrand in three variables
LanglandsTunnell.CubicInduction.integrable_jacquetIntegrand3_dilate_mul_quasiChar1 below · cited by 2 · depth 21 - The congruence set K₁(𝔭ᵥᶜ) in GL₃ is inverse-closed
LanglandsTunnell.CubicInduction.inv_mem_congruenceK10 below · cited by 3 · depth 21 - diag(g,1) lies in K₁(𝔭ᶜ) iff integral
LanglandsTunnell.CubicInduction.iotaGL_mem_congruenceK1_iff0 below · cited by 4 · depth 21 - Level-one g at v gives diag(g,1) integral in GL₃
LanglandsTunnell.CubicInduction.iotaGL_mem_localMaximalCompact3_of_mem_localLevelOne0 below · cited by 4 · depth 21 - Compactness of the congruence set K₁(mathfrak pᵥᶜ) in GL₃(mathbb Qᵥ)
LanglandsTunnell.CubicInduction.isCompact_congruenceK11 below · cited by 3 · depth 21 - Cuspidality along P_{2,1} of the mirabolic Whittaker series
LanglandsTunnell.CubicInduction.isCuspidalAlongP21_mirabolicSeries1 below · cited by 2 · depth 21 - Dual of a ψ-Whittaker function on GL₃
LanglandsTunnell.CubicInduction.isGL3PsiWhittakerFn_dualWhittakerFn30 below · cited by 11 · depth 21 - Torus dilation law for the GL₃ Jacquet vector
LanglandsTunnell.CubicInduction.jacquetVector3_iotaGL_diagUnitGL2_mul0 below · cited by 11 · depth 21 - Jacquet vector at a real diagonal torus element, unfolded
LanglandsTunnell.CubicInduction.jacquetVector3_iota_upperUnit_eq_integral_godementInner3_mulShift0 below · cited by 1 · depth 21 - Squared archimedean constant equals square of λ-product
LanglandsTunnell.CubicInduction.lamSqArch_eq_prod_lambdaArch_sq0 below · cited by 1 · depth 21 - Local central character at -1 and at pinning elements
LanglandsTunnell.CubicInduction.localChar_centralChar_neg_one_and_pin_eq_finprod_of_eq_finprod_mul_of_isCubicInductionDataOn7 below · cited by 6 · depth 21 - Sign-flip transport of the local GL₃ package, gauge edition
LanglandsTunnell.CubicInduction.localPackage_psiLocal_inv_comp_mul_diagonal_of_localPackage_psiLocal_of_gauge2 below · cited by 2 · depth 21 - Conjugation by diag(1,-1,1) of local GL₃ zeta integrals
LanglandsTunnell.CubicInduction.localZeta_conj_diagonal_signFlip2 below · cited by 6 · depth 21 - Vanishing of a θ-isotypic vector in a (1,2) Jacquet module
LanglandsTunnell.CubicInduction.mem_span_rightTranslate_radicalP12_sub_of_forall_apply_mul_diagonal30 below · cited by 2 · depth 21 - Vanishing along the (2,1) radical of a θ-isotypic principal series vector
LanglandsTunnell.CubicInduction.mem_span_rightTranslate_radicalP21_sub_of_forall_apply_mul_diagonal31 below · cited by 1 · depth 21 - Mirabolic series equals its dual from radical coefficients
LanglandsTunnell.CubicInduction.mirabolicSeries_eq_dual_of_radicalCoefficient_eq7 below · cited by 1 · depth 21 - Place separation for local zeta quotients at a bad place
LanglandsTunnell.CubicInduction.mul_eq_mul_localZeta30_localZetaDual31_polynomial_of_isCubicInductionDataOn_of_forall_mem_bad512 below · cited by 1 · depth 21 - The congruence set K₁(𝔭ᵥᶜ) in GL₃ is multiplicatively closed
LanglandsTunnell.CubicInduction.mul_mem_congruenceK10 below · cited by 3 · depth 21 - Ramified places force pinned exponent sum at least one
LanglandsTunnell.CubicInduction.one_le_finsum_inertiaDeg_mul_pinnedExp_of_isRamifiedIn9 below · cited by 4 · depth 21 - Integrability of the dual S-part zeta integrand on GL₃
LanglandsTunnell.CubicInduction.sPartDual_integrable_of_isCubicInductionDataOn_of_isGaugeMajorised329 below · cited by 1 · depth 21 - S-part factorisation of a GL₃ zeta integral
LanglandsTunnell.CubicInduction.sPart_eq_arch_mul_localZeta_v_mul_badPlacesPart_archDetermined_of_isCubicInductionDataOn4 below · cited by 6 · depth 21 - Euler factorisation of the S-part zeta integral at v
LanglandsTunnell.CubicInduction.sPart_eq_arch_mul_localZeta_v_mul_badPlacesPart_archTwisted_of_isCubicInductionDataOn4 below · cited by 3 · depth 21 - Convergence of the S-part zeta integral for cubic induction data
LanglandsTunnell.CubicInduction.sPart_integrable_of_isCubicInductionDataOn_of_isGaugeMajorised325 below · cited by 1 · depth 21 - Torus values of a spherical GL₃ Whittaker function
LanglandsTunnell.CubicInduction.sphericalTorusValue_eq_of_isCosetEigenfunction_of_isGL3PsiWhittakerFn3 below · cited by 6 · depth 21 - Character dilation equals diagonal translation for GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.whittaker3_eq_whittaker3_globalPointsGL_mul_of_forall_apply_eq_apply_mul4 below · cited by 1 · depth 21 - Mirabolic series recovers its GL₃ Whittaker function
LanglandsTunnell.CubicInduction.whittaker3_mirabolicSeries_eq1 below · cited by 1 · depth 21 - Non-vanishing of archimedean and local Whittaker factors
LanglandsTunnell.CubicInduction.whittakerArch_ne_zero_and_whittakerLoc_ne_zero_of_isCubicInductionDataOn_of_form_ne_zero0 below · cited by 1 · depth 21 - Measurability of the dual S-part zeta integrands
LanglandsTunnell.CubicInduction.aestronglyMeasurable_sPartDual_integrand_of_isCubicInductionDataOn2 below · cited by 1 · depth 22 - Vanishing of a level-b principal series vector on the (2,1) parabolic
LanglandsTunnell.CubicInduction.apply_eq_zero_of_apply_two_eq_zero_of_mem_principalSeries3_of_level1 below · cited by 3 · depth 22 - Whittaker functions: torus conjugation of the (1,2) unipotent
LanglandsTunnell.CubicInduction.apply_iotaGL_diagUnitGL2_mul_upperUnipotent3_mul_of_isGL3PsiWhittakerFn0 below · cited by 1 · depth 22 - Multiplicity bound for torus-equivariant functionals on a GL₃ principal series
LanglandsTunnell.CubicInduction.card_le_ncard_of_linearIndependent_of_upperUnipotent3_of_diagonal35 below · cited by 1 · depth 22 - At most one such functional on the trivial principal series of GL₃
LanglandsTunnell.CubicInduction.card_le_one_of_linearIndependent_of_upperUnipotent3_of_diagonal3_one3 below · cited by 1 · depth 22 - Whittaker coefficients as sums of translated Jacquet–Whittaker functions
LanglandsTunnell.CubicInduction.coefficientFn_eq_sum_jacquetWhittaker3_of_isWhittakerFunctional3_inv15 below · cited by 3 · depth 22 - Contragredient of a spherical GL₃ Whittaker function
LanglandsTunnell.CubicInduction.dualWhittakerFn3_spherical_and_iotaTorusLocal_eq_of_torusValues1 below · cited by 1 · depth 22 - Vanishing of K₁-invariant Whittaker functions off dominant diagonals
LanglandsTunnell.CubicInduction.eq_zero_of_coe_eq_diagonal_of_valued_lt_of_isGL3PsiWhittakerFn_of_mem_congruenceK10 below · cited by 1 · depth 22 - Finite Fourier inversion on a valuation shell
LanglandsTunnell.CubicInduction.eq_zero_of_forall_setIntegral_valuationShell_addChar_mul_eq_zero0 below · cited by 1 · depth 22 - Threefold twisted differences of torus Jacquet values vanish near 0
LanglandsTunnell.CubicInduction.eventually_threefold_twistedDifference_torusJacquetValueFn_eq_zero13 below · cited by 1 · depth 22 - Entire norm-≥ 1 part of a GL₃× GL₁ zeta integral
LanglandsTunnell.CubicInduction.exists_differentiable_boundedOnStrips_globalZeta30_eq_add_of_integrable26 below · cited by 1 · depth 22 - Entire norm-≥ 1 part of the GL(3)× GL(1) zeta integral
LanglandsTunnell.CubicInduction.exists_differentiable_boundedOnStrips_globalZeta31_eq_add_of_integrable26 below · cited by 1 · depth 22 - Local functional equation at v matches induced Euler polynomials
LanglandsTunnell.CubicInduction.exists_eval_mul_eq_mul_eval_of_forall_localZeta31_fe_one_of_isCubicInductionDataOn_of_addCharLevel493 below · cited by 1 · depth 22 - Ramified place: local functional-equation datum matches induced Euler polynomials
LanglandsTunnell.CubicInduction.exists_eval_mul_eq_mul_eval_of_forall_localZeta31_fe_one_of_isCubicInductionDataOn_of_isRamifiedIn493 below · cited by 1 · depth 22 - Finite level-n transversals modulo level-m congruence in GL₃
LanglandsTunnell.CubicInduction.exists_finset_localMaximalCompact3_eq_mul_of_level_le1 below · cited by 4 · depth 22 - Whittaker function as Jacquet integrals of a flat family
LanglandsTunnell.CubicInduction.exists_flatSection_jacquetIntegral_eq_finsum_cpow_of_embedding_principalSeries214 below · cited by 1 · depth 22 - Torus line of a smooth Whittaker function on GL₃ vanishes for large |a|
LanglandsTunnell.CubicInduction.exists_forall_apply_iotaGL_diagUnitGL2_mul_eq_zero_of_lt_valued_of_isGL3PsiWhittakerFn0 below · cited by 1 · depth 22 - Finiteness of admissible Whittaker functions along the GL₃ torus
LanglandsTunnell.CubicInduction.exists_forall_apply_iotaGL_diagZ_mul_scalarPi_zpow_eq_sum_of_isGL3PsiWhittakerFn0 below · cited by 3 · depth 22 - Deep-torus vanishing of unipotent coboundaries of Whittaker functions
LanglandsTunnell.CubicInduction.exists_forall_apply_iotaGL_torus_eq_zero_of_mem_span_radical_of_isGL3PsiWhittakerFn0 below · cited by 3 · depth 22 - Characters agreeing on local units differ by an unramified twist
LanglandsTunnell.CubicInduction.exists_forall_eq_mul_modulus_cpow_of_forall_eq_of_mem_adicCompletionIntegers1 below · cited by 1 · depth 22 - Annihilator of local translates of a cuspidal GL₃ eigenform
LanglandsTunnell.CubicInduction.exists_forall_sum_smul_translate_eq_zero_of_isCuspidalAlong335 below · cited by 1 · depth 22 - Gauge majorant for cyclic translates of principal-series Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_gauge_of_mem_gl3CyclicSubspace_coefficientFn_principalSeries323 below · cited by 1 · depth 22 - Two-point global-to-local zeta factorisation at a bad place
LanglandsTunnell.CubicInduction.exists_globalZeta30_eq_mul_localZeta30_and_globalZetaDual31_eq_mul_of_isCubicInductionDataOn508 below · cited by 1 · depth 22 - Conductor bound for ωᵥ at a ramified dyadic place
LanglandsTunnell.CubicInduction.exists_hasConductorExponentAt_le_finsum_addCharLevel_of_eulerCoeff_eq_inducedE3_one_cofinite_of_valued_two_lt_one_of_finsum_le_two326 below · cited by 1 · depth 22 - Non-vanishing archimedean zeta of a block-harmonic Jacquet vector
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_blockHarmonicOne_colHarmonic_gaussian3_of_weightZero38 below · cited by 1 · depth 22 - Non-vanishing archimedean zeta for the conjugate block-harmonic section
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_conjBlockHarmonicOne_colHarmonic_gaussian357 below · cited by 1 · depth 22 - Non-vanishing of the weight-zero minor-section archimedean zeta integral
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_minorSection_gaussian337 below · cited by 1 · depth 22 - Admissible idele class character of ℚ with prescribed component at v and parity
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_isUnramifiedCharAt_localChar_eq_isArchCompAt_of_hasConductorExponentAt8 below · cited by 1 · depth 22 - Convergence of the dual archimedean GL₃ zeta integral at the trivial twist
LanglandsTunnell.CubicInduction.exists_isArchZeta31ConvergentAbove_dualWhittakerFn3_whittakerArch_of_isCubicInductionDataOn0 below · cited by 1 · depth 22 - Convergence of local GL₃× GL₁ zeta integrals and their duals
LanglandsTunnell.CubicInduction.exists_isLocalZeta30ConvergentAbove_and_dual_of_isGL3PsiWhittakerFn2 below · cited by 3 · depth 22 - Level-pᵈ Whittaker vector in a unitary principal series of GL₃
LanglandsTunnell.CubicInduction.exists_isWhittakerFunctional3_coefficientFn_ne_zero_forall_deepTwist_eq_of_forall_higherUnitsAt_of_pos11 below · cited by 1 · depth 22 - Laurent form of the dual local zeta integral at v
LanglandsTunnell.CubicInduction.exists_laurent_localZetaDual31_one_sub_eq_of_norm_eq_one17 below · cited by 1 · depth 22 - Local zeta integral factors through three local L-factors
LanglandsTunnell.CubicInduction.exists_laurent_localZeta_eq_prod_localLFactorAt_mul_of_twistedDifference1 below · cited by 2 · depth 22 - Dual section of the GL₂ principal series
LanglandsTunnell.CubicInduction.exists_modulus_det_mul_apply_antidiagonal_mul_transposeInvN_mem_principalSeries21 below · cited by 3 · depth 22 - Jacquet's lemma in polynomial recurrence form for GL₃
LanglandsTunnell.CubicInduction.exists_polynomial_sum_coeff_smul_rightTranslate_pow_mem_span_radical_of_admissible1 below · cited by 3 · depth 22 - Common middle of the local GL₃timesGL₂ functional equation
LanglandsTunnell.CubicInduction.exists_rsLocalIntegral_mul_eq_and_dual_mul_eq_middle_of_dominant_of_forall_localZeta31_fe_of_gauge76 below · cited by 1 · depth 22 - Invariant positive Hermitian form on the unitary principal series of GL₃
LanglandsTunnell.CubicInduction.exists_sesqForm_gl3AmbientRightTranslate_invariant_of_norm_eq_one5 below · cited by 1 · depth 22 - Whittaker model of a unitary principal series of GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.exists_whittaker_localLevelOne_centralChar_admissible_principalSeries2_of_norm_eq_one_of_higherUnitsAt39 below · cited by 1 · depth 22 - Conductor bound for the characters of a principal series
LanglandsTunnell.CubicInduction.forall_higherUnitsAt_eq_one_of_mem_principalSeries2_of_forall_mem_localLevelOne_pow1 below · cited by 1 · depth 22 - Unramified twist shifts the local (3,1) functional equation
LanglandsTunnell.CubicInduction.forall_localZeta31_fe_of_twist_modulus_cpow0 below · cited by 3 · depth 22 - Continuity and decay of the Godement inner integral
LanglandsTunnell.CubicInduction.godementInner3_mulShift_polyGauss3_continuousOn_and_decay0 below · cited by 1 · depth 22 - Vanishing of e₃ at a twist-ramified place
LanglandsTunnell.CubicInduction.inducedE3_inducedCoeff_eq_zero_of_isTwistRamifiedAbove0 below · cited by 2 · depth 22 - Contragredient Euler parameters at a good place
LanglandsTunnell.CubicInduction.inducedE_inducedCoeff_inv_eq_of_not_isBadPlace0 below · cited by 1 · depth 22 - Positivity of the induced level at a ramified place
LanglandsTunnell.CubicInduction.inducedLevelAt_pos3 below · cited by 2 · depth 22 - Integrability of local zeta integrands killed by three twisted differences
LanglandsTunnell.CubicInduction.integrable_mul_charExt_mul_modulus_cpow_of_twistedDifference1 below · cited by 3 · depth 22 - Lower unipotent support of a spherical Whittaker function
LanglandsTunnell.CubicInduction.integral_of_iotaGL_diagUnitGL2_mul_lowerUnipotent21_ne_zero1 below · cited by 1 · depth 22 - Convergence of the dual GL₃ local zeta integral
LanglandsTunnell.CubicInduction.isLocalZeta31ConvergentAbove_dualWhittakerFn3_of_norm_eq_one16 below · cited by 1 · depth 22 - Weight law for the Jacquet vector of a Gaussian section
LanglandsTunnell.CubicInduction.jacquetVector3_mul_iota_eq_archWeightChar_inv_mul_of_colHarmonic_gaussian30 below · cited by 2 · depth 22 - Equivariance of the Jacquet vector under ι of row isometries
LanglandsTunnell.CubicInduction.jacquetVector3_mul_iota_eq_archWeightChar_inv_mul_of_conjBlockHarmonic_colHarmonic_gaussian30 below · cited by 2 · depth 22 - Weight-one K-type of the minor-section Jacquet vector
LanglandsTunnell.CubicInduction.jacquetVector3_mul_iota_eq_archWeightOne_inv_mul_of_minorSection_gaussian30 below · cited by 1 · depth 22 - Local functional equation for the GL₃ zeta integrals
LanglandsTunnell.CubicInduction.localZetaDual31_one_sub_eq_mul_localZeta30_of_mem_strip44 below · cited by 1 · depth 22 - Haar scaling on the unipotent subgroup: dilating the integral ball
LanglandsTunnell.CubicInduction.measure_unipotentEntry_preimage_mul_eq0 below · cited by 6 · depth 22 - Generating series and convergence disc for cubic torus values
LanglandsTunnell.CubicInduction.mk_sphericalTorusValue_mul_coe_eq_one_and_hasSum0 below · cited by 1 · depth 22 - Unipotent invariance of the dual Rankin–Selberg integrand
LanglandsTunnell.CubicInduction.mul_dual_eq_of_isGL3PsiWhittakerFn_inv_of_unipotent0 below · cited by 2 · depth 22 - Vanishing Mellin transform forces vanishing shell integrals
LanglandsTunnell.CubicInduction.setIntegral_valuationShell_eq_zero_of_forall_integral_mul_modulus_cpow_eq_zero3 below · cited by 1 · depth 22 - Vanishing of middle-cell sums for deeply ramified I(χ) vectors
LanglandsTunnell.CubicInduction.sum_apply_weylPrime3_mul_radicalP21_mul_iotaGL_eq_zero_of_level2 below · cited by 1 · depth 22 - Open-cell sums vanish for level-fixed GL₃ principal series vectors
LanglandsTunnell.CubicInduction.sum_sum_apply_longWeyl3_mul_iotaGL_mul_radicalP21_eq_zero_of_level2 below · cited by 1 · depth 22 - Absolute Jacquet integral: finiteness and transformation law
LanglandsTunnell.CubicInduction.absoluteJacquetIntegral_lt_top_and_unipotent_and_diagonal2_and_bounded_of_mem_principalSeries23 below · cited by 5 · depth 23 - Parity of the torus profile at weight zero
LanglandsTunnell.CubicInduction.archDatumR_W_diagOne_neg_eq_of_weightZero11 below · cited by 20 · depth 23 - Local constancy, large-modulus vanishing and finite-sum dual Jacquet slices
LanglandsTunnell.CubicInduction.dualJacquetValueSlices_eventually_eq_and_eq_zero_and_eq_mul_sum12 below · cited by 3 · depth 23 - Equivariant functionals on Bruhat steps of a GL₃ principal series
LanglandsTunnell.CubicInduction.eq_torusChar3_mul_halfModulus3_of_linearIndependent_domRestrict_of_le_card3 below · cited by 1 · depth 23 - Vanishing on GL₂(𝒪) for level below both conductors
LanglandsTunnell.CubicInduction.eq_zero_of_valuation_le_one_of_diagUnits2_mul_of_mul_level0 below · cited by 3 · depth 23 - Threefold twisted differences of dual Jacquet values vanish near 0
LanglandsTunnell.CubicInduction.eventually_threefold_twistedDifference_dualJacquetValueFn_eq_zero14 below · cited by 2 · depth 23 - Weight-one torus profile as Gaussian multiplicative convolution
LanglandsTunnell.CubicInduction.exists_archDatumR_W_diagOne_add_eq_mul_mulConvGaussian_of_weightOne29 below · cited by 6 · depth 23 - Discrete-series torus profile of a real archimedean Whittaker datum
LanglandsTunnell.CubicInduction.exists_archDatumR_W_diagOne_eq_mul_exp_and_eq_zero_of_discrete16 below · cited by 5 · depth 23 - Weight-zero torus profile is a Gaussian multiplicative convolution
LanglandsTunnell.CubicInduction.exists_archDatumR_W_diagOne_eq_mul_mulConvGaussian_of_weightZero16 below · cited by 6 · depth 23 - Archimedean zeta of the weight-zero Jacquet vector as Γ_ℝ times a Mellin transform
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_GammaR_mul_mellin_of_blockHarmonicOne_colHarmonic_gaussian3_of_weightZero11 below · cited by 1 · depth 23 - Archimedean zeta of the block-harmonic Jacquet vector as a Mellin transform
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_GammaR_mul_mellin_of_conjBlockHarmonicOne_colHarmonic_gaussian312 below · cited by 1 · depth 23 - Archimedean zeta of the minor-section Jacquet vector as Mellin transform
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_GammaR_mul_mellin_of_minorSection_gaussian3_of_weightZero10 below · cited by 1 · depth 23 - Twisted translated Jacquet–Whittaker function: admissible, unitary central, gauged
LanglandsTunnell.CubicInduction.exists_detTwist_jacquetWhittaker3_translate_whittaker_smooth_central_admissible_gauge23 below · cited by 2 · depth 23 - Existence of a dual middle datum at a finite place
LanglandsTunnell.CubicInduction.exists_dualMiddleDatum_rsLocalIntegral_dual_mul_eq_of_iotaGL_invariant_of_dominant72 below · cited by 1 · depth 23 - Uniqueness of ψ-Whittaker functionals on GL₂ principal series
LanglandsTunnell.CubicInduction.exists_eq_smul_of_forall_apply_principalSeries2Rep_upperUnipotent2_eq_mul0 below · cited by 2 · depth 23 - Admissibility of the principal series of GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.exists_finset_forall_mem_principalSeries2_invariant_mem_span1 below · cited by 2 · depth 23 - Truncated Jacquet integrals of a flat section: Laurent polynomial in q^u
LanglandsTunnell.CubicInduction.exists_finset_forall_setIntegral_flatSection_antidiagonal_unipotentGL2_addChar_eq_sum_cpow3 below · cited by 2 · depth 23 - Non-vanishing of the partial twisted induced Euler product
LanglandsTunnell.CubicInduction.exists_forall_ne_zero_of_hasProd_inducedEulerPoly_eval_inv2 below · cited by 1 · depth 23 - Non-orthogonality of GL₃ cusp forms with equal Hecke eigenvalues
LanglandsTunnell.CubicInduction.exists_inner_toL2_translateRight_ne_zero_of_isCosetEigenfunction331 below · cited by 1 · depth 23 - Invariant positive functional on the GL₃ principal series
LanglandsTunnell.CubicInduction.exists_invariant_pos_linearMap_of_torusChar3_eq_halfModulus34 below · cited by 1 · depth 23 - Non-vanishing archimedean zeta integral for the weight-zero quadratic section
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_detPow_blockQuadratic_colHarmonicTwo_gaussian3_of_weightZero38 below · cited by 1 · depth 23 - An admissible twist with non-vanishing archimedean GL₃ zeta integral
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_detPow_colHarmonic_gaussian362 below · cited by 1 · depth 23 - Compact smoothing operator on the GL₃ cuspidal subspace
LanglandsTunnell.CubicInduction.exists_isCompactOperator_cuspidalSubspace_smoothingOperator15 below · cited by 15 · depth 23 - A Jacquet–Whittaker functional on the principal series of GL₃(ℚᵥ)
LanglandsTunnell.CubicInduction.exists_isWhittakerFunctional3_psiLocal_and_inv_eq_jacquetValue_and_eq_sum9 below · cited by 6 · depth 23 - Extension of ψ-Whittaker functionals on the principal series
LanglandsTunnell.CubicInduction.exists_linearMap_forall_apply_principalSeries2Rep_upperUnipotent2_eq_mul_and_apply_eq_of_forall_mem0 below · cited by 2 · depth 23 - Stabilised Jacquet functional on the GL₂(ℚₚ) principal series
LanglandsTunnell.CubicInduction.exists_linearMap_stabilised_jacquetIntegral_principalSeries25 below · cited by 2 · depth 23 - Uniform rapid decay of smoothed cuspidal L² classes on GL₃
LanglandsTunnell.CubicInduction.exists_mem_cuspFunctions_toL2_eq_and_norm_le_of_mem_cuspidalSubspace14 below · cited by 10 · depth 23 - A K₁(N)-fixed vector in the principal series I(θ₀,θ₁)
LanglandsTunnell.CubicInduction.exists_mem_principalSeries2_ne_zero_forall_localLevelOne_mul_eq_of_higherUnitsAt1 below · cited by 1 · depth 23 - Adelic Siegel-set reduction for GL₃ over ℚ
LanglandsTunnell.CubicInduction.exists_mul_eq_unipotent_mul_diagonal_mul_compact5 below · cited by 20 · depth 23 - Moderate growth of the Jacquet integral along torus shells
LanglandsTunnell.CubicInduction.exists_norm_apply_diagZ_mul_le_of_stabilised_jacquetIntegral_of_norm_eq_one1 below · cited by 1 · depth 23 - Primal middle datum for the local GL₃timesGL₂ integral
LanglandsTunnell.CubicInduction.exists_primalMiddleDatum_rsLocalIntegral_mul_eq_of_iotaGL_invariant_of_dominant61 below · cited by 1 · depth 23 - Rationality of torus-shell averages of admissible Whittaker functions
LanglandsTunnell.CubicInduction.exists_rational_torusShellAverage_and_dual_of_admissible_of_isGL3PsiWhittakerFn11 below · cited by 2 · depth 23 - Right translates of slab cuspidal functions: isometry and continuity
LanglandsTunnell.CubicInduction.exists_translateRight_mem_cuspFunctions_norm_toL2_eq_and_continuous19 below · cited by 19 · depth 23 - Flat sections of the principal series and the Iwasawa height
LanglandsTunnell.CubicInduction.flatSection_mem_principalSeries2_and_iwasawaHeight_mul_eq1 below · cited by 2 · depth 23 - Stability of the GL₃× GL₁ local functional equation
LanglandsTunnell.CubicInduction.forall_localZetaDual31_eq_mul_localZeta30_and_exists_localZeta30_ne_zero_of_ne_zero_of_principalCongruence_of_two_mul_le20 below · cited by 1 · depth 23 - Integrability of Jacquet's integrand in the dominant range
LanglandsTunnell.CubicInduction.integrable_apply_antidiagonal_mul_unipotentGL2_mul_addChar_of_mem_principalSeries21 below · cited by 5 · depth 23 - Integrality of torus parameters of an invariant Whittaker function
LanglandsTunnell.CubicInduction.integral_of_iotaGL_diagUnits2_ne_zero0 below · cited by 1 · depth 23 - Unfolded dual and primal (3,2) local integrals agree
LanglandsTunnell.CubicInduction.integral_transposeInvN_mul_integral_integral_diagUnits2_eq_integral_upperUnipotent2_mul_of_mem_principalSeries23 below · cited by 1 · depth 23 - Smoothing and translation preserve level and Hecke eigenvalues at p
LanglandsTunnell.CubicInduction.isRightInvariant_and_isCosetEigenfunction_translateRight_smoothingOperator1 below · cited by 8 · depth 23 - Weight zero of the block-quadratic Gaussian Jacquet vector
LanglandsTunnell.CubicInduction.jacquetVector3_mul_iota_eq_archWeightChar_inv_mul_of_detPow_blockQuadratic_gaussian30 below · cited by 1 · depth 23 - At most one unipotent-invariant functional on each corner-filtration step
LanglandsTunnell.CubicInduction.le_one_of_linearIndependent_domRestrict_of_upperUnipotent33 below · cited by 1 · depth 23 - Twisting a local GL₃ Whittaker function by χ∘det
LanglandsTunnell.CubicInduction.localZeta30_localZetaDual31_twist_det0 below · cited by 1 · depth 23 - Tate's local functional equation at a finite place of ℚ
LanglandsTunnell.CubicInduction.localZeta_tateFourier_mul_localLFactorAt_eq20 below · cited by 2 · depth 23 - Rational left translations preserve the adelic GL₃ determinant slab
LanglandsTunnell.CubicInduction.measurePreserving_mul_left_globalPointsGL_restrict_setOf_ideleNorm_det_mem_Icc4 below · cited by 14 · depth 23 - Unitary principal series for GL₂(ℚₚ): every non-zero vector is cyclic
LanglandsTunnell.CubicInduction.mem_span_range_translate_of_mem_principalSeries2_of_ne_zero_of_norm_eq_one31 below · cited by 1 · depth 23 - Primal–dual middle datum comparison: a γ-factor identity
LanglandsTunnell.CubicInduction.middleDatum_compare_of_primalMiddleDatum_of_dualMiddleDatum_of_ne_zero10 below · cited by 1 · depth 23 - Smoothing a cuspidal GL₃ function yields a slab cusp function
LanglandsTunnell.CubicInduction.smoothingOperator_mem_cuspFunctions_of_isCuspidalAlong10 below · cited by 4 · depth 23 - Local zeta integral as limit of truncated coupled integrals
LanglandsTunnell.CubicInduction.tendsto_localZeta_truncChar_mul_coupled_localZeta_of_jacquetValue19 below · cited by 1 · depth 23 - Dual local zeta integral as a limit of truncated coupled integrals
LanglandsTunnell.CubicInduction.tendsto_localZeta_truncPsi_mul_coupled_of_forall_eq_integral_jacquetValue24 below · cited by 1 · depth 23 - Three Casimir scalars on an irreducible cuspidal subspace
LanglandsTunnell.CubicInduction.SlabL2.exists_casimir_eq_smul_of_irreducible_cuspidal43 below · cited by 1 · depth 24 - Existence of a minimal cusp-generated stable subspace for GL₃
LanglandsTunnell.CubicInduction.SlabL2.exists_le_minimal_of_stable_translateRight_smoothingOperator34 below · cited by 1 · depth 24 - Closed translation- and smoothing-stable hull of a cusp form
LanglandsTunnell.CubicInduction.SlabL2.exists_submodule_toL2_mem_and_forall_inner_toL2_translateRight_eq_zero21 below · cited by 1 · depth 24 - Finite Haar measure of a GL₃(A_ℚ) Siegel set slab
LanglandsTunnell.CubicInduction.adelicGLHaar_siegelSet_inter_setOf_ideleNorm_det_mem_Icc_lt_top2 below · cited by 11 · depth 24 - Bruhat relation for GL₃ Whittaker values at u ↦ u⁻¹
LanglandsTunnell.CubicInduction.apply_iotaGL_diagUnits2_mul_longWeyl3_upperUnipotent3_weylPrime3_eq_central_mul_of_isGL3PsiWhittakerFn0 below · cited by 2 · depth 24 - Archimedean zeta integral Z₀ at 1 as a dy/|y| integral
LanglandsTunnell.CubicInduction.archZeta30_one_eq_mul_integral_quasiChar_of_isArchCompAt0 below · cited by 7 · depth 24 - Discreteness of GL₃(ℚ) in GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.discreteTopology_range_globalPointsGL_three_rat0 below · cited by 3 · depth 24 - Dual section in the principal series and its Jacquet integral
LanglandsTunnell.CubicInduction.dualSection_mem_principalSeries2_and_jacquetIntegral_eq0 below · cited by 3 · depth 24 - Unipotent-fixed stable subspace of a unitary principal series vanishes
LanglandsTunnell.CubicInduction.eq_bot_of_stable_of_forall_principalSeries2Rep_upperUnipotent2_eq_of_norm_eq_one0 below · cited by 1 · depth 24 - Unipotent-cotrivial stable subspace of a principal series is everything
LanglandsTunnell.CubicInduction.eq_top_of_stable_of_forall_principalSeries2Rep_upperUnipotent2_sub_mem_of_norm_eq_one12 below · cited by 1 · depth 24 - Archimedean GL₃× GL₁ zeta integral as a Mellin transform
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_GammaR_mul_mellin_of_detPow_blockQuadratic_colHarmonicTwo_gaussian3_of_weightZero11 below · cited by 1 · depth 24 - Mellin formula for the archimedean GL₃timesGL₁ zeta integral
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_GammaR_mul_mellin_of_detPow_colHarmonic_gaussian311 below · cited by 1 · depth 24 - Rational points cover finite adelic GL₃ over ℚ
LanglandsTunnell.CubicInduction.exists_forall_componentAt3_mul_mem_localMaximalCompact33 below · cited by 1 · depth 24 - Flat families of stabilised GL₃ Jacquet–Whittaker functions
LanglandsTunnell.CubicInduction.exists_forall_jacquetWhittaker3_twistFamily_eq_finsum_and_forall_le_jacquetTruncated3_eq13 below · cited by 1 · depth 24 - Uniform tail bound outside an annulus for the dual cubic zeta integral
LanglandsTunnell.CubicInduction.exists_forall_norm_dualZetaRemainder_outside_annulus_le15 below · cited by 1 · depth 24 - Windowed Jacquet integral small outside large balls, uniformly
LanglandsTunnell.CubicInduction.exists_forall_norm_setIntegral_annulus_setIntegral_compl_ball_jacquetWindow_le15 below · cited by 1 · depth 24 - Shell integrals of ψ(-t) η(t) |t|^z vanish for large radius
LanglandsTunnell.CubicInduction.exists_forall_setIntegral_shell_psiLocal_mul_charExt_mul_cpow_eq_zero8 below · cited by 4 · depth 24 - Lower support bound for torus-shell averages of GL₃ Whittaker data
LanglandsTunnell.CubicInduction.exists_forall_torusShellAverage_and_dual_eq_zero_of_lt_of_isGL3PsiWhittakerFn3 below · cited by 1 · depth 24 - Hecke-matched GL₃ cusp forms non-orthogonal after right translation
LanglandsTunnell.CubicInduction.exists_inner_toL2_translateRight_ne_zero_of_isCosetEigenfunction_of_isCentreFinite311 below · cited by 1 · depth 24 - Level-uniform integrable dominants for a cubic-induction inner integral
LanglandsTunnell.CubicInduction.exists_integrable_levelUniform_dominant_coupledInner10 below · cited by 3 · depth 24 - Non-vanishing archimedean zeta of a flat-section Jacquet vector
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_conjBlockHarmonic_pow_colHarmonic_gaussian389 below · cited by 1 · depth 24 - Uniform right-invariance of cell sections along a twist family
LanglandsTunnell.CubicInduction.exists_isOpen_forall_cellSectionOf_twistFamily_mul_eq10 below · cited by 2 · depth 24 - Pointwise rational functional equation for Z₁ from Z₀
LanglandsTunnell.CubicInduction.exists_localZeta31_fe_of_forall_mem_gl3CyclicSubspace_exists_localZeta30_localZetaDual31_fe2 below · cited by 4 · depth 24 - Hecke-eigen cusp function inside an invariant L² subspace
LanglandsTunnell.CubicInduction.exists_mem_cuspFunctions_toL2_mem_and_isCosetEigenfunction_of_forall_inner_eq_zero30 below · cited by 1 · depth 24 - Torus-shell averages are values of cyclic smooth vectors
LanglandsTunnell.CubicInduction.exists_mem_gl3CyclicSubspace_forall_torusShellAverage_eq_apply4 below · cited by 1 · depth 24 - Non-vanishing Jacquet integral on the GL₂ principal series
LanglandsTunnell.CubicInduction.exists_mem_principalSeries2_integral_antidiagonal_mul_unipotentGL2_mul_addChar_ne_zero0 below · cited by 1 · depth 24 - Siegel set covering of GL₃(ℝ) modulo GL₃(ℤ)
LanglandsTunnell.CubicInduction.exists_mul_eq_unipotent_mul_diagonal_mul_orthogonal_real0 below · cited by 1 · depth 24 - Rationality of two-variable torus-shell Whittaker series
LanglandsTunnell.CubicInduction.exists_mvPolynomial_forall_tsum_torusShellAverage_mul_eval_eq_and_dual_of_admissible_of_isGL3PsiWhittakerFn13 below · cited by 1 · depth 24 - Rationality of the two-variable torus series of a Whittaker vector
LanglandsTunnell.CubicInduction.exists_mvPolynomial_forall_tsum_torus_apply_mul_eval_eq_of_mem_gl3CyclicSubspace4 below · cited by 3 · depth 24 - Uniform bound for truncated local ball integrals
LanglandsTunnell.CubicInduction.exists_norm_setIntegral_ball_truncChar_mul_charExt_mul_cpow_le9 below · cited by 3 · depth 24 - Non-zero s-independent local Rankin–Selberg integral at a finite place
LanglandsTunnell.CubicInduction.exists_rsLocalIntegral_sum_translate_eq_const_of_apply_one_ne_zero3 below · cited by 1 · depth 24 - Spherical vector in an unramified principal series of GL₂
LanglandsTunnell.CubicInduction.exists_spherical_mem_principalSeries2_of_unramified0 below · cited by 2 · depth 24 - Shell expansion of the local GL₃ zeta integral
LanglandsTunnell.CubicInduction.hasSum_shell_localZeta30_one_iotaGL_scalarPi_zpow_of_iotaGL_invariant5 below · cited by 2 · depth 24 - Integrability of the coupled cubic-induction integrand on 0<Re s<1
LanglandsTunnell.CubicInduction.integrable_coupledIntegrand1 below · cited by 2 · depth 24 - Middle identity for a double torus Whittaker integral
LanglandsTunnell.CubicInduction.integral_integral_diagUnits2_longWeyl3_upperUnipotent3_weylPrime3_eq_mul_of_central0 below · cited by 3 · depth 24 - Whittaker law descends to the coefficients Eᵢ
LanglandsTunnell.CubicInduction.isGL3PsiWhittakerFn_of_forall_isGL3PsiWhittakerFn_finsum_cpow_mul1 below · cited by 1 · depth 24 - Level sets of level data are compact open in GL₃(A_ℚ^f)
LanglandsTunnell.CubicInduction.isOpen_and_isCompact_setOf_forall_componentAt3_finEmbedN_mem0 below · cited by 15 · depth 24 - Openness of the congruence set K₁(𝔭ᵥᶜ) in GL₃(Kᵥ)
LanglandsTunnell.CubicInduction.isOpen_congruenceK11 below · cited by 1 · depth 24 - Jacquet integral of the spherical vector: Casselman–Shalika formula for GL₂
LanglandsTunnell.CubicInduction.jacquetIntegral_spherical_laws_of_unramified_of_norm_lt8 below · cited by 2 · depth 24 - Jacquet vector at 1 of the harmonic Gaussian section, weight zero
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_of_blockHarmonicOne_colHarmonic_gaussian3_of_weightZero7 below · cited by 1 · depth 24 - Jacquet vector at 1 of a conjugate block-harmonic Gaussian section
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_of_conjBlockHarmonicOne_colHarmonic_gaussian37 below · cited by 1 · depth 24 - Minor-section Jacquet vector at 1 as an explicit double integral
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_of_minorSection_gaussian3_of_weightZero6 below · cited by 1 · depth 24 - Modulus twist of the GL₃ Jacquet–Whittaker function
LanglandsTunnell.CubicInduction.jacquetWhittaker3_mul_eq_modulus_det_cpow_mul_jacquetWhittaker30 below · cited by 1 · depth 24 - Four-term decomposition of the dual local zeta integral
LanglandsTunnell.CubicInduction.localZeta_primedDual_eq_pieces_add_setIntegral_annulus_of_forall_eq_zero7 below · cited by 1 · depth 24 - Truncated Jacquet zeta integral factors through two Tate integrals
LanglandsTunnell.CubicInduction.localZeta_truncPsi_mul_coupled_eq_localZeta_of_forall_eq_integral_jacquetWindow4 below · cited by 1 · depth 24 - Non-vanishing of the local Euler polynomials E and E^∨
LanglandsTunnell.CubicInduction.ne_zero_and_ne_zero_of_forall_localZeta30_eq_inv_eval_mul_and_localZetaDual31_eq_inv_eval_mul6 below · cited by 1 · depth 24 - No twisted Whittaker functionals forces trivial unipotent action
LanglandsTunnell.CubicInduction.principalSeries2Rep_upperUnipotent2_eq_self_of_forall_whittaker_functional_eq_zero2 below · cited by 1 · depth 24 - Unipotent radical acts trivially modulo a translation-stable subspace
LanglandsTunnell.CubicInduction.principalSeries2Rep_upperUnipotent2_sub_mem_of_forall_whittaker_functional_eq_zero2 below · cited by 1 · depth 24 - Torus integral of the box-truncated Jacquet integral unfolded
LanglandsTunnell.CubicInduction.setIntegral_tallBoxJacquet_div_norm_mul_charExt_mul_cpow_eq2 below · cited by 1 · depth 24 - Window-truncated Jacquet integrals converge to the Jacquet value
LanglandsTunnell.CubicInduction.tendsto_setIntegral_annulus_setIntegral_ball_jacquetWindow_sub_jacquetValue12 below · cited by 1 · depth 24 - Whittaker dichotomy for a stable subspace of I(θ)
LanglandsTunnell.CubicInduction.whittaker_functional_eq_zero_or_eq_zero_of_forall_mem_eq_zero_of_stable2 below · cited by 1 · depth 24 - Pole of the GL₃ Epstein integral against |φ|²
LanglandsTunnell.CubicInduction.AdelicEpstein.integrable_and_tendsto_sub_one_mul_integral_epstein_of_pureTensor16 below · cited by 6 · depth 25 - Casimir eigenvalue equations descend from all smoothings to F
LanglandsTunnell.CubicInduction.SlabL2.casimir_eq_smul_of_forall_isSmoothingKernel_casimir_smoothingOperator_eq_smul5 below · cited by 1 · depth 25 - Casimir eigenvalues pass to smoothings of cusp functions in a closed span
LanglandsTunnell.CubicInduction.SlabL2.casimir_smoothingOperator_eq_smul_of_toL2_mem_topologicalClosure_span_casimir_eq_smul27 below · cited by 1 · depth 25 - Spectrally invariant closed subspaces: cusp generation and translation stability
LanglandsTunnell.CubicInduction.SlabL2.eq_topologicalClosure_and_stable_of_eq_map_of_invariant_spectralOperators331 below · cited by 1 · depth 25 - Irreducible cuspidal piece generated by smooth Casimir eigenvectors
LanglandsTunnell.CubicInduction.SlabL2.eq_topologicalClosure_span_casimir_eq_smul_of_irreducible_cuspidal29 below · cited by 1 · depth 25 - A joint Casimir eigenfunction in a smoothing-stable cuspidal subspace
LanglandsTunnell.CubicInduction.SlabL2.exists_casimir_eq_smul_smooth_cuspFunction_of_ne_bot_of_stable_smoothingOperator35 below · cited by 1 · depth 25 - A compact smoothing lift not annihilating a given cuspidal vector
LanglandsTunnell.CubicInduction.SlabL2.exists_isSmoothingKernel_isCompactOperator_isCuspLift3_apply_ne_zero24 below · cited by 4 · depth 25 - Existence of a minimal closed invariant subspace in the cuspidal L²
LanglandsTunnell.CubicInduction.SlabL2.exists_le_minimal_of_isClosed_of_invariant_spectralOperators326 below · cited by 1 · depth 25 - Adjoint of a smoothing operator on cuspidal L²(GL₃)
LanglandsTunnell.CubicInduction.SlabL2.isSmoothingKernel_star_inv_and_inner_toL2_smoothingOperator_eq25 below · cited by 3 · depth 25 - Cusp-generated stable subspaces are cuspidal and spectrally invariant
LanglandsTunnell.CubicInduction.SlabL2.le_cuspidalSubspace_and_isClosed_and_invariant_spectralOperators3_of_eq_comap27 below · cited by 2 · depth 25 - Vanishing of χ∘det-equivariant functionals on a principal series
LanglandsTunnell.CubicInduction.eq_zero_of_forall_apply_principalSeries2Rep_eq_det_mul_of_norm_eq_one6 below · cited by 1 · depth 25 - Continuous automorphic form with vanishing L²-class is zero
LanglandsTunnell.CubicInduction.eq_zero_of_toL2_eq_zero_of_continuous10 below · cited by 5 · depth 25 - Non-vanishing of dual (3,1) local zeta integrals at large real points
LanglandsTunnell.CubicInduction.exists_forall_exists_localZetaDual31_ne_zero_of_irreducible5 below · cited by 1 · depth 25 - Truncated Jacquet integrals of a flat cell-section family
LanglandsTunnell.CubicInduction.exists_forall_jacquetTruncated3_cellSectionOf_twistFamily_eq_finsum0 below · cited by 1 · depth 25 - Universal torus table for GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_forall_whittaker3_mul_iotaGL_zpow_eq_mul5 below · cited by 7 · depth 25 - Non-orthogonal right translates of two cuspidal GL₃ forms
LanglandsTunnell.CubicInduction.exists_inner_toL2_translateRight_ne_zero_of_forall_whittakerBlock_one_mul_eq60 below · cited by 6 · depth 25 - Admissible twist with non-vanishing archimedean zeta, discrete branch
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_conjBlockHarmonic_pow_colHarmonic_gaussian3_of_discreteLevi39 below · cited by 1 · depth 25 - Weight-one Levi branch: non-vanishing archimedean zeta of a Jacquet vector
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_conjBlockHarmonic_pow_colHarmonic_gaussian3_of_weightOneLevi38 below · cited by 1 · depth 25 - Non-vanishing archimedean zeta integral, weight-zero Levi branch
LanglandsTunnell.CubicInduction.exists_isAdmissibleTwist_archZeta30_jacquetVector3_ne_zero_of_conjBlockHarmonic_pow_colHarmonic_gaussian3_of_weightZeroLevi54 below · cited by 1 · depth 25 - Smoothness of principal series vectors for GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.exists_isOpen_forall_mul_eq_of_mem_principalSeries20 below · cited by 13 · depth 25 - Det-equivariant functional on a proper stable subspace
LanglandsTunnell.CubicInduction.exists_ne_zero_forall_apply_principalSeries2Rep_eq_det_mul_of_ne_top_of_forall_sub_mem4 below · cited by 1 · depth 25 - Full Whittaker integral as a factor Λ(σ)≥ 1 times its block
LanglandsTunnell.CubicInduction.exists_one_le_and_lintegral_quotientMeasure_eq_mul_whittakerBlock9 below · cited by 6 · depth 25 - Unfolding a GL₃ Epstein integral to the Whittaker quotient
LanglandsTunnell.CubicInduction.exists_pos_lt_top_lintegral_slab_eq_mul_pow_three_mul_lintegral_quotientMeasure29 below · cited by 7 · depth 25 - Bounded Whittaker block for a centre-finite cusp form on GL₃
LanglandsTunnell.CubicInduction.exists_sum_translate_ne_zero_and_whittakerBlock_le_of_isCentreFinite310 below · cited by 1 · depth 25 - Non-zero limit of (σ-1)Λ(σ) and blocks at σ=1
LanglandsTunnell.CubicInduction.exists_tendsto_sub_one_mul_and_whittakerBlock_one_mul_eq_of_whittakerBlock_le1 below · cited by 6 · depth 25 - Inner Godement integral of a column-harmonic Gaussian section
LanglandsTunnell.CubicInduction.godementInner3_eq_mul_exp_of_blockPoly_mul_colHarmonic_gaussian32 below · cited by 7 · depth 25 - Godement inner integral of a column-linear Gaussian section
LanglandsTunnell.CubicInduction.godementInner3_eq_mul_exp_of_blockPoly_mul_colLinear_gaussian32 below · cited by 8 · depth 25 - Unfolding the weighted dual zeta integral on GL₃
LanglandsTunnell.CubicInduction.integrable_and_integral_weight_mul_jacquetWindow_eq_integral_unfolded3 below · cited by 4 · depth 25 - Tate–Mellin evaluation of a Godement inner integral
LanglandsTunnell.CubicInduction.integral_cpow_mul_godementInner3_mulShift_eq_mul_Gamma_of_blockPoly_mul_colHarmonic_gaussian32 below · cited by 4 · depth 25 - Tate–Mellin evaluation of a linear-section Godement integral
LanglandsTunnell.CubicInduction.integral_cpow_mul_godementInner3_mulShift_eq_mul_Gamma_of_blockPoly_mul_colLinear_gaussian32 below · cited by 1 · depth 25 - Dual-configuration Godement integral of a harmonic Gaussian section
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_blockHarmonicOne_colHarmonic_gaussian32 below · cited by 1 · depth 25 - Dual-configuration integral of a conjugate-block harmonic Gaussian
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_conjBlockHarmonicOne_colHarmonic_gaussian32 below · cited by 1 · depth 25 - Dual-configuration integral of the minor Gaussian section
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_minorSection_gaussian32 below · cited by 1 · depth 25 - Dual Jacquet vector at a Siegel upper-unit torus point
LanglandsTunnell.CubicInduction.jacquetVector3_longWeyl3_transposeInv3_iota_upperUnit_eq0 below · cited by 6 · depth 25 - Torus unfolding of a GL₃ Jacquet vector at the identity
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_of_detPow_blockQuadratic_colHarmonicTwo_gaussian3_of_weightZero7 below · cited by 1 · depth 25 - Torus unfolding of the Jacquet vector at the identity
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_of_detPow_colHarmonic_gaussian37 below · cited by 1 · depth 25 - Determinant-one elements act trivially modulo a unipotent-trivial subspace
LanglandsTunnell.CubicInduction.principalSeries2Rep_sub_mem_of_det_eq_one_of_forall_upperUnipotent2_sub_mem0 below · cited by 1 · depth 25 - Simple pole at σ=1 of the adelic Epstein integral on GL₃
LanglandsTunnell.CubicInduction.AdelicEpstein.exists_forall_epstein_eq_div_sub_one_add_of_pureTensor13 below · cited by 1 · depth 26 - Archimedean derivative of a smoothing operator along Eᵢⱼ
LanglandsTunnell.CubicInduction.SlabL2.archDeriv_smoothingOperator1 below · cited by 11 · depth 26 - Casimir operators through the archimedean chart at the identity
LanglandsTunnell.CubicInduction.SlabL2.casimir_apply_eq_kernelCasimir_archChart0 below · cited by 1 · depth 26 - Casimir operators on archimedean convolutions, and weak eigenfunction equations
LanglandsTunnell.CubicInduction.SlabL2.casimir_archConvN_and_eq_of_forall_integral_kernelCasimir0 below · cited by 1 · depth 26 - Casimir operators commute with smoothing on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.SlabL2.casimir_smoothingOperator2 below · cited by 4 · depth 26 - Continuous maps on the cuspidal subspace determined by cusp classes
LanglandsTunnell.CubicInduction.SlabL2.continuousLinearMap_eq_of_forall_toL2_eq0 below · cited by 2 · depth 26 - Continuity and archimedean smoothness of the smoothing operator
LanglandsTunnell.CubicInduction.SlabL2.continuous_and_isArchSmooth3_smoothingOperator2 below · cited by 11 · depth 26 - Compact operator non-vanishing on a closed invariant subspace
LanglandsTunnell.CubicInduction.SlabL2.exists_isCompactOperator_of_ne_bot_of_invariant_spectralOperators325 below · cited by 1 · depth 26 - Smoothing operators as archimedean convolutions; level sets near 1
LanglandsTunnell.CubicInduction.SlabL2.smoothingOperator_eq_archConvN_and_exists_levelSet_subset2 below · cited by 1 · depth 26 - Casimir right-differential operators commute with elementary derivatives
LanglandsTunnell.CubicInduction.WhittakerBlock.casimir_archDeriv_and_comm0 below · cited by 4 · depth 26 - Casimir operators commute with right translation
LanglandsTunnell.CubicInduction.WhittakerBlock.casimir_translateRight0 below · cited by 5 · depth 26 - Right mathfrakgl₃-derivatives at infinity: smoothness, linearity, commutators, translation
LanglandsTunnell.CubicInduction.WhittakerBlock.isArchSmooth3_archDeriv_and_archDeriv_add_smul_comm_translate0 below · cited by 13 · depth 26 - Contragredient involution maps I(μ₀,μ₁) to I(μ₁⁻¹,μ₀⁻¹)
LanglandsTunnell.CubicInduction.conj_transposeInvN_mem_principalSeries20 below · cited by 4 · depth 26 - Vanishing of a Whittaker product on GL₃ over ℚ
LanglandsTunnell.CubicInduction.conj_whittaker3_mul_whittaker3_eq_zero_of_forall_integral_conj_mul_eq_zero58 below · cited by 1 · depth 26 - Principal-series vectors vanishing at 1 and on the big cell
LanglandsTunnell.CubicInduction.eq_zero_of_apply_one_eq_zero_of_forall_apply_antidiagonal2_mul_upperUnipotent2_eq_zero0 below · cited by 1 · depth 26 - Explicit Hermite sum for a degree-m archimedean GL₃ zeta integral
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_sum_GammaR_mul_mellin_of_conjBlockHarmonic_pow_colHarmonic_gaussian3_of_weightZeroLevi16 below · cited by 1 · depth 26 - Hermite expansion of the archimedean GL₃ zeta integral
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_sum_GammaR_mul_mellin_twoSheet_of_conjBlockHarmonic_pow_colHarmonic_gaussian316 below · cited by 3 · depth 26 - Unfolding an adelic Epstein integral along the rational mirabolic subgroup
LanglandsTunnell.CubicInduction.exists_isFundamentalDomain_mirabolic_and_lintegral_domainMeasure_eq_mul_lintegral13 below · cited by 1 · depth 26 - Big-cell vectors in a principal series of GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.exists_principalSeries2_apply_one_eq_zero_apply_antidiagonal2_mul_upperUnipotent2_eq_indicator0 below · cited by 1 · depth 26 - Translate combination with non-zero Whittaker coefficient and bounded block
LanglandsTunnell.CubicInduction.exists_sum_translate_whittaker_ne_zero_and_whittakerBlock_empty_le_of_isCentreFinite309 below · cited by 1 · depth 26 - Fibration of the GL₃ Whittaker block over bottom rows
LanglandsTunnell.CubicInduction.exists_whittakerBlock_one_eq_lintegral_and_eq_smul_map_withDensity_haar7 below · cited by 1 · depth 26 - Flip by diag(1,-1) in the local Godement integral
LanglandsTunnell.CubicInduction.godementDock_diagFlip_eq4 below · cited by 1 · depth 26 - Godement–Whittaker function of a pure tensor at ι(g)
LanglandsTunnell.CubicInduction.godementWhittaker3_iotaGL_eq_of_pureTensor0 below · cited by 1 · depth 26 - Integrability of the Godement integrand against a GL₂ Jacquet integral
LanglandsTunnell.CubicInduction.integrable_rowFourier23_jacquet_godementIntegrand_of_principalSeries214 below · cited by 1 · depth 26 - Planar Hecke–Bochner identity for (u₀+ε i u₁)^m
LanglandsTunnell.CubicInduction.integral_colHarmonic_pow_mul_gaussian_mul_fourierChar_fin_two1 below · cited by 9 · depth 26 - Dual-configuration Godement integral of a quadratic harmonic section
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_detPow_blockQuadratic_colHarmonic_gaussian31 below · cited by 1 · depth 26 - Dual-configuration Godement integral of det^δ times harmonic Gaussian
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_detPow_colHarmonic_gaussian32 below · cited by 1 · depth 26 - Dual Jacquet integral of a principal-series vector
LanglandsTunnell.CubicInduction.integral_psiLocal_mul_transposeInvN_eq_mul_integral_psiLocal_mul_dual0 below · cited by 3 · depth 26 - Box sheet is a fundamental domain for rational unipotent points
LanglandsTunnell.CubicInduction.isFundamentalDomain_boxSheet_rationalUnipotent31 below · cited by 1 · depth 26 - Column Fourier transform preserves Schwartz–Bruhat functions
LanglandsTunnell.CubicInduction.isSchwartzBruhat_colFourier231 below · cited by 7 · depth 26 - Schwartz–Bruhat functions on M₂ are stable under the matrix Fourier transform
LanglandsTunnell.CubicInduction.isSchwartzBruhat_matFourier222 below · cited by 12 · depth 26 - Smoothness and Whittaker laws of a GL₂ Jacquet integral
LanglandsTunnell.CubicInduction.jacquetIntegral_principalSeries2_smooth_law_central_flip1 below · cited by 3 · depth 26 - Godement-section realisation of the GL₃ Jacquet–Whittaker function
LanglandsTunnell.CubicInduction.jacquetWhittaker3_diagonal3_mul_eq_mul_godementWhittaker3_of_chamber27 below · cited by 1 · depth 26 - Parseval identity for the GL₃ box-conditioned Whittaker expansion
LanglandsTunnell.CubicInduction.lintegral_box_norm_sq_radicalCoefficient_eq_tsum_norm_sq_whittaker3_diag10 below · cited by 2 · depth 26 - Parseval identity along the (2,1) unipotent radical of GL₃
LanglandsTunnell.CubicInduction.lintegral_box_norm_sq_radicalP21_eq_tsum_norm_sq_radicalCoefficient10 below · cited by 2 · depth 26 - Left translation covariance of the 2×2 matrix Fourier transform
LanglandsTunnell.CubicInduction.matFourier22_comp_inv_mul_eq3 below · cited by 8 · depth 26 - Fourier transform of a pure tensor on M_{2× 3}
LanglandsTunnell.CubicInduction.matFourier23_leftBlock_mul_lastCol_mul_const0 below · cited by 1 · depth 26 - Big-cell ball vectors span the kernel of evaluation at 1
LanglandsTunnell.CubicInduction.mem_span_of_apply_one_eq_zero_of_forall_apply_antidiagonal2_mul_upperUnipotent2_eq_indicator1 below · cited by 1 · depth 26 - Transport of radical Fourier coefficients along GL₂(ℚ)
LanglandsTunnell.CubicInduction.radicalCoefficient_eq_radicalCoefficient_psi_neg_iotaGL_globalPoints_mul4 below · cited by 2 · depth 26 - Closed R₀-stable subspaces absorb all right translates of cusp functions
LanglandsTunnell.CubicInduction.toL2_translateRight_mem_of_mem_of_isClosed20 below · cited by 2 · depth 26 - Unipotent equivariance and norm invariance of W_f on GL₃
LanglandsTunnell.CubicInduction.whittaker3_upperUnipotent3_mul_and_norm_whittaker3_unipotentSubgroup3_mul0 below · cited by 18 · depth 26 - Idele norm of a positive archimedean scalar idele
LanglandsTunnell.CubicInduction.AdelicEpstein.ideleNorm_archIdele4 below · cited by 1 · depth 27 - Idele norm of det(zI₃ g) is ‖z‖³‖det g‖
LanglandsTunnell.CubicInduction.AdelicEpstein.ideleNorm_det_centralScalarGL_mul0 below · cited by 1 · depth 27 - Torus–compact coordinates for the zeroth shell of GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.WhittakerBlock.setLIntegral_zerothShell_eq_mul_lintegral_torus12 below · cited by 1 · depth 27 - Big-cell GL₃ section as a GL₂ Godement integral
LanglandsTunnell.CubicInduction.cellSectionOf_antidiagonal3_mul_mul_eq_integral_godementDatum4 below · cited by 1 · depth 27 - Covariance of the column Fourier transform under column-wise substitutions
LanglandsTunnell.CubicInduction.colFourier23_comp_colwise_eq2 below · cited by 3 · depth 27 - Weight-zero fold of the degree-m Jacquet vector's archimedean zeta
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_integral_hermiteBracket_of_conjBlockHarmonic_pow_colHarmonic_gaussian3_of_weightZeroLevi11 below · cited by 1 · depth 27 - Folded two-sheet Hermite formula for the degree-m archimedean zeta
LanglandsTunnell.CubicInduction.exists_archZeta30_jacquetVector3_eq_mul_integral_twoSheet_hermiteBracket_of_conjBlockHarmonic_pow_colHarmonic_gaussian311 below · cited by 1 · depth 27 - Reproducing identity at the infinite place on GL₃
LanglandsTunnell.CubicInduction.exists_contDiff_hasCompactSupport_eq_integral_archRealLift30 below · cited by 2 · depth 27 - Finite pure-tensor decomposition of the local Godement datum
LanglandsTunnell.CubicInduction.exists_finset_pureTensor_godementDatum4 below · cited by 1 · depth 27 - Uniform exponent θ₀>1/2 for ray decay of GL₃ Whittaker integrals
LanglandsTunnell.CubicInduction.exists_one_half_lt_forall_rayOrder_whittaker3_of_isCentreFinite_of_isRightInvariant298 below · cited by 1 · depth 27 - Godement slot vectors: principal series membership and support
LanglandsTunnell.CubicInduction.godementDatum_mem_principalSeries2_and_support2 below · cited by 2 · depth 27 - Per-monomial torus integral as a^l tfrac12Γ_ℝ· Mellin transform
LanglandsTunnell.CubicInduction.integrable_and_setIntegral_torusMonomial_eq_mul_GammaR_mul_mellin_of_archDatumR1 below · cited by 1 · depth 27 - Two-sheet torus monomial: integrability and tfrac12Γ_ℝcdotM H evaluation
LanglandsTunnell.CubicInduction.integrable_and_setIntegral_twoSheet_torusMonomial_eq_mul_GammaR_mul_mellin_of_archDatumR1 below · cited by 1 · depth 27 - Affine Fourier duality for 2×3 frames over ℚᵥ
LanglandsTunnell.CubicInduction.integral_frame23_mul_eq_integral_matFourier23_dualFrame23_mul14 below · cited by 1 · depth 27 - Unipotent fibre integration of the GL₃ Godement integrand
LanglandsTunnell.CubicInduction.integral_godementIntegrand_mul_unipotent_eq_mul_integral_frame2 below · cited by 1 · depth 27 - Jacquet unfolding of a Godement section on GL₃
LanglandsTunnell.CubicInduction.integral_godementSection_upperUnipotent3_eq_godementWhittaker3_of_continuous4 below · cited by 1 · depth 27 - Matrix Fourier transform of a Godement datum: dual datum
LanglandsTunnell.CubicInduction.isSchwartzBruhat_and_law_matFourier23_dualDatum2 below · cited by 1 · depth 27 - Jacquet–Whittaker function at diag(1,-1,1)Y as a ψ-integral
LanglandsTunnell.CubicInduction.jacquetWhittaker3_diagonal3_mul_eq_mul_integral_psiLocal_cellSectionOf15 below · cited by 1 · depth 27 - Fourier transform on M_{2× 3} under left and right translation
LanglandsTunnell.CubicInduction.matFourier23_comp_mul_mul_eq7 below · cited by 1 · depth 27 - Whittaker decay on the full diagonal torus from ray bounds
LanglandsTunnell.CubicInduction.norm_whittaker3_sum_translate_diag_le_of_forall_rayOrder20 below · cited by 1 · depth 27 - Automorphy conditions pass to iterated archimedean derivatives
LanglandsTunnell.CubicInduction.conditions_foldr_archDeriv0 below · cited by 2 · depth 28 - Continuity and uniform moderate growth of archimedean derivatives
LanglandsTunnell.CubicInduction.continuous_and_norm_iterate_archDeriv_sum_translate_le_of_isCentreFinite1 below · cited by 8 · depth 28 - Derivative words of a right translate on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.exists_continuous_coeff_foldr_archDeriv_mul_right_eq_sum1 below · cited by 5 · depth 28 - Principal series embedding when the Jacquet module is non-zero
LanglandsTunnell.CubicInduction.exists_linearMap_principalSeries2_of_jacquet_ne_top1 below · cited by 1 · depth 28 - Simple-root derivatives of the GL₃ Whittaker coefficient at a diagonal point
LanglandsTunnell.CubicInduction.exists_ne_zero_forall_mul_whittaker3_diag_eq_whittaker3_archDeriv2 below · cited by 2 · depth 28 - Uniform ray exponent >1/2 for GL₃ Whittaker derivative words
LanglandsTunnell.CubicInduction.exists_one_half_lt_forall_foldr_archDeriv_rayOrder_whittaker3_of_casimir_relations_of_isRightInvariant290 below · cited by 1 · depth 28 - Right translation of archimedean derivative words on GL₃
LanglandsTunnell.CubicInduction.foldr_archDeriv_mul_right_eq_sum0 below · cited by 3 · depth 28 - Symplectic Fourier swap for Godement–Whittaker integrals on GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.godementWhittaker2_symplecticFourier_swap_eq_godementWhittaker2_of_weight17 below · cited by 2 · depth 28 - Jacquet integral for GL₃ cell sections in the positive chamber
LanglandsTunnell.CubicInduction.integrable_and_jacquetWhittaker3_eq_integral_of_norm_eq_rpow_of_lt14 below · cited by 1 · depth 28 - Jacquet integral of a Godement section on GL₂(ℚₚ)
LanglandsTunnell.CubicInduction.integral_godementSection_antidiagonal_mul_unipotentGL2_mul_psiLocal_eq_godementWhittaker2_of_chamber3 below · cited by 2 · depth 28 - Right translates preserve archimedean smoothness on GL₃
LanglandsTunnell.CubicInduction.isArchSmooth3_mul_right0 below · cited by 11 · depth 28 - Torus unfolding of the degree-m Jacquet vector with x-moment
LanglandsTunnell.CubicInduction.jacquetVector3_one_eq_integral_xMoment_of_conjBlockHarmonic_pow_colHarmonic_gaussian37 below · cited by 2 · depth 28 - Fourier inversion for `matFourier22` on M₂(ℚₚ)
LanglandsTunnell.CubicInduction.matFourier22_matFourier22_psiLocal_eq_comp_neg_of_isSchwartzBruhat17 below · cited by 3 · depth 28 - Covariance of the 2× 3 Fourier transform under transvections
LanglandsTunnell.CubicInduction.matFourier23_comp_mul_transvection_eq3 below · cited by 1 · depth 28 - Uniform two-variable torus bound for GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.norm_whittaker3_diag_le_of_isCentreFinite_of_forall_rayOrder12 below · cited by 1 · depth 28 - Transport of the GL₃ cusp package under g↦(g^{mathsf T})⁻¹
LanglandsTunnell.CubicInduction.rayOrder_transport_transposeInv3_of_isCentreFinite_of_isRightInvariant16 below · cited by 1 · depth 28 - Transport of automorphy conditions under g↦^tg⁻¹
LanglandsTunnell.CubicInduction.archPackage_comp_transposeInv3_of_isCentreFinite4 below · cited by 1 · depth 29 - Vanishing of subcritical coefficients in a GL₃ Whittaker expansion
LanglandsTunnell.CubicInduction.coeff_eq_zero_of_re_le_one_half_of_lintegral_torus_whittaker3_sq_le2 below · cited by 1 · depth 29 - Column Fourier transforms in distinct columns commute
LanglandsTunnell.CubicInduction.colFourier23_colFourier23_comm0 below · cited by 3 · depth 29 - Linear central element acts by a scalar
LanglandsTunnell.CubicInduction.exists_casimir1_eq_smul_of_isArchSmooth30 below · cited by 1 · depth 29 - Whittaker expansion on GL₃ from two Casimir relations
LanglandsTunnell.CubicInduction.exists_exponents_whittaker3_diag_expansion_of_casimir_relations7 below · cited by 1 · depth 29 - Two-variable Whittaker decay on GL₃ from regular-singular systems
LanglandsTunnell.CubicInduction.exists_forall_isCompact_orth3_norm_whittaker3_le_of_systems2 below · cited by 1 · depth 29 - Simple-pole bound for torus mean squares of GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_lintegral_torus_whittaker3_sq_le_div_sub_one_of_isCuspidalAlong_of_isRightInvariant121 below · cited by 1 · depth 29 - Regular-singular diagonal systems for GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_words_whittaker3_diag_hasDerivAt_systems_of_casimir_relations_natDegree_le2 below · cited by 5 · depth 29 - Vanishing of the logarithm-free coefficient at exponent of real part 1/2
LanglandsTunnell.CubicInduction.expCoeff_eq_zero_of_re_eq_one_half_of_mem_span_archDeriv_translate272 below · cited by 1 · depth 29 - Whittaker bound on orthogonal compacta extends to all compacta
LanglandsTunnell.CubicInduction.forall_isCompact_norm_whittaker3_le_of_forall_isCompact_orth30 below · cited by 1 · depth 29 - Absolute convergence of Jacquet integrals of GL₃ cell sections
LanglandsTunnell.CubicInduction.integrable_cellSectionOf_antidiagonal3_mul_upperUnipotent3_mul_of_norm_eq_rpow_of_lt12 below · cited by 1 · depth 29 - Dual-configuration Gaussian integral for the degree-m flat section
LanglandsTunnell.CubicInduction.integral_dualConfig_eq_of_conjBlockHarmonic_pow_colHarmonic_gaussian32 below · cited by 1 · depth 29 - Derivative words of translates stay cuspidal along both parabolics
LanglandsTunnell.CubicInduction.isCuspidalAlong_foldr_archDeriv_sum_translate3 below · cited by 2 · depth 29 - Finite-adelic invariance of derivative words of translate combinations
LanglandsTunnell.CubicInduction.isRightInvariant_foldr_archDeriv_sum_translate0 below · cited by 1 · depth 29 - Jacquet–Whittaker function as an absolutely convergent Jacquet integral
LanglandsTunnell.CubicInduction.jacquetWhittaker3_eq_integral_of_integrable9 below · cited by 1 · depth 29 - Right translation covariance of the 2×2 matrix Fourier transform
LanglandsTunnell.CubicInduction.matFourier22_comp_mul_right_eq7 below · cited by 2 · depth 29 - Matrix Fourier transform sees only nonsingular matrices
LanglandsTunnell.CubicInduction.matFourier22_congr_of_forall_det_ne_zero1 below · cited by 1 · depth 29 - Transpose–inverse involution exchanges the two GL₃ Whittaker rays
LanglandsTunnell.CubicInduction.norm_whittaker3_archRealLift3_diag_mul_eq_norm_whittaker3_comp_transposeInv35 below · cited by 1 · depth 29 - Uniform moderate-growth bound for the GL₃ Whittaker coefficient on the diagonal
LanglandsTunnell.CubicInduction.norm_whittaker3_archRealLift3_diag_mul_le_of_isCompact1 below · cited by 5 · depth 29 - Uniform simple-pole bound for the adelic Epstein pairing on a slab
LanglandsTunnell.CubicInduction.AdelicEpstein.exists_forall_sub_one_mul_lintegral_nnnorm_sq_mul_epsteinPlus_le_of_decay9 below · cited by 1 · depth 30 - Existence of slab fundamental domains in adelic GL₃/ℚ
LanglandsTunnell.CubicInduction.SlabL2.exists_isSlabDomain3 below · cited by 3 · depth 30 - Transpose-inverse involution negates archimedean derivatives, swapping indices
LanglandsTunnell.CubicInduction.archDeriv_comp_transposeInv3_of_isArchSmooth30 below · cited by 2 · depth 30 - Column Fourier transforms in the two columns of a 2×2 matrix commute
LanglandsTunnell.CubicInduction.colFourier22_colFourier22_comm3 below · cited by 1 · depth 30 - Transition-stable harmonic families vanish once their bottom degree does
LanglandsTunnell.CubicInduction.compactPicture_eq_bot_of_transitionStable_of_bottom_eq_bot2 below · cited by 1 · depth 30 - Torus slices of squared Whittaker coefficients dominated on compacta
LanglandsTunnell.CubicInduction.exists_lintegral_torus_whittaker3_sq_le_mul_lintegral_quotientMeasure10 below · cited by 1 · depth 30 - Bounded test function on A_ℚ³ positive on a third-row window
LanglandsTunnell.CubicInduction.exists_measurable_bounded_compactArch_integral_pos_on_thirdRow_window1 below · cited by 1 · depth 30 - Mean-square bound on a box for the Whittaker expansion terms
LanglandsTunnell.CubicInduction.exists_nhds_lintegral_sum_cpow_log_sq_le_of_lintegral_torus_whittaker3_sq_le0 below · cited by 1 · depth 30 - Smoothing submodule carrying the leading Whittaker coefficient at exponent 1/2
LanglandsTunnell.CubicInduction.exists_smoothingSubmodule_leadingCoeff_form_of_expCoeff_re_eq_one_half_centreFinite_mg146 below · cited by 1 · depth 30 - Vanishing of coefficients below the order of decay
LanglandsTunnell.CubicInduction.expLogSum_coeff_eq_zero_of_re_lt_of_norm_le_rpow0 below · cited by 18 · depth 30 - Unimodularity of rational determinants; Borel determinant slabs
LanglandsTunnell.CubicInduction.ideleNorm_det_globalPointsGL_eq_one_and_measurableSet_ideleNormDetSlab4 below · cited by 5 · depth 30 - Fourier-slice identity along lower-triangular fibres in M₂(ℚₚ)
LanglandsTunnell.CubicInduction.integral_matFourier22_lowerTriangular_eq_integral_integral_upperTriangular_mul_psiLocal13 below · cited by 1 · depth 30 - Centre-finiteness preserved under the transpose-inverse involution on GL₃
LanglandsTunnell.CubicInduction.isCentreFinite_comp_transposeInv3_of_isArchSmooth32 below · cited by 1 · depth 30 - Vanishing leading coefficient, or transition-stable harmonic families
LanglandsTunnell.CubicInduction.leadingCoeff_eq_zero_or_exists_transitionStable_family_ne_bot_of_smoothingSubmodule_re136 below · cited by 1 · depth 30 - Borel measurability of the gauge on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.measurable_gauge30 below · cited by 1 · depth 30 - Cuspidal moderate-growth functions on GL₃ are slab cusp functions
LanglandsTunnell.CubicInduction.mem_cuspFunctions_of_isCuspidalAlong_of_archDeriv_growth109 below · cited by 1 · depth 30 - Rapid decay on Siegel sets for GL₃ cusp forms
LanglandsTunnell.CubicInduction.norm_mul_gauge3_pow_le_of_siegel_of_isCuspidalAlong_of_archDeriv_growth107 below · cited by 3 · depth 30 - Archimedean derivatives commute with the GL₃ Whittaker integral
LanglandsTunnell.CubicInduction.whittaker3_iterate_archDeriv_eq_iterate_archDeriv_whittaker30 below · cited by 8 · depth 30 - Left GL₃(ℚ)-invariance of the adelic Epstein integral
LanglandsTunnell.CubicInduction.AdelicEpstein.epsteinPlus_globalPointsGL_mul3 below · cited by 1 · depth 31 - Gauge bound for the adelic Epstein integral on GL₃
LanglandsTunnell.CubicInduction.AdelicEpstein.epsteinPlus_le_mul_gauge3_rpow_div_sub_one2 below · cited by 1 · depth 31 - Whittaker expansions on GL₃ persist under right smoothing
LanglandsTunnell.CubicInduction.SlabL2.exists_expansion_whittaker3_smoothingOperator99 below · cited by 5 · depth 31 - Left O(3)-finite smoothing kernels concentrating at the identity
LanglandsTunnell.CubicInduction.SlabL2.exists_isSmoothingKernel_leftOrthFinite_setIntegral_compl_le99 below · cited by 1 · depth 31 - Mass-concentration approximate identity for right smoothing on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.SlabL2.exists_nhds_one_forall_norm_smoothingOperator_sub_le_of_setIntegral_compl_le0 below · cited by 1 · depth 31 - Gelfand central operators commute with right derivatives; reversed cubic identity
LanglandsTunnell.CubicInduction.WhittakerBlock.casimir_commute_archDeriv_and_sum_reversed_cubic_eq0 below · cited by 2 · depth 31 - Reverse-cyclic cubic equals C₃+C₁²-3C₂
LanglandsTunnell.CubicInduction.WhittakerBlock.sum_archDeriv_rev_eq_casimir3_add_casimir1_casimir1_sub_three_smul_casimir21 below · cited by 2 · depth 31 - Casimir eigenvalues of a ν-equivariant function on GL₃
LanglandsTunnell.CubicInduction.casimir_eq_smul_of_upperTriangular_equivariant9 below · cited by 1 · depth 31 - Transition-stable subspaces contain the bottom K-type generator
LanglandsTunnell.CubicInduction.compactPicture_bottom_mem_of_transitionStable_degree_le_two0 below · cited by 1 · depth 31 - No harmonic polynomial of degree ≥ 3 in both lowering kernels
LanglandsTunnell.CubicInduction.compactPicture_eq_zero_of_lowering_eq_zero_of_three_le0 below · cited by 2 · depth 31 - Borel equivariance of the top-logarithmic double Whittaker coefficient
LanglandsTunnell.CubicInduction.doubleSlotCoeff_upperTriangular_equivariant_of_joint_expansion_top3 below · cited by 2 · depth 31 - Shape of a complex triple from power-sum reality conditions
LanglandsTunnell.CubicInduction.exists_eq_neg_half_add_mul_I_of_powerSum_symmetry_of_re_eq_neg_half0 below · cited by 1 · depth 31 - Joint two-variable Whittaker expansion with leading-slice non-triviality
LanglandsTunnell.CubicInduction.exists_exponents_whittaker3_diag_joint_expansion_nontrivial_of_casimir_relations40 below · cited by 1 · depth 31 - Unipotent displacement on a Siegel set in GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.exists_forall_norm_sub_radical_mul_le_div_archRoot_of_archDeriv_le_of_siegel12 below · cited by 1 · depth 31 - Uniform gauge bound on a Siegel set determinant slab
LanglandsTunnell.CubicInduction.exists_gauge3_le_mul_archRoot_mul_archRoot_sq_of_siegel_of_ideleNorm_det_mem_Icc10 below · cited by 1 · depth 31 - Joint Casimir eigenvector with non-vanishing coefficient functional
LanglandsTunnell.CubicInduction.exists_joint_casimir_eigenvector_apply_ne_zero_of_positive_skew_form3 below · cited by 1 · depth 31 - Derivative words inherit automorphy, cuspidality and a common level
LanglandsTunnell.CubicInduction.exists_level_forall_foldr_archDeriv_invariant_cuspidal_archSmooth100 below · cited by 1 · depth 31 - Spectral parameter of a Casimir triple and reality conditions
LanglandsTunnell.CubicInduction.exists_spectralParameter_of_casimir_scalars_and_powerSum_symmetry0 below · cited by 1 · depth 31 - Induced-picture package from a top-slot double leading Whittaker coefficient
LanglandsTunnell.CubicInduction.exists_submodule_inducedPicture_package_of_doubleSlotCoeff_top27 below · cited by 1 · depth 31 - Transition-stable harmonic families from an induced-picture package
LanglandsTunnell.CubicInduction.exists_transitionStable_families_ne_bot_of_inducedPicture_package_top82 below · cited by 1 · depth 31 - Vanishing of parameter-dependent coefficients below the remainder order
LanglandsTunnell.CubicInduction.expLogSum_coeff_eq_zero_of_re_lt_of_forall_norm_le_rpow1 below · cited by 2 · depth 31 - Adjoints of the three mathfrakgl₃ central words under a skew hermitian form
LanglandsTunnell.CubicInduction.form_casimir_eq_of_skew_archDeriv0 below · cited by 1 · depth 31 - Gauge bounds: right compact translation and smoothing on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.gauge3_mul_le_of_isCompact_and_norm_smoothingOperator_le_gauge3_pow1 below · cited by 5 · depth 31 - Oscillation bound for cuspidal functions on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.norm_le_of_isCuspidalAlong_of_arch_oscillation_le10 below · cited by 1 · depth 31 - Regularity package for derivative words of centre-finite translates
LanglandsTunnell.CubicInduction.seed_package_of_mem_span_archDeriv_translate9 below · cited by 5 · depth 31 - Monic Casimir relations on the GL₃ smoothing module
LanglandsTunnell.CubicInduction.smoothingModule_casimir_relations111 below · cited by 1 · depth 31 - Smoothing module on GL₃: leading-coefficient functional and its properties
LanglandsTunnell.CubicInduction.smoothingModule_expansion_leadingCoeff118 below · cited by 2 · depth 31 - Orthogonal finiteness and derivative stability of the smoothing module
LanglandsTunnell.CubicInduction.smoothingModule_orthFinite_and_archDeriv_mem117 below · cited by 3 · depth 31 - Regularity and gauge growth in the GL₃ smoothing module
LanglandsTunnell.CubicInduction.smoothingModule_regularity_and_growth116 below · cited by 3 · depth 31 - The slab form on the GL₃ smoothing module
LanglandsTunnell.CubicInduction.smoothingModule_slabForm139 below · cited by 1 · depth 31 - Simple-pole bound for the adelic Epstein integral on GL₃
LanglandsTunnell.CubicInduction.AdelicEpstein.epsteinPlus_le_div_sub_one_of_forall_exists_le_norm_vecMul0 below · cited by 1 · depth 32 - Smoothing kernels on GL₃(A_ℚ) are continuous with compact support
LanglandsTunnell.CubicInduction.SlabL2.continuous_and_hasCompactSupport_of_isSmoothingKernel99 below · cited by 4 · depth 32 - Isometric strongly continuous lift of right translation to cuspidal L²
LanglandsTunnell.CubicInduction.SlabL2.exists_isCuspLift3_translateRight_norm_eq21 below · cited by 2 · depth 32 - Archimedean derivatives of smoothed coefficients on GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.SlabL2.hasDerivAt_integral_mul_comp_archRealLift3_smoothingKernel5 below · cited by 4 · depth 32 - Skew-adjointness of archimedean derivatives on the slab
LanglandsTunnell.CubicInduction.SlabL2.integral_archDeriv_smoothingOperator_mul_conj_eq_neg122 below · cited by 1 · depth 32 - Left orthogonal finiteness passes to archimedean derivative kernels
LanglandsTunnell.CubicInduction.SlabL2.leftOrthFinite_archDerivKernel99 below · cited by 3 · depth 32 - Mean-value bound along a square-zero direction in GL₃(A_ℚ)
LanglandsTunnell.CubicInduction.WhittakerBlock.norm_sub_le_sum_abs_mul_of_mul_self_eq_zero_of_archDeriv_le0 below · cited by 1 · depth 32 - Casimir eigenvalues from left diagonal first-order data on GL₃
LanglandsTunnell.CubicInduction.casimir_eq_smul_of_deriv_archRealLift3_mul_eq7 below · cited by 1 · depth 32 - Square-zero conjugate of a radical element past a Siegel point
LanglandsTunnell.CubicInduction.conj_radical_sub_one_mul_self_eq_zero_and_norm_le_div_archRoot_of_siegel10 below · cited by 1 · depth 32 - First-order derivatives of an upper-triangular-equivariant function on GL₃
LanglandsTunnell.CubicInduction.deriv_archRealLift3_mul_eq_of_upperTriangular_equivariant0 below · cited by 1 · depth 32 - No non-zero linear form is read in the split sign class
LanglandsTunnell.CubicInduction.eq_zero_of_isHomogeneous_one_of_read_signIsotypic_linear_of_inducedPicture_package47 below · cited by 1 · depth 32 - Vanishing of degree-zero reads in a constant sign class
LanglandsTunnell.CubicInduction.eq_zero_of_isHomogeneous_zero_of_read_signIsotypic_const_of_inducedPicture_package47 below · cited by 1 · depth 32 - Archimedean splitting of GL₃ of the adeles of ℚ
LanglandsTunnell.CubicInduction.exists_eq_archRealLift3_mul_of_archComponent3_eq_one0 below · cited by 1 · depth 32 - Joint two-variable expansion of a GL₃ Whittaker coefficient
LanglandsTunnell.CubicInduction.exists_exponents_whittaker3_diag_joint_expansion_of_casimir_relations15 below · cited by 3 · depth 32 - Gauge lower bound for adelically moved rational vectors
LanglandsTunnell.CubicInduction.exists_inv_mul_gauge3_le_norm_vecMul_of_forall_mem_adicCompletionIntegers0 below · cited by 1 · depth 32 - Existence of a congruence level for a smooth adelic function
LanglandsTunnell.CubicInduction.exists_level_forall_mul_finEmbedN_eq_of_isRightInvariant_of_isOpen99 below · cited by 1 · depth 32 - Sign-isotypic splitting of a B⁺-equivariant adelic function space
LanglandsTunnell.CubicInduction.exists_signIsotypic_submodules_of_upperTriangular_equivariant_of_orthogonalRightStable0 below · cited by 1 · depth 32 - Vanishing of a leading first-ratio Whittaker coefficient on GL₃
LanglandsTunnell.CubicInduction.exists_threshold_firstRatioCoeff_eq_zero_of_forall_secondRatioCoeff_eq_zero_of_casimir_relations34 below · cited by 1 · depth 32 - Threshold vanishing of second-ratio Whittaker coefficients on GL₃
LanglandsTunnell.CubicInduction.exists_threshold_secondRatioCoeff_eq_zero_of_forall_firstRatioCoeff_eq_zero_of_casimir_relations34 below · cited by 1 · depth 32 - Transition-stable family of harmonic polynomials read on O(3)
LanglandsTunnell.CubicInduction.exists_transitionStable_family_of_signIsotypic_submodule19 below · cited by 2 · depth 32 - Odd sign classes: vanishing of sign-isotypic leading coefficients
LanglandsTunnell.CubicInduction.forall_apply_orthogonal_eq_zero_of_signIsotypic_odd_of_inducedPicture_package71 below · cited by 1 · depth 32 - Double-slot coefficients intertwine the archimedean flow derivative
LanglandsTunnell.CubicInduction.hasDerivAt_doubleSlotCoeff_archFlow_of_joint_expansion_archDeriv12 below · cited by 2 · depth 32 - Archimedean smoothness from Iwasawa equivariance and orthogonal finiteness
LanglandsTunnell.CubicInduction.isArchSmooth3_of_continuous_of_upperTriangular_equivariant_of_orthogonalFinite4 below · cited by 1 · depth 32 - Vanishing Whittaker integral kills all joint-expansion coefficients
LanglandsTunnell.CubicInduction.joint_expansion_coeff_eq_zero_of_forall_whittaker3_diag_mul_eq_zero1 below · cited by 4 · depth 32 - Right translation acts on joint Whittaker expansions on GL₃
LanglandsTunnell.CubicInduction.joint_expansion_comp_mul_right0 below · cited by 2 · depth 32 - Joint Whittaker expansions are linear in the form
LanglandsTunnell.CubicInduction.joint_expansion_sum_mul0 below · cited by 2 · depth 32 - Low-degree members of harmonic families with pure sign type
LanglandsTunnell.CubicInduction.le_span_and_eq_bot_of_signType_of_isHomogeneous0 below · cited by 1 · depth 32 - Leibniz rule for the slab pairing along an archimedean direction
LanglandsTunnell.CubicInduction.SlabL2.hasDerivAt_integral_smoothingOperator_comp_archRealLift3_mul_conj118 below · cited by 1 · depth 33 - Right archimedean translation preserves the slab L² pairing
LanglandsTunnell.CubicInduction.SlabL2.integral_smoothingOperator_comp_archRealLift3_mul_conj_eq24 below · cited by 1 · depth 33 - Archimedean real lift: identity at 1, idele norm of determinant
LanglandsTunnell.CubicInduction.WhittakerBlock.archRealLift3_one_and_ideleNorm_det_archRealLift34 below · cited by 3 · depth 33 - Continuity of the real lift into GL₃(A_ℚ) on det ≠ 0
LanglandsTunnell.CubicInduction.WhittakerBlock.continuousOn_archRealLift30 below · cited by 2 · depth 33 - Derivative at arbitrary s along an archimedean elementary flow
LanglandsTunnell.CubicInduction.WhittakerBlock.hasDerivAt_apply_mul_archRealLift3_of_isArchSmooth30 below · cited by 5 · depth 33 - Sign type of a polynomial read off a sign-isotypic space
LanglandsTunnell.CubicInduction.aeval_signTwist_eq_smul_of_read_signIsotypic0 below · cited by 1 · depth 33 - Conjugating an archimedean real matrix past an adelic point of GL₃
LanglandsTunnell.CubicInduction.archRealLift3_mul_eq_mul_archRealLift3_conj0 below · cited by 6 · depth 33 - Casimir values as reversed left-translation flow derivatives
LanglandsTunnell.CubicInduction.casimir_apply_eq_sum_deriv_archRealLift3_mul5 below · cited by 1 · depth 33 - Archimedean real lift in GL₃(A_ℚ): components and orthogonality
LanglandsTunnell.CubicInduction.componentAt3_archRealLift3_eq_one_and_realMat_archComponent3_eq0 below · cited by 3 · depth 33 - A separated sign-projection read forces p=0
LanglandsTunnell.CubicInduction.eq_zero_of_read_signProjection_of_separating_stable_submodule13 below · cited by 2 · depth 33 - Linear double-slot coefficient functional on a smoothing module
LanglandsTunnell.CubicInduction.exists_linearMap_doubleSlotCoeff_of_smoothingSubmodule15 below · cited by 3 · depth 33 - Degree-one odd sign classes lie in a coordinate line
LanglandsTunnell.CubicInduction.exists_ne_and_le_span_X_of_signType_one_of_odd0 below · cited by 2 · depth 33 - A non-zero harmonic polynomial read from a sign-isotypic space
LanglandsTunnell.CubicInduction.exists_ne_zero_isHomogeneous_harmonic_read_of_signIsotypic_apply_ne_zero3 below · cited by 1 · depth 33 - Polynomial model for a separating stable submodule
LanglandsTunnell.CubicInduction.exists_polynomialModel_positive_actSkew_form_of_separating_stable_submodule7 below · cited by 1 · depth 33 - Lowering the degree by one preserves readability from V_ε
LanglandsTunnell.CubicInduction.exists_read_lowerOne_xi_of_read_signIsotypic7 below · cited by 1 · depth 33 - Lowering a read harmonic polynomial by two degrees
LanglandsTunnell.CubicInduction.exists_read_lowerTwo_xi_of_read_signIsotypic6 below · cited by 1 · depth 33 - Degree-two transition preserves readability from a sign-isotypic space
LanglandsTunnell.CubicInduction.exists_read_sameTwo_xi_of_read_signIsotypic7 below · cited by 1 · depth 33 - Separating O(3)-stable submodule for the sign-projected double coefficient
LanglandsTunnell.CubicInduction.exists_stable_submodule_separating_signProjection_of_doubleSlotCoeffMap27 below · cited by 3 · depth 33 - Vanishing of leading flat Whittaker coefficients at orthogonal translates
LanglandsTunnell.CubicInduction.exists_threshold_firstRatioCoeff_eq_zero_orth_of_flat_of_leading_of_casimir_relations29 below · cited by 1 · depth 33 - Flat leading Whittaker coefficients vanish at orthogonal translates
LanglandsTunnell.CubicInduction.exists_threshold_secondRatioCoeff_eq_zero_orth_of_flat_of_leading_of_casimir_relations29 below · cited by 1 · depth 33 - Uniform Iwasawa transport of GL₃ Whittaker coefficients on a compact set
LanglandsTunnell.CubicInduction.exists_whittaker3_diag_mul_transport_of_isCompact1 below · cited by 1 · depth 33 - From orthogonal to all translates for a leading Whittaker coefficient
LanglandsTunnell.CubicInduction.firstRatioCoeff_eq_zero_of_forall_orth_eq_zero_of_leading_of_casimir_relations3 below · cited by 1 · depth 33 - Flatness climbs the logarithmic chain (first ratio)
LanglandsTunnell.CubicInduction.firstRatioCoeff_flat_of_le_of_flat_of_casimir_relations2 below · cited by 1 · depth 33 - No positive invariant form on an odd sign class
LanglandsTunnell.CubicInduction.forall_eval_orthogonal_eq_zero_of_odd_signClass_of_positive_actSkew_form19 below · cited by 1 · depth 33 - Normal forms of the quadratic and cubic mathfrakgl₃ Casimirs
LanglandsTunnell.CubicInduction.gl3_casimir_normalForm0 below · cited by 1 · depth 33 - Smoothness, linearity and bracket law for archimedean left-flow derivatives
LanglandsTunnell.CubicInduction.isArchSmooth3_deriv_archRealLift3_mul_and_linear_and_bracket0 below · cited by 2 · depth 33 - Lowering by one preserves harmonicity and homogeneity
LanglandsTunnell.CubicInduction.isHomogeneous_sub_one_and_sum_pderiv_pderiv_eq_zero_lowerOne_xi0 below · cited by 2 · depth 33 - Harmonicity and degree ℓ-2 of lower₂(Xi_ν p)
LanglandsTunnell.CubicInduction.isHomogeneous_sub_two_and_sum_pderiv_pderiv_eq_zero_lowerTwo_xi0 below · cited by 2 · depth 33 - Degree-two same-degree transition polynomial is harmonic
LanglandsTunnell.CubicInduction.isHomogeneous_two_and_sum_pderiv_pderiv_eq_zero_sameTwo_xi0 below · cited by 1 · depth 33 - Leading Whittaker coefficient: from orthogonal translates to all translates
LanglandsTunnell.CubicInduction.secondRatioCoeff_eq_zero_of_forall_orth_eq_zero_of_leading_of_casimir_relations3 below · cited by 1 · depth 33 - Flatness of second-ratio coefficients climbs the logarithmic chain
LanglandsTunnell.CubicInduction.secondRatioCoeff_flat_of_le_of_flat_of_casimir_relations2 below · cited by 1 · depth 33 - Odd sign classes: transition-stable families reach degree one
LanglandsTunnell.CubicInduction.transitionStable_family_one_ne_bot_of_odd_signClass1 below · cited by 2 · depth 33 - Smoothed GL₃ cusp forms are bounded on determinant slabs
LanglandsTunnell.CubicInduction.SlabL2.exists_forall_norm_smoothingOperator_le_of_ideleNorm_det_mem_Icc114 below · cited by 1 · depth 34 - Finiteness of a slab fundamental domain measure for GL₃
LanglandsTunnell.CubicInduction.SlabL2.isFiniteMeasure_domainMeasure14 below · cited by 1 · depth 34 - Rotation-flow derivative as difference of elementary arch derivatives
LanglandsTunnell.CubicInduction.WhittakerBlock.hasDerivAt_comp_archRealLift3_rotation_of_isArchSmooth30 below · cited by 3 · depth 34 - Induced-picture operators preserve vanishing on real O(3)
LanglandsTunnell.CubicInduction.eval_inducedPicture_act_eq_zero_of_forall_eval_orthogonal_eq_zero0 below · cited by 2 · depth 34 - Rotation Casimir acts by -ℓ(ℓ+1) on harmonic column realisations
LanglandsTunnell.CubicInduction.eval_rotationCasimir_det_pow_mul_columnRealisation_eq_of_harmonic1 below · cited by 1 · depth 34 - Degree-two same-level transition identity at orthogonal matrices
LanglandsTunnell.CubicInduction.eval_sum_act_quadric_realise_eulerShift_eq_twelfth_eval_realise_sameTwo0 below · cited by 1 · depth 34 - A degree-lowering identity on O(3) for the symmetrised action
LanglandsTunnell.CubicInduction.eval_sum_act_quadric_realise_pderiv_pderiv_eq_half_eval_realise_lowerTwo0 below · cited by 4 · depth 34 - Symmetrised action on column realisations matches tfrac12 lower₁(Xi_ν p)
LanglandsTunnell.CubicInduction.eval_sum_act_quadric_realise_rot_pderiv_eq_half_eval_realise_lowerOne0 below · cited by 2 · depth 34 - Right O(3)-stable spaces contain det^α times a column harmonic
LanglandsTunnell.CubicInduction.exists_det_pow_mul_columnRealisation_mem_of_finiteDimensional_of_orthogonalRightStable0 below · cited by 2 · depth 34 - Odd sign class forces a det-twisted row entry
LanglandsTunnell.CubicInduction.exists_mem_eval_eq_det_pow_mul_entry_of_odd_signClass_of_actStable15 below · cited by 1 · depth 34 - Quadric read-outs of a twisted column realisation stay in V
LanglandsTunnell.CubicInduction.exists_mem_forall_det_pow_mul_eval_sum_quadric_mul_columnRealisation_eq1 below · cited by 3 · depth 34 - Two archimedean derivative identities for GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_ne_zero_forall_whittaker3_archDeriv_archDeriv_diag_eq_mul_and_whittaker3_archDeriv_corner_eq_zero5 below · cited by 1 · depth 34 - Separating stable submodule inside an equivariant stable submodule
LanglandsTunnell.CubicInduction.exists_separating_stable_submodule_of_equivariant_stable_submodule26 below · cited by 1 · depth 34 - Flat regular-singular system for leading GL₃ Whittaker coefficients
LanglandsTunnell.CubicInduction.exists_threshold_firstRatioCoeff_flat_regularSingular_system_of_leading_of_casimir_relations26 below · cited by 1 · depth 34 - Flat regular–singular coefficient system at a leading exponent
LanglandsTunnell.CubicInduction.exists_threshold_secondRatioCoeff_flat_regularSingular_system_of_leading_of_casimir_relations26 below · cited by 1 · depth 34 - Distinct rotation-Casimir eigenvalues force β-orthogonality
LanglandsTunnell.CubicInduction.form_eq_zero_of_rotationCasimir_eigen_ne_of_actSkew0 below · cited by 1 · depth 34 - Derivative of B-equivariant F along E_{cd} in the induced picture
LanglandsTunnell.CubicInduction.hasDerivAt_archFlow_eq_eval_inducedPicture_act_of_upperTriangular_equivariant1 below · cited by 5 · depth 34 - Induced-picture operators commute with multiplication by det X
LanglandsTunnell.CubicInduction.inducedPicture_act_det_mul0 below · cited by 7 · depth 34 - Diagonal action on a row entry and its harmonic splitting
LanglandsTunnell.CubicInduction.inducedPicture_act_diag_row_eq_and_harmonic_split0 below · cited by 1 · depth 34 - Quadratic operator annihilating constants and one row in the induced picture
LanglandsTunnell.CubicInduction.inducedPicture_quantisedMinor_annihilates_lowest_of_matched0 below · cited by 1 · depth 34 - Sign projection: equivariance preserved, ε-isotypic
LanglandsTunnell.CubicInduction.upperTriangular_equivariant_and_signIsotypic_signProjection0 below · cited by 3 · depth 34 - Admissibility of the cyclic hull at a fixed type
LanglandsTunnell.CubicInduction.exists_finiteDimensional_forall_mem_hull_of_rotationType_of_smoothingSubmodule19 below · cited by 1 · depth 35 - Isotypic projectors for the lowest orthogonal types, with naturality
LanglandsTunnell.CubicInduction.exists_isotypicProjector_natural_of_orthFinite_of_derivStable2 below · cited by 1 · depth 35 - Invariant complement for a positive skew-symmetric form
LanglandsTunnell.CubicInduction.exists_stable_submodule_inf_eq_bot_and_sub_mem_of_positive_skew_form0 below · cited by 1 · depth 35 - Sphere evaluations span the dual of degree-ℓ forms
LanglandsTunnell.CubicInduction.exists_sum_mul_eval_sphere_eq_of_isHomogeneous0 below · cited by 2 · depth 35 - Transition-stable family read in a sign-isotypic polynomial space
LanglandsTunnell.CubicInduction.exists_transitionStable_family_read_of_actStable_signIsotypic11 below · cited by 1 · depth 35 - Central relations pass to derivative words of right translates
LanglandsTunnell.CubicInduction.isArchSmooth3_and_sum_smul_iterate_casimir_eq_zero_of_mem_span_foldr_archDeriv_mul_right3 below · cited by 2 · depth 35 - Kernel of the sign-projected read is translation- and derivative-stable
LanglandsTunnell.CubicInduction.signProjection_read_kernel_stable_of_doubleSlotCoeffMap2 below · cited by 1 · depth 35 - Sign type of a polynomial read in a χ_ε-isotypic space
LanglandsTunnell.CubicInduction.aeval_signTwist_eq_C_mul_of_read_polynomial_signIsotypic0 below · cited by 1 · depth 36 - Archimedean derivatives of a right translate: conjugation by c(k)
LanglandsTunnell.CubicInduction.archDeriv_comp_mul_right_eq_sum_realMat_of_isArchSmooth36 below · cited by 1 · depth 36 - Sign-isotypic stable polynomial space reads a harmonic homogeneous polynomial
LanglandsTunnell.CubicInduction.exists_ne_zero_isHomogeneous_harmonic_read_polynomial_of_signIsotypic_eval_ne_zero1 below · cited by 1 · depth 36 - Rotation invariants in the mathfrakgl₃-span of f are finite-dimensional
LanglandsTunnell.CubicInduction.finiteDimensional_invariants_gKSpan_of_isCentreFinite0 below · cited by 1 · depth 36 - Admissibility at the three-dimensional orthogonal type
LanglandsTunnell.CubicInduction.finiteDimensional_ker_rotationCasimir_add_two_gKSpan_of_isCentreFinite7 below · cited by 1 · depth 36 - Reads of the two lowering transitions stay inside W
LanglandsTunnell.CubicInduction.read_lowerTwo_xi_and_read_lowerOne_xi_of_read_polynomial_actStable5 below · cited by 1 · depth 36 - Homogeneous symbols split off invariants modulo the diagonal Casimir
LanglandsTunnell.CubicInduction.tmul_mem_span_invariant_mul_sup_span_diagCasimir_of_isHomogeneous6 below · cited by 1 · depth 37 - Skew-Hermitian diagonal action on homogeneous symbols tensored with W
LanglandsTunnell.CubicInduction.exists_posDef_hermitian_skew_diagAction_tensor_homogeneous_and_casimir_tmul0 below · cited by 2 · depth 38 - Kernel of Ω+2 on degree n+1 symbols
LanglandsTunnell.CubicInduction.mem_span_invariant_mul_of_diagCasimir_add_two_eq_zero4 below · cited by 1 · depth 38 - Adapted basis for a type-one unitary mathfrakso₃-module
LanglandsTunnell.CubicInduction.exists_basis_forall_apply_eq_sum_single_sub_single_smul_of_typeOne0 below · cited by 1 · depth 39 - SO(3)-equivariant matrix polynomials on symmetric 3× 3 matrices
LanglandsTunnell.CubicInduction.exists_eq_sum_aeval_trace_pow_smul_pow_of_matrix_map_rotationDerivation_eq_commutator0 below · cited by 1 · depth 39 - Type-one mathfrakso₃-triples: kernel identities and splitting
LanglandsTunnell.CubicInduction.so3Triple_typeOne_ker_identities_and_decomposition0 below · cited by 1 · depth 39
LanglandsTunnell.CubicLambda 9
- Parity of a Dirichlet character with Jacobi-symbol values
LanglandsTunnell.CubicLambda.dirichletChar_neg_one_eq_of_forall_eq_jacobiSym0 below · cited by 2 · depth 17 - Jacobi symbol detects splitting in a quadratic field
LanglandsTunnell.CubicLambda.exists_int_jacobiSym_eq_of_finrank_eq_two0 below · cited by 2 · depth 17 - Order-ℓ Hecke character attached to a prime-degree Galois extension
LanglandsTunnell.CubicLambda.exists_isFiniteOrderHeckeChar_eulerCoeff_of_isGalois_of_prime_finrank256 below · cited by 3 · depth 17 - Resolvent Hecke character of a non-normal cubic field
LanglandsTunnell.CubicLambda.exists_resolventChar261 below · cited by 2 · depth 17 - Euler factor of ζ_K for a non-normal cubic field
LanglandsTunnell.CubicLambda.zetaEulerPoly_eq_of_resolvent0 below · cited by 2 · depth 17 - Ramified Euler factor identity for a non-normal cubic field
LanglandsTunnell.CubicLambda.zetaEulerPoly_eq_of_resolvent_of_ramified0 below · cited by 2 · depth 17 - Finite-order Hecke character attached to a prime-degree Galois field
LanglandsTunnell.CubicLambda.exists_isFiniteOrderHeckeChar_eulerCoeff_and_localChar_eq_one_of_factorization_discr_le_of_isGalois_rat_of_prime_finrank297 below · cited by 1 · depth 24 - Hecke character of a prime-degree abelian extension, via the Artin map
LanglandsTunnell.CubicLambda.exists_isFiniteOrderHeckeChar_eulerCoeff_and_eq_comp_idelicArtinMap_of_isGalois_of_prime_finrank256 below · cited by 1 · depth 25 - Triviality of χ∘ r on higher units at q
LanglandsTunnell.CubicLambda.localChar_comp_idelicArtinMap_eq_one_of_mem_higherUnitsAt_of_factorization_discr_le_of_prime_finrank292 below · cited by 1 · depth 25
LanglandsTunnell.ExplicitLift 4
- A multiplicative lift GL₂(mathbb F₃)toGL₂(ℤ[√-2])
LanglandsTunnell.ExplicitLift.exists_monoidHom_map_red_eq2 below · cited by 3 · depth 8 - Lifting invertible matrices over 𝔽₃ to the monoid generated by S,T
LanglandsTunnell.ExplicitLift.exists_mem_closure_map_red_eq_of_det_ne_zero0 below · cited by 3 · depth 9 - Reduction mod 3 is injective on the monoid generated by S, T
LanglandsTunnell.ExplicitLift.map_red_injOn_closure0 below · cited by 4 · depth 9 - Trace and determinant agreement with the explicit octahedral lift
LanglandsTunnell.ExplicitLift.trace_det_eq_of_agreeUpToPartner_of_conjPow0 below · cited by 1 · depth 13
LanglandsTunnell.HeckeTate 5
- Hecke–Tate: the degree-one L-datum of an idele class character is nicely pinned
LanglandsTunnell.HeckeTate.isNicePinned_heckeDatum92 below · cited by 6 · depth 17 - Archimedean root numbers of induced data over a quadratic extension
LanglandsTunnell.HeckeTate.exists_archRootNumber_induced_of_finrank_eq_two0 below · cited by 1 · depth 18 - Trivial root number and pinned exponent at almost all places
LanglandsTunnell.HeckeTate.finite_setOf_stdRootNumberAt_ne_one_and_finite_setOf_pinnedExp_ne_zero19 below · cited by 9 · depth 18 - Euler factors above p of a norm-twisted idele character
LanglandsTunnell.HeckeTate.finprod_euler_comp_X_pow_inertiaDeg_eq_inducedEulerPoly_comp5 below · cited by 6 · depth 21 - Pinned functional equation for a non-normic character twisted from ℚ
LanglandsTunnell.HeckeTate.isNicePinned_heckeDatum_mul_comp_idelicNorm_of_not_exists_eq_pow_inertiaDeg108 below · cited by 6 · depth 21
LanglandsTunnell.P2 21
- Norm compatibility of the order-4 and order-8 Artin ray symbols
LanglandsTunnell.P2.raySymbol_artinValue4_eq_raySymbol_artinValue_relNorm_of_inertia_rat2 below · cited by 1 · depth 14 - Artin ray symbol trivial on narrow principal ideals
LanglandsTunnell.P2.raySymbol_artinValue_span_eq_one71 below · cited by 2 · depth 14 - Ramified primes divide a fixed nonzero ideal
LanglandsTunnell.P2.Artin.exists_ne_bot_forall_inertia_ne_bot_dvd0 below · cited by 3 · depth 15 - Artin reciprocity for the cyclic octic layer L/K'
LanglandsTunnell.P2.raySymbol_artinValue_span_eq_one_of_artinPairCore1 below · cited by 1 · depth 15 - Admissible multiple of a modulus containing the ramification
LanglandsTunnell.P2.Artin.exists_dvd_and_isAdmissibleModulusOfDegree_of_ramified_dvd0 below · cited by 16 · depth 16 - Ramified primes divide a common nonzero ideal
LanglandsTunnell.P2.Artin.exists_ne_bot_forall_inertia_primeAbove_ne_bot_dvd0 below · cited by 11 · depth 16 - Narrow ray class character of a prime-degree cyclic extension
LanglandsTunnell.P2.Artin.exists_rayClassChar_orderOf_eq_inertiaDeg_of_finrank_eq_prime117 below · cited by 2 · depth 16 - Ideal-theoretic and idelic norm indices agree
LanglandsTunnell.P2.Artin.normRaySubgroup_index_eq_of_anchors0 below · cited by 3 · depth 16 - Reciprocity for the cyclic octic subextension of a GL₂(𝔽₃)-extension
LanglandsTunnell.P2.raySymbol_artinValue_span_eq_one_of_artinFieldCore0 below · cited by 1 · depth 16 - Artin symbol of an idèle's content as a Frobenius product
LanglandsTunnell.P2.Artin.artinSymbol_fadContentHom3 below · cited by 3 · depth 17 - Existence of an admissible modulus supported at inertia-ramified places
LanglandsTunnell.P2.Artin.exists_admissibleModulus_supported0 below · cited by 7 · depth 17 - Idèle classes modulo norms as ray classes modulo the norm-ray subgroup
LanglandsTunnell.P2.Artin.exists_mulEquiv_quotient_normRaySubgroup_apply_eq_contents_of_anchors14 below · cited by 2 · depth 17 - Unit idèles at an admissible modulus are idelic norms
LanglandsTunnell.P2.Artin.unitIdeles_le_range_idelicNorm_of_isAdmissibleModulusOfDegree3 below · cited by 2 · depth 17 - Ramified place yields local unit outside the reciprocity kernel
LanglandsTunnell.P2.Artin.exists_localUnit_notMem_principalIdeles_sup_range_idelicNorm_of_inertia_ne_bot253 below · cited by 4 · depth 18 - Everywhere-unit 1-adjusted idèles lie in U_f
LanglandsTunnell.P2.Artin.mem_unitIdeles_of_placeOrd_eq_zero_of_isAdjuster_one0 below · cited by 2 · depth 18 - Unit idèles of an admissible modulus are idelic norms
LanglandsTunnell.P2.Artin.unitIdeles_le_range_idelicNorm_of_dvd_twentyFour3 below · cited by 1 · depth 19 - A modulus whose unit idèles are idelic norms
LanglandsTunnell.P2.Artin.exists_ne_bot_and_unitIdeles_le_range_idelicNorm4 below · cited by 2 · depth 24 - Restriction of the Frobenius idèle symbol under base change
LanglandsTunnell.P2.Artin.resHom_finprod_artinFrob_pow_placeOrd_map_eq_finprod_pow_finrank2 below · cited by 1 · depth 24 - Monotonicity of unit idèle congruence subgroups in the level
LanglandsTunnell.P2.Artin.unitIdeles_le_unitIdeles_of_dvd0 below · cited by 1 · depth 24 - Descent of the Frobenius product formula to level f
LanglandsTunnell.P2.Artin.eq_finprod_artinFrob_pow_placeOrd_of_isAdjuster_one_of_dvd14 below · cited by 1 · depth 25 - Functoriality of Artin Frobenius elements in a tower
LanglandsTunnell.P2.Artin.resHom_artinFrob_eq_artinFrob_pow_inertiaDeg0 below · cited by 1 · depth 25
LanglandsTunnell.RankinSelberg 339
- Pinned niceness of Rankin–Selberg L-data over a cubic field
LanglandsTunnell.RankinSelberg.exists_isNicePinned_rsDatum_isArchCompAt_of_isArithGenuineCuspRealizable2,501 below · cited by 1 · depth 15 - Pinned niceness passes from Rankin–Selberg datum to twisted base change
LanglandsTunnell.RankinSelberg.isNicePinned_twistedDatum_formalBaseChange_of_isNicePinned_rsDatum1 below · cited by 2 · depth 15 - Rigidity of idele class characters under base change to ℚ
LanglandsTunnell.RankinSelberg.eq_comp_idelicNorm_of_forall_under_notMem_uniformizerIdele_eq_pow_inertiaDeg10 below · cited by 6 · depth 16 - Rigidity of idele class characters over ℚ
LanglandsTunnell.RankinSelberg.eq_comp_idelicNorm_of_forall_uniformizerIdele_eq_pow_inertiaDeg10 below · cited by 2 · depth 16 - Logarithm of a unitary self-dual Rankin–Selberg local factor
LanglandsTunnell.RankinSelberg.exists_nonneg_exp_tsum_mul_pow_eq_inv_eval_rsEulerPoly_self_of_norm_eq_one0 below · cited by 1 · depth 16 - Norm-one unit idèle nontrivial at a prescribed place above p₀
LanglandsTunnell.RankinSelberg.exists_unitIdele_over_idelicNorm_eq_one_and_apply_ne_one_of_ne2 below · cited by 2 · depth 16 - Niceness of the pinned Rankin–Selberg datum of a cubic twist
LanglandsTunnell.RankinSelberg.isNicePinned_rsDatum_of_centralInduced_of_localWhittaker_of_not_exists_eq_pow_inertiaDeg_of_normPin_archTrivial2,488 below · cited by 2 · depth 16 - Rankin–Selberg Euler polynomial of a cubic automorphic induction
LanglandsTunnell.RankinSelberg.rsEulerPoly_induced_eq_finprod_twist_formalBaseChange0 below · cited by 2 · depth 16 - Entire pair for the cubic Rankin–Selberg datum
LanglandsTunnell.RankinSelberg.exists_entire_boundedOnStrips_eq_archFactor_mul_lFun_rsDatum_of_le_conductorExponentAt_of_centralInduced_of_localSpaceAt_of_normPin_archTrivial2,487 below · cited by 1 · depth 17 - Well-formedness, convergence and positive conductor for a twisted Rankin–Selberg datum
LanglandsTunnell.RankinSelberg.wellFormed_and_converges_rsDatum_and_finiteConductor_pos_of_le_conductorExponentAt_of_not_exists_eq_pow_inertiaDeg30 below · cited by 3 · depth 17 - Dual-side family identity in the GL₂timesGL₃ entire-pair assembly
LanglandsTunnell.RankinSelberg.EntirePairAssembly.dual_identity_family24 below · cited by 1 · depth 18 - Convergence of the Rankin–Selberg L-datum under Satake root bounds
LanglandsTunnell.RankinSelberg.converges_rsDatum_of_summable_of_forall_exists_norm_lt_sqrt3 below · cited by 2 · depth 18 - Archimedean holomorphy and non-vanishing from a torus Γ-factor identity
LanglandsTunnell.RankinSelberg.differentiableOn_and_rsArchIntegral_ne_zero_of_torusPair_eq_gammaFactor5 below · cited by 1 · depth 18 - Holomorphy of Rankin–Selberg L-functions beyond the abscissa
LanglandsTunnell.RankinSelberg.differentiableOn_lFun_rsDatum_of_summable_of_exists_norm_lt_sqrt4 below · cited by 1 · depth 18 - Entirety and strip-boundedness of a GL₂timesGL₃ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.differentiable_rsGlobalIntegral_and_boundedOnStrips_of_hasIotaMoments2 below · cited by 1 · depth 18 - Local relations at p for the dual translate of W_f
LanglandsTunnell.RankinSelberg.dualTranslate_finWhittaker_local_relations3 below · cited by 1 · depth 18 - Archimedean GL₂timesGL₃ torus-pair identity for the cubic induction
LanglandsTunnell.RankinSelberg.exists_archWhittaker_torusPair_eq_gammaFactor_of_archWhittakerDatum324 below · cited by 1 · depth 18 - Half-plane integrability of archimedean GL₂timesGL₃ Rankin–Selberg integrands
LanglandsTunnell.RankinSelberg.exists_forall_integrable_archWhittaker_torusPair_rpow_det7 below · cited by 1 · depth 18 - Rankin–Selberg integral as archimedean times finite integral times partial L-function
LanglandsTunnell.RankinSelberg.exists_forall_rsGlobalIntegral_eq_mul_rsArchIntegral_mul_rsFinIntegral_mul_lFun24 below · cited by 2 · depth 18 - Finite GL₃-translate family: constant integral and dual root number
LanglandsTunnell.RankinSelberg.exists_gl3Translates_sum_rsFinIntegral_cells_eq_const_and_dual_eq_rootNumberMonomial_of_finWhittaker_one_ne_zero_of_localSpaceAt_of_member_of_fe32_normPin_twisted_offSQ_archPsi_bump_levelShift_global982 below · cited by 1 · depth 18 - Two-sided unfolding of the GL₂timesGL₃ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_ne_zero_forall_rsGlobalIntegral_eq_mul_integral_unipotentQuotient_whittakerCoefficient_mul_of_hasSum_mirabolicTranslate_and_dual13 below · cited by 1 · depth 18 - Inverted coefficient triple at an unramified prime of a cubic field
LanglandsTunnell.RankinSelberg.inducedE_inv_eq_of_finrank_eq_three0 below · cited by 1 · depth 18 - Integrability of the unfolded GL₂timesGL₃ Rankin–Selberg integrand
LanglandsTunnell.RankinSelberg.integrable_unipotentQuotient_whittakerCoefficient_mul_of_hasSum_mirabolicTranslate_and_dual13 below · cited by 1 · depth 18 - Functional equation of the GL₂timesGL₃ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.rsGlobalIntegral_eq_rsGlobalIntegral_one_sub_transposeInvN_dualForm4 below · cited by 1 · depth 18 - Evaluating the induced Euler polynomial in degree at most three
LanglandsTunnell.RankinSelberg.eval_inducedEulerPoly_eq_of_finrank_le_three0 below · cited by 17 · depth 19 - Half-plane integrability of pure-tensor Rankin–Selberg cell integrands
LanglandsTunnell.RankinSelberg.exists_forall_integrable_rsFinCellIntegrand_pureTensorTerm_dual_and_hybrid_of_depth_twisted_torusFinite_central_growth_of_principalLevel_of_gammaHyp136 below · cited by 1 · depth 19 - Integrability of the twisted Rankin–Selberg finite-cell integrands
LanglandsTunnell.RankinSelberg.exists_forall_integrable_rsFinCellIntegrand_translate_and_dual_twisted116 below · cited by 1 · depth 19 - Half-plane integrability of an archimedean torus profile
LanglandsTunnell.RankinSelberg.exists_forall_lintegral_norm_torusProfile_mul_rpow_lt_top0 below · cited by 1 · depth 19 - One-place factorisation of the finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_forall_rsFinIntegral_eq_const_mul_rsLocalIntegral_of_factorsAt11 below · cited by 3 · depth 19 - Normalised K₁(p^ℓ)-invariant vector with mirabolic bump support
LanglandsTunnell.RankinSelberg.exists_mem_gl3CyclicSubspace_congruenceK1_invariant_iotaGL_eq_bump_of_localZeta31_fe_one107 below · cited by 2 · depth 19 - Archimedean GL₂× GL₃ torus-pair Gamma identity, minimal type
LanglandsTunnell.RankinSelberg.exists_mem_polyGauss3_jacquetVector3_torusPair_eq_gammaFactor_of_archWhittakerDatum_of_isCasimirEigen_of_minimalType281 below · cited by 1 · depth 19 - Convergence of the Rankin–Selberg GL₂timesGL₂ Euler product
LanglandsTunnell.RankinSelberg.exists_multipliable_differentiableOn_tprod_inv_eval_rsEulerPoly_of_norm_le_rpow1 below · cited by 4 · depth 19 - Unipotent-compact bump: finite Rankin–Selberg integral is c W(1)F(1)
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsFinIntegral_eq_mul_of_support_subset_unipotent_mul1 below · cited by 1 · depth 19 - Local Rankin–Selberg integral of a unipotent-supported bump integrand
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsLocalIntegral_eq_mul_of_support_subset_unipotent_mul1 below · cited by 2 · depth 19 - Rational local γ at a level prime, archimedean nonvanishing edition
LanglandsTunnell.RankinSelberg.exists_rational_gamma_rsLocalIntegral_member_twisted_of_finiteFamily_arch_deep_archPsi489 below · cited by 1 · depth 19 - Torus finiteness for the cyclic space of a deep twist
LanglandsTunnell.RankinSelberg.forall_mem_gl3CyclicSubspace_twist_det_torusFinite_of_principalLevel_of_admissible_of_deepTwist12 below · cited by 1 · depth 19 - Value form of the local GL₂timesGL₃ functional equation at p
LanglandsTunnell.RankinSelberg.forall_mem_span_rsLocalIntegral_dual_eq_stdRootNumber_mul_of_localZeta31_identified_of_torusFinite_of_centralChar_of_gauge_of_admissible_of_principalNormPin_adm_gamma_bump_levelShift_global514 below · cited by 1 · depth 19 - Determinant twists cancel in the local GL₃× GL₂ Rankin–Selberg data
LanglandsTunnell.RankinSelberg.gl3CyclicSubspace_detTwist_and_rsIntegrand_detTwist_eq0 below · cited by 1 · depth 19 - Cell expansion of the local Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.hasSum_cell_terms_rsLocalIntegral1 below · cited by 10 · depth 19 - Modulus of a real Whittaker function on torus times O(2)
LanglandsTunnell.RankinSelberg.norm_archWhittaker_upperUnit_mul_rowIsometry0 below · cited by 1 · depth 19 - Sign identity for the cubic root-number block
LanglandsTunnell.RankinSelberg.prod_sq_mul_finprod_localChar_neg_one_mul_neg_one_pow_eq_one_of_finprod_sq_mul_lamSqArch_eq_one_of_not_isBadPlace3 below · cited by 1 · depth 19 - Partial L-function factors out of the finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.rsFinIntegral_eq_LFun_rsDatum_mul_rsFinIntegral_indicator12 below · cited by 2 · depth 19 - Global realisation of local Rankin–Selberg pairs at p
LanglandsTunnell.RankinSelberg.exists_factor_fundamentalDomain_forall_rsGlobalIntegral_realisation_member_twisted_of_finiteFamily_arch_of_archNonvanishing467 below · cited by 1 · depth 20 - Cut-off remainder integrands of the dual finite cell are integrable
LanglandsTunnell.RankinSelberg.exists_forall_integrable_cutoff_remainder_mul_finprod_away113 below · cited by 1 · depth 20 - Half-plane integrability of primal and dual finite cell integrands
LanglandsTunnell.RankinSelberg.exists_forall_integrable_rsFinCellIntegrand_translate_and_dual104 below · cited by 2 · depth 20 - Local GL₃× GL₂ gamma factor from a global realisation
LanglandsTunnell.RankinSelberg.exists_forall_mem_span_rsLocalIntegral_dual_mul_eq_mul_of_rsGlobalIntegral_realisation6 below · cited by 1 · depth 20 - Unfolding the archimedean torus pairing of the GL₃ Jacquet vector
LanglandsTunnell.RankinSelberg.exists_forall_torusPair_jacquetVector3_eq_integral_quasiChar_mul_torusIntegral_mul_godementMellin6 below · cited by 2 · depth 20 - Pinned Rankin–Selberg niceness for a cubic base change
LanglandsTunnell.RankinSelberg.exists_isNicePinned_rsDatum_archOfParam_isArchCompAt_of_whittaker_link_of_isArithGenuineCuspRealizable_of_localWhittaker2,501 below · cited by 1 · depth 20 - Unfolded archimedean GL₂× GL₃ torus-pair identity at minimal type
LanglandsTunnell.RankinSelberg.exists_mem_polyGauss3_jacquetVector3_unfoldedTorusPair_eq_gammaFactor_of_archWhittakerDatum_of_isCasimirEigen_of_minimalType280 below · cited by 1 · depth 20 - A non-vanishing rational local Rankin–Selberg pair at a level prime
LanglandsTunnell.RankinSelberg.exists_mem_rsLocalIntegral_ne_zero_and_rational_member_twisted_of_finiteFamily_arch_deep58 below · cited by 1 · depth 20 - Normalised K₁(mathfrak pᵥ^ℓ)-newvector from a trivial-Euler functional equation
LanglandsTunnell.RankinSelberg.exists_normalisedNewvector_of_isLocalWhittakerDatum_of_localFE32_spherical_of_eulerPoly_eq_one21 below · cited by 1 · depth 20 - Finiteness, continuity and unit phase of dual Whittaker products
LanglandsTunnell.RankinSelberg.finite_mulSupport_and_continuous_and_exists_phase_finprod_dualWhittakerFn3_away1 below · cited by 3 · depth 20 - Spherical Rankin–Selberg periods determine torus values
LanglandsTunnell.RankinSelberg.forall_apply_diagZ_mul_scalarPi_pow_eq_ite_of_forall_rsLocalIntegral_spherical_eq_measure6 below · cited by 1 · depth 20 - Vanishing of K₁-invariant GL₃ Whittaker values off the dominant cone
LanglandsTunnell.RankinSelberg.forall_apply_iotaGL_diagZ_mul_scalarPi_zpow_eq_zero_of_isGL3PsiWhittakerFn_of_congruenceK16 below · cited by 1 · depth 20 - Pair stability of the GL₃timesGL₂ local functional equation
LanglandsTunnell.RankinSelberg.forall_mem_span_rsLocalIntegral_dual_eq_mul_of_forall_localZeta31_fe_of_deepTwist_of_principalLevel_of_admissible_of_gammaFactor_of_forall_localZeta31_fe_of_bump_levelShift_global489 below · cited by 1 · depth 20 - Convergence and rationality of local GL₃timesGL₂ Rankin–Selberg integrals
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_integrable_and_eq_laurent_of_torusFinite_of_centralChar_of_shellGrowth20 below · cited by 1 · depth 20 - Third induced Euler coefficient at an unramified prime of a cubic field
LanglandsTunnell.RankinSelberg.inducedE3_eq_neg_one_pow_mul_finprod_of_not_isRamifiedIn_of_finrank_eq_three0 below · cited by 4 · depth 20 - Swapping the S_Q-slots: dual and hybrid pure-tensor integrability
LanglandsTunnell.RankinSelberg.integrable_pureTensorTerm_dual_and_hybrid_of_integrable_cutoff_of_forall_lintegral_lt_top15 below · cited by 1 · depth 20 - Support and normalisation of a local ψ-bump on GL₂
LanglandsTunnell.RankinSelberg.localLevelOne_bump_of_forall_apply_diagZ_mul_scalarPi_zpow_eq_ite1 below · cited by 1 · depth 20 - Local Euler factor splits the finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.rsFinIntegral_eq_inv_eval_rsEulerPoly_mul_rsFinIntegral_indicator11 below · cited by 1 · depth 20 - Depth floor above p gives the bound 2e(w∣ p)b+1
LanglandsTunnell.RankinSelberg.two_mul_ramificationIdx_mul_add_one_le_conductorExponentAt_of_depth_floor1 below · cited by 2 · depth 20 - Entirety and strip-boundedness of the GL₂timesGL₃ global integral
LanglandsTunnell.RankinSelberg.differentiable_and_boundedOnStrips_rsGlobalIntegral_of_hasIotaMoments0 below · cited by 2 · depth 21 - Convergence of finite-adelic big-cell Rankin–Selberg integrals under a gauge bound
LanglandsTunnell.RankinSelberg.exists_forall_integrable_bigCell_indicator_mul_finprod_iotaGL_of_gauge18 below · cited by 1 · depth 21 - Convergence of the dual local GL₃timesGL₂ Rankin–Selberg integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_dual_rsLocalIntegrand_of_gauge9 below · cited by 3 · depth 21 - Integrability of the translated split dual finite cell integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_translate_rsFinCellIntegrand_dual_split_of_dualFactor_phase109 below · cited by 1 · depth 21 - Integrability of the unfolded archimedean torus-pair integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_unfoldedTorusPairIntegrand_jacquetVector34 below · cited by 1 · depth 21 - Purified p-slot splitting of Whittaker coefficients of p-adic translates
LanglandsTunnell.RankinSelberg.exists_frozen_forall_sum_translate_purified_whittakerCoefficient_eq_mul_pSlot_of_finiteFamily_arch351 below · cited by 3 · depth 21 - p-slot factorisation of GL₃ Whittaker functions along ι
LanglandsTunnell.RankinSelberg.exists_frozen_forall_sum_translate_whittaker_iota_eq_mul_pSlot_of_finiteFamily_arch42 below · cited by 1 · depth 21 - Haar splitting of finite-adelic GL₂ at one place
LanglandsTunnell.RankinSelberg.exists_isHaarMeasure_map_eq_prod_localAt1 below · cited by 3 · depth 21 - Local Rankin–Selberg integrals evaluating a finite Whittaker family
LanglandsTunnell.RankinSelberg.exists_mem_gl3CyclicSubspace_forall_rsLocalIntegral_eq_mul_apply_of_finite11 below · cited by 2 · depth 21 - Level 3B bump vector in a twisted principal-series Whittaker model
LanglandsTunnell.RankinSelberg.exists_mem_gl3CyclicSubspace_twist_coefficientFn_principalSeries3_congruenceK1_invariant_iotaGL_bump_of_pos_of_level157 below · cited by 1 · depth 21 - Unfolded archimedean torus pair and its dual Γ-factors
LanglandsTunnell.RankinSelberg.exists_mem_polyGauss3_iotaWeight_archZeta30_ne_zero_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_archWhittakerDatum_of_isCasimirEigen272 below · cited by 1 · depth 21 - Non-degenerate test pair for the local GL₃× GL₂ integral
LanglandsTunnell.RankinSelberg.exists_mem_span_forall_rsLocalIntegral_eq_const_ne_zero_of_isGL3PsiWhittakerFn13 below · cited by 1 · depth 21 - Unfolding the global GL₂timesGL₃ Rankin–Selberg integral, primal and dual
LanglandsTunnell.RankinSelberg.exists_ne_zero_forall_rsGlobalIntegral_eq_mul_integral_unipotentQuotient_whittakerCoefficient_mul_of_hasSum_mirabolicTranslate_and_dual_rpow13 below · cited by 2 · depth 21 - Non-negative coefficients of the local Rankin–Selberg factor P(y)⁻¹
LanglandsTunnell.RankinSelberg.exists_nonneg_hasSum_mul_pow_inv_eval_rsEulerPoly_conj_self2 below · cited by 1 · depth 21 - Laurent polynomiality of the dual local Rankin–Selberg integral at level vᵇ
LanglandsTunnell.RankinSelberg.exists_polynomial_forall_rsLocalIntegral_dualWhittakerFn3_iotaGL_eq_of_forall_torusShell_transposeInvN_eq_zero9 below · cited by 3 · depth 21 - Local Rankin–Selberg integral is a Laurent polynomial in q^{-s}
LanglandsTunnell.RankinSelberg.exists_polynomial_forall_rsLocalIntegral_iotaGL_eq_of_forall_torusShell_localLevelOne_pow_eq_zero9 below · cited by 3 · depth 21 - A principal-series GL₃ Whittaker model with prescribed central character
LanglandsTunnell.RankinSelberg.exists_principalSeries3_whittaker_deepTwist_centralChar_of_higherUnitsAt_unitary_shallow12 below · cited by 1 · depth 21 - Non-vanishing far right of a reference Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_pureTranslates_combination_forall_rsGlobalIntegral_ne_zero_member_twisted_of_finiteFamily_arch_of_archNonvanishing463 below · cited by 1 · depth 21 - Rationality of local Rankin–Selberg integrals for GL₃ principal series
LanglandsTunnell.RankinSelberg.exists_rational_rsLocalIntegral_and_dual_of_principalSeries363 below · cited by 1 · depth 21 - Unisolvence points, reference points and cut-off subgroups at S_Q
LanglandsTunnell.RankinSelberg.exists_unisolvence_refPoint_cutoff_of_linearIndependent_slots1 below · cited by 1 · depth 21 - Transfer of a Laurent functional equation to cleared rational forms
LanglandsTunnell.RankinSelberg.forall_cleared_fe_of_laurent_fe_of_rational_forms1 below · cited by 1 · depth 21 - Local level invariance at p depends only on vₚ(N)
LanglandsTunnell.RankinSelberg.forall_mem_localLevelOne_pow_mul_eq_of_forall_mem_localLevelOne_mul_eq0 below · cited by 7 · depth 21 - Multiplicativity of the GL₃timesGL₂ local γ-factor in principal series
LanglandsTunnell.RankinSelberg.forall_mem_span_rsLocalIntegral_dual_eq_mul_of_forall_localZeta31_fe_of_principalSeries273 below · cited by 1 · depth 21 - Deep twist: GL₃timesGL₂ local integrals are Laurent polynomials
LanglandsTunnell.RankinSelberg.forall_mem_span_rsLocalIntegral_eq_laurent_of_deepTwist_of_principalLevel_of_admissible20 below · cited by 2 · depth 21 - Pair stability at (3,2): transfer of the cleared functional equation
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_clearedFE_of_forall_rsLocalIntegral_clearedFE_of_centralChar_eq_of_deepTwist_pairStability32_of_bump59 below · cited by 1 · depth 21 - Multiplicativity of the local GL₃× GL₂ functional equation
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_clearedFE_prod_of_principalSeries3_of_forall_torusZeta_fe_multiplicativity3_ed3305 below · cited by 1 · depth 21 - Transfer of the local functional equation between rational forms
LanglandsTunnell.RankinSelberg.forall_rsLocal_fe32_of_rsLocal_fe32_of_eq_rational0 below · cited by 1 · depth 21 - Regrouping an Euler product over K along the primes of F
LanglandsTunnell.RankinSelberg.hasProd_inv_eval_inducedEulerPoly_of_hasProd0 below · cited by 3 · depth 21 - Integrability transfer at one place for Rankin–Selberg cell integrals
LanglandsTunnell.RankinSelberg.integrable_finCell_of_integrable_of_factorsAt11 below · cited by 2 · depth 21 - Integrability of the unfolded Rankin–Selberg integrand, primal and dual
LanglandsTunnell.RankinSelberg.integrable_unipotentQuotient_whittakerCoefficient_mul_of_hasSum_mirabolicTranslate_and_dual_rpow13 below · cited by 2 · depth 21 - Measurability and isolation identity for pure-tensor remainders
LanglandsTunnell.RankinSelberg.measurable_remainder_and_dualFactor_translate_mul_prod_eq_of_pureTensor_expansion2 below · cited by 1 · depth 21 - Two-row Cauchy identity inverting the GL₂timesGL₃ Euler polynomial
LanglandsTunnell.RankinSelberg.mk_twoRowCauchySum_mul_coe_rsEulerPoly_eq_one0 below · cited by 6 · depth 21 - Transpose–inverse invariance of the GL₂timesGL₃ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.rsGlobalIntegral_eq_rsGlobalIntegral_transposeInvN_dualForm_and_isFundamentalDomain_preimage0 below · cited by 1 · depth 21 - Uncountable non-vanishing of the cut finite Rankin–Selberg factor
LanglandsTunnell.RankinSelberg.exists_finTranslate_not_countable_rsFinIntegral_indicator_ne_zero_of_purifier_of_finiteFamily_arch93 below · cited by 1 · depth 22 - Integrability of a real Whittaker torus profile against |t|^{s-1/2}t⁻²
LanglandsTunnell.RankinSelberg.exists_forall_integrable_Wr_mul_abs_cpow_mul_inv_sq0 below · cited by 1 · depth 22 - Convergence of two intermediate GL₃timesGL₂ local integrals
LanglandsTunnell.RankinSelberg.exists_forall_integrable_flatSection_mul_whittaker_iotaGL_diagUnits2_longWeyl3_of_gauge1 below · cited by 1 · depth 22 - Convergence of the unfolded GL₃timesGL₂ local integral
LanglandsTunnell.RankinSelberg.exists_forall_integrable_whittaker_iotaGL_mul_principalSeries2_antidiagonal_of_gauge10 below · cited by 3 · depth 22 - Half-plane finiteness of a gauge-majorised local GL₃timesGL₂ integral
LanglandsTunnell.RankinSelberg.exists_forall_lintegral_enorm_comp_iotaGL_mul_modulus_cpow_lt_top_of_gauge6 below · cited by 1 · depth 22 - Local GL₃× GL₁ functional equation for deeply twisted principal series
LanglandsTunnell.RankinSelberg.exists_forall_localZeta31_fe_one_twist_coefficientFn_principalSeries3_of_exactConductor59 below · cited by 1 · depth 22 - Frozen complements: explicit p-slot splitting of GL₃ Whittaker functions
LanglandsTunnell.RankinSelberg.exists_frozen_forall_sum_translate_whittaker_iota_eq_mul_pSlot_of_finiteFamily_arch_explicit42 below · cited by 2 · depth 22 - Bump test vector for the local GL₃× GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_mem_gl3CyclicSubspace_forall_rsLocalIntegral_eq_mul_setIntegral_translate9 below · cited by 1 · depth 22 - Archimedean Rankin–Selberg pair outside weight-one GL₂ parameters
LanglandsTunnell.RankinSelberg.exists_mem_polyGauss3_iotaWeight_archZeta30_ne_zero_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_archWhittakerDatum_of_isCasimirEigen_of_not_weightOne206 below · cited by 1 · depth 22 - Factorisation of the purified reference Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_ne_zero_forall_rsGlobalIntegral_reference_eq_mul_rsArchIntegral_mul_rsFinIntegral_indicator_mul_of_finiteFamily_arch410 below · cited by 1 · depth 22 - Non-negative log coefficients of a degree-nine Rankin–Selberg factor
LanglandsTunnell.RankinSelberg.exists_nonneg_exp_tsum_mul_pow_eq_inv_one_sub_mul_std_mul_contragredient_mul_eval_rsEulerPoly_self_of_norm_eq_one0 below · cited by 1 · depth 22 - Local Rankin–Selberg integral as a Laurent polynomial in q^{-s}
LanglandsTunnell.RankinSelberg.exists_polynomial_forall_rsLocalIntegral_eq_of_forall_setIntegral_torusShell_eq_zero7 below · cited by 2 · depth 22 - A p-adic purifier with pure-tensor Whittaker coefficient
LanglandsTunnell.RankinSelberg.exists_purifier_whittakerCoefficient_eq_mul_pSlot_of_finiteFamily_arch25 below · cited by 2 · depth 22 - Rationality of principal-series Rankin–Selberg local integrals at p
LanglandsTunnell.RankinSelberg.exists_rational_rsLocalIntegral_and_dual_of_jacquetWhittaker3_ed257 below · cited by 2 · depth 22 - Non-degenerate local datum realising pair 2's cleared functional equation
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral_clearedFE_datum_of_centralChar_eq_of_deepTwist_pairStability32_of_bump56 below · cited by 1 · depth 22 - Specialising a flat family of local Rankin–Selberg functional equations
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral_dual_eq_mul_finsum_of_forall_re_rsLocalIntegral_dual_eq_mul_finsum_cpow_of_torusShell8 below · cited by 1 · depth 22 - Local GL₃timesGL₂ functional equation for a Jacquet-integral section
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral_jacquetIntegral_dual_eq_mul_of_forall_localZeta31_fe_of_integrable_setIntegral_localLevelOne_of_torusShell49 below · cited by 1 · depth 22 - Weight-one unfolded torus-pair identities with Γ-factors
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_weightOne_of_blockHarmonicOne_colHarmonic_gaussian348 below · cited by 1 · depth 22 - Weight-one torus-pair identities for the conjugate-block Gaussian section
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian373 below · cited by 1 · depth 22 - Weight-one minor-section torus-pair identities with archimedean Γ-factors
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_weightOne_of_minorSection_gaussian347 below · cited by 1 · depth 22 - Linear assembly of cleared local functional equations
LanglandsTunnell.RankinSelberg.forall_clearedFE_sum_smul_of_forall_clearedFE_of_forall_rational0 below · cited by 1 · depth 22 - Local Rankin–Selberg functional equation spreads to both spans
LanglandsTunnell.RankinSelberg.forall_mem_span_rsLocalIntegral_dual_eq_mul_of_forall_rightTranslate_rsLocalIntegral_dual_eq_mul6 below · cited by 1 · depth 22 - Cleared Rankin–Selberg functional equation for one Jacquet–Whittaker vector
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_clearedFE_prod_of_jacquetWhittaker3_of_forall_torusZeta_fe301 below · cited by 1 · depth 22 - Cleared (3,2) functional equation from a common middle
LanglandsTunnell.RankinSelberg.rsLocal_fe32_of_mul_eq_middle_of_eq_rational0 below · cited by 1 · depth 22 - Explicit dual archimedean torus pair: root number times π(-1)ᶜρ times Γ-factor
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_weightOne_of_blockHarmonicOne_colHarmonic_gaussian3_of_profile36 below · cited by 1 · depth 23 - Folded dual torus pair on the discrete branch, explicit constant
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian3_of_discrete_profile28 below · cited by 1 · depth 23 - Folded dual torus pair: root number, explicit constant, dual Γ-factors
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian3_of_weightOne_profile28 below · cited by 1 · depth 23 - Dual minor-section archimedean torus pair equals ε_∞ times Γ-factors
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_weightOne_of_minorSection_gaussian3_of_profile37 below · cited by 1 · depth 23 - Finite shell expansion of a local Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_finset_forall_rsLocalIntegral_eq_sum_mul_setIntegral_of_forall_setIntegral_torusShell_eq_zero7 below · cited by 1 · depth 23 - Local integrability of the Rankin–Selberg integrand at p
LanglandsTunnell.RankinSelberg.exists_forall_integrable_iotaGL_mul_of_mem_span_localSpaceAt_of_mem_gl3CyclicSubspace_twist_of_finiteFamily_arch40 below · cited by 1 · depth 23 - Non-vanishing of a Hecke-local Whittaker function at a point trivial outside S_Q
LanglandsTunnell.RankinSelberg.exists_forall_localAt_eq_one_and_ne_zero_of_heckeLocal_of_levelOne_invariant9 below · cited by 1 · depth 23 - Non-vanishing of a local GL₃× GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_mem_gl3CyclicSubspace_forall_rsLocalIntegral_ne_zero_of_ne_zero13 below · cited by 1 · depth 23 - Archimedean GL₃× GL₂ pair identity: discrete-series case
LanglandsTunnell.RankinSelberg.exists_mem_polyGauss3_iotaWeight_archZeta30_ne_zero_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_archWhittakerDatum_of_isCasimirEigen_of_discreteSeries141 below · cited by 1 · depth 23 - Euler factorisation of the cut finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_ne_zero_forall_rsFinIntegral_indicator_purified_eq_mul_sum_prod_rsLocalIntegral36 below · cited by 1 · depth 23 - Unfolding the GL₃timesGL₂ local integral at a principal-series section
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsLocalIntegral_iotaGL_jacquetIntegral_eq_mul_integral_localZeta316 below · cited by 4 · depth 23 - Test vectors with equal local integrals, one constant
LanglandsTunnell.RankinSelberg.exists_testVectors_rsLocalIntegral_eq_and_eq_const_of_centralChar_eq_of_deepTwist_of_bump55 below · cited by 1 · depth 23 - Even principal parameter: primal and dual unfolded torus-pair identities
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_evenPrincipal_of_detPow_blockQuadratic_colHarmonicTwo_gaussian351 below · cited by 1 · depth 23 - Even principal torus-pair identities for a weight-zero Gaussian section
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian378 below · cited by 1 · depth 23 - Explicit unfolded archimedean torus pair, weight one, block-harmonic section
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_weightOne_of_blockHarmonicOne_colHarmonic_gaussian3_of_profile31 below · cited by 1 · depth 23 - Discrete-branch unfolded torus pair equals explicit Gamma-factor product
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian3_of_discrete_profile24 below · cited by 1 · depth 23 - Weight-one unfolded torus pair as explicit Γ-factor product
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian3_of_weightOne_profile25 below · cited by 1 · depth 23 - Explicit primal torus pair for the minor-section Jacquet vector
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_weightOne_of_minorSection_gaussian3_of_profile31 below · cited by 1 · depth 23 - Rationality of local GL₃× GL₂ Rankin–Selberg integrals and duals
LanglandsTunnell.RankinSelberg.forall_exists_rational_rsLocalIntegral_and_dual_of_shellRecurrence_of_centralChar_of_rationalTorusShell_of_gauge21 below · cited by 1 · depth 23 - Local GL₃× GL₂ cleared functional equation from torus equations
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_clearedFE_prod_of_jacquetWhittaker3_of_forall_torusZeta_fe_core287 below · cited by 1 · depth 23 - Primal transport of the local GL₃timesGL₁ functional equation
LanglandsTunnell.RankinSelberg.integral_principalSeries2_mul_whittaker_iotaGL_diagUnits2_longWeyl3_eq_mul_of_forall_integral_localZeta31_eq_of_torusShell25 below · cited by 1 · depth 23 - Dual transport of the GL₃timesGL₁ functional equation
LanglandsTunnell.RankinSelberg.mul_integral_transposeInvN_mul_whittaker_iotaGL_diagUnits2_longWeyl3_eq_of_forall_integral_localZeta31_dualWhittakerFn3_eq_of_torusShell23 below · cited by 1 · depth 23 - Translating the first factor of a local Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.rsLocalIntegral_finset_sum_translate_eq_sum_mul_rsLocalIntegral_of_le_localLevelOne8 below · cited by 1 · depth 23 - Dual torus pair unfolded for the block-harmonic section
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_setIntegral_dualConfig_of_weightOne_of_blockHarmonicOne_colHarmonic_gaussian35 below · cited by 1 · depth 24 - Unfolded dual torus pair for the conjugate-harmonic weight-one section
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_setIntegral_dualConfig_of_weightOne_of_conjBlockHarmonicOne_colHarmonic_gaussian35 below · cited by 2 · depth 24 - Dual torus pair of the minor-section Jacquet vector, unfolded
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_setIntegral_dualConfig_of_weightOne_of_minorSection_gaussian35 below · cited by 1 · depth 24 - Cleared local Rankin–Selberg functional equation at the family centre
LanglandsTunnell.RankinSelberg.exists_cleared_rsLocalIntegral_fe_of_forall_lt_cleared_fe_finsum_cpow_of_isGL3PsiWhittakerFn14 below · cited by 1 · depth 24 - Dual torus pair with explicit constant 2π(-1)ᵇρ
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_blockQuadratic_colHarmonicTwo_gaussian3_of_profile37 below · cited by 1 · depth 24 - Dual torus pair identity, discrete Levi branch
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_discrete_profile25 below · cited by 1 · depth 24 - Dual archimedean torus pair, weight-one Levi branch
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_weightOne_profile25 below · cited by 1 · depth 24 - Dual torus pair, even principal type, weight-zero Levi branch
LanglandsTunnell.RankinSelberg.exists_dualTorusPair_eq_archRootNumber_mul_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_weightZero_profile36 below · cited by 1 · depth 24 - Polar decomposition and functional equation of Godement–Eisenstein series
LanglandsTunnell.RankinSelberg.exists_entire_sub_polarPart_godementEisenstein_isUniformlySiegelBounded_fe_of_mem_schwartzBruhat285 below · cited by 1 · depth 24 - Polar decomposition and functional equation of (1,1) Godement–Eisenstein series
LanglandsTunnell.RankinSelberg.exists_entire_sub_polarPart_godementEisenstein_one_one_isUniformlySiegelBounded_fe_of_mem_schwartzBruhat285 below · cited by 1 · depth 24 - Cleared local GL₃× GL₂ integrals along a flat twist family
LanglandsTunnell.RankinSelberg.exists_forall_lt_rsLocalIntegral_jacquetWhittaker3_twistFamily_mul_centralTate_eq_cpow_mul_eval144 below · cited by 1 · depth 24 - Archimedean Rankin–Selberg integral of a discrete-series torus profile
LanglandsTunnell.RankinSelberg.exists_forall_rsArchIntegral_gaussian_eq_mul_Gamma_mul_Gamma_of_discreteSeries_torusPair3 below · cited by 1 · depth 24 - Archimedean Rankin–Selberg integral against the Gaussian for torus profiles
LanglandsTunnell.RankinSelberg.exists_forall_rsArchIntegral_gaussian_eq_mul_Gamma_mul_mellin_of_torusProfile4 below · cited by 1 · depth 24 - Rankin–Selberg side conditions for GL₂timesGL₂ over ℚ
LanglandsTunnell.RankinSelberg.exists_forall_summable_integrable_rs22_sideConditions_of_measurable_rat129 below · cited by 1 · depth 24 - Unfolded torus pair in Iwasawa coordinates, block-harmonic Gaussian section
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_setIntegral_iwasawa_tateM_of_blockHarmonic_colHarmonic_gaussian34 below · cited by 1 · depth 24 - Iwasawa–Tate evaluation of an unfolded archimedean torus integral
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_setIntegral_iwasawa_tateM_of_conjBlockHarmonic_colHarmonic_gaussian34 below · cited by 3 · depth 24 - Iwasawa and Tate–Mellin form of the minor-section torus pair
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_setIntegral_iwasawa_tateM_of_minorSection_gaussian34 below · cited by 1 · depth 24 - Peeling unramified Euler factors off the finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_hasProd_rsFinIntegral_eq_rsFinIntegral_indicator_mul_of_torus_law14 below · cited by 1 · depth 24 - Jacquet–Shalika test vectors with non-vanishing unit-shell pairing
LanglandsTunnell.RankinSelberg.exists_mem_span_schwartzBruhat_fourier_unitShell_pairing_ne_zero_of_deepTwist_of_conductor_le40 below · cited by 1 · depth 24 - Euler factorisation of the cut finite Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_ne_zero_forall_integrable_and_rsFinIntegral_indicator_eq_mul_finprod_rsLocalIntegral_of_pure_of_measurable19 below · cited by 1 · depth 24 - Dual local GL₃timesGL₁ functional equation in Laurent form
LanglandsTunnell.RankinSelberg.exists_polynomial_localZeta31_dualWhittakerFn3_iotaGL_eq_and_mul_localZeta30_eq_of_forall_mem_gl3CyclicSubspace_fe_of_torusShell6 below · cited by 1 · depth 24 - Local GL₃timesGL₁ functional equation at ι(h), polynomial form
LanglandsTunnell.RankinSelberg.exists_polynomial_localZeta31_iotaGL_eq_and_localZeta30_dualWhittakerFn3_eq_mul_of_forall_mem_gl3CyclicSubspace_fe_of_torusShell6 below · cited by 1 · depth 24 - Dual Rankin–Selberg integral of a smoothed GL₃ bump vector
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsLocalIntegral_dual_longWeyl3_smoothedBump_eq_mul_setIntegral_unitShell12 below · cited by 1 · depth 24 - Unfolding the GL₂timesGL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_rs22GlobalIntegral_godementEisenstein_eq_mul_rs22WhittakerIntegral_of_isUnitaryChar_of_re_pos_of_forall_summable_of_integrable101 below · cited by 1 · depth 24 - Archimedean GL₃× GL₂ torus-pair identity: discrete series, flat section
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian391 below · cited by 1 · depth 24 - Unfolded torus pair equals 2π(-1)ᵇρ times Gamma factors
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_blockQuadratic_colHarmonicTwo_gaussian3_of_profile32 below · cited by 1 · depth 24 - Unfolded torus pair equals (-1)ᵇ(π/2)ρ times Γ-product
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_discrete_profile19 below · cited by 1 · depth 24 - Explicit unfolded torus pair: (-1)ᵇ(π/2)ρ times the twisted Γ-product
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_weightOne_profile18 below · cited by 1 · depth 24 - Unfolded archimedean torus pair in the weight-zero Levi branch
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_eq_explicit_mul_gammaFactor_of_evenPrincipal_of_detPow_colHarmonic_gaussian3_of_weightZero_profile29 below · cited by 1 · depth 24 - Rationality of the dual local GL₃× GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.forall_exists_rational_rsLocalIntegral_dual_translate_of_shellRecurrence_of_centralChar_of_rationalTorusShell_of_gauge16 below · cited by 1 · depth 24 - Rationality of one local GL₃× GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.forall_exists_rational_rsLocalIntegral_translate_of_shellRecurrence_of_centralChar_of_rationalTorusShell_of_gauge16 below · cited by 1 · depth 24 - Cleared GL₃× GL₂ functional equation in the positive chamber
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral_clearedFE_prod_of_jacquetWhittaker3_of_forall_torusZeta_fe_core_of_chamber270 below · cited by 1 · depth 24 - Entirety and vertical boundedness of the GL₂× GL₂ global integral
LanglandsTunnell.RankinSelberg.integrableOn_and_differentiable_and_boundedOnStrips_rs22GlobalIntegral_of_isUniformlySiegelBounded0 below · cited by 1 · depth 24 - Integrability of the unfolded Rankin–Selberg integrand in Bruhat coordinates
LanglandsTunnell.RankinSelberg.integrable_principalSeries2_mul_whittaker_iotaGL_diagUnitGL2_mul_lowerUnipotent21_of_integrable_whittaker_iotaGL_mul_principalSeries24 below · cited by 4 · depth 24 - Equal smoothed Whittaker integrals along ι(GL₂)w₃ at level K₁(p^f)
LanglandsTunnell.RankinSelberg.integral_integral_iotaGL_mul_longWeyl3_mul_upperUnipotent3_eq_of_congruenceK1_of_centralChar_of_iotaGL_bump1 below · cited by 1 · depth 24 - Unipotent smoothing of a K₁(p^f)-invariant function on GL₃
LanglandsTunnell.RankinSelberg.integral_integral_upperUnipotent3_translate_mem_gl3CyclicSubspace_of_congruenceK1_invariant0 below · cited by 1 · depth 24 - Rankin–Selberg unfolded integral over ℚ factorises into carriers
LanglandsTunnell.RankinSelberg.rs22WhittakerIntegral_rat_eq_rsArchIntegral_mul_rsFinIntegral_of_eq_mul8 below · cited by 1 · depth 24 - Rescaled conjugate Rankin–Selberg Euler factor at qX
LanglandsTunnell.RankinSelberg.rsEulerPoly_rescale_conj_eval_mul_eq_rsEulerPoly_contragredient_eval0 below · cited by 2 · depth 24 - One-sided Whittaker profile for a discrete-series archimedean parameter
LanglandsTunnell.RankinSelberg.archWhittaker_profile_eq_zero_and_eq_two_mul_cpow_mul_exp_of_discrete2 below · cited by 1 · depth 25 - Archimedean Whittaker value at a reflected dual torus point
LanglandsTunnell.RankinSelberg.archWhittaker_w0R_mul_transposeInv_upperUnit_eq_mul_archProfile0 below · cited by 6 · depth 25 - Cleared functional equation for a sum of products
LanglandsTunnell.RankinSelberg.clearedFE_of_sum_mul_of_termwise_clearedFE0 below · cited by 1 · depth 25 - Dual torus pair unfolded for a quadratic Schwartz section
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_setIntegral_dualConfig_of_evenPrincipal_of_detPow_blockQuadratic_colHarmonicTwo_gaussian34 below · cited by 1 · depth 25 - Dual unfolding of the even-type archimedean torus pair
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_setIntegral_dualConfig_of_evenPrincipal_of_detPow_colHarmonic_gaussian35 below · cited by 3 · depth 25 - Cleared local Rankin–Selberg integral is a two-variable Laurent polynomial
LanglandsTunnell.RankinSelberg.exists_finset_forall_rsLocalIntegral_finsum_mul_eq_sum_cpow_of_forall_lt_cleared_laurent_of_torusShell7 below · cited by 1 · depth 25 - Integrability of the folded Rankin–Selberg integrand over ℚ
LanglandsTunnell.RankinSelberg.exists_forall_integrableOn_norm_mul_godementSection_majorant_rat78 below · cited by 1 · depth 25 - Integrability of the archimedean Rankin–Selberg integrand over ℚ
LanglandsTunnell.RankinSelberg.exists_forall_integrable_archWhittaker_gaussian_rpow_det_rat4 below · cited by 3 · depth 25 - Integrability of the finite Rankin–Selberg integrand over ℚ
LanglandsTunnell.RankinSelberg.exists_forall_integrable_finWhittaker_rpow_ideleNorm_det_rat29 below · cited by 3 · depth 25 - Integrability of the Rankin–Selberg integrand on NbackslashGL₂(A_ℚ)
LanglandsTunnell.RankinSelberg.exists_forall_integrable_norm_whittakerCoefficient_mul_rs22Kernel_unipotentQuotient_rat40 below · cited by 1 · depth 25 - Integrability of the split Rankin–Selberg integrand over ℚ
LanglandsTunnell.RankinSelberg.exists_forall_integrable_prod_archWhittaker_finWhittaker_rpow_rat34 below · cited by 1 · depth 25 - Absolute convergence of the Bruhat series of a Godement section
LanglandsTunnell.RankinSelberg.exists_forall_summable_norm_godementSection_bruhat_one_one_rat85 below · cited by 1 · depth 25 - Unfolded torus pair in Iwasawa coordinates with Tate–Mellin evaluation
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_setIntegral_iwasawa_tateM_of_colHarmonic_gaussian34 below · cited by 3 · depth 25 - Iwasawa form of the unfolded torus pair, quadratic-block Gaussian datum
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_setIntegral_iwasawa_tateM_of_detPow_blockQuadratic_colHarmonic_gaussian34 below · cited by 1 · depth 25 - Kirillov bump in a twisted Whittaker translate span
LanglandsTunnell.RankinSelberg.exists_mem_span_twist_det_kirillov_eq_indicator_shell_of_localLevelOne8 below · cited by 1 · depth 25 - Laurent-polynomial form of the local (3,1) zeta integral at ι(h)
LanglandsTunnell.RankinSelberg.exists_polynomial_forall_localZeta31_iotaGL_eq_of_forall_setIntegral_iotaGL_diagUnitGL2_mul_eq_zero2 below · cited by 2 · depth 25 - Local Rankin–Selberg integral supported on a unipotent orbit of U
LanglandsTunnell.RankinSelberg.exists_pos_forall_integrable_and_rsLocalIntegral_eq_mul_setIntegral_of_support_subset_unipotent_mul3 below · cited by 1 · depth 25 - Finite Rankin–Selberg integral constant in s and positive
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsFinIntegral_eq_const_of_ideleNorm_det_eq_one_of_nonneg0 below · cited by 1 · depth 25 - Local big-cell Rankin–Selberg integral is a positive constant
LanglandsTunnell.RankinSelberg.exists_pos_forall_rsLocalIntegral_indicator_unipotent_mul_localLevelOne_eq5 below · cited by 1 · depth 25 - Unfolding the Godement section over Z(F)N(A)backslashGL₂(A)
LanglandsTunnell.RankinSelberg.exists_pos_integrable_and_integral_rationalCentreUnipotentQuotient_godementSection_eq_mul_rs22WhittakerIntegral70 below · cited by 1 · depth 25 - Positivity of the finite-adelic big cell volume
LanglandsTunnell.RankinSelberg.exists_pos_integral_indicator_forall_localAt_unipotent_mul_localLevelOne_withDensity_eq5 below · cited by 1 · depth 25 - Rationality in q^{-s} of a local GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_rational_rsLocalIntegral_of_shellGauge_of_rationalTorusShell_of_shellRecurrence_of_central15 below · cited by 3 · depth 25 - Rationality of the dual GL₂× GL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral22_dual_mul_one_sub_eq_cpow_mul_eval_of_principalSeries2_of_forall_torusZeta_polynomial40 below · cited by 3 · depth 25 - Rationality of the local GL₂timesGL₂ Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral22_mul_one_sub_eq_cpow_mul_eval_of_principalSeries2_of_forall_torusZeta_polynomial39 below · cited by 3 · depth 25 - Unfolded GL₃× GL₂ Rankin–Selberg integrals, primal and dual
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral_jacquetWhittaker3_iotaGL_eq_sum_and_dual_eq_mul_sum_of_chamber_ed2111 below · cited by 2 · depth 25 - Cleared local GL₃× GL₂ Rankin–Selberg integrals in a chamber
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral_jacquetWhittaker3_mul_centralTate_eq_cpow_mul_eval_and_dual_of_chamber141 below · cited by 1 · depth 25 - One-place disintegration of a finite-adelic unipotent quotient integral
LanglandsTunnell.RankinSelberg.exists_sFinite_forall_lintegral_withDensity_density_eq_lintegral_lintegral_mul_finFactor_placeEmbed11 below · cited by 1 · depth 25 - Primal and dual torus integrals, discrete Levi branch with one complex place
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_oneComplex_discreteLevi71 below · cited by 1 · depth 25 - Torus-pair unfolding equals twisted Γ-factors: one complex place, k_ℂ=0
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_oneComplex_weightOneLevi74 below · cited by 1 · depth 25 - Primal and dual torus pairs: three real places, opposite signs
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_threeReal_oppSign74 below · cited by 1 · depth 25 - Torus pairs for three real places with equal Levi signs
LanglandsTunnell.RankinSelberg.exists_unfoldedTorusPair_and_dualTorusPair_eq_gammaFactor_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_threeReal_sameSign58 below · cited by 1 · depth 25 - Whittaker functions agreeing on ι(GL₂) agree on ι(GL₂)N₃Z₃K₁
LanglandsTunnell.RankinSelberg.forall_apply_iotaGL_mul_upperUnipotent3_mul_scalar_mul_eq_of_forall_apply_iotaGL_eq0 below · cited by 1 · depth 25 - Godement–Jacquet zeta integrals for GL₂: cleared functional equation
LanglandsTunnell.RankinSelberg.forall_godementZeta2_clearedFE_of_forall_torusZeta_fe167 below · cited by 1 · depth 25 - Measurability of the unfolded Rankin–Selberg integrand over ℚ
LanglandsTunnell.RankinSelberg.forall_measurable_whittakerCoefficient_mul_rs22Kernel_rat2 below · cited by 1 · depth 25 - Cleared GL₂× GL₂ local functional equation: principal series case
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral22_schwartz_clearedFE_of_principalSeries2_of_forall_torusZeta_fe_ed2219 below · cited by 1 · depth 25 - Unconditional inverse-Iwasawa change of variables on M₂(ℝ)
LanglandsTunnell.RankinSelberg.integral_matrixTwo_eq_setIntegral_iwasawaInv_unconditional0 below · cited by 19 · depth 25 - Schwartz–Bruhat cut-off kernels with prescribed local Fourier transforms
LanglandsTunnell.RankinSelberg.isSchwartzBruhat_and_tateFourier_shellKernels_of_conductor_le15 below · cited by 1 · depth 25 - Rankin–Selberg identity for two Hecke recursion sequences
LanglandsTunnell.RankinSelberg.mk_heckeRecursionSeq_mul_heckeRecursionSeq_mul_coe_rsEulerPoly_eq_and_hasSum0 below · cited by 1 · depth 25 - Constancy of the blind product on the cut set
LanglandsTunnell.RankinSelberg.mul_eq_mul_one_and_ideleNorm_det_eq_finprod_of_mem_cut_of_blind6 below · cited by 1 · depth 25 - Twist invariance of the contragredient Rankin–Selberg Euler polynomial
LanglandsTunnell.RankinSelberg.rsEulerPoly_contragredient_twist_eq0 below · cited by 1 · depth 25 - Local dual Rankin–Selberg integrand of a smoothed bump vector
LanglandsTunnell.RankinSelberg.rsIntegrand_dual_longWeyl3_smoothedBump_invariant_support_bound_and_bigCell_eq3 below · cited by 1 · depth 25 - Non-vanishing of a unit-shell Whittaker–Fourier pairing
LanglandsTunnell.RankinSelberg.setIntegral_unitShell_pairing_ne_zero_of_kirillov_shell_of_deepTwist_of_conductor_le34 below · cited by 1 · depth 25 - Unipotent invariance of a product of two Whittaker coefficients
LanglandsTunnell.RankinSelberg.whittakerCoefficient_mul_whittakerCoefficient_inv_unipotent_mul_rat0 below · cited by 2 · depth 25 - Twisted contragredient of a Whittaker vector is again Whittaker
LanglandsTunnell.RankinSelberg.dualPartner_block_of_admissible2 below · cited by 2 · depth 26 - Dual of the mixed-model GL₃ Whittaker function
LanglandsTunnell.RankinSelberg.dualWhittakerFn3_godementWhittaker3_eq_godementWhittaker3_matFourier23_dual28 below · cited by 1 · depth 26 - Uniform radial profile of a Schwartz–Bruhat function on bottom rows
LanglandsTunnell.RankinSelberg.exists_forall_apply_row_localLevelOne_eq_zero_and_eq_apply_zero_of_isLocallyConstant_of_hasCompactSupport0 below · cited by 2 · depth 26 - Closed form of the dual torus integral, discrete Levi branch
LanglandsTunnell.RankinSelberg.exists_forall_dualTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_discreteLevi29 below · cited by 1 · depth 26 - Closed form of the dual torus pair, weight-one Levi branch
LanglandsTunnell.RankinSelberg.exists_forall_dualTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_weightOneLevi32 below · cited by 2 · depth 26 - Closed form of the dual torus pair: weight-zero Levi branch
LanglandsTunnell.RankinSelberg.exists_forall_dualTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_weightZeroLevi32 below · cited by 1 · depth 26 - Convergence of the dual GL₂× GL₂ Rankin–Selberg integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_dual_rsIntegrand22_withDensity_of_admissible_of_chamber33 below · cited by 3 · depth 26 - Absolute convergence of the unfolded local Godement integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_godementUnfold_of_principalSeries2_of_admissible_ed236 below · cited by 1 · depth 26 - Integrability of the folded local Rankin–Selberg integrand in the chamber
LanglandsTunnell.RankinSelberg.exists_forall_integrable_jacquetIntegral_mul_whittaker_mul_row_mul_cpow_withDensity_of_principalSeries2_of_chamber28 below · cited by 3 · depth 26 - Vanishing of deep dual torus shells over K₀
LanglandsTunnell.RankinSelberg.exists_forall_le_setIntegral_localLevelOne_dualJacquet_mul_partner_mul_eq_zero_of_dualTorusZeta_polynomial12 below · cited by 1 · depth 26 - Ω-averaged evaluations represent invariant functionals on admissible translate spaces
LanglandsTunnell.RankinSelberg.exists_forall_mem_span_apply_eq_sum_mul_setIntegral_translate_of_invariant_of_admissible1 below · cited by 5 · depth 26 - Bruhat-series majorant for Godement sections on rational Siegel sets
LanglandsTunnell.RankinSelberg.exists_forall_norm_godementSection_add_tsum_le_mul_archHeight_pow_of_mem_integralWindowedSiegelSet_rat76 below · cited by 1 · depth 26 - Torus-shell expansion of a local Rankin–Selberg product integral
LanglandsTunnell.RankinSelberg.exists_forall_setIntegral_torusShell_eq_sum_mul_torusShellArray_of_shellRecurrence_of_central2 below · cited by 1 · depth 26 - Separation of variables for Ω-averages of local translates
LanglandsTunnell.RankinSelberg.exists_forall_setIntegral_translate_eq_mul_sum_linearMap_of_admissible4 below · cited by 3 · depth 26 - Closed form of the unfolded torus pair: discrete Levi branch
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_discreteLevi_ed224 below · cited by 1 · depth 26 - Unfolded torus pair in closed form: discrete series against principal Levi
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_weightOneLevi_ed227 below · cited by 2 · depth 26 - Closed form of the unfolded torus pair: weight-zero Levi branch
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_closedForm_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian3_of_weightZeroLevi_ed227 below · cited by 1 · depth 26 - Twisted torus Mellin transform as Laurent polynomial in q^{-s}
LanglandsTunnell.RankinSelberg.exists_polynomial_forall_integral_diagUnitGL2_mul_eq_of_forall_setIntegral_diagUnitGL2_mul_eq_zero1 below · cited by 1 · depth 26 - Central average of the quotient density over a norm slab
LanglandsTunnell.RankinSelberg.exists_pos_forall_lintegral_adelicUnipotent_lintegral_indicator_slab_mul_density_centralScalar_inv_mul_eq66 below · cited by 1 · depth 26 - Rationality of the local (2,2) Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.exists_rsLocalIntegral22_mul_one_sub_eq_cpow_mul_eval_of_principalSeries2_of_forall_torusZeta_polynomial_core38 below · cited by 1 · depth 26 - Open compact subgroup adapted to φ₁ and χ
LanglandsTunnell.RankinSelberg.exists_subgroup_isOpen_isCompact_forall_apply_mul_eq_and_det_eq_one_and_transposeInv_mem0 below · cited by 2 · depth 26 - Finite sums of N^{ms}P(N^{-s}) are again of that form
LanglandsTunnell.RankinSelberg.exists_sum_cpow_mul_eval_eq_cpow_mul_eval0 below · cited by 1 · depth 26 - Summable exponential bounds for per-place shell sums
LanglandsTunnell.RankinSelberg.exists_summable_forall_tsum_shell_le_exp_of_norm_le_rpow3 below · cited by 1 · depth 26 - Archimedean torus profile and reciprocal for Rankin–Selberg over ℚ
LanglandsTunnell.RankinSelberg.exists_torusProfile_archRecip_of_realArchParam_mellin_of_diagOne_eq_rat11 below · cited by 1 · depth 26 - Half-plane integrability of local Godement–Jacquet integrals on GL₂
LanglandsTunnell.RankinSelberg.forall_exists_integrable_godementZeta2_coefficient36 below · cited by 1 · depth 26 - Godement–Jacquet zeta integrals of GL₂ matrix coefficients
LanglandsTunnell.RankinSelberg.forall_exists_laurent_godementZeta2_coefficient_of_forall_torusZeta_fe47 below · cited by 1 · depth 26 - Rationality of Whittaker Godement–Jacquet zeta integrals on GL₂
LanglandsTunnell.RankinSelberg.forall_exists_rational_godementZeta2_whittaker_of_forall_torusZeta_fe48 below · cited by 3 · depth 26 - Cleared local Godement–Jacquet functional equation for Whittaker coefficients
LanglandsTunnell.RankinSelberg.forall_godementZeta2_whittaker_clearedFE_of_forall_torusZeta_fe166 below · cited by 2 · depth 26 - Local GL₂× GL₂ functional equation for Laurent numerators
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral22_schwartz_centralCleared_laurentFE_of_principalSeries2_of_forall_torusZeta_fe206 below · cited by 1 · depth 26 - Integrability of the local Rankin–Selberg integrand from its unfolding
LanglandsTunnell.RankinSelberg.integrable_rsIntegrand_godementSlot_of_integrable_unfold9 below · cited by 1 · depth 26 - Tonelli peel of the finite Rankin–Selberg lower integral
LanglandsTunnell.RankinSelberg.lintegral_enorm_mul_rpow_ideleNorm_det_eq_tprod_tsum_mul_lintegral_indicator_of_torus_law12 below · cited by 1 · depth 26 - Finiteness of a big-cell integral against ‖det‖^τ
LanglandsTunnell.RankinSelberg.lintegral_indicator_bigCell_enorm_mul_rpow_ideleNorm_det_lt_top_of_support13 below · cited by 2 · depth 26 - Haar measure of the unipotent part of the level-one group
LanglandsTunnell.RankinSelberg.measure_setOf_mem_localLevelOne_top_pos_and_lt_top1 below · cited by 1 · depth 26 - Central unfolding of the Rankin–Selberg Godement section
LanglandsTunnell.RankinSelberg.mul_mul_rs22Kernel_centralScalar_mul_eq_and_mul_godementSection_eq_integral0 below · cited by 1 · depth 26 - Unfolding of a Godement-section Rankin–Selberg local integral
LanglandsTunnell.RankinSelberg.rsLocalIntegral_godementWhittaker_iotaGL_eq_sum_rsLocalIntegral_mul_godementZeta9 below · cited by 1 · depth 26 - Twisted contragredient transport of a separated family
LanglandsTunnell.RankinSelberg.setIntegral_translate_transposeTwist_eq_mul_sum_of_forall_setIntegral_translate_eq2 below · cited by 1 · depth 26 - Measurability of the unfolded Godement double integrand
LanglandsTunnell.RankinSelberg.aestronglyMeasurable_godementUnfold_integrand3 below · cited by 1 · depth 27 - Entire reciprocal of the principal-series archimedean Rankin–Selberg factor
LanglandsTunnell.RankinSelberg.exists_entire_apply_zero_eq_zero_mul_Gamma_mul_mellin_besselProfile_eq_one4 below · cited by 1 · depth 27 - Entire reciprocal of the discrete-series archimedean factor
LanglandsTunnell.RankinSelberg.exists_entire_apply_zero_eq_zero_mul_Gamma_mul_mellin_discreteSeriesProfile_eq_one1 below · cited by 1 · depth 27 - Dual torus pair for a discrete-series profile: Gamma factors times Laplace–Mellin
LanglandsTunnell.RankinSelberg.exists_forall_dualTorusPair_eq_const_mul_setIntegral_W_diagOne_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian328 below · cited by 3 · depth 27 - Half-plane integrability of the local GL₂timesGL₂ integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_jacquetIntegral_mul_whittaker_mul_row_withDensity_of_admissible_of_chamber25 below · cited by 2 · depth 27 - Two-exponent asymptotics of chamber Jacquet integrals on small torus
LanglandsTunnell.RankinSelberg.exists_forall_jacquetIntegral_diagOne_mul_eq_sqrt_modulus_mul_add_of_mem_principalSeries2_of_chamber6 below · cited by 2 · depth 27 - Inner bound for the local Rankin–Selberg N₂backslash GL₂ integral
LanglandsTunnell.RankinSelberg.exists_forall_lintegral_enorm_jacquetIntegral_mul_whittaker_mul_translate_mul_row_le_of_admissible_of_chamber23 below · cited by 2 · depth 27 - Gauge bound and far-out vanishing for a GL₂ Jacquet integral
LanglandsTunnell.RankinSelberg.exists_forall_norm_jacquetIntegral_principalSeries2_diagUnits2_mul_le_and_eq_zero_of_chamber8 below · cited by 2 · depth 27 - Unfolded torus pair for a discrete-series GL₂ profile
LanglandsTunnell.RankinSelberg.exists_forall_unfoldedTorusPair_eq_const_mul_setIntegral_W_diagOne_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian323 below · cited by 3 · depth 27 - Torus-shell series of Jacquet and Whittaker integrals sums to q^{ms}P(q^{-s})
LanglandsTunnell.RankinSelberg.exists_hasSum_torusShells_jacquetIntegral_mul_whittaker_mul_row_eq_cpow_mul_eval_of_forall_torusZeta_polynomial_ed217 below · cited by 1 · depth 27 - Big-cell support of W· F inside N· K'
LanglandsTunnell.RankinSelberg.exists_isCompact_bigCell_inter_support_subset_finUnipotent_mul1 below · cited by 2 · depth 27 - Iwasawa integration formula for Haar measure on GL₂(ℚₚ)
LanglandsTunnell.RankinSelberg.exists_pos_forall_lintegral_eq_mul_lintegral_prod_lintegral_unipotent_diagUnits220 below · cited by 4 · depth 27 - Iwasawa integration formula for the unipotent density on GL₂(ℚₚ)
LanglandsTunnell.RankinSelberg.exists_pos_forall_lintegral_withDensity_density_eq_mul_lintegral_prod_diagUnits210 below · cited by 4 · depth 27 - Local integrability of a shifted Godement–Jacquet zeta integrand
LanglandsTunnell.RankinSelberg.forall_exists_integrable_godementZeta2_whittaker_shift29 below · cited by 8 · depth 27 - Laurent Godement–Jacquet integrals of GL₂ Whittaker vectors
LanglandsTunnell.RankinSelberg.forall_exists_laurent_godementZeta2_whittaker_of_forall_torusZeta_fe42 below · cited by 4 · depth 27 - Rationality in q^{-s} of local Godement zeta integrals
LanglandsTunnell.RankinSelberg.forall_exists_rational_godementZeta2_whittaker_shift44 below · cited by 1 · depth 27 - Cleared Godement–Jacquet functional equation for a Whittaker coefficient
LanglandsTunnell.RankinSelberg.forall_godementZeta2_whittaker_clearedFE_of_forall_torusZeta_fe_of_borelEigenfunctional92 below · cited by 2 · depth 27 - Godement–Jacquet functional equation for cuspidal Whittaker coefficients
LanglandsTunnell.RankinSelberg.forall_godementZeta2_whittaker_clearedFE_of_forall_torusZeta_fe_of_cuspidal121 below · cited by 1 · depth 27 - Centre-cleared local GL₂× GL₂ functional equation, principal-series branch
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral22_schwartz_centralCleared_laurentFE_of_principalSeries2_of_forall_torusZeta_fe_of_borelEigenfunctional187 below · cited by 1 · depth 27 - Centre-cleared local functional equation for GL₂× GL₂: cuspidal branch
LanglandsTunnell.RankinSelberg.forall_rsLocalIntegral22_schwartz_centralCleared_laurentFE_of_principalSeries2_of_forall_torusZeta_fe_of_cuspidal196 below · cited by 1 · depth 27 - Transpose-inverse symmetry of the local GL₂ Godement zeta integral
LanglandsTunnell.RankinSelberg.godementZeta2_comp_transposeInvN_eq_godementZeta2_conj_of_central0 below · cited by 2 · depth 27 - Torus-shell expansion of a local Rankin–Selberg integral
LanglandsTunnell.RankinSelberg.hasSum_torusShells_rsLocalIntegral22_jacquetIntegral_schwartz_of_integrable9 below · cited by 1 · depth 27 - Kirillov vanishing or Borel eigenfunctional dichotomy for GL₂(ℚₚ)
LanglandsTunnell.RankinSelberg.kirillov_vanish_near_zero_or_exists_borelEigenfunctional_of_irreducible_admissible4 below · cited by 4 · depth 27 - One-place torus peel for a Rankin–Selberg lower integral
LanglandsTunnell.RankinSelberg.lintegral_enorm_mul_rpow_ideleNorm_det_eq_tsum_mul_lintegral_indicator_of_torus_law_at11 below · cited by 1 · depth 27 - Torus profile of a Whittaker function twisted by |det|^{-e}
LanglandsTunnell.RankinSelberg.mul_conj_mul_abs_det_rpow_upperUnit_eq_abs_rpow_mul_norm_sq_of_diagOne_eq0 below · cited by 1 · depth 27 - Unfolded dual torus pair as 4π i^m times scaled-shape integral
LanglandsTunnell.RankinSelberg.dualTorusPair_eq_const_mul_setIntegral_scaledShape_of_discreteSeries_of_conjBlockHarmonic_colHarmonic_gaussian310 below · cited by 1 · depth 28 - Uniqueness of the γ-factor: rational equals monomial
LanglandsTunnell.RankinSelberg.eval_mul_cpow_eq_mul_cpow_mul_eval_of_laurent_fe_of_rational_fe0 below · cited by 1 · depth 28 - Product integrability of a local Rankin–Selberg kernel in the chamber
LanglandsTunnell.RankinSelberg.exists_forall_integrable_prod_row_mul_row_mul_cpow_mul_whittaker_diagOne_mul_cpow_of_admissible_of_chamber35 below · cited by 1 · depth 28 - Product integrability of the local Rankin–Selberg kernel in a chamber
LanglandsTunnell.RankinSelberg.exists_forall_integrable_prod_row_mul_row_mul_cpow_mul_whittaker_diagOne_mul_cpow_of_chamber35 below · cited by 2 · depth 28 - Iwasawa integral of the degree-m conjugate-block torus pair
LanglandsTunnell.RankinSelberg.exists_forall_iwasawaIntegral_eq_const_mul_oneSided_torusPair_add_mirror_of_discreteProfile_conjBlockHarmonic_colHarmonic7 below · cited by 1 · depth 28 - Vanishing of deep torus shells against a Whittaker vector
LanglandsTunnell.RankinSelberg.exists_forall_le_setIntegral_localLevelOne_mul_diagZ_mul_eq_zero_of_sqrt_modulus_tail_of_forall_torusZeta_polynomial7 below · cited by 1 · depth 28 - Rationality and cleared functional equation for local GL₂ zeta integrals
LanglandsTunnell.RankinSelberg.exists_gamma_forall_rational_godementZeta2_principalSeries2_and_clearedFE66 below · cited by 1 · depth 28 - Principal series vectors as Godement sections on primitive vectors
LanglandsTunnell.RankinSelberg.exists_godementDatum_primitive_of_mem_principalSeries23 below · cited by 2 · depth 28 - Ω-averaged Whittaker coefficients are compactly supported modulo the centre
LanglandsTunnell.RankinSelberg.exists_isCompact_forall_setIntegral_translate_ne_zero_of_cuspidal25 below · cited by 3 · depth 28 - Constant η-twisted local torus zeta integrals for Whittaker models
LanglandsTunnell.RankinSelberg.exists_mem_span_forall_torusZeta_twist_eq_const_and_dual_of_irreducible_admissible6 below · cited by 3 · depth 28 - Local Godement zeta integral as a Rankin–Selberg row-slice integral
LanglandsTunnell.RankinSelberg.exists_pos_forall_godementZeta2_eq_mul_rsLocalIntegral_rowSlice0 below · cited by 2 · depth 28 - Refolding the dual local Rankin–Selberg integral along N
LanglandsTunnell.RankinSelberg.exists_pos_forall_integral_dual_jacquetIntegral_godementSection_mul_row_eq_mul_integral_rot_row_mul_row_mul_dualTorusZeta4 below · cited by 2 · depth 28 - Haar measure on GL₂(ℚₚ) as κ |det X|⁻² dX
LanglandsTunnell.RankinSelberg.exists_pos_forall_integral_haar_eq_mul_integral_pi_det_inv_sq14 below · cited by 3 · depth 28 - Local Rankin–Selberg integral against a Godement section
LanglandsTunnell.RankinSelberg.exists_pos_forall_integral_mul_jacquetIntegral_godementSection_mul_row_eq_mul_integral_row_mul_row_mul_torusZeta4 below · cited by 2 · depth 28 - Weighted Whittaker unfolding of a local GL₂ zeta integral
LanglandsTunnell.RankinSelberg.exists_pos_forall_integral_weightedGodementWhittaker2_mul_row_eq_mul_integral_row_mul_row_mul_weightedTorusZeta6 below · cited by 2 · depth 28 - Whittaker Godement zeta equals that of its principal-series image
LanglandsTunnell.RankinSelberg.exists_schwartz_godementZeta2_whittaker_eq_godementZeta2_section_and_dual_of_equivariant_embedding18 below · cited by 1 · depth 28 - Local test function matching Godement–Jacquet and torus zeta integrals
LanglandsTunnell.RankinSelberg.exists_schwartz_godementZeta2_whittaker_eq_mul_torusZeta_and_dual_of_integrable22 below · cited by 1 · depth 28 - One-shell Kirillov function under the Weyl element, cuspidal case
LanglandsTunnell.RankinSelberg.forall_apply_diagOne_mul_weylJ_eq_of_apply_diagOne_eq_inv_mul_indicator_shell_of_cuspidal36 below · cited by 3 · depth 28 - Uniform abscissa for local Godement–Jacquet Whittaker integrals
LanglandsTunnell.RankinSelberg.forall_exists_forall_integrable_godementZeta2_whittaker_shift_of_isLocallyConstant30 below · cited by 2 · depth 28 - Uniform integrability of dual local Godement–Jacquet integrals
LanglandsTunnell.RankinSelberg.forall_exists_forall_integrable_godementZeta2_whittaker_transposeInvN_shift_of_isLocallyConstant32 below · cited by 1 · depth 28 - Integrability of the dual local Godement–Jacquet zeta integrand
LanglandsTunnell.RankinSelberg.forall_exists_integrable_godementZeta2_whittaker_transposeInvN_shift31 below · cited by 1 · depth 28 - Laurent polynomiality of shifted Godement–Jacquet zeta integrals
LanglandsTunnell.RankinSelberg.forall_exists_laurent_godementZeta2_whittaker_shift_of_torusLaurent38 below · cited by 1 · depth 28 - Pull-down of a local det-zeta integral identity
LanglandsTunnell.RankinSelberg.forall_integral_mul_modulus_det_cpow_eq_finsum_of_eqOn_of_forall_integrable1 below · cited by 1 · depth 28 - Parseval identity for the Fourier transform on M₂(ℚₚ)
LanglandsTunnell.RankinSelberg.integral_matFourier22_mul_eq_integral_mul_matFourier228 below · cited by 3 · depth 28 - Left averaging over a compact open subgroup under a Haar integral
LanglandsTunnell.RankinSelberg.integral_mul_eq_integral_mul_setAverage_of_forall_mul_left_eq1 below · cited by 2 · depth 28 - Iwasawa cell series for an N-K-invariant local integral
LanglandsTunnell.RankinSelberg.lintegral_eq_tsum_cellMass_mul_apply_torus_of_invariant1 below · cited by 2 · depth 28 - Fourier transform of cuspidal Kirillov matrix-coefficient functions on M₂
LanglandsTunnell.RankinSelberg.matFourier22_kirillov_det_mul_coefficient_eq_of_cuspidal75 below · cited by 3 · depth 28 - The locus X₀₀=0 or det X=0 is null in M₂(ℚₚ)
LanglandsTunnell.RankinSelberg.measure_pi_selfDualHaarAt_setOf_apply_eq_zero_or_det_eq_zero1 below · cited by 1 · depth 28 - Shell gauge and rational torus shells for twisted row slices
LanglandsTunnell.RankinSelberg.rowSlice_twist_shellGauge_and_rationalTorusShell11 below · cited by 1 · depth 28 - Godement–Eisenstein series: explicit poles, entire part, functional equation
LanglandsTunnell.RankinSelberg.exists_entire_sub_polarPart_godementEisenstein_isUniformlySiegelBounded_fe_poles_of_mem_schwartzBruhat285 below · cited by 1 · depth 29 - Kirillov decomposition of cuspidal Whittaker vectors into shell–character vectors
LanglandsTunnell.RankinSelberg.exists_finset_eq_sum_smul_shell_character_kirillov_of_cuspidal15 below · cited by 2 · depth 29 - Godement unfolding of a GL₂ principal-series zeta integral
LanglandsTunnell.RankinSelberg.exists_forall_integrable_and_godementZeta2_eq_mul_twoVarZeta_slice_of_mem_principalSeries26 below · cited by 2 · depth 29 - Godement unfolding of the contragredient local GL₂ zeta integral
LanglandsTunnell.RankinSelberg.exists_forall_integrable_and_godementZeta2_transposeInv_matFourier22_eq_mul_twoVarZeta_fourierSlice_of_mem_principalSeries235 below · cited by 1 · depth 29 - Local integrability of the unfolded Rankin–Selberg integrand
LanglandsTunnell.RankinSelberg.exists_forall_integrable_whittaker_mul_principalSeries2_antidiagonal2_mul_row_mul_cpow_of_admissible_of_chamber28 below · cited by 2 · depth 29 - Vanishing of deep torus shells in the unfolded zeta integrand
LanglandsTunnell.RankinSelberg.exists_forall_setIntegral_localLevelOne_rowSlice_whittaker_shell_eq_zero_of_le17 below · cited by 1 · depth 29 - Weyl element on shells: twisted local functional equation
LanglandsTunnell.RankinSelberg.exists_forall_setIntegral_units_apply_diagUnitGL2_mul_weylJ_eq_mul_setIntegral_of_cuspidal27 below · cited by 3 · depth 29 - Compact-open averages as θ₀-twisted Kirillov pairings
LanglandsTunnell.RankinSelberg.exists_mem_forall_setIntegral_translate_eq_kirillov_pairing_of_cuspidal49 below · cited by 1 · depth 29 - Non-negative Godement section for real norm-power characters
LanglandsTunnell.RankinSelberg.exists_normPowChar_godementSection_abs_mem_principalSeries2_of_lt4 below · cited by 2 · depth 29 - Shell profile of a unipotent row-slice integral over ℚₚ
LanglandsTunnell.RankinSelberg.exists_rowSlice_shell_profile_of_isLocallyConstant_of_hasCompactSupport10 below · cited by 3 · depth 29 - Godement zeta on a box as constant times torus zeta
LanglandsTunnell.RankinSelberg.godementZeta2_boxIndicator_eq_mul_torusZeta_of_isOpen_of_chart3 below · cited by 1 · depth 29 - Dual Godement–Jacquet zeta of the Fourier-transformed torus box
LanglandsTunnell.RankinSelberg.godementZeta2_transposeInv_matFourier22_boxIndicator_eq_mul_torusZeta_dual_of_integrable_of_chart14 below · cited by 1 · depth 29 - Tonelli regrouping of the local GL₂× F^× Rankin–Selberg kernel
LanglandsTunnell.RankinSelberg.lintegral_prod_enorm_row_mul_row_mul_cpow_mul_whittaker_diagOne_eq_lintegral_shear_and_integrable_of_lt_top6 below · cited by 2 · depth 29 - Big-cell Weyl coordinates push additive Haar onto M₂(ℚₚ)
LanglandsTunnell.RankinSelberg.map_bigCellWeyl_withDensity_eq_pi_selfDualHaarAt2 below · cited by 1 · depth 29 - Big-cell coordinates: |a/b| measure pushes to |det X|⁻² dX
LanglandsTunnell.RankinSelberg.map_bigCell_withDensity_eq_pi_withDensity_det_inv_sq2 below · cited by 1 · depth 29 - Central transformation law extends to the span of right translates
LanglandsTunnell.RankinSelberg.apply_scalar_mul_eq_mul_of_mem_span_translate0 below · cited by 1 · depth 30 - Non-degeneracy of the θ₀-twisted Kirillov pairing
LanglandsTunnell.RankinSelberg.eq_zero_of_forall_integral_kirillov_pairing_eq_zero16 below · cited by 1 · depth 30 - Vanishing of deep torus shell integrals
LanglandsTunnell.RankinSelberg.exists_forall_localLevelOne_setIntegral_units_whittaker_diagUnitGL2_eq_zero_of_le_of_torusLaurent10 below · cited by 1 · depth 30 - Local functional equation with vector-independent γ-factor
LanglandsTunnell.RankinSelberg.exists_rational_forall_torusZeta_fe_twist_of_irreducible_admissible26 below · cited by 1 · depth 30 - Bochner Iwasawa integration formula on GL₂(ℚₚ)
LanglandsTunnell.RankinSelberg.forall_integrable_prod_and_integral_eq_mul_setIntegral_unipotent_diagUnits2_of_forall_lintegral_eq2 below · cited by 1 · depth 30 - θ₀-invariance of the Kirillov pairing under translation
LanglandsTunnell.RankinSelberg.integral_kirillov_pairing_translate_eq_centralChar_det_mul_of_cuspidal38 below · cited by 2 · depth 30 - K-averaging commutes with the θ₀-twisted Kirillov pairing
LanglandsTunnell.RankinSelberg.integral_setIntegral_translate_kirillov_pairing_eq_mul_of_invariant_of_cuspidal42 below · cited by 1 · depth 30 - Local Godement slice is locally constant with compact support
LanglandsTunnell.RankinSelberg.isLocallyConstant_and_hasCompactSupport_slice_of_mem_principalSeries21 below · cited by 1 · depth 30 - Unit-torus average over the local level-one subgroup
LanglandsTunnell.RankinSelberg.setIntegral_localLevelOne_eq_setIntegral_setIntegral_units_diagUnitGL2_mul_of_isLocallyConstant3 below · cited by 1 · depth 30 - Weyl element on a pure Kirillov vector of one shell
LanglandsTunnell.RankinSelberg.forall_apply_diagOne_mul_weylJ_eq_of_apply_diagOne_eq_shell_character22 below · cited by 2 · depth 31 - Rationality of twisted torus zeta integrals over ℚₚ
LanglandsTunnell.RankinSelberg.forall_mem_span_exists_rational_torusZeta_twist_and_dual_of_irreducible_admissible9 below · cited by 1 · depth 31 - Shell constants multiply to θ₀(-1) when w_J²=-1
LanglandsTunnell.RankinSelberg.shell_constants_mul_eq_centralChar_neg_one_of_weylJ_sq25 below · cited by 1 · depth 31
LanglandsTunnell.RealArchParam 1
- Entire ratio of archimedean factors forces the signs to agree
LanglandsTunnell.RealArchParam.eq_of_archFactor_twist_mul_eq_archFactor_twist_mul_entire0 below · cited by 4 · depth 20
LanglandsTunnell.TateLocal 91
- Level of the local standard character equals the exponent of the different
LanglandsTunnell.TateLocal.addCharLevel_psiLocal_eq_count_differentIdeal8 below · cited by 28 · depth 16 - Complex archimedean zeta integral equals Tate's local zeta integral
LanglandsTunnell.TateLocal.complexZeta_eq_localZeta0 below · cited by 7 · depth 16 - Characters of large exact conductor exponent avoiding a prescribed value
LanglandsTunnell.TateLocal.exists_continuous_hasConductorExponentAt_apply_ne1 below · cited by 2 · depth 16 - Continuous characters of Kᵥ^× have a conductor exponent
LanglandsTunnell.TateLocal.exists_hasConductorExponentAt_of_continuous1 below · cited by 54 · depth 16 - The level of an additive character of Kᵥ is attained
LanglandsTunnell.TateLocal.forall_eq_one_and_exists_ne_one_of_addCharLevel0 below · cited by 100 · depth 16 - Conductor exponent of a character composed with a cubic norm
LanglandsTunnell.TateLocal.hasConductorExponentAt_comp_norm_of_finrank_eq_three6 below · cited by 11 · depth 16 - Conductor exponent of a product of characters with distinct exponents
LanglandsTunnell.TateLocal.hasConductorExponentAt_mul_of_hasConductorExponentAt_of_lt0 below · cited by 12 · depth 16 - Symmetry of Tate's local zeta functional ratio
LanglandsTunnell.TateLocal.localZeta_fourier_mul_symm0 below · cited by 9 · depth 16 - The Tate-local modulus on ℂ is |z|²
LanglandsTunnell.TateLocal.modulus_complex_eq_nnnorm_sq0 below · cited by 1 · depth 16 - The standard local character is trivial on 𝒪ᵥ
LanglandsTunnell.TateLocal.psiLocal_eq_one_of_mem_integers0 below · cited by 116 · depth 16 - Nontriviality of the standard local additive character ψ_{K,v}
LanglandsTunnell.TateLocal.psiLocal_ne_one3 below · cited by 144 · depth 16 - Existence of characters with prescribed exact conductor exponent
LanglandsTunnell.TateLocal.exists_continuous_hasConductorExponentAt0 below · cited by 2 · depth 17 - Conductor exponent is preserved by norm when e(w∣ v)=1
LanglandsTunnell.TateLocal.hasConductorExponentAt_comp_norm_of_ramificationIdx_eq_one3 below · cited by 10 · depth 17 - Conductor exponent is invariant under unramified twist
LanglandsTunnell.TateLocal.hasConductorExponentAt_mul_of_hasConductorExponentAt_zero0 below · cited by 9 · depth 17 - The modulus at the real place is the absolute value
LanglandsTunnell.TateLocal.modulus_real_eq_nnnorm0 below · cited by 7 · depth 17 - At ℝ: archimedean zeta integral equals Tate local zeta integral
LanglandsTunnell.TateLocal.realZeta_eq_localZeta0 below · cited by 13 · depth 17 - Unramified twist of the standard local root number
LanglandsTunnell.TateLocal.stdRootNumberAt_mul_of_hasConductorExponentAt_zero3 below · cited by 21 · depth 17 - Level of ψ_w equals level of ψᵥ when e(w∣ v)=1
LanglandsTunnell.TateLocal.addCharLevel_psiLocal_eq_of_ramificationIdx_eq_one9 below · cited by 13 · depth 18 - The standard character of ℚₚ has level 0
LanglandsTunnell.TateLocal.addCharLevel_psiLocal_rat0 below · cited by 87 · depth 18 - Conductor exponent 0 is preserved by the local norm
LanglandsTunnell.TateLocal.hasConductorExponentAt_comp_norm_zero1 below · cited by 8 · depth 18 - Norm surjects onto higher unit groups when e=1
LanglandsTunnell.TateLocal.image_norm_higherUnitsAt_eq_of_ramificationIdx_eq_one2 below · cited by 4 · depth 18 - Unramified local zeta integral equals μ(𝒪ᵥ^×) Lᵥ(χ,s)
LanglandsTunnell.TateLocal.localZeta_stdTestFunAt_eq_of_unramified1 below · cited by 7 · depth 18 - Local zeta integral of a ramified standard test function
LanglandsTunnell.TateLocal.localZeta_stdTestFunAt_eq_real_image_higherUnitsAt1 below · cited by 17 · depth 18 - Non-vanishing of the unramified local zeta integral
LanglandsTunnell.TateLocal.localZeta_stdTestFunAt_ne_zero_of_unramified3 below · cited by 2 · depth 18 - Ramified local ε-factor: Tate zeta integral of the standard test function
LanglandsTunnell.TateLocal.localZeta_tateFourier_stdTestFunAt10 below · cited by 15 · depth 18 - Haar modulus equals the v-adic norm on Kᵥ
LanglandsTunnell.TateLocal.modulus_adicCompletion_eq_nnnorm0 below · cited by 258 · depth 18 - Self-dual volume of the higher unit group Uᵥ⁽ᵃ⁾, a≥ 1
LanglandsTunnell.TateLocal.selfDualHaarAt_real_image_higherUnitsAt2 below · cited by 17 · depth 18 - Self-dual Haar volume of the local units at a finite place
LanglandsTunnell.TateLocal.selfDualHaarAt_real_units_eq1 below · cited by 7 · depth 18 - Invariance of the local root number under norm at a place with e=f=1
LanglandsTunnell.TateLocal.stdRootNumberAt_comp_norm_of_inertiaDeg_eq_one27 below · cited by 8 · depth 18 - Root number under composition with an unramified quadratic norm
LanglandsTunnell.TateLocal.stdRootNumberAt_comp_norm_of_inertiaDeg_eq_two27 below · cited by 8 · depth 18 - Unramified standard local root number at v equals 1
LanglandsTunnell.TateLocal.stdRootNumberAt_eq_one_of_hasConductorExponentAt_zero2 below · cited by 14 · depth 18 - Standard local root number of the trivial character is 1
LanglandsTunnell.TateLocal.stdRootNumberAt_one6 below · cited by 11 · depth 18 - Fourier transform of a ball indicator over Kᵥ
LanglandsTunnell.TateLocal.tateFourier_indicator_setOf_valued_sub_le0 below · cited by 21 · depth 18 - Dual and primal unit Gauss-integral series agree up to ε²
LanglandsTunnell.TateLocal.tsum_mul_setIntegral_psiLocal_neg_mul_charExt_eq_mul_charExt_sq_mul_stdRootNumberAt_sq16 below · cited by 1 · depth 18 - Conductor exponent of μ∘ N bounded by e· a
LanglandsTunnell.TateLocal.exists_hasConductorExponentAt_comp_norm_and_le_ramificationIdx_mul0 below · cited by 3 · depth 19 - Quadratic characters of ℚₚ^× have conductor exponent ≤ 3
LanglandsTunnell.TateLocal.exists_hasConductorExponentAt_le_three_of_pow_two_eq_one0 below · cited by 1 · depth 19 - Ramified local zeta integral of a Fourier-transformed ball indicator
LanglandsTunnell.TateLocal.localZeta_tateFourier_indicator_setOf_valued_sub_one_le3 below · cited by 1 · depth 19 - Haar volume of the valuation balls of Kᵥ
LanglandsTunnell.TateLocal.measureReal_setOf_valued_le_exp1 below · cited by 13 · depth 19 - Tate local root numbers of unitary characters have modulus one
LanglandsTunnell.TateLocal.norm_stdRootNumberAt_eq_one_of_hasConductorExponentAt21 below · cited by 3 · depth 19 - Self-dual local Haar measure: μ(mathcal Oᵥ) μ(mathfrak pᵥ⁻ⁿ)=1
LanglandsTunnell.TateLocal.selfDualHaarAt_real_integers_mul_real_setOf_valued_le_exp_addCharLevel2 below · cited by 7 · depth 19 - Vanishing of unit-shell Gauss integrals away from the critical valuation
LanglandsTunnell.TateLocal.setIntegral_addChar_mul_charExt_eq_zero_of_valued_ne8 below · cited by 3 · depth 19 - Product of unit Gauss integrals of χ and χ⁻¹
LanglandsTunnell.TateLocal.setIntegral_addChar_mul_charExt_mul_setIntegral_inv_mul_pow_eq9 below · cited by 7 · depth 19 - Non-vanishing of the standard local root number at a ramified character
LanglandsTunnell.TateLocal.stdRootNumberAt_ne_zero_of_hasConductorExponentAt13 below · cited by 12 · depth 19 - Additive duality on the upper conductor filtration
LanglandsTunnell.TateLocal.exists_forall_mem_higherUnitsAt_apply_eq_psiLocal_mul_sub_one_of_hasConductorExponentAt6 below · cited by 12 · depth 20 - Non-triviality of χ² and χ³ on units of level m
LanglandsTunnell.TateLocal.exists_mem_higherUnitsAt_pow_two_ne_one_and_pow_three_ne_one_of_hasConductorExponentAt0 below · cited by 1 · depth 20 - The measure dx/|x|ᵥ is Haar on Kᵥ^×
LanglandsTunnell.TateLocal.isHaarMeasure_comap_val_mulMeasure1 below · cited by 68 · depth 20 - Stability of Tate's local root number under small twists
LanglandsTunnell.TateLocal.stdRootNumberAt_mul_of_two_mul_conductorExponent_le14 below · cited by 8 · depth 20 - Root numbers of χ and χ⁻¹ multiply to χ(-1)
LanglandsTunnell.TateLocal.stdRootNumberAt_mul_stdRootNumberAt_inv_eq_apply_neg_one15 below · cited by 4 · depth 20 - Transport of a character's local pin along the norm, e=1
LanglandsTunnell.TateLocal.comp_norm_apply_eq_psiLocal_algebraMap_mul_sub_one_of_ramificationIdx_eq_one2 below · cited by 6 · depth 21 - Weighted conductor exponents above p sum to 3c
LanglandsTunnell.TateLocal.exists_forall_finsum_inertiaDeg_mul_conductorExponent_add_addCharLevel_eq7 below · cited by 6 · depth 21 - Unitarity of the standard local root number at a ramified place
LanglandsTunnell.TateLocal.norm_stdRootNumberAt_eq_one18 below · cited by 11 · depth 21 - Root number of a character composed with an unramified cubic norm
LanglandsTunnell.TateLocal.stdRootNumberAt_comp_norm_of_inertiaDeg_eq_three33 below · cited by 7 · depth 21 - Dilation rule for the local Tate Fourier transform
LanglandsTunnell.TateLocal.tateFourier_comp_mul_left0 below · cited by 4 · depth 21 - Shell calculus for the multiplicative Haar measure on ℚₚ^×
LanglandsTunnell.TateLocal.hasSum_setIntegral_shell_comap_val_mulMeasure_and_modulus_eq_of_valued_eq2 below · cited by 36 · depth 22 - Shell expansion of a local twisted Mellin integral
LanglandsTunnell.TateLocal.integrable_and_hasSum_setIntegral_shell_of_isLocallyConstant_of_norm_le2 below · cited by 3 · depth 22 - Inversion in an additive Haar integral: d(x⁻¹)=|x|⁻² dx
LanglandsTunnell.TateLocal.integral_comp_inv_eq_integral_modulus_inv_sq_mul_adicCompletion1 below · cited by 7 · depth 22 - Uniform finite Fourier expansion on local units mod U⁽ᵇ⁾
LanglandsTunnell.TateLocal.exists_finset_forall_exists_eq_sum_of_forall_mem_higherUnitsAt0 below · cited by 3 · depth 23 - Haar volumes of valuation balls and local Gauss-type integrals over ℚᵥ
LanglandsTunnell.TateLocal.addHaar_ball_eq_and_setIntegral_psiLocal_inv_mul_rat8 below · cited by 3 · depth 24 - Fourier expansion of a Uᵥ⁽ᵇ⁾-invariant function on the local units
LanglandsTunnell.TateLocal.exists_finset_hasConductorExponentAt_le_eq_sum_of_forall_mem_higherUnitsAt0 below · cited by 2 · depth 24 - Self-duality of ℚₚ: every smooth non-trivial character is a dilate of ψₚ
LanglandsTunnell.TateLocal.exists_forall_eq_psiLocal_mul_of_ne_one_rat10 below · cited by 1 · depth 24 - Polar decomposition of a local character of conductor exponent c
LanglandsTunnell.TateLocal.exists_unitary_mul_modulus_cpow_of_hasConductorExponentAt1 below · cited by 3 · depth 24 - Fourier transform preserves Schwartz–Bruhat functions on Kᵥ
LanglandsTunnell.TateLocal.isSchwartzBruhat_tateFourier0 below · cited by 23 · depth 24 - Local norms in an unramified extension of completions
LanglandsTunnell.TateLocal.mem_range_unitsMap_norm_iff_inertiaDeg_dvd_of_ramificationIdx_eq_one3 below · cited by 2 · depth 24 - Localisation of a twisted unit integral on the critical shell
LanglandsTunnell.TateLocal.setIntegral_addChar_mul_mul_charExt_eq_apply_mul_setIntegral_of_forall_mem_higherUnitsAt2 below · cited by 1 · depth 24 - Tate's Gauss-sum formula for the standard local root number
LanglandsTunnell.TateLocal.setIntegral_psiLocal_mul_charExt_inv_mul_cpow_eq_charExt_mul_stdRootNumberAt13 below · cited by 1 · depth 24 - Local Fourier inversion over ℚₚ for the standard character
LanglandsTunnell.TateLocal.tateFourier_tateFourier_psiLocal_selfDualHaarAt_rat9 below · cited by 6 · depth 25 - Fourier inversion on ℚₚ × ℚₚ for Schwartz–Bruhat functions
LanglandsTunnell.TateLocal.isSchwartzBruhat_prodFourier_and_prodFourier_prodFourier_selfDualHaarAt_eq14 below · cited by 5 · depth 26 - Local Fourier inversion at a finite place
LanglandsTunnell.TateLocal.tateFourier_tateFourier_eq_of_isSchwartzBruhat1 below · cited by 1 · depth 26 - Vanishing of deep shell integrals from a Laurent-polynomial Mellin transform
LanglandsTunnell.TateLocal.exists_forall_le_setIntegral_units_mul_zpow_eq_zero_of_mellin_eq_cpow_mul_eval_of_re_lt4 below · cited by 1 · depth 27 - Fourier inversion at v for the self-dual measure
LanglandsTunnell.TateLocal.tateFourier_tateFourier_selfDualHaarAt_of_isSchwartzBruhat11 below · cited by 4 · depth 27 - Box decomposition of Schwartz–Bruhat functions on ℚₚⁿ
LanglandsTunnell.TateLocal.exists_finset_forall_eq_sum_mul_prod_indicator_ball_of_isLocallyConstant_of_hasCompactSupport0 below · cited by 4 · depth 28 - Unit-shell integrals from a rational local Mellin transform
LanglandsTunnell.TateLocal.forall_sum_coeff_mul_sqrt_zpow_mul_setIntegral_units_eq_coeff_of_mellin_mul_eval_eq_cpow_mul_eval4 below · cited by 3 · depth 28 - Shell integrals recovered from a rational Mellin transform
LanglandsTunnell.TateLocal.forall_sum_coeff_mul_sqrt_zpow_mul_setIntegral_units_eq_coeff_of_mellin_mul_eval_eq_cpow_mul_eval_of_re_lt4 below · cited by 2 · depth 28 - Derivative at s=1 of Tate's local zeta integral
LanglandsTunnell.TateLocal.hasDerivAt_localZeta_one_one_integral_mul_log_modulus_of_continuous_of_hasCompactSupport1 below · cited by 1 · depth 28 - Fourier tail of a negative-exponent quasi-character
LanglandsTunnell.TateLocal.integrableOn_charExt_mul_norm_inv_and_exists_forall_setIntegral_psiLocal_mul_eq_add_mul_inv_of_lt_zero2 below · cited by 1 · depth 28 - Tate local zeta integral at trivial character and s=1
LanglandsTunnell.TateLocal.localZeta_one_one_eq_integral1 below · cited by 1 · depth 28 - Local integrability of log|y|ᵥ on a nonarchimedean completion
LanglandsTunnell.TateLocal.locallyIntegrable_log_modulus1 below · cited by 5 · depth 28 - Fourier inversion for Schwartz–Bruhat functions on Kᵥ
LanglandsTunnell.TateLocal.tateFourier_tateFourier_of_isSchwartzBruhat4 below · cited by 1 · depth 28 - Holomorphy of Tate's local zeta integral for Re s>0
LanglandsTunnell.TateLocal.differentiableOn_localZeta_one_of_continuous_of_hasCompactSupport2 below · cited by 1 · depth 29 - Deep unit-shell integrals vanish when the Mellin transform is polynomial
LanglandsTunnell.TateLocal.exists_forall_le_setIntegral_units_mul_zpow_eq_zero_of_mellin_eq_cpow_mul_eval4 below · cited by 1 · depth 29 - Two-variable local zeta integrals: rationality and cleared functional equation
LanglandsTunnell.TateLocal.exists_gamma_forall_twoVarZeta_rational_and_clearedFE36 below · cited by 1 · depth 29 - Multiplicative Fourier inversion at a finite place of ℚ
LanglandsTunnell.TateLocal.integrable_unitsFourier_and_integral_mul_psiLocal_eq_inv_modulus_mul_of_shell_window13 below · cited by 1 · depth 29 - Partial Fourier transform of the symplectic Fourier transform swaps coordinates
LanglandsTunnell.TateLocal.integral_symplecticFourier_mul_psiLocal_eq_integral_swap_mul_psiLocal11 below · cited by 1 · depth 29 - Symplectic Fourier integral under right translation by GL₂
LanglandsTunnell.TateLocal.symplecticFourier_comp_rowAction_eq_inv_modulus_det_mul_symplecticFourier3 below · cited by 1 · depth 29 - Double Tate transform of a ball indicator on Kᵥ
LanglandsTunnell.TateLocal.tateFourier_tateFourier_indicator_setOf_valued_sub_le3 below · cited by 1 · depth 29 - Product of two local Tate zeta integrals: rationality and cleared functional equation
LanglandsTunnell.TateLocal.exists_gamma_forall_localZeta_mul_localZeta_rational_and_clearedFE31 below · cited by 1 · depth 30 - Finite linear combinations preserve rationality and cleared functional equations
LanglandsTunnell.TateLocal.exists_rational_clearedFE_finset_sum0 below · cited by 1 · depth 30 - Rational local Tate zeta integrals and cleared functional equation
LanglandsTunnell.TateLocal.exists_gamma_forall_localZeta_rational_and_clearedFE30 below · cited by 1 · depth 31 - Tate's local functional equation with φ-uniform γ-factor
LanglandsTunnell.TateLocal.exists_gamma_forall_localZeta_tateFourier_mul_eq_of_strip23 below · cited by 1 · depth 32 - Convergence and rationality of local Tate zeta integrals over ℚ
LanglandsTunnell.TateLocal.exists_rational_localZeta_of_isSchwartzBruhat_of_logb_lt_re7 below · cited by 1 · depth 32 - Polar decomposition χ = η |·|ᵥ^t of a local quasi-character
LanglandsTunnell.TateLocal.exists_norm_eq_one_and_hasConductorExponentAt_and_eq_mul_modulus_cpow1 below · cited by 1 · depth 33