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Namespace ResidualGaloisRep 92 theorems

Landmarks here: Odd irreducible residual representations are absolutely irreducible

— 89 · IsAbsolutelyIrreducible 3

directly in ResidualGaloisRep 89

• Base change along the identity residue-field map ResidualGaloisRep.baseChangeAlong_residueFieldMap_algebraMap_self_isEquiv 0 below · cited by 2 · depth 7
• landmark Odd irreducible residual representations are absolutely irreducible ResidualGaloisRep.isAbsolutelyIrreducible_of_isIrreducible_of_isOdd 0 below · cited by 20 · depth 7
• Characteristic polynomials under base change of residual representations ResidualGaloisRep.charpoly_baseChangeAlong 0 below · cited by 37 · depth 8
• Charpolys agree everywhere from agreement at unramified Frobenii ResidualGaloisRep.charpoly_eq_of_charpoly_frobenius_eq 4 below · cited by 20 · depth 8
• Absolute irreducibility equals spanning of End_k(V) ResidualGaloisRep.isAbsolutelyIrreducible_iff_span_eq_top 2 below · cited by 27 · depth 8
• Equal characteristic polynomials give equivalence when absolutely irreducible ResidualGaloisRep.isEquiv_of_isAbsolutelyIrreducible_of_charpoly_eq 5 below · cited by 31 · depth 8
• Absolute irreducibility transfers to the matrix representation ResidualGaloisRep.isAbsolutelyIrreducible_iff_matrixRepresentation 5 below · cited by 2 · depth 9
• Equal traces imply equivalence for absolutely irreducible residual representations ResidualGaloisRep.isEquiv_of_isAbsolutelyIrreducible_of_trace_eq 4 below · cited by 1 · depth 9
• Characteristic polynomial of a residual Galois representation ResidualGaloisRep.charpoly_eq 1 below · cited by 2 · depth 10
• Taylor–Wiles primes with power-series presentation of R_Q ResidualGaloisRep.exists_taylorWilesPrimes_mvPowerSeries_surjective_strictOrdinary 1,855 below · cited by 3 · depth 10
• No G-stable line: transfer along a coefficient field map ResidualGaloisRep.forall_indexTwo_stable_eq_bot_or_top_baseChangeAlong 0 below · cited by 2 · depth 10
• Absolute irreducibility transfers along equal characteristic polynomials ResidualGaloisRep.isAbsolutelyIrreducible_of_isAbsolutelyIrreducible_of_charpoly_eq 3 below · cited by 41 · depth 10
• Attachment via Frobenius trace and determinant ResidualGaloisRep.isAttachedTo_iff_trace_det 2 below · cited by 1 · depth 10
• Transitivity of coefficient base change for residual Galois representations ResidualGaloisRep.isEquiv_baseChangeAlong_baseChangeAlong 0 below · cited by 4 · depth 10
• Descent of an absolutely irreducible residual representation to a finite subfield ResidualGaloisRep.exists_baseChangeAlong_subtype_isEquiv_of_forall_charpoly_coeff_mem 6 below · cited by 1 · depth 11
• Existence of an auxiliary prime for Taylor–Wiles systems ResidualGaloisRep.exists_prime_not_dvd_sub_one_trace_frobenius_sq_ne 26 below · cited by 2 · depth 11
• Taylor–Wiles primes bounding first-order deformation classes ResidualGaloisRep.exists_taylorWilesPrimes_finrank_span_dualNumberClasses_le_strictOrdinary 1,836 below · cited by 1 · depth 11
• Descent of local decomposition-irreducibility along coefficient base change ResidualGaloisRep.forall_decompositionStable_eq_bot_or_top_of_isEquiv_baseChangeAlong 0 below · cited by 3 · depth 11
• Base change along the identity preserves a residual representation ResidualGaloisRep.isEquiv_baseChangeAlong_id 0 below · cited by 1 · depth 11
• Index-two restrictions of an odd irreducible residual representation ResidualGaloisRep.restrict_index_two_of_isIrreducible_of_isOdd 0 below · cited by 1 · depth 11
• Traces determined by Frobenius traces outside a finite set ResidualGaloisRep.trace_eq_of_trace_frobenius_eq 17 below · cited by 5 · depth 11
• Equality of H¹(ℚ,ad⁰ρ̄) classes versus strict conjugacy of dual lifts ResidualGaloisRep.H1Pi_adZero_eq_iff_exists_dualNumber_conj 1 below · cited by 1 · depth 12
• Diagonalising inertia with a swap from a moved eigencharacter ResidualGaloisRep.exists_inertia_diagonal_swap_of_eigenvector 0 below · cited by 1 · depth 12
• Taylor–Wiles primes killing the dual Selmer group ResidualGaloisRep.exists_taylorWilesPrimes_card_eq_finrank_continuousH1S_dualTwist 77 below · cited by 1 · depth 12
• Greenberg–Wiles count for ad⁰ρ̄ at Taylor–Wiles level ResidualGaloisRep.finrank_strictSelmer_adZero_le_card_taylorWilesPrimes_add_finrank_dualSelmer 1,215 below · cited by 1 · depth 12
• Transfer of local irreducibility along Frobenius characteristic polynomials ResidualGaloisRep.forall_decompositionStable_eq_bot_or_top_of_charpoly_frobenius_map_eq 30 below · cited by 1 · depth 12
• Decomposition-stable submodules are trivial for swapped diagonal inertia ResidualGaloisRep.forall_decompositionStable_eq_bot_or_top_of_inertia_diagonal_of_swap 0 below · cited by 1 · depth 12
• Stable-subspace irreducibility equals `Representation.IsIrreducible` ResidualGaloisRep.isIrreducible_iff_representationIsIrreducible 0 below · cited by 1 · depth 12
• Cyclotomic determinant forces detρ̄(c)=-1 ResidualGaloisRep.det_complexConjugation_eq_neg_one_of_detIsCyclotomic 0 below · cited by 1 · depth 13
• Intertwiners between ρ̄ and its twist ρ̄⊗χ are scalar ResidualGaloisRep.exists_eq_smul_one_of_forall_mul_eq_smul_mul 0 below · cited by 1 · depth 13
• Frobenius without eigenvalue 1 at primes ℓ ≡ 1 (mod N) ResidualGaloisRep.exists_prime_modEq_one_isFrobeniusAt_eval_charpoly_ne_zero_of_isAbsolutelyIrreducible 20 below · cited by 22 · depth 13
• A Taylor–Wiles prime at which a given H¹ class survives ResidualGaloisRep.exists_taylorWilesPrime_map_ne_zero_of_mem_continuousH1S 50 below · cited by 1 · depth 13
• Existence of Taylor–Wiles primes of depth n outside a finite set ResidualGaloisRep.exists_taylorWilesPrime_notMem_of_isAbsolutelyIrreducible 37 below · cited by 1 · depth 13
• Flat local classes at p: dimension at most h⁰+1 ResidualGaloisRep.finiteDimensional_localFlatClasses_and_finrank_le 753 below · cited by 1 · depth 13
• Involutions of determinant -1 fix a line in ad⁰ ResidualGaloisRep.finrank_invariants_adZero_res_zpowers_eq_one_of_det_eq_neg_one 3 below · cited by 1 · depth 13
• Regular semisimple σ fixes a line in ad⁰ρ̄ ResidualGaloisRep.finrank_ker_adZeroRep_sub_one_eq_one_of_charpoly_eq 2 below · cited by 2 · depth 13
• Absolute irreducibility over ℚ(ζ_{pⁿ}) for odd p ResidualGaloisRep.baseChange_submodule_eq_bot_or_eq_top_of_forall_apply_eq_self 3 below · cited by 2 · depth 14
• Non-zero classes do not vanish on Gal(ℚ̄/Fₙ) ResidualGaloisRep.exists_apply_eq_self_and_adZeroRep_eq_one_and_cocycles_apply_ne_zero 21 below · cited by 1 · depth 14
• Matrix model of a base-changed residual Galois representation ResidualGaloisRep.exists_basis_monoidHom_toMatrix_eq_baseChangeAlong 0 below · cited by 1 · depth 14
• Absolutely irreducible residual representations are non-Eisenstein ResidualGaloisRep.exists_prime_modEq_one_isFrobeniusAt_trace_ne_add_one_of_isAbsolutelyIrreducible 18 below · cited by 12 · depth 14
• Existence of a Taylor–Wiles prime detecting a cocycle ResidualGaloisRep.exists_taylorWilesPrime_map_H1_ne_zero_of_notMem_range 26 below · cited by 1 · depth 14
• Taylor–Wiles primes avoiding a finite set, in Frobenius form ResidualGaloisRep.exists_taylorWilesPrime_notMem_of_seed 28 below · cited by 1 · depth 14
• Flat bound for H¹_f(ℚₚ,adρ̄), p odd ResidualGaloisRep.finiteDimensional_localFlatClassesAd_and_finrank_le 729 below · cited by 1 · depth 14
• Trace splitting: dim(adρ̄)^G=dim(ad⁰ρ̄)^G+1 ResidualGaloisRep.finrank_invariants_res_adRep_eq_finrank_invariants_res_adZero_add_one 0 below · cited by 1 · depth 14
• Fixed points of ad⁰ρ̄(σ) as trace-zero commuting matrices ResidualGaloisRep.finrank_ker_adZeroRep_sub_one_eq 0 below · cited by 1 · depth 14
• Local flat classes: from ad⁰ to ad, one dimension gained ResidualGaloisRep.finrank_localFlatClasses_add_one_le_finrank_localFlatClassesAd 45 below · cited by 1 · depth 14
• Local flatness of a cocycle forces flatness of ρ̄⊕ρ̄ ResidualGaloisRep.isLocallyFlatCocycleAd_zero_of_isLocallyFlatCocycle 2 below · cited by 1 · depth 14
• Non-zero trace on inertia coinvariants of ordinary residual representations ResidualGaloisRep.trace_inertiaCoinvariants_ne_zero_of_isOrdinaryAt_of_detIsCyclotomic 2 below · cited by 3 · depth 14
• Ramified residual representations give no eigenvector in H¹(Γ₀(M),k) ResidualGaloisRep.eq_zero_of_forall_heckeT_eq_smul_of_not_isUnramifiedAt 1,382 below · cited by 1 · depth 15
• No nonzero χ-equivariant functional on ad⁰ρ̄ ResidualGaloisRep.eq_zero_of_forall_map_adZeroRep_eq_smul 0 below · cited by 1 · depth 15
• Distinct split eigenvalues of Frobenius at a Taylor–Wiles prime ResidualGaloisRep.exists_charpoly_eq_mul_of_isTaylorWilesPrime 6 below · cited by 1 · depth 15
• Residual inertia-fixed line and reduction of the Frobenius scalar ResidualGaloisRep.exists_finrank_inertiaFixed_eq_one_and_frobenius_sub_smul_mem_of_isEquiv_residual_of_stableLine 2 below · cited by 1 · depth 15
• A non-zero locally flat scalar cocycle for ad ρ̄ ResidualGaloisRep.exists_isLocallyFlatCocycleAd_smul_one 41 below · cited by 1 · depth 15
• Unipotent, connected or ordinary trichotomy at p for finite flat ρ̄ ResidualGaloisRep.exists_unipotent_or_connected_model_or_ordinary_of_isLocallyFlatCocycleAd 45 below · cited by 1 · depth 15
• Flat local bound for connected models of ad ρ̄ ResidualGaloisRep.finiteDimensional_localFlatClassesAd_and_finrank_le_of_isLocalRing_baseChange 446 below · cited by 1 · depth 15
• Unipotent flat local bound: dim H¹_f ≤ h⁰ + 1 ResidualGaloisRep.finiteDimensional_localFlatClassesAd_and_finrank_le_of_isLocalRing_cartierDual 438 below · cited by 2 · depth 15
• Flat local bound for ordinary ρ̄ at p ResidualGaloisRep.finiteDimensional_localFlatClassesAd_and_finrank_le_of_ordinary 332 below · cited by 1 · depth 15
• Injectivity of H¹(G,ad⁰ρ̄)→ H¹(G,ad ρ̄) ResidualGaloisRep.injective_map_H1_of_adZero_le_adRep 0 below · cited by 1 · depth 15
• Flat classes for ad⁰ map into flat classes for ad ResidualGaloisRep.map_localFlatClasses_le_localFlatClassesAd 0 below · cited by 1 · depth 15
• Trace vanishing for cocycles coming from ad⁰ρ̄ ResidualGaloisRep.trace_apply_eq_zero_of_mem_range_map_H1 0 below · cited by 1 · depth 15
• Determinant at Frobenius of a weight-two residual representation ResidualGaloisRep.det_eq_natCast_of_isFrobeniusAt_of_charpoly_frobenius_eq 28 below · cited by 1 · depth 16
• Dual-lift module of a scalar cocycle as equivariant quotient ResidualGaloisRep.exists_cocycle_smul_one_surjective_pi_dualLiftModuleActAd 0 below · cited by 1 · depth 16
• Finite flat model for ̄ V from a flat ad-cocycle ResidualGaloisRep.exists_finiteFlat_padicInt_model_of_isLocallyFlatCocycleAd 1 below · cited by 1 · depth 16
• Honda-system model bounding local flat classes of ad ρ̄ ResidualGaloisRep.exists_hondaSystem_finrank_endHonda_le_injective_of_isLocalRing_cartierDual 433 below · cited by 1 · depth 16
• Vanishing traces at ℓ ≡ 2 (mod 3) force a stable line ResidualGaloisRep.exists_index_two_stable_line_of_trace_frobenius_eq_zero_of_modEq_two 20 below · cited by 1 · depth 16
• Cyclotomic inertia subspace, or a model with local Cartier dual ResidualGaloisRep.exists_submodule_inertia_eq_smul_and_unipotent_model_of_eq_bot 37 below · cited by 1 · depth 16
• Connectedness criterion for a finite flat model of ̄ V⊕̄ V ResidualGaloisRep.exists_submodule_inertia_sub_mem_and_connected_model_of_eq_top 33 below · cited by 1 · depth 16
• Cartier-dual unipotent model and isomorphic local flat classes ResidualGaloisRep.exists_unipotent_model_and_linearEquiv_localFlatClassesAd_of_isLocalRing_baseChange 20 below · cited by 1 · depth 16
• Dimension bound for ordinary unit classes in H¹(ℚₚ,ad ρ̄) ResidualGaloisRep.finiteDimensional_ordinaryUnitClassesAd_and_finrank_le 281 below · cited by 1 · depth 16
• Finiteness of k from a finite flat trivial deformation ResidualGaloisRep.finite_of_isLocallyFlatCocycleAd_zero 0 below · cited by 1 · depth 16
• Vanishing inertia coinvariants under local irreducibility ResidualGaloisRep.iSup_range_sub_one_eq_top_and_trace_quotient_eq_zero_of_forall_stable 0 below · cited by 2 · depth 16
• Flat classes are ordinary unit classes at p ResidualGaloisRep.unitRootInertia_trivial_and_localFlatClassesAd_le_ordinaryUnitClassesAd 81 below · cited by 1 · depth 16
• The cyclotomically twisted dual of a residual representation ResidualGaloisRep.exists_dualTwist_linearEquiv_dual 0 below · cited by 1 · depth 17
• Local flat classes inject into Honda self-extensions ResidualGaloisRep.exists_injective_localFlatClassesAd_selfExt_of_hondaSystem_model 428 below · cited by 1 · depth 17
• Flat cocycles are cohomologous to ordinary flat cocycles ResidualGaloisRep.exists_isOrdinaryCocycleAd_of_isLocallyFlatCocycleAd 58 below · cited by 1 · depth 17
• Unipotent finite flat model of ̄ V from one of ̄ V⊕̄ V ResidualGaloisRep.exists_unipotent_model_V_of_isLocalRing_cartierDual 97 below · cited by 1 · depth 17
• Unipotent flat model for the twisted dual ̄ V^∨(1) ResidualGaloisRep.exists_unipotent_model_dualTwist_of_isLocalRing_baseChange 16 below · cited by 1 · depth 17
• Honda system endomorphisms bounded by local invariants of ad ρ̄ ResidualGaloisRep.finrank_endHonda_le_finrank_invariants_of_hondaSystem_model 362 below · cited by 1 · depth 17
• Local invariants of ad ρ̄ under Cartier dual twist ResidualGaloisRep.finrank_invariants_adRep_eq_of_dualTwist 0 below · cited by 1 · depth 17
• Flatness at p descends along an extension of coefficients ResidualGaloisRep.isFlatAt_ofResidualGaloisRep_of_baseChangeAlong 1 below · cited by 3 · depth 17
• Flat classes in ad ρ̄ and in its cyclotomic dual twist ResidualGaloisRep.nonempty_localFlatClassesAd_linearEquiv_of_dualTwist 13 below · cited by 1 · depth 17
• Absolutely irreducible residual representations are not Eisenstein ResidualGaloisRep.not_isAbsolutelyIrreducible_of_charpoly_frobenius_eisenstein 46 below · cited by 1 · depth 17
• Flat classes in H¹(ℚₚ,adρ̄) inject into Honda self-extensions ResidualGaloisRep.exists_injective_flatClassSet_selfExt_of_hondaSystem_model 423 below · cited by 1 · depth 18
• Locally flat ad ρ̄-cocycles are closed under addition ResidualGaloisRep.isLocallyFlatCocycleAd_add 4 below · cited by 1 · depth 18
• Descent of strict ordinarity at odd p along a coefficient field map ResidualGaloisRep.isStrictOrdinaryAt_ofResidualGaloisRep_of_baseChangeAlong 1 below · cited by 1 · depth 18
• Fontaine–Conrad presentation of a locally flat ad-cocycle ResidualGaloisRep.exists_fontaineConradPresentation_of_isLocallyFlatCocycleAd 206 below · cited by 1 · depth 19
• Non-Eisenstein Frobenius trace at primes ℓ≡ 1 mod M ResidualGaloisRep.exists_prime_modEq_one_trace_frobenius_ne_of_isAbsolutelyIrreducible 22 below · cited by 1 · depth 19
• Unipotent models of locally flat first-order deformations of ρ̄ ResidualGaloisRep.exists_unipotent_model_of_isLocallyFlatCocycleAd_of_isLocalRing_cartierDual 71 below · cited by 1 · depth 20
• Trace equality transfers absolute irreducibility in dimension 2 ResidualGaloisRep.isAbsolutelyIrreducible_of_isAbsolutelyIrreducible_of_trace_eq 4 below · cited by 1 · depth 20

ResidualGaloisRep.IsAbsolutelyIrreducible 3

• Absolute irreducibility is preserved by coefficient extension ResidualGaloisRep.IsAbsolutelyIrreducible.baseChangeAlong 3 below · cited by 58 · depth 7
• Absolute irreducibility transfers along an equivalence ResidualGaloisRep.IsAbsolutelyIrreducible.of_isEquiv 0 below · cited by 5 · depth 10
• Absolute irreducibility implies irreducibility ResidualGaloisRep.IsAbsolutelyIrreducible.isIrreducible 5 below · cited by 2 · depth 11