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Namespace Ihara 36 theorems

• Ihara's lemma for Γ₀(N), group-theoretic form Ihara.exists_coprime_forall_mem_Gamma_apply_eq_zero 30 below · cited by 3 · depth 11
• Trivial Schur multiplier of SL₂(𝔽_q), q≥ 5 Ihara.hasTrivialSchurMultiplier_SL2_ZMod_prime 2 below · cited by 1 · depth 11
• Perfectness of SL₂(ℤ/qⁿ) for q≥ 5 Ihara.isPerfect_SL2_ZMod_prime_pow 0 below · cited by 5 · depth 11
• Injectivity of Ihara's amalgam homomorphism Ihara.amalgamToGamma0Away_injective 0 below · cited by 4 · depth 12
• Surjectivity of Ihara's amalgam map onto Γ₀ away from q Ihara.amalgamToGamma0Away_surjective 0 below · cited by 6 · depth 12
• Central extensions of SL₂(ℤ/q): commutator kernel is q-torsion Ihara.exists_pow_prime_pow_eq_one_of_sl2_stem 1 below · cited by 3 · depth 12
• Congruence subgroup property of SL₂(ℤ[1/q]) Ihara.exists_principalCongruenceAway_le_of_finiteIndex 22 below · cited by 2 · depth 12
• Finiteness of the abelianisation of Γ₀(N) in SL₂(ℤ[1/q]) Ihara.finite_abelianization_gamma0Away 24 below · cited by 1 · depth 12
• Trivial Schur multiplier for SL₂(ℤ/qⁿ), q odd prime Ihara.hasTrivialSchurMultiplier_SL2_ZMod_odd_prime_pow 2 below · cited by 2 · depth 12
• Trivial Schur multiplier of SL₂(𝔽_q), all primes Ihara.hasTrivialSchurMultiplier_SL2_ZMod_of_prime 2 below · cited by 1 · depth 12
• Mod-3 kernel pairs for Γ₀(Nq) are Eisenstein Ihara.heckeOperatorHom_eisenstein_mod_three_of_parabolic_levelRaisingKernel 34 below · cited by 1 · depth 12
• Characters of Γ₀(N) over ℤ[1/q] factor through (ℤ/N)^× Ihara.gamma0Away_hom_factor 23 below · cited by 2 · depth 13
• Dicyclic groups have trivial Schur multiplier Ihara.hasTrivialSchurMultiplier_of_dicyclic 0 below · cited by 1 · depth 13
• Schur's Sylow criterion for trivial Schur multiplier Ihara.hasTrivialSchurMultiplier_of_sylow 0 below · cited by 1 · depth 13
• Ihara's lemma for Γ₀(N) in homomorphism form Ihara.ihara_hom_factor 24 below · cited by 1 · depth 13
• Central extensions of a dicyclic pair meet the commutator trivially Ihara.ker_inf_commutator_eq_bot_of_dicyclic_closure_pair 0 below · cited by 1 · depth 13
• Mennicke's congruence subgroup property for SL₂(ℤ[1/q]) Ihara.mennickeCSP_of_prime 21 below · cited by 1 · depth 13
• Stem extensions of SL(2,ℤ/2ᵃ) have kernel of order dividing 2 Ihara.card_ker_dvd_two_of_stemExtension_SL2_ZMod_two_pow 0 below · cited by 1 · depth 14
• Perfectness and trivial Schur multiplier of SL₂(ℤ/m) Ihara.commutator_eq_top_and_hasTrivialSchurMultiplier_SL2_ZMod 0 below · cited by 2 · depth 14
• Homomorphisms of Γ₀(N;ℤ/M) factor through the diagonal character Ihara.gamma0Fin_hom_factor 0 below · cited by 1 · depth 14
• Trivial Schur multiplier of SL₂(ℤ/qⁿ) for q≥ 5 Ihara.hasTrivialSchurMultiplier_SL2_ZMod_prime_pow 8 below · cited by 2 · depth 14
• Mennicke's Lemma 2.2 for SL₂(ℤ[1/q]), unconditionally Ihara.ihxw14_dio_lemma22_statement_unconditional 3 below · cited by 2 · depth 14
• Mennicke's congruence-subgroup property at a coprime level Ihara.mennickeCSP_of_coprime_of_stem 3 below · cited by 2 · depth 14
• Mennicke's Lemma 2.1: U centralises N_m modulo Q_m Ihara.mennickeLemma21 1 below · cited by 5 · depth 14
• Mennicke's composite step: Q_{m''} lies in [Q_{m''},Q_{m''}]∨ Q_{m'm''} Ihara.mennickeQ_le_commutator_sup_mennickeQ_mul 2 below · cited by 2 · depth 14
• Normal closure of the Mennicke generator is all of SL₂(ℤ[1/q]) Ihara.normalClosure_mennickeA_eq_top 0 below · cited by 3 · depth 14
• An inductive step in Mennicke's congruence subgroup property Ihara.pow_card_mem_mennickeQ_mul 4 below · cited by 1 · depth 14
• Primes dividing #SL₂(ℤ/2ᵃ3ᵇ) are 2 or 3 Ihara.prime_dvd_card_SL2_ZMod_two_pow_mul_three_pow 0 below · cited by 1 · depth 14
• Mennicke: Γ(m) inside commutator subgroup joined with Q_m Ihara.principalCongruenceAway_le_commutator_sup_mennickeQ 0 below · cited by 2 · depth 14
• Mennicke replacement of the upper-left entry modulo A^m Ihara.exists_replacement_lowerUnip 0 below · cited by 1 · depth 15
• Trivial Schur multiplier for SL₂(ℤ/q²) Ihara.hasTrivialSchurMultiplier_SL2_ZMod_sq 3 below · cited by 1 · depth 15
• Inductive step for the Schur multiplier of SL₂(ℤ/q^m) Ihara.hasTrivialSchurMultiplier_SL2_ZMod_step 0 below · cited by 1 · depth 15
• Mennicke's Lemma 2.2 at saturated levels, conditional form Ihara.mennickeZ_eq_top_of_corrected_reading_of_diophantine 2 below · cited by 1 · depth 15
• Trivial central kernel from an abelian layer and cyclic fibre Ihara.ker_eq_bot_of_stem_of_fibre 0 below · cited by 1 · depth 16
• Odd exponent descends to the kernel of a stem extension Ihara.ker_pow_eq_one_of_stem 0 below · cited by 1 · depth 16
• Congruence-kernel preimage in a central extension of SL₂(ℤ/q²) is abelian Ihara.sl2_zmod_sq_congruence_preimage_commute 0 below · cited by 1 · depth 16