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Mathlib.NumberTheory.ModularForms.CongruenceSubgroups

Congruence subgroups #

This defines congruence subgroups of SL(2, ℤ) such as Γ(N), Γ₀(N) and Γ₁(N) for N a natural number.

It also contains basic results about congruence subgroups.

@[simp]
theorem SL_reduction_mod_hom_val (N : ) (γ : Matrix.SpecialLinearGroup (Fin 2) ) (i : Fin 2) (j : Fin 2) :
((Matrix.SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) i j = (γ i j)

The full level N congruence subgroup of SL(2, ℤ) of matrices that reduce to the identity modulo N.

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    @[simp]
    theorem CongruenceSubgroup.Gamma_mem (N : ) (γ : Matrix.SpecialLinearGroup (Fin 2) ) :
    γ CongruenceSubgroup.Gamma N (γ 0 0) = 1 (γ 0 1) = 0 (γ 1 0) = 0 (γ 1 1) = 1

    The congruence subgroup of SL(2, ℤ) of matrices whose lower left-hand entry reduces to zero modulo N.

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      theorem CongruenceSubgroup.Gamma0_det (N : ) (A : (CongruenceSubgroup.Gamma0 N)) :
      (A).det = 1

      The group homomorphism from Gamma0 to ZMod N given by mapping a matrix to its lower right-hand entry.

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        The congruence subgroup Gamma1 (as a subgroup of Gamma0) of matrices whose bottom row is congruent to (0,1) modulo N.

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          theorem CongruenceSubgroup.Gamma1_to_Gamma0_mem (N : ) (A : (CongruenceSubgroup.Gamma0 N)) :
          A CongruenceSubgroup.Gamma1' N (A 0 0) = 1 (A 1 1) = 1 (A 1 0) = 0

          The congruence subgroup Gamma1 of SL(2, ℤ) consisting of matrices whose bottom row is congruent to (0,1) modulo N.

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            @[simp]
            theorem CongruenceSubgroup.Gamma1_mem (N : ) (A : Matrix.SpecialLinearGroup (Fin 2) ) :
            A CongruenceSubgroup.Gamma1 N (A 0 0) = 1 (A 1 1) = 1 (A 1 0) = 0

            A congruence subgroup is a subgroup of SL(2, ℤ) which contains some Gamma N for some (N : ℕ+).

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