Atkin–Lehner matrices and the Γ₀(N) double cosets

An Atkin–Lehner matrix W for a divisor Q of N, read in GL(2, ℚ), normalizes the image of Γ₀(N). This file shows that conjugation by W moreover fixes every double coset Γ₀(N) α Γ₀(N) with α ∈ Δ₀(N) of determinant coprime to Q:

W⁻¹ α W ∈ Γ₀(N) α Γ₀(N).

Writing W = !![Q a, b; N c, Q d] and α = !![p, q; N r, s], the conjugate is the integral matrix B with W B = α W. Its lower-left entry is divisible by N, and its upper-left entry is congruent to s modulo Q and to p modulo N / Q, hence a unit modulo N (s is a unit modulo Q because det α ≡ p s is); so W⁻¹ α W ∈ Δ₀(N). It has the determinant of α, and the same common divisors of entries (a common divisor of either matrix is coprime to Q, and Q • B = adj W · α · W), so the two lie in the same Γ₀(N)-double coset (HeckeRing.GL2.mem_doubleCoset_of_det_eq_of_dvd_iff). No coprimality with N / Q is needed, so this covers the U_p double cosets at the primes p ∣ N / Q.

These are the two hypotheses under which the slash by W commutes with the Hecke operator of Γ₀(N) α Γ₀(N) (HeckeRing.GL2.heckeSlashSum_slash_of_mem_normalizer).

Main results

• TauCeti.IsAtkinLehnerMatrix.mem_normalizer_map_mapGL: W normalizes the image of Γ₀(N) in GL(2, ℚ).
• TauCeti.IsAtkinLehnerMatrix.inv_mul_mul_mem_doubleCoset: for α ∈ Δ₀(N) of determinant coprime to Q, W⁻¹ α W ∈ Γ₀(N) α Γ₀(N).

References

• G. Shimura, Introduction to the arithmetic theory of automorphic functions, Proposition 3.32.
• A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(m), Math. Ann. 185 (1970), 134–160.

theorem TauCeti.IsAtkinLehnerMatrix.mem_normalizer_map_mapGL {N Q : ℕ} {M : Matrix (Fin 2) (Fin 2) ℤ} {w : GL (Fin 2) ℚ} (hQN : Q ∣ N) (h : IsAtkinLehnerMatrix N Q M) (hw : ↑w = M.map Int.cast) : w ∈ Subgroup.normalizer ↑(Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) (CongruenceSubgroup.Gamma0 N))

An Atkin–Lehner matrix normalizes Γ₀(N) in GL(2, ℚ).

theorem TauCeti.IsAtkinLehnerMatrix.inv_mul_mul_mem_doubleCoset {N Q : ℕ} {M : Matrix (Fin 2) (Fin 2) ℤ} {w : GL (Fin 2) ℚ} [NeZero N] (hQN : Q ∣ N) (h : IsAtkinLehnerMatrix N Q M) (hw : ↑w = M.map Int.cast) {α : GL (Fin 2) ℚ} (hα : α ∈ HeckeRing.GL2.Delta0 N) {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : ↑α = A.map Int.cast) (hAdet : A.det.gcd ↑Q = 1) : w⁻¹ * α * w ∈ DoubleCoset.doubleCoset α ↑(Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) (CongruenceSubgroup.Gamma0 N)) ↑(Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) (CongruenceSubgroup.Gamma0 N))

Conjugation by an Atkin–Lehner matrix fixes a Γ₀(N) double coset of determinant coprime to Q. If W is an Atkin–Lehner matrix for Q ∣ N and α ∈ Δ₀(N) has an integral matrix A with determinant coprime to Q, then W⁻¹ α W lies in the double coset Γ₀(N) α Γ₀(N).