Atkin–Lehner operators preserve the new subspace

Let Q ∥ N be an exact divisor and let W be an Atkin–Lehner matrix for Q. The slash operator by W on S_k(Γ₁(N)) preserves the old subspace and scales the Petersson product by Q ^ (k - 2). Its square is that nonzero scalar times a diamond operator, and diamond operators also preserve the old subspace. These facts show that W_Q preserves the Petersson-orthogonal complement, hence the new subspace.

Together with the character transport from TauCeti/NumberTheory/ModularForms/AtkinLehner/Gamma1.lean, this gives the general-nebentypus stability needed for Atkin and Li's pseudo-eigenvalues: W_Q carries the new part of S_k(N, χ) into the new part of S_k(N, χ ∘ ι_Q).

Main results

• TauCeti.atkinLehnerOperatorGamma1Cusp_mem_cuspFormsNew: an Atkin–Lehner operator on S_k(Γ₁(N)) preserves the new subspace.
• TauCeti.atkinLehnerOperatorGamma1Cusp_mem_cuspFormsNew_inf_cuspFormCharSpace: on a nebentypus component it preserves newness and transports the character by ι_Q.

References

• A. O. L. Atkin and W.-C. W. Li, Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math. 48 (1978), 221–243, §1.
• F. Diamond and J. Shurman, A First Course in Modular Forms, §5.8.

theorem TauCeti.atkinLehnerOperatorGamma1Cusp_mem_cuspFormsNew {N Q : ℕ} [NeZero N] {W : Matrix (Fin 2) (Fin 2) ℤ} {k : ℤ} (hQ : 0 < Q) (hQN : Q ∣ N) (hW : IsAtkinLehnerMatrix N Q W) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormsNew N k) : (atkinLehnerOperatorGamma1Cusp hQ hQN hW k) f ∈ cuspFormsNew N k

An Atkin–Lehner operator on S_k(Γ₁(N)) preserves the new subspace. This holds in every integral weight k and for every Atkin–Lehner matrix W of every exact divisor Q ∥ N.

theorem TauCeti.atkinLehnerOperatorGamma1Cusp_mem_cuspFormsNew_inf_cuspFormCharSpace {N Q : ℕ} [NeZero N] {W : Matrix (Fin 2) (Fin 2) ℤ} {k : ℤ} (hQ : 0 < Q) (hQN : Q ∣ N) (hW : IsAtkinLehnerMatrix N Q W) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} {χ : (ZMod N)ˣ →* ℂˣ} (hf : f ∈ cuspFormsNew N k ⊓ cuspFormCharSpace k χ) : (atkinLehnerOperatorGamma1Cusp hQ hQN hW k) f ∈ cuspFormsNew N k ⊓ cuspFormCharSpace k (χ.comp ↑⋯.unitsInvPart)

Atkin–Lehner stability with the nebentypus transport. The operator W_Q carries the new part of S_k(N, χ) into the new part of S_k(N, χ ∘ ι_Q), where ι_Q inverts the residue modulo Q and fixes it modulo N / Q.