Hecke equivariance of the period map

Let Γ ≤ SL(2, ℤ) be a subgroup of finite index and D = Γ' δ Γ' = ⊔ᵥ Γ' aᵥ a double coset, with Γ' = Γ.map (mapGL ℚ) and δ an integral matrix of positive determinant. The same double coset acts on cusp forms of weight k = w + 2 on Γ, by f ↦ ∑ᵥ f ∣[k] aᵥ (HeckeRing.GL2.heckeSlashCuspFormEnd), and on the modular symbols 𝕄_w(Γ; R), by {α, β} ⊗ P ↦ ∑ᵥ {aᵥα, aᵥβ} ⊗ (P ∣ adj aᵥ) (TauCeti.ModularSymbols.heckeSymbol). This file proves that the period map TauCeti.ModularSymbols.periodMap intertwines the two actions:

periodMap (T_D f) = periodMap f ∘ T_D.

Since the period map sends a cusp form to a functional on the symbols, the symbol-side operator appears by precomposition, as a transpose. The proof is the substitution z ↦ aᵥ z in each summand (TauCeti.ModularSymbols.cuspIntegral_periodIntegrand_slash): ∫_β^α (f ∣[k] aᵥ)(z) P(z, 1) dz = ∫_{aᵥβ}^{aᵥα} f(z) (P ∣ adj aᵥ)(z, 1) dz, where the determinant factors of the slash action and of the adjugate cancel. Both sides are computed with the same representatives aᵥ = DoubleCoset.rightCosetRep D v, so no comparison of coset decompositions is needed.

Specialized to Γ = Γ₁(N) and the double coset of diag(1, n), this is the equivariance periodMap (T_n f) = periodMap f ∘ T_n of the period map for the Hecke operators T_n. This is the input, alongside the injectivity of the period map, through which integrality of the Hecke operators on modular symbols is transferred to the Hecke operators on cusp forms.

Main results

• TauCeti.ModularSymbols.periodMap_heckeSlashCuspFormEnd: the period map intertwines the operator of a double coset on cusp forms with the transpose of its operator on modular symbols.
• TauCeti.ModularSymbols.periodMap_heckeTCuspNat: the same for the Hecke operators T_n at level Γ₁(N).

Provenance

The statement corresponds to periodMap'_heckeEnd in the AINTLIB LeanModularForms project (HeckeRIngs/GL2/ModularSymbols/PeriodHecke.lean, Apache-2.0); no code is transcribed.

References

• G. Shimura, Introduction to the arithmetic theory of automorphic functions, §8.2.
• W. Stein, Modular Forms: A Computational Approach, Graduate Studies in Mathematics 79, American Mathematical Society, 2007, §8.5.

theorem TauCeti.ModularSymbols.periodMap_heckeSlashCuspFormEnd {R : Type u_1} [CommRing R] [Algebra R ℂ] {k : ℤ} {w : ℕ} {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} [Γ.FiniteIndex] {Δ : Submonoid (GL (Fin 2) ℚ)} (D : HeckeCoset Δ (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) Γ) (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) Γ)) [Finite (DoubleCoset.DecompQuotient (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) Γ) (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℚ) Γ) (↑(Quotient.out D))⁻¹)] (hDpos : ↑(Quotient.out D) ∈ Matrix.GLPos (Fin 2) ℚ) (hDint : ↑(Quotient.out D) ∈ HeckeRing.GLn.intEntries 2) (hk : k = ↑w + 2) (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) Γ) k) : (periodMap R Γ hk) ((HeckeRing.GL2.heckeSlashCuspFormEnd k D hDpos) f) = (periodMap R Γ hk) f ∘ₗ heckeSymbol Γ Γ D hDint

Hecke equivariance of the period map. For a double coset D = Γ' δ Γ' with δ an integral matrix of positive determinant, the period map intertwines the operator of D on cusp forms of weight w + 2 on Γ with the transpose of its operator on 𝕄_w(Γ; R): ⟪T_D f, x⟫ = ⟪f, T_D x⟫.

theorem TauCeti.ModularSymbols.periodMap_heckeTCuspNat {R : Type u_1} [CommRing R] [Algebra R ℂ] {k : ℤ} {w : ℕ} (N : ℕ) [NeZero N] (n : ℕ) [NeZero n] (hk : k = ↑w + 2) (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : (periodMap R (CongruenceSubgroup.Gamma1 N) hk) ((HeckeRing.GL2.heckeTCuspNat k n) f) = (periodMap R (CongruenceSubgroup.Gamma1 N) hk) f ∘ₗ heckeTSymbol R w N n

Hecke equivariance of the period map for T_n. At level Γ₁(N), the period map intertwines the Hecke operator T_n on cusp forms of weight w + 2 with the transpose of the Hecke operator T_n on 𝕄_w(Γ₁(N); R): ⟪T_n f, x⟫ = ⟪f, T_n x⟫.