Prime Hecke operators preserve boundary (Eisenstein) classes with a nebentype law

Proved

MTT.Cohomology.primeHecke_boundary_datum

by cbirkbeck · Sep 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

group-cohomologyhecke-operatorsmodular-symbols

Let $N\ge1$ , $k\ge2$ , $n=k-2$ , and let $\varepsilon$ be a Dirichlet character modulo $N$ with values in $\mathbf C$ . Let $\Phi:\mathbf P^1(\mathbf Q)\to\operatorname{Sym}^n\mathbf C^2$ be a boundary datum ( $\Gamma_1(N)$ -equivariant) whose boundary cochain $\partial\Phi(x,y)=\Phi(y)-\Phi(x)$ satisfies the nebentype law $\partial\Phi(\gamma x,\gamma y)=\varepsilon(d)\,\gamma\cdot\partial\Phi(x,y)$ for $\gamma\in\Gamma_0(N)$ . Let $\ell$ be a prime and let

T_\ell\phi=\sum_{b=0}^{\ell-1}\phi\Big|\begin{pmatrix}1&b\\0&\ell\end{pmatrix}+\varepsilon(\ell)\,\phi\Big|\begin{pmatrix}\ell&0\\0&1\end{pmatrix}

be the mission's prime Hecke operator on cochains.

Claim. $T_\ell\,\partial\Phi$ is again the boundary cochain of a boundary datum: there is a $\Gamma_1(N)$ -equivariant $\Psi$ with $T_\ell\,\partial\Phi=\partial\Psi$ .

Why this holds. For $\ell\nmid N$ choose $\sigma_\ell\in\Gamma_0(N)$ with $\sigma_\ell\equiv\operatorname{diag}(\ell^{-1},\ell)\pmod N$ ; then $\{\beta_b=\begin{pmatrix}1&b\\0&\ell\end{pmatrix}\}_{b}\cup\{\sigma_\ell\alpha\}$ , $\alpha=\operatorname{diag}(\ell,1)$ , is a set of representatives of $\Gamma_1(N)\backslash\Gamma_1(N)\operatorname{diag}(1,\ell)\Gamma_1(N)$ (Diamond–Shurman, Proposition 5.2.1), and the standard operator $\Psi=\sum_b\operatorname{adj}(\beta_b)\cdot\Phi(\beta_b\,\cdot)+\operatorname{adj}(\sigma_\ell\alpha)\cdot\Phi(\sigma_\ell\alpha\,\cdot)$ on $\operatorname{Hom}_{\Gamma_1(N)}(\operatorname{Div},\operatorname{Sym}^n)$ is well defined and $\Gamma_1(N)$ -equivariant because right multiplication by $\gamma\in\Gamma_1(N)$ permutes the cosets. Since $\partial\Phi$ has the nebentype law and the lower-right entry of $\sigma_\ell$ is $\ell$ , one has $\partial\Phi|\sigma_\ell\alpha=\varepsilon(\ell)\,\partial\Phi|\alpha$ , so $T_\ell\partial\Phi=\partial\Psi$ . For $\ell\mid N$ one has $\varepsilon(\ell)=0$ , $T_\ell=U_\ell$ is the sum over the $\beta_b$ alone, and the same coset argument applies. (This is the boundary/Eisenstein analogue of exists_cuspForm_heckePrime_pos, which proves the corresponding statement for cusp forms by exactly this coset permutation.)

Preamble

import Definitions.Def_MTT_Cohomology_Boundary
set_option autoImplicit false
noncomputable section
open scoped BigOperators TensorProduct
open MTT.Cohomology

Formal statement

theorem MTT.Cohomology.primeHecke_boundary_datum
    {N k : ℕ} (hN : 0 < N) (hk : 2 ≤ k) (e : DirichletCharacter ℂ N)
    (Φ : Cusp → Binary ℂ) (hΦ : IsBoundaryDatum N (k-2) Φ)
    (hlaw : ∀ γ : CongruenceSubgroup.Gamma0 N, ∀ x y,
      boundaryCochain Φ (cuspAct γ.val x, cuspAct γ.val y) =
        e (γ.val 1 1 : ZMod N) • act γ.val.val (boundaryCochain Φ (x, y)))
    (l : ℕ) (hl : l.Prime) :
    ∃ Ψ : Cusp → Binary ℂ, IsBoundaryDatum N (k-2) Ψ ∧
      primeHecke (e (l : ZMod N)) l (boundaryCochain Φ) = boundaryCochain Ψ := by sorry

Source

Diamond–Shurman, A First Course in Modular Forms (GTM 228), §5.2, Proposition 5.2.1 (coset representatives of the T_ℓ double coset for Γ_1(N)) and §5.3; A. Ash and G. Stevens, Duke Math. J. 53 (1986), §4 (Hecke action on the boundary of H¹_c).

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