Parabolic group cohomology in degree one, with Hecke cochain operator #

Definition: Gamma0CoeffCohomology (077b1ce7-afc1-5e0a-bf2d-900e8ab21871), by Claude, 2026-09-05.

Over a commutative ring $K$, for a group $G$ and a representation $\rho \colon G \to \mathrm{GL}_K(V)$ on a $K$-module $V$, the module HeckeEis.coeffCocycles is the submodule of $K$-valued functions $z \colon G \to V$ satisfying the inhomogeneous cocycle identity $z(gh) = z(g) + \rho(g)\,z(h)$ for all $g,h$, and HeckeEis.coeffCoboundaryMap is the $K$-linear map sending $v \in V$ to the function $g \mapsto \rho(g)v - v$, whose range is HeckeEis.coeffCoboundaries; the coboundaries lie in the cocycles. In the parabolic section $\Gamma$ is a subgroup of $\mathrm{SL}_2(\mathbb Z)$ and $\rho$ a representation of $\Gamma$. The predicate HeckeEis.IsParabolicCocycle applies to an arbitrary function $z \colon \Gamma \to V$ and asserts that for every $\gamma \in \Gamma$ whose underlying integer matrix has $\operatorname{tr}(\gamma)^2 = 4$ the value $z(\gamma)$ lies in the image of the endomorphism $\rho(\gamma) - 1$. Conjoining this with the cocycle condition gives the submodule HeckeEis.coeffParabolicCocycles, which contains the coboundaries and is contained in the cocycles. The $K$-module HeckeEis.coeffH1par is defined as the quotient of coeffParabolicCocycles by the preimage of coeffCoboundaries under the inclusion, with HeckeEis.coeffH1parMk the quotient map: it is surjective, and a parabolic cocycle has trivial class exactly when it is a coboundary.

Finally, for $N, \ell$ natural numbers with $\ell \neq 0$, a representation $\rho$ of $\Gamma_0(N)$, a $K$-linear endomorphism $a$ of $V$ and a function $z \colon \Gamma_0(N) \to V$, the function HeckeEis.coeffHeckeFun is defined by $$(T z)(g) = \sum_{q \in \Gamma_0(N)/\Gamma_\alpha} \rho\bigl(r_{g\cdot q}\bigr)\, a\bigl(z(\alpha\, r_{g\cdot q}^{-1} g\, r_q\, \alpha^{-1})\bigr),$$ where $\Gamma_\alpha =$ heckeUpper N ℓ is the finite-index subgroup of $\Gamma_0(N)$ cut out by $\ell \mid b$, the chosen coset representatives $r_q$ and the transfer element $r_{g\cdot q}^{-1} g\, r_q \in \Gamma_\alpha$ come from transferAux, and conjugation by $\alpha = \operatorname{diag}(1,\ell)$ is heckeConj N ℓ. This is the cochain-level operator only; HeckeEis.coeffHeckeFun_trivial identifies it, for the trivial one-dimensional representation on $K$ and $a = \mathrm{id}$, with the additive-character operator HeckeEis.heckeOperatorHom N ℓ K.

Relation to Mathlib. Mathlib develops group cohomology for objects of Rep; here the degree-one inhomogeneous cocycles and coboundaries are set up directly as submodules of $G \to V$ attached to a Representation K G V, and the parabolic condition, the parabolic subspace and the quotient $H^1_{\mathrm{par}}$ have no Mathlib counterpart.

Where it is used. These modules provide the group-cohomological model of modular symbols for $\Gamma_0(N)$ used to carry Hecke actions; in the trivial-coefficient case the parabolic classes are the parabolic additive characters of $\Gamma_0(N)$, and the cochain operator here is the coefficient version of the Hecke operator on those characters.