The Eichler-Shimura isomorphism \[ S_k(\Gamma_1(N), {\mathbb C}) \oplus \overline{S_k(\Gamma_1(N), {\mathbb C})} \cong H^1_{\text{par}}(\Gamma_1(N), \text{ Sym}^{k-2}({\mathbb C}^2)) \] relates the space of holomorphic cusp forms of weight \(k \geq 2\) and level \(\Gamma_1(N)\), \(N\geq 1\), and the parabolic group cohomology of \(\Gamma_1(N)\) with coefficients in the \((k-2)\)-th symmetric power of \({\mathbb C}^2\) with the standard \(\text{SL}_2({\mathbb Z})\)-action. There are compatible actions of Hecke operators on both sides, and it follows immediately that the cohomology group on the right hand side is a free module of rank \(2\) over the Hecke algebra \({\mathbb T}_{\mathbb C}\), and in particular is a faithful \({\mathbb T}_{\mathbb C}\)-module.
Over finite fields, one has the space \(S_k(\Gamma_1(N), {\mathbb F}_p)\) of cuspidal Katz modular forms of weight \(k\geq 2\) and level \(\Gamma_1(N)\); we denote by \({\mathbb T}_{\mathbb F_p}\) its Hecke algebra. There is no analogue of the Eichler-Shimura isomorphism in this case, but one may still ask whether (a) \(H^1_{\text{par}}(\Gamma_1(N), \text{ Sym}^{k-2}({\mathbb F}_p^2))\) is a faithful \({\mathbb T}_{\mathbb F_p}\)-module, or (b) even is a free \({\mathbb T}_{\mathbb F_p}\)-module of rank \(2\). There are partial results of Edixhoven, and of Emerton, Pollack and Weston in this direction.
The main results of the paper at hand are as follows: Let \(2 \leq k \leq p+1\), \(N \geq 5\) and assume \(p\nmid N\). Let \(\mathfrak P\) be a maximal ideal of \({\mathbb T}_{\mathbb F_p}\) corresponding to an ordinary normalized eigenform \(f\). Then there is an isomorphism \({\mathbb T}_{\mathbb F_p}(S_k(\Gamma_1(N), {\mathbb F}_p)_{\mathfrak P}) \cong {\mathbb T}_{\mathbb F_p}(H^1_{\text{par}}(\Gamma_1(N), \text{ Sym}^{k-2}(F_p^2))_{\mathfrak P})\), so “(a) is true at \(\mathfrak P\)”. Under an additional, somewhat technical assumption, a similar result for cuspidal modular forms for \(\Gamma_1(N)\) with a Dirichlet character holds.
If the answer to (a) above is yes, then one can compute the Hecke algebra \({\mathbb T}_{\mathbb F_p}\) via the Hecke operators on the (finite-dimensional) \({\mathbb F}_p\)-vector space \(H^1_{\text{par}}(\Gamma_1(N), \text{ Sym}^{k-2}(F_p^2))\). This computation only requires linear algebra methods and can be done on a computer.