On the faithfulness of parabolic cohomology as a Hecke module over a finite field

In this article we prove that under certain conditions the Hecke algebra of cuspidal modular forms over F_p coincides with the Hecke algebra of a certain parabolic group cohomology group with coefficients in F_p. These results can e.g. be used to compute Katz modular forms of weight one over an algebraic closure of F_p with methods of linear algebra over F_p.