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

In this article we prove conditions under which a certain parabolic group cohomology space over a finite field F is a faithful module for the Hecke algebra of Katz modular forms over an algebraic closure of F. These results can e.g. be used to compute Katz modular forms of weight one with methods of linear algebra over F. This is essentially Chapter 3 of my thesis.