Hasse invariant and group cohomology

Abstract

Let p ≥ 5 be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between eigenforms of weights 2 and p + 1, in terms of group cohomology. We also illustrate how our method works for inert primes p ≥ 5 in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields, cyclic and of even degree over the rationals.