Hasse invariant and group cohomology

Abstract

Let $p\ge 5$ 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 $p\ge 5$ 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.