Hasse Invariant and Group Cohomology

Let $p\geq5$ 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 $p\geq5$ 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.

2000 Mathematics Subject Classification: 11F33

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