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# Mathematical Research Letters

## Volume 22 (2015)

### Number 5

### Weight shiftings for automorphic forms on definite quaternion algebras, and Grothendieck ring

Pages: 1459 – 1490

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n5.a9

#### Author

#### Abstract

Let $p$ be a prime number and let $\mathbb{F}$ be a finite field of characteristic $p$. We investigate the interplay between the algebraic structure of the Grothendieck ring of finitely generated $\mathbb{F}[GL_2 (\mathbb{F})]$-modules, and the existence of cohomological operators producing congruences modulo $p$ between automorphic forms on definite quaternion algebras over a totally real field.

#### Keywords

congruences between automorphic forms, modular representations of $GL_2$, Grothendieck ring

#### 2010 Mathematics Subject Classification

11F33, 20C33