On the reduction modulo p of representations of a quaternion division algebra over a p-adic field

TL;DR

This paper investigates how irreducible smooth representations of a quaternion division algebra and the Weil group over a p-adic field behave under reduction modulo p, revealing compatibility issues with established correspondences.

Contribution

It explicitly computes the reduction modulo p of these representations and analyzes their compatibility with the local Langlands and Jacquet-Langlands correspondences.

Findings

Reduction modulo p is computed for specific representations.

Compatibility between mod p representations and classical correspondences is limited.

The results extend understanding of mod p representation theory over p-adic fields.

Abstract

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we first compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands and Jacquet-Langlands correspondence are not compatible with the reduction, except in the cases considered by Vigneras.