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

Abstract

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we 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 correspondence and the local Jacquet–Langlands correspondence are not compatible with the reduction, except in the cases considered by Vignéras.