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

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 modp 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.