Correspondance de Jacquet-Langlands locale et congruences modulo l

Abstract

Let F be a non-Archimedean local field of residual characteristic p, and be a prime number different from p. We consider the local Jacquet-Langlands correspondence between -adic discrete series of GLn(F) and an inner form GLm(D). We show that it respects the relationship of congruence modulo . More precisely, we show that two integral -adic discrete series of GLn(F) are congruent modulo if and only if the same holds for their Jacquet- Langlands transfers to GLm(D). We also prove that the Langlands-Jacquet morphism from the Grothendieck group of finite length -adic representations of GLn(F) to that of GLm(D) defined by Badulescu is compatible with reduction mod l.