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Let F be a non-Archimedean local field of residual characteristic p, and{\ell} be a prime number different from p. We consider the localJacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F)and an inner form GL(m,D). We show that it respects the relationship ofcongruence modulo {\ell}. More precisely, we show that two integral {\ell}-adicdiscrete series of GL(m,D) are congruent modulo {\ell} if and only if the sameholds for their Jacquet-Langlands transfers to GL(m,D).

The local Jacquet--Langlands correspondence and congruences modulo {\ell}