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

TL;DR

This paper demonstrates that the local Jacquet-Langlands correspondence preserves congruences modulo a prime , establishing a key compatibility between representations of GL(n,F) and its inner forms over non-Archimedean local fields.

Contribution

It proves that the Jacquet-Langlands correspondence respects congruences modulo  for ll-adic discrete series, a novel result linking representation theory and congruence relations.

Findings

Congruences modulo  are preserved under Jacquet-Langlands transfer.

The correspondence is compatible with integral ll-adic structures.

The result applies to non-Archimedean local fields with residual characteristic p.

Abstract

Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-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 of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).