Inventiones mathematicae

, Volume 208, Issue 2, pp 553–631 | Cite as

Correspondance de Jacquet–Langlands locale et congruences modulo \(\ell \)

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Abstract

Let \(\mathrm{F}\) 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 \(\mathrm{GL}_n(\mathrm{F})\) and an inner form \(\mathrm{GL}_m(\mathrm{D})\). We show that it respects the relationship of congruence modulo \(\ell \). More precisely, we show that two integral \(\ell \)-adic discrete series of \(\mathrm{GL}_n(\mathrm{F})\) are congruent modulo \(\ell \) if and only if the same holds for their Jacquet–Langlands transfers to \(\mathrm{GL}_m(\mathrm{D})\). We also prove that the Langlands–Jacquet morphism from the Grothendieck group of finite length \(\ell \)-adic representations of \(\mathrm{GL}_n(\mathrm{F})\) to that of \(\mathrm{GL}_m(\mathrm{D})\) defined by Badulescu is compatible with reduction mod \(\ell \).

Mathematics Subject Classification

22E50 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu, Paris Rive GaucheUniversité Pierre et Marie CurieParisFrance
  2. 2.Laboratoire de Mathématiques de Versailles, UVSQ, CNRSUniversité Paris-SaclayVersaillesFrance

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