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 \(\ell \).
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Le séjour de V. Sécherre à l’University of East Anglia (Norwich) en mai 2014, durant lequel une partie de ce travail a été effectuée, a été financé par l’EPSRC grant EP/H00534X/1. Il remercie chaleureusement Shaun Stevens pour son invitation et les discussions à propos de ce travail. A. Mínguez remercie João Pedro Dos Santos et Erez Lapid pour les discussions à propos de ce travail.
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Mínguez, A., Sécherre, V. Correspondance de Jacquet–Langlands locale et congruences modulo \(\ell \) . Invent. math. 208, 553–631 (2017). https://doi.org/10.1007/s00222-016-0696-y
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DOI: https://doi.org/10.1007/s00222-016-0696-y