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Représentations localement analytiques de $ {GL}_2(\mathbf{Q}_p)$ et $ (\varphi,\Gamma)$-modules

Author: Pierre Colmez
Journal: Represent. Theory 20 (2016), 187-248
MSC (2010): Primary 22E50
Published electronically: July 15, 2016
MathSciNet review: 3522263
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Abstract: We extend the $ p$-adic local Langlands correspondence for $ {\bf GL}_2(\mathbf {Q}_p)$ to a correspondence $ \Delta \mapsto \Pi (\Delta )$ between $ (\varphi ,\Gamma )$-modules of rank $ 2$ over the Robba ring and certain locally analytic representations of $ {\bf GL}_2(\mathbf {Q}_p)$. If $ \Delta $ is isocline, one uses the existing correspondence; in the remaining cases one builds a $ {\bf GL}_2(\mathbf {Q}_p)$-module from parabolically induced locally analytic representations and their duals. This construction extends to $ {\bf GL}_2(F)$ if $ F$ is a finite extension of $ \mathbf {Q}_p$, which suggests that the same should be true for the correspondence $ \Delta \mapsto \Pi (\Delta )$.

RéSUMé. Nous étendons la correspondance de Langlands locale $ p$-adique pour $ {\bf GL}_2(\mathbf {Q}_p)$ en une correspondance $ \Delta \mapsto \Pi (\Delta )$ entre les $ (\varphi ,\Gamma )$-modules de rang 2 sur l'anneau de Robba et certaines représentations localement analytiques de $ {\bf GL}_2(\mathbf {Q}_p)$. Si $ \Delta $ est isocline, on se ramène à la correspondance déjà établie ; dans le cas contraire, on construit un $ {\bf GL}_2(\mathbf {Q}_p)$-module formé d'induites paraboliques localement analytiques et de leurs duales. Cette construction s'étend à $ {\bf GL}_2(F)$, si $ F$ est une extension finie de $ \mathbf {Q}_p$, ce qui suggère qu'il en est de même de la correspondance $ \Delta \mapsto \Pi (\Delta )$.

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Pierre Colmez
Affiliation: Institut de mathématiques de Jussieu, Centre National de la Recherche Scientifique, 4 place Jussieu, 75005 Paris, France

Received by editor(s): November 28, 2015
Received by editor(s) in revised form: June 10, 2016
Published electronically: July 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society