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


Author: Pierre Colmez
Journal: Represent. Theory 20 (2016), 187-248
MSC (2010): Primary 22E50
DOI: https://doi.org/10.1090/ert/484
Published electronically: July 15, 2016
MathSciNet review: 3522263
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Abstract:

We extend the $p$-adic local Langlands correspondence for $\textbf {GL}_2(\mathbf {Q}_p)$ to a correspondence $\Delta \mapsto \Pi (\Delta )$ between $(\varphi ,\Gamma )$-modules of rank $2$ over the Robba ring and certain locally analytic representations of $\textbf {GL}_2(\mathbf {Q}_p)$. If $\Delta$ is isocline, one uses the existing correspondence; in the remaining cases one builds a $\textbf {GL}_2(\mathbf {Q}_p)$-module from parabolically induced locally analytic representations and their duals. This construction extends to $\textbf {GL}_2(F)$ 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 $\textbf {GL}_2(\mathbf {Q}_p)$ 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 $\textbf {GL}_2(\mathbf {Q}_p)$. Si $\Delta$ est isocline, on se ramène à la correspondance déjà établie ; dans le cas contraire, on construit un $\textbf {GL}_2(\mathbf {Q}_p)$-module formé d’induites paraboliques localement analytiques et de leurs duales. Cette construction s’étend à $\textbf {GL}_2(F)$, 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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Additional Information

Pierre Colmez
Affiliation: Institut de mathématiques de Jussieu, Centre National de la Recherche Scientifique, 4 place Jussieu, 75005 Paris, France
MR Author ID: 50720
Email: pierre.colmez@imj-prg.fr

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