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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Représentations localement analytiques de $\textbf {GL}_2(\mathbf {Q}_p)$ et $(\varphi ,\Gamma )$-modules
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by Pierre Colmez PDF
Represent. Theory 20 (2016), 187-248 Request permission


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:
  • Received by editor(s): November 28, 2015
  • Received by editor(s) in revised form: June 10, 2016
  • Published electronically: July 15, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Represent. Theory 20 (2016), 187-248
  • MSC (2010): Primary 22E50
  • DOI:
  • MathSciNet review: 3522263