Triangulable 𝒪F-analytic (φq,Γ)-modules of rank 2

Fourquaux, Lionel; Xie, Bingyong

doi:10.2140/ant.2013.7.2545

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Triangulable $\mathcal O_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2

Lionel Fourquaux and Bingyong Xie

Vol. 7 (2013), No. 10, 2545–2592

DOI: 10.2140/ant.2013.7.2545

Abstract

The theory of $(φ_{q}, Γ)$ -modules is a generalization of Fontaine’s theory of $(φ, Γ)$ -modules, which classifies $G_{F}$ -representations on $O_{F}$ -modules and $F$ -vector spaces for any finite extension $F$ of $ℚ_{p}$ . In this paper following Colmez’s method we classify triangulable $O_{F}$ -analytic $(φ_{q}, Γ)$ -modules of rank $2$ . In the process we establish two kinds of cohomology theories for $O_{F}$ -analytic $(φ_{q}, Γ)$ -modules. Using them, we show that if $D$ is an étale $O_{F}$ -analytic $(φ_{q}, Γ)$ -module such that $D^{φ_{q} = 1, Γ = 1} = 0$ (i.e., $V^{G_{F}} = 0$ , where $V$ is the Galois representation attached to $D$ ), then any overconvergent extension of the trivial representation of $G_{F}$ by $V$ is $O_{F}$ -analytic. In particular, contrary to the case of $F = ℚ_{p}$ , there are representations of $G_{F}$ that are not overconvergent.

Keywords

triangulable, analytic

Mathematical Subject Classification 2010

Primary: 11S20

Milestones

Received: 9 October 2012

Revised: 11 March 2013

Accepted: 11 April 2013

Published: 18 January 2014

Authors

Lionel Fourquaux
Université Rennes 1 35042 Rennes France

Bingyong Xie
Department of Mathematics East China Normal University Dongchuan Road, 500 Shanghai, 200241 PR China

Vol. 7, No. 10, 2013