Vol. 7, No. 10, 2013

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Triangulable $\mathcal O_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2

Lionel Fourquaux and Bingyong Xie

Vol. 7 (2013), No. 10, 2545–2592
Abstract

The theory of (φq,Γ)-modules is a generalization of Fontaine’s theory of (φ,Γ)-modules, which classifies GF-representations on OF-modules and F-vector spaces for any finite extension F of p. In this paper following Colmez’s method we classify triangulable OF-analytic (φq,Γ)-modules of rank 2. In the process we establish two kinds of cohomology theories for OF-analytic (φq,Γ)-modules. Using them, we show that if D is an étale OF-analytic (φq,Γ)-module such that Dφq=1,Γ=1 = 0 (i.e., V GF = 0, where V is the Galois representation attached to D), then any overconvergent extension of the trivial representation of GF by V is OF-analytic. In particular, contrary to the case of F = p, there are representations of GF 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