Open Access
2013 Triangulable $\mathcal O_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2
Lionel Fourquaux, Bingyong Xie
Algebra Number Theory 7(10): 2545-2592 (2013). DOI: 10.2140/ant.2013.7.2545


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., VGF=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.


Download Citation

Lionel Fourquaux. Bingyong Xie. "Triangulable $\mathcal O_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2." Algebra Number Theory 7 (10) 2545 - 2592, 2013.


Received: 9 October 2012; Revised: 11 March 2013; Accepted: 11 April 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1297.11145
MathSciNet: MR3194651
Digital Object Identifier: 10.2140/ant.2013.7.2545

Primary: 11S20

Keywords: analytic , triangulable

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 10 • 2013
Back to Top