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Article Dans Une Revue Algebra & Number Theory Année : 2013

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

Résumé

The theory of $(\varphi_q,\Gamma)$-modules is a generalization of Fontaine's theory of $(\varphi,\Gamma)$-modules, which classifies $G_F$-representations on $\CO_F$-modules and $F$-vector spaces for any finite extension $F$ of $\BQ_p$. In this paper following Colmez's method we classify triangulable $\CO_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2. In this process we establish two kinds of cohomology theories for $\CO_F$-analytic $(\varphi_q,\Gamma)$-modules. Using them we show that, if $D$ is an $\CO_F$-analytic $(\varphi_q,\Gamma)$-module such that $D^{\varphi_q=1,\Gamma=1}=0$, then any extension of the trivial representation of $G_F$ by the representation attached to $D$ that is overconvergent is $\CO_F$-analytic. In particular, contrarily to the case of $F=\BQ_p$, there are representations of $G_F$ that are not overconvergent.

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Dates et versions

hal-00709786 , version 1 (19-06-2012)

Identifiants

Citer

Lionel Fourquaux, Bingyong Xie. Triangulable $\CO_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2. Algebra & Number Theory, 2013, 7 (10), pp.2545-2592. ⟨10.2140/ant.2013.7.2545⟩. ⟨hal-00709786⟩
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