1469 articles – 5618 references  [version française]
 HAL: hal-00709786, version 1
 arXiv: 1206.2102
 Triangulable $\CO_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2
 (2012-06-11)
 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.
 1: Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Research team: Géométrie algébrique
 Subject : Mathematics/Number Theory