#### Vol. 7, No. 10, 2013

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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
##### Abstract

The theory of $\left({\phi }_{q},\Gamma \right)$-modules is a generalization of Fontaine’s theory of $\left(\phi ,\Gamma \right)$-modules, which classifies ${G}_{\phantom{\rule{0.3em}{0ex}}F}$-representations on ${\mathsc{O}}_{F}$-modules and $F$-vector spaces for any finite extension $F$ of ${ℚ}_{p}$. In this paper following Colmez’s method we classify triangulable ${\mathsc{O}}_{F}$-analytic $\left({\phi }_{q},\Gamma \right)$-modules of rank $2$. In the process we establish two kinds of cohomology theories for ${\mathsc{O}}_{F}$-analytic $\left({\phi }_{q},\Gamma \right)$-modules. Using them, we show that if $D$ is an étale ${\mathsc{O}}_{F}$-analytic $\left({\phi }_{q},\Gamma \right)$-module such that ${D}^{{\phi }_{q}=1,\Gamma =1}=0$ (i.e., ${V}^{{G}_{\phantom{\rule{0.3em}{0ex}}F}}=0$, where $V$ is the Galois representation attached to $D$), then any overconvergent extension of the trivial representation of ${G}_{\phantom{\rule{0.3em}{0ex}}F}$ by $V$ is ${\mathsc{O}}_{F}$-analytic. In particular, contrary to the case of $F={ℚ}_{p}$, there are representations of ${G}_{\phantom{\rule{0.3em}{0ex}}F}$ that are not overconvergent.

##### Keywords
triangulable, analytic
Primary: 11S20