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

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 $G_F$-representations on ${\mathcal{O}}_F$-modules and $F$-vector spaces for any finite extension $F$ of ${\mathbb{Q}}_p$. In this paper following Colmez’s method we classify triangulable ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-modules of rank $2$. In the process we establish two kinds of cohomology theories for ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-modules. Using them, we show that if $D$ is an étale ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-module such that $D^{{\phi }_q=1,\Gamma =1}=0$ (i.e., $V^{G_F}=0$, where $V$ is the Galois representation attached to $D$), then any overconvergent extension of the trivial representation of $G_F$ by $V$ is ${\mathcal{O}}_F$-analytic. In particular, contrary to the case of $F={\mathbb{Q}}_p$, there are representations of $G_F$ that are not overconvergent.