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

TL;DR

This paper classifies a special class of rank 2 $(_q,)$-modules over $_O_F$, introduces new cohomology theories for them, and explores their implications for Galois representations and overconvergence.

Contribution

It provides a classification of triangulable $_O_F$-analytic $(_q,)$-modules of rank 2 and develops two cohomology theories for these modules.

Findings

Established cohomology theories for $_O_F$-analytic $(_q,)$-modules.

Proved conditions under which overconvergent extensions are $_O_F$-analytic.

Showed existence of non-overconvergent Galois representations for $F eq Q_p$.

Abstract

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 $D^{\varphi_q=1,\Gamma=1}=0$ 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 $\CO_F$-analytic. In particular, contrarily to the case of $F=\BQ_p$, there are representations of $G_F$ that are not overconvergent.