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Triangulable $\CO_F$-analytic $(φ_q,Γ)$-modules of rank 2

The theory of $(φ_q,Γ)$-modules is a generalization of Fontaine's theory of $(φ,Γ)$-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 $(φ_q,Γ)$-modules of rank 2. In this process we establish two kinds of cohomology theories for $\CO_F$-analytic $(φ_q,Γ)$-modules. Using them we show that, if $D$ is an $\CO_F$-analytic $(φ_q,Γ)$-module such that $D^{φ_q=1,Γ=1}=0$ 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.