Triangulable O_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 CV-modules and F-vector spaces for any finite extension F of Q_p. In this paper following Colmez's method we classify triangulable CV-analytic (φ_q, Γ)-modules of rank 2. In the process we establish two kinds of cohomology theories for Of-analytic (φ_q, Γ)-modules. Using them, we show that if D is an étale CV-analytic (φ_q, Γ)-module such that D^{φ q=1, Γ=1} = 0 (i.e., V^GF = 0, where V is the Galois representation attached to D), then any overconvergent extension of the trivial representation of Gf by V is CV-analytic. In particular, contrary to the case of F = Q_p, there are representations of Gf that are not overconvergent.