## Triangulable $$\mathcal O_F$$-analytic $$(\varphi_q,\Gamma)$$-modules of rank 2.(English)Zbl 1297.11145

Summary: The theory of $$(\varphi _{q},\Gamma )$$-modules is a generalization of J.-M. Fontaine’s theory of $$(\varphi ,\Gamma )$$-modules [in: Jacobiennes généralisées globale relatives. The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 249–309 (1990; Zbl 0743.11066)], 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 P. Colmez’s method [in: $$p$$-adic representations of $$p$$-adic groups II: Representations of $$\text{GL}_2 (\mathbb Q_p)$$ et $$(\varphi, \gamma)$$-modules. Paris: Société Mathématique de France. Astérisque 330, 281–509 (2010; Zbl 1218.11107)] we classify triangulable $$\mathcal O_{F}$$-analytic $$(\varphi _{q},\Gamma )$$-modules of rank 2. In the process we establish two kinds of cohomology theories for $$\mathcal O_{F}$$-analytic $$(\varphi _{q},\Gamma )$$-modules. Using them, we show that if D is an étale $$\mathcal O_{F}$$-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 $$\mathcal O_{F}$$-analytic. In particular, contrary to the case of $$F = \mathbb Q_{p}$$, there are representations of $$G_{F}$$ that are not overconvergent.

### MSC:

 11S20 Galois theory

### Citations:

Zbl 0743.11066; Zbl 1218.11107
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