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.


11S20 Galois theory
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