Triangulable $\CO_F$-analytic $({\phi }_q,\Gamma )$-modules of rank 2

Abstract

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

35 pages

References in corpus (1)

• Classification of two dimensional split trianguline representations of $p$-adic fields

Cited by in corpus (4)

• Multivariable $(\phi ,\Gamma )$-modules and locally analytic vectors
• A note on multivariable $(\phi ,\Gamma )$-modules
• Finiteness of analytic cohomology of Lubin-Tate $({\phi }_L,{\Gamma }_L)$-modules
• $\epsilon$-isomorphisms for rank one $(\phi ,\Gamma )$-modules over Lubin-Tate Robba rings