The theory of -modules is a generalization of Fontaine’s theory of -modules, which classifies -representations on -modules and -vector spaces for any finite extension of . In this paper following Colmez’s method we classify triangulable -analytic -modules of rank . In the process we establish two kinds of cohomology theories for -analytic -modules. Using them, we show that if is an étale -analytic -module such that (i.e., , where is the Galois representation attached to ), then any overconvergent extension of the trivial representation of by is -analytic. In particular, contrary to the case of , there are representations of that are not overconvergent.
"Triangulable $\mathcal O_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2." Algebra Number Theory 7 (10) 2545 - 2592, 2013. https://doi.org/10.2140/ant.2013.7.2545