Analyse p-adique et repr\'esentations Galoisiennes

Abstract

The notion of a p-adic de Rham representation of the absolute Galois group of a p-adic field was introduced about twenty years ago (see e.g. [Fo93]). Three important results for this theory have been obtained recently: The structure theorem for the {\it almost $C_p$-representations}, the theorem {\it weakly admissible implies admissible} and the theorem {\it de Rham implies potentially semi-stable}. The proofs of the first two theorems are closely related to the study of a new kind of analytic groups, the {\it Banach-Colmez spaces} and the proof of the third uses deep results on p-{\it adic differential equations on the Robba ring}.