Représentations cristallines de torsion

Abstract

In this paper, we study representations of a Galois group of a local field absolutely unramified whose residue field is perfect of characteristic $p$ and give two ways of recognize crystalline representations with weights between $r$ and $r+p-1$. The first one is the description of terms of weakly admissible filtered modules proved by J.-M. Fontaine and G. Laffaille (Ann Sc ENS 1982): here a new proof of this result is proposed (second chapter). The second criterion (third chapter) is the equivalence between crystalline representations with weights between $r$ and $r+p-1$ and representations of finite ‘cr-height’ ${\leqslant} p-1$, result announced by J.-M. Fontaine in a paper edited a few years ago where he introduced the category of $(\varphi, \Gamma)$-modules.