Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$, valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

Let K be a finite unramified extension of Qp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional -representations of GK and characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example $F=\mathbb{Q}_p$), and let GF be the absolute Galois group of F. The main result of this article is the following: a p-adic representation of GF, which is a limit of subquotients of crystalline representations with Hodge–Tate weights in an interval [a; b], is itself crystalline with Hodge–Tate weights in [a; b]. In order to show this, we study the $(\phi,\Gamma)$-modules attached to crystalline representations, which allows us to improve some results of Fontaine, Wach and Colmez.

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.