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.

For p≥3 an odd prime and a nonnegative integer r≤p−2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the anti-equivalence of categories between the category of strongly divisible lattices of weight r and the category of Galois stable -lattices in semi-stable p-adic Galois representations with Hodge–Tate weights in {0,…,r}.

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.