The p-adic local Langlands correspondence for GL 2 (Qp) is given by an exact functor from unitary Banach representations of GL 2 (Qp) to representations of the absolute Galois group G Qp of Qp. We prove, using characteristic 0 methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of GL 2 (Qp) and absolutely irreducible 2-dimensional representations of G Qp . This had already been proved, by characteristic p methods, but only for p ≥ 5.1 18 Contrary to [17], all results of [21] are proved for all primes p.

We compute the p p -adic geometric étale cohomology of the coverings of the Drinfeld half-plane, and we show that, if the base field is Q p \mathbf {Q}_p , this cohomology encodes the p p -adic local Langlands correspondence for 2 2 -dimensional de Rham representations (of weight 0 0 and 1 1 ).

Soit Π une représentation unitaire de GL 2 (Qp), topologiquement de longueur finie. Nous décrivons la sous-représentation Π an de ses vecteurs localement analytiques, et sa filtration par rayon d'analyticité, en termes du (ϕ, Γ)-module qui lui est associé via la correspondance de Langlands locale p-adique, et nous en déduisons que le complété universel de Π an n'est autre que Π.Abstract. -Let Π be a unitary representation of GL 2 (Qp), topologically of finite length. We describe the sub-representation Π an made of its locally analytic vectors, and its filtration by radius of analyticity, in terms of the (ϕ, Γ)-module attached to Π via the p-adic local Langlands correspondence, and we deduce that the universal completion of Π an is Π itself.1. δ est automatiquement localement analytique, donc la définition a un sens.

RésuméLet V be a two-dimensional absolutely irreducible Qp-representation of Gal(Qp/Qp) and let Π(V ) be the GL2(Qp) Banach representation associated by Colmez's p-adic Langlands correspondence. We establish a link between the action of the Lie algebra of GL2(Qp) on the locally analytic vectors Π(V )an of Π(V ), the connection ∇ on the (ϕ, Γ)-module associated to V and the Sen polynomial of V . This answers a question of Harris, concerning the infinitesimal character of Π(V )an . Using this result, we give a new proof of a theorem of Colmez, stating that Π(V ) has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate weights.