Extensions de repr\'esentations de de Rham et vecteurs localement alg\'ebriques

Abstract

Let $\Pi$ be an irreducible unitary completion of a locally algebraic ${\rm GL}_2(\qp)$-representation. We describe those first-order deformations of $\Pi$ which are themselves completions of a locally algebraic representation. This answers a question of Paskunas and has direct applications to the Breuil-M\'ezard conjecture.