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

TL;DR

This paper characterizes certain deformations of locally algebraic representations of GL_2(Q_p) that remain within the same class, providing insights relevant to the Breuil-Mézard conjecture.

Contribution

It describes first-order deformations of irreducible unitary completions of locally algebraic GL_2(Q_p) representations that are also completions of locally algebraic representations, answering Paskunas' question.

Findings

Identifies specific first-order deformations preserving local algebraicity.

Provides a classification of such deformations.

Offers applications to the Breuil-Mézard conjecture.

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.