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Extensions de représentations de de Rham et vecteurs localement algébriques
Let ${\rm\Pi}$ be an irreducible unitary completion of a locally algebraic $\text{GL}_{2}(\mathbf{Q}_{p})$ -representation. We describe those first-order deformations of ${\rm\Pi}$ which are themselves completions of a locally algebraic representation. This answers a question of Paškūnas and has direct applications to the Breuil–Mézard conjecture.