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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.

For V a two-dimensional p-adic representation of Gℚp, we denote by B(V ) the admissible unitary representation of GL2(ℚp) attached to V under the p-adic local Langlands correspondence of GL2(ℚp) initiated by Breuil. In this paper, building on the works of Berger–Breuil and Colmez, we determine the locally analytic vectors B(V )an of B (V ) when V is irreducible, crystabelian and Frobenius semisimple with distinct Hodge–Tate weights; this proves a conjecture of Breuil. Using this result, we verify Emerton’s conjecture that dim Ref η⊗ψ (V )=dim Exp η∣⋅∣⊗xψ (B (V )an ⊗(x∣⋅∣∘det )) for those V which are irreducible, crystabelian and Frobenius semisimple.

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Extensions de représentations de de Rham et vecteurs localement algébriques

Locally analytic vectors of some crystabelian representations of GL2(ℚp)

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2 results

Extensions de représentations de de Rham et vecteurs localement algébriques

Locally analytic vectors of some crystabelian representations of GL2(ℚp)