Locally analytic vectors of some crystabelian representations of GL_2(Qp)

TL;DR

This paper characterizes the locally analytic vectors of certain irreducible, crystabelian 2-dimensional p-adic Galois representations of G_Qp, confirming a conjecture of Breuil and verifying Emerton's conjecture in specific cases.

Contribution

It explicitly determines the locally analytic vectors of B(V) for a class of irreducible, crystabelian Galois representations, proving Breuil's conjecture and verifying Emerton's conjecture in these cases.

Findings

Determined B(V)an for irreducible, crystabelian, Frobenius semi-simple V.

Proved Breuil's conjecture on locally analytic vectors.

Verified Emerton's conjecture for certain V.

Abstract

For V a 2-dimensional p-adic representation of G_Qp, we denote by B(V) the admissible unitary representation of GL_2(Qp) attached to V under the p-adic local Langlands correspondence of GL_2(Qp) initiated by Breuil. In this article, 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 semi-simple with Hodge-Tate weights (0,k-1) for some integer k>=2; this proves a conjecture of Breuil. Using this result, we verify Emerton's conjecture that dim Ref^{\eta\otimes\psi}(V)=dim Exp^{\eta|\cdot|\otimes x\psi}(B(V)an\otimes(x|\cdot|\circ\det)) for those V which are irreducible, crystabelian and not exceptional.