Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

TL;DR

This paper explicitly describes the rigid analytic vectors within the locally analytic vectors of certain crystalline Galois representations associated with $p$-adic Langlands, confirming their existence and non-triviality.

Contribution

It provides an explicit description of rigid analytic vectors in the locally analytic representation $ extbf{B}(V)_{ ext{la}}$ and proves their non-triviality, advancing understanding in $p$-adic Langlands.

Findings

Existence of non-zero rigid analytic vectors in $ extbf{B}(V)_{ ext{la}}$

Explicit description of these vectors using rigid analytic subgroups of $GL(2)$

Identification of a rigid analytic representation inside the locally analytic one

Abstract

Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\mathbf{B}(V)_{\mathrm{la}}$. In particular, we show the existence of rigid analytic vectors inside $\mathbf{B}(V)_{\mathrm{la}}$ and prove that its non-null. This gives us a rigid analytic representation (in the sense of Emerton) lying inside the locally analytic representation $\mathbf{B}(V)_{\mathrm{la}}$.