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

Abstract

Let $\mathbf{B}(V)$ be the admissible unitary $G{L}_2({\mathbb{Q}}_p)$-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 $\mathbf{B}(V{)}_{\mathrm{la}}$.