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

Abstract

Let B(V) be the admissible unitary GL2(Qp)-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 B(V)la of 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 B(V)la. In particular, we show the existence of rigid analytic vectors inside B(V)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 B(V)la.