Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$

Abstract

Let A be an affinoid Q p -algebra, in the sense of Tate. We develop a theory of locally convex A-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in A-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma. In the case of a field this has been substantially developed by Kohlhaase.As an application we propose a p-adic Langlands correspondence in families: For a regular trianguline (ϕ, Γ)-module of dimension 2 over the relative Robba ring R A we construct a locally analytic GL 2 (Q p )representation in A-modules.