Strong local-global compatibility in the p-adic Langlands program for U(2)

Abstract

We prove that certain Galois-isotypic parts of the completed cohomology group for U(2) can be written as a completed tensor product of a representation coming from the p-adic Langlands correspondence for $G{L}_2({\mathbb{Q}}_p)$ and a representation arising via the local Langlands correspondence in famillies of Emerton and Helm.