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

Abstract

For certain mod $p$ Galois representations $\bar{\rho}$, arising from modular forms on definite unitary groups in two variables, we express the $\bar{\rho}$-part of completed cohomology $\widehat{H}_{\bar{\rho}}^0$ (away from $\Sigma=\Sigma_p\cup \Sigma_0$) as a tensor product $\Pi_p\otimes \Pi_{\Sigma_0}$. Here $\Pi_p$ is attached to the universal deformation $\rho^{univ}$ via the $p$-adic local Langlands correspondence for GL$_2(\mathbb Q_p)$, and $\Pi_{\Sigma_0}$ is given by the local Langlands correspondence in families, of Emerton and Helm.