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 ${\hat{H}}_{\bar{\rho }}^0$ (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.