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

Abstract

We study the completed cohomology $\widehat{H}^0$ of a definite unitary group $G$ in two variables associated with a CM-extension $\mathcal K/F$. When the prime $p$ splits, we prove that (under technical asumptions) the $p$-adic local Langlands correspondence for GL$_2(\mathbb Q_p)$ occurs in $\widehat{H}^0$. As an application, we obtain a result towards the Fontaine–Mazur conjecture over $\mathcal K$. If $x$ is a point on the eigenvariety such that $\rho_x$ is geometric (and satisfying additional hypotheses which we suppress), then $x$ must be a classical point. Thus, not only is $\rho_x$ modular, but the weight of $x$ defines an accessible refinement. This follows from a recent result of Colmez (which describes the locally analytic vectors in $p$-adic unitary principal series), knowing that $\rho_x$ admits a triangulation compatible with the weight.