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

TL;DR

This paper investigates the presence of the p-adic local Langlands correspondence within the completed cohomology of certain unitary groups, providing insights into the Fontaine-Mazur conjecture over CM fields.

Contribution

It demonstrates the occurrence of the p-adic local Langlands correspondence in the cohomology for U(2) and extends understanding of the Fontaine-Mazur conjecture in this context.

Findings

p-adic local Langlands correspondence appears in cohomology when p splits

Provides evidence supporting Fontaine-Mazur conjecture over CM fields

Connects Emerton's work for GL(2) to unitary groups U(2)

Abstract

Inspired by Emerton's work for GL(2), we study the completed cohomology of the tower of finite sets associated with a definite unitary group in two variables. When p splits (and other technical assumptions are fulfilled), we show that the p-adic local Langlands correspondence for GL(2) (over Q_p) occurs in the cohomology. We give an application to the Fontaine-Mazur conjecture over CM fields.