  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.

Number Theory;Representation Theory