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We study the completed cohomology of a definite unitary group in two variables associated with a CM-extension . When the prime splits, we prove that (under technical asumptions) the -adic local Langlands correspondence for GL occurs in . As an application, we obtain a result towards the Fontaine–Mazur conjecture over . If is a point on the eigenvariety such that is geometric (and satisfying additional hypotheses which we suppress), then must be a classical point. Thus, not only is modular, but the weight of defines an accessible refinement. This follows from a recent result of Colmez (which describes the locally analytic vectors in -adic unitary principal series), knowing that admits a triangulation compatible with the weight.
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Przemyslaw Chojecki, Claus Sorensen, Weak local-global compatibility in the -adic Langlands program for . Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 101–133