JournalsrsmupVol. 137falsepp. 101–133

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

  • Przemyslaw Chojecki

    Oxford University, UK
  • Claus Sorensen

    University of California, San Diego, USA
Weak local-global compatibility in the $p$-adic Langlands program for $U(2)$ cover

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We study the completed cohomology H^0\widehat{H}^0 of a definite unitary group GG in two variables associated with a CM-extension K/F\mathcal K/F. When the prime pp splits, we prove that (under technical asumptions) the pp-adic local Langlands correspondence for GL2(Qp)_2(\mathbb Q_p) occurs in H^0\widehat{H}^0. As an application, we obtain a result towards the Fontaine–Mazur conjecture over K\mathcal K. If xx is a point on the eigenvariety such that ρx\rho_x is geometric (and satisfying additional hypotheses which we suppress), then xx must be a classical point. Thus, not only is ρx\rho_x modular, but the weight of xx defines an accessible refinement. This follows from a recent result of Colmez (which describes the locally analytic vectors in pp-adic unitary principal series), knowing that ρx\rho_x admits a triangulation compatible with the weight.

Cite this article

Przemyslaw Chojecki, Claus Sorensen, Weak local-global compatibility in the pp-adic Langlands program for U(2)U(2). Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 101–133

DOI 10.4171/RSMUP/137-6