Weak local-global compatibility in the p-adic Langlands program for U(2)
Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 101-133.

We study the completed cohomology H ^ 0 of a definite unitary group G in two variables associated with a CM-extension 𝒦/F. When the prime p splits, we prove that (under technical asumptions) the p-adic local Langlands correspondence for GL 2 ( p ) occurs in H ^ 0 . As an application, we obtain a result towards the Fontaine–Mazur conjecture over 𝒦. If x is a point on the eigenvariety such that ρ x is geometric (and satisfying additional hypotheses which we suppress), then x must be a classical point. Thus, not only is ρ 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 ρ x admits a triangulation compatible with the weight.

DOI : 10.4171/RSMUP/137-6
Classification : 11
Mots clés : Galois representations, automorphic forms, $p$-adic Langlands program
Chojecki, Przemyslaw 1 ; Sorensen, Claus 2

1 Oxford University, OXFORD, UNITED KINGDOM
2 University of California, San Diego (UCSD), LA JOLLA, UNITED STATES
@article{RSMUP_2017__137__101_0,
     author = {Chojecki, Przemyslaw and Sorensen, Claus},
     title = {Weak local-global compatibility in the $p$-adic {Langlands} program for $U(2)$},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {101--133},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {137},
     year = {2017},
     doi = {10.4171/RSMUP/137-6},
     mrnumber = {3652871},
     zbl = {1428.11082},
     url = {http://www.numdam.org/articles/10.4171/RSMUP/137-6/}
}
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Chojecki, Przemyslaw; Sorensen, Claus. Weak local-global compatibility in the $p$-adic Langlands program for $U(2)$. Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 101-133. doi : 10.4171/RSMUP/137-6. http://www.numdam.org/articles/10.4171/RSMUP/137-6/

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