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.
Mots-clés : Galois representations, automorphic forms, $p$-adic Langlands program
@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/} }
TY - JOUR AU - Chojecki, Przemyslaw AU - Sorensen, Claus TI - Weak local-global compatibility in the $p$-adic Langlands program for $U(2)$ JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2017 SP - 101 EP - 133 VL - 137 PB - European Mathematical Society Publishing House PP - Zuerich, Switzerland UR - http://www.numdam.org/articles/10.4171/RSMUP/137-6/ DO - 10.4171/RSMUP/137-6 ID - RSMUP_2017__137__101_0 ER -
%0 Journal Article %A Chojecki, Przemyslaw %A Sorensen, Claus %T Weak local-global compatibility in the $p$-adic Langlands program for $U(2)$ %J Rendiconti del Seminario Matematico della Università di Padova %D 2017 %P 101-133 %V 137 %I European Mathematical Society Publishing House %C Zuerich, Switzerland %U http://www.numdam.org/articles/10.4171/RSMUP/137-6/ %R 10.4171/RSMUP/137-6 %F RSMUP_2017__137__101_0
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/
Cité par Sources :