On the cohomology of compact unitary group Shimura varieties at ramified split places
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Shin_On the cohomology.pdf
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Author(s)
Scholze, Peter
Shin, Sug Woo
Date Issued
August 2012
Journal
Journal of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Scholze, Peter, and Sug Woo Shin. “On the cohomology of compact unitary group Shimura varieties at ramified split places.” Journal of the American Mathematical Society 26, no. 1 (January 1, 2013): 261-294. © 2012 American Mathematical Society
Version
Final published version
Abstract
In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases.
We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in the article http://dx.doi.org/10.1090/S0894-0347-2012-00753-X, following the stabilization given by Kottwitz. This introduces endoscopic transfers of the functions φτ,h introduced in the above article. We state a general conjecture relating these endoscopic transfers with Langlands parameters.
We verify this conjecture in all cases of EL type and deduce new results about the endoscopic part of the cohomology of Shimura varieties. This allows us to simplify the construction of Galois representations attached to conjugate self-dual regular algebraic cuspidal automorphic representations of GL[subscript n].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
http://dx.doi.org/10.1090/s0894-0347-2012-00752-8
