Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



On the cohomology of compact unitary group Shimura varieties at ramified split places

Authors: Peter Scholze and Sug Woo Shin
Journal: J. Amer. Math. Soc. 26 (2013), 261-294
MSC (2010): Primary 11F70, 11F80, 11G18, 11R39, 11S37; Secondary 14G35, 11F72, 22E50, 22E55
Published electronically: August 20, 2012
MathSciNet review: 2983012
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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 10.1090/S0894-0347-2012-00753-X, following the stabilization given by Kottwitz. This introduces endoscopic transfers of the functions $ \phi _{\tau ,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 $ \mathrm {GL}_n$.

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Additional Information

Peter Scholze
Affiliation: Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Sug Woo Shin
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 – and – Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Received by editor(s): November 9, 2011
Received by editor(s) in revised form: July 23, 2012
Published electronically: August 20, 2012
Additional Notes: This work was written while the first author was a Clay Research Fellow.
The second author’s work was supported by the National Science Foundation during his stay at the Institute for Advanced Study under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.