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

Scholze, Peter; Shin, Sug

doi:10.1090/S0894-0347-2012-00752-8

On the cohomology of compact unitary group Shimura varieties at ramified split places
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by Peter Scholze and Sug Woo Shin

J. Amer. Math. Soc. 26 (2013), 261-294

DOI: https://doi.org/10.1090/S0894-0347-2012-00752-8

Published electronically: August 20, 2012

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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 http://dx.doi.org/ 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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