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Local models of Shimura varieties and a conjecture of Kottwitz

Pappas, G. and Zhu, X. (2013) Local models of Shimura varieties and a conjecture of Kottwitz. Inventiones Mathematicae, 194 (1). pp. 147-254. ISSN 0020-9910.

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We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

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Additional Information:© 2012 Springer. Received: 30 June 2012 / Accepted: 12 November 2012 / Published online: 14 December 2012. Erratum to: Local models of Shimura varieties and a conjecture of Kottwitz. Inventiones mathematicae, October 2013, Volume 194, Issue 1, p 255. G. P partially supported by NSF grant DMS11-02208. X. Z partially supported by NSF grant DMS10-01280. The authors would like to warmly thank M. Rapoport, B. Conrad, T. Haines and B. Levin for useful discussions and comments.
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Subject Keywords:Shimura variety, affine flag variety, local model, group scheme, nearby cycles
Issue or Number:1
Record Number:CaltechAUTHORS:20140919-152324978
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49869
Deposited By: George Porter
Deposited On:23 Sep 2014 00:38
Last Modified:03 Oct 2019 07:17

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