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Local models for Weil-restricted groups

Part of: Arithmetic problems. Diophantine geometry Special varieties Arithmetic algebraic geometry

Published online by Cambridge University Press:  27 September 2016

Brandon Levin
Brandon Levin*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, IL 60637, USA email bwlevin@math.uchicago.edu
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Abstract

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We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194 (2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when $p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

Keywords

algebraic groupsaffine Grassmanniansnearby cyclesShimura varieties

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 14M15: Grassmannians, Schubert varieties, flag manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2563 - 2601
Copyright
© The Author 2016 

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