Local models of Shimura varieties and a conjecture of Kottwitz
Georgios Pappas,Xinwen Zhu +1 more
TLDR
In this paper, a group theoretic definition of local models of Grassmannian degenerations of Shimura varieties has been given, which are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general reductive groups.Abstract:
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.read more
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Towards a theory of local Shimura varieties
Michael Rapoport,Eva Viehmann +1 more
TL;DR: In this article, a survey article that advertises the idea that there should exist a theory of p-adic local analogues of Shimura varieties is presented, and the towers of rigid-analytic spaces defined by Rapoport-Zink spaces are reviewed.
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On the p-adic cohomology of the Lubin-Tate tower
TL;DR: In this article, a finiteness result for the p-adic cohomology of the Lubin-Tate tower is proved for any n ≥ 1 and a padic field F, where F is the absolute Galois group of F and D/F is the central division algebra of invariant 1/n.
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Integral models of Shimura varieties with parahoric level structure
Mark Kisin,Georgios Pappas +1 more
TL;DR: In this article, the authors construct integral models over $p > 2$>>\s for Shimura varieties with parahoric level structure, attached to Shimura data of abelian type, such that $G$ ≥ 2$¯¯ splits over a tamely ramified extension of ${\mathbf {Q}}_{\,p}$¯¯.
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The stable Bernstein center and test functions for Shimura varieties
TL;DR: In this article, the stable Bernstein center of a PEL Shimura variety with arbitrary level structure is used to express the Hasse-Weil zeta function in terms of automorphic $L$-functions.
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Toroidal compactifications of integral models of Shimura varieties of Hodge type
TL;DR: In this paper, the authors construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type, and show that these compactifications are canonical in the sense that they cover all previously known cases of PEL type, as well as all Hodge cycles involving parahoric level structures.
References
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Linear Algebraic Groups
TL;DR: A linear algebraic group over an algebraically closed field k is a subgroup of a group GL n (k) of invertible n × n-matrices with entries in k, whose elements are precisely the solutions of a set of polynomial equations in the matrix coordinates as mentioned in this paper.
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Groupes réductifs sur un corps local : I. Données radicielles valuées
TL;DR: In this paper, the authors present a set of conditions générales d'utilisation of commercial or impression systématique, i.e., the copie ou impression de ce fichier doit contenir la présente mention de copyright.