Topological flatness of orthogonal local models in the split, even case. I

Smithling, Brian

doi:10.1007/s00208-010-0562-y

Cited by 13 publications

(41 citation statements)

References 19 publications

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“…This coincides with the description of M loc above and we have M loc = M(G, {µ 1 }) K ⊔ M(G, {µ 2 }) K . This local model was studied in the Iwahori case by Smithling in [67]. In general, the suggested correction of the naive local model in this orthogonal case [50] involves the so-called "spin condition" whose main virtue is that it also attempts to give a moduli theoretic interpretation.…”

Section: B2mentioning

confidence: 99%

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Local models of Shimura varieties and a conjecture of Kottwitz

2012

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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.

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Section: B2mentioning

confidence: 99%

“…As an example of a result we can obtain when G = G ♭ , let us suppose that G ♭ is a split even orthogonal similitude group as above and that L gives a maximal self-dual lattice chain (the Iwahori case considered in [67]). Then the special fiber of P ♭ is connected.…”

Section: B2mentioning

confidence: 99%

Local models of Shimura varieties and a conjecture of Kottwitz

2012

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“…The strongest combinatorial results relating local models and Adm(µ) are due to Brian Smithling, see e.g. [Sm1,Sm2,Sm3].…”

Section: Explicit Computation Of the Test Functionsmentioning

confidence: 99%

The stable Bernstein center and test functions for Shimura varieties

Haines

2014

Automorphic Forms and Galois Representations

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We elaborate the theory of the stable Bernstein center of a p-adic group G, and apply it to state a general conjecture on test functions for Shimura varieties due to the author and R. Kottwitz. This conjecture provides a framework by which one might pursue the Langlands-Kottwitz method in a very general situation: not necessarily PEL Shimura varieties with arbitrary level structure at p. We give a concrete reinterpretation of the test function conjecture in the context of parahoric level structure. We also use the stable Bernstein center to formulate some of the transfer conjectures (the "fundamental lemmas") that would be needed if one attempts to use the test function conjecture to express the local Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions.

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“…In § 1 we define the unitary Shimura varieties and reduce our structure problem to the case of local models. Actually, as this paper is being revised to its final version, we mention that Smithling has recently initiated a detailed study [32] of the even orthogonal local models introduced in § 8. In § 3 we recall results from [27] on affine flag varieties, and state the Local models.…”

Section: Is This Scheme Reduced (In Which Case It Is Normal With Ratmentioning

confidence: 99%

Local models in the ramified case. III Unitary groups

Pappas¹,

Rapoport²

2009

J. Inst. Math. Jussieu

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We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe 'good' p-adic integral models of these Shimura varieties and study theirétale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

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Topological flatness of orthogonal local models in the split, even case. I

Cited by 13 publications

References 19 publications

Local models of Shimura varieties and a conjecture of Kottwitz

Local models of Shimura varieties and a conjecture of Kottwitz

The stable Bernstein center and test functions for Shimura varieties

Local models in the ramified case. III Unitary groups

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