Integral models of Shimura varieties with parahoric level structure

Kisin, Mark; Pappas, G.

doi:10.1007/s10240-018-0100-0

Cited by 63 publications

(139 citation statements)

References 55 publications

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“…Then the model [46], and under the assumption p ∤ |π 1 (G der )|. The statement above is for the modified local models of this paper and can be deduced by the results in [30]. Part (b) follows from [30, Thm.…”

Section: Shimura Varietiesmentioning

confidence: 91%

“…is a bijection. b) ( [30]) Assume that (G, X) is of Hodge type, that K is the stabilizer of a point in the Bruhat-Tits building of G, and that p does not divide |π 1 (G der )|. Then the model [46], and under the assumption p ∤ |π 1 (G der )|.…”

Section: Shimura Varietiesmentioning

confidence: 99%

“…semistable, reduction of a specific class of p-integral models of Shimura varieties. Such a specific class has been established in recent years for Shimura varieties with level structure which is parahoric at p, the most general result being due to M. Kisin and the second author [30]. The main point of these models is that their singularities are modeled by their associated local models, cf.…”

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confidence: 99%

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Good and semi-stable reductions of Shimura varieties

He¹,

Pappas²,

Rapoport³

2020

Journal De L’École Polytechnique — Mathématiques

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We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.Date: March 16, 2020. 1 2 X. HE, G. PAPPAS, AND M. RAPOPORTtwo examples above exhaust all possibilities (this statement has to be interpreted correctly, by considering the natural compactification of the modular curve). This comes down to a statement about the spectral decomposition under the action of the Hecke algebra of the ℓ-adic cohomology of modular curves. Unfortunately, the generalization of this statement to other Shimura varieties seems out of reach at the moment.The other possible interpretation of the question is to ask for good, resp. semistable, reduction of a specific class of p-integral models of Shimura varieties. Such a specific class has been established in recent years for Shimura varieties with level structure which is parahoric at p, the most general result being due to M. Kisin and the second author [30]. The main point of these models is that their singularities are modeled by their associated local models, cf. [40]. These are projective varieties which are defined in a certain sense by linear algebra, cf. [21,47]. More precisely, for every closed point of the reduction modulo p of the p-integral model of the Shimura variety, there is an isomorphism between the strict henselization of its local ring and the strict henselization of the local ring of a corresponding closed point in the reduction modulo p of the local model. Very often every closed point of the local model is attained in this way. In this case, the model of the Shimura variety has good, resp. semi-stable, reduction if and only if the local model has this property. Even when this attainment statement is not known, we deduce that if the local model has good, resp. semi-stable, reduction, then so does the model of the Shimura variety. Therefore, the emphasis of the present paper is on the structure of the singularities of the local models and our results determine local models which have good, resp. semi-stable reduction.Let us state now the main results of the paper, as they pertain to local models. See Section 3 for corresponding results for Shimura varieties, and Section 4 for results on Rapoport-Zink spaces. Local models are associated to local model triples. Here a LM triple over a finite extension F of Q p is a triple (G, {µ}, K) consisting of a reductive group G over F , a conjugacy class of cocharacters {µ} of G over an algebraic closure of F , and a parahoric group K of G. We sometimes write G for the affine smooth group scheme over O F corresponding to K. It is assumed that the cocharacter {µ} is minuscule (i.e., any root takes values in {0, ±1} on {µ}). The reflex field of the LM triple (G, {µ}, K) is the field of definition of the conjugacy class {µ}. One would like to associate to (G, {µ}, K) a local model M loc K (G, {µ}), a flat projective scheme over the ring...

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Section: Shimura Varietiesmentioning

confidence: 91%

Section: Shimura Varietiesmentioning

confidence: 99%

mentioning

confidence: 99%

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Good and semi-stable reductions of Shimura varieties

He¹,

Pappas²,

Rapoport³

2020

Journal De L’École Polytechnique — Mathématiques

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“…This is deduced in [Ki 2, 3.4.13] from Deligne's results [De 2], except for the claim in (2) that (G, X) can be chosen so that Z G is a torus, and the final claim in (3) that (G, X) may be chosen so that E(G, X) p = E(G ad , X ad ) p . That (G, X) may be chosen to satisfy the former condition is shown in [KP,4.6.22]. The construction of (G, X) in (2) uses [De 2, 2.3.10].…”

Section: Proofmentioning

confidence: 99%

Mod $p$ points on Shimura varieties of abelian type

Kisin

2017

J. Amer. Math. Soc.

215

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“…In joint work with Rapoport [32], we purposed an axiomatic approach to the study of these characteristic subsets in a general Shimura variety. We formulated five axioms, based on the existence of integral models of Shimura varieties (which have been established in various cases by the work of Rapoport and Zink [66], Kisin and Pappas [40]), the existence of the following commutative diagram and some compatibility conditions:…”

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

Some Results on Affine Deligne–lusztig Varieties

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.2010 Mathematics Subject Classification. 14L05, 20G25.Affine Deligne-Lusztig varieties are schemes locally of finite type over F q . Also the varieties are isomorphic if the element b is replaced by another element b ′ in the same σ-conjugacy class.A major difference between affine Deligne-Lusztig varieties and classical Deligne-Lusztig varieties is that affine Deligne-Lusztig varieties have the second parameter:

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Integral models of Shimura varieties with parahoric level structure

Cited by 63 publications

References 55 publications

Good and semi-stable reductions of Shimura varieties

Good and semi-stable reductions of Shimura varieties

Mod $p$ points on Shimura varieties of abelian type

Some Results on Affine Deligne–lusztig Varieties

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