Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

He, Xuhua

doi:10.24033/asens.2305

Cited by 43 publications

(33 citation statements)

References 15 publications

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“…To prove the converse, using flatness of M D Zp , it suffices to prove M D Zp (k) = ∅. Via the identification (4.3), this follows from [25].…”

Section: Rapoport-zink Spacesmentioning

confidence: 99%

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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“…To prove the converse, using flatness of M D Zp , it suffices to prove M D Zp (k) = ∅. Via the identification (4.3), this follows from [25].…”

Section: Rapoport-zink Spacesmentioning

confidence: 99%

Good and semi-stable reductions of Shimura varieties

He¹,

Pappas²,

Rapoport³

2020

Journal De L’École Polytechnique — Mathématiques

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“…We have the following additivity on the admissible sets. It is proved in [21,Theorem 5.1] for Iwahori subgroups and in [15,Theorem 1.4] in the general case. Theorem 6.1.…”

Section: 11mentioning

confidence: 99%

Fully Hodge–Newton Decomposable Shimura Varieties

Görtz

Nie

2019

Peking Math J

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The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties.We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms.Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of [25]) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions.While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure.We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.

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“…This was conjectured by Kottwitz and Rapoport in [47] and [62] and was proved in [27]. This fact allows one to reduce many questions (e.g.…”

Section: Connected Componentsmentioning

confidence: 74%

“…The desired decomposition of K ′ Adm(µ)K ′ for an arbitrary parahoric subgroup K ′ was given in [27] as…”

Section: 2mentioning

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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Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

Cited by 43 publications

References 15 publications

Good and semi-stable reductions of Shimura varieties

Good and semi-stable reductions of Shimura varieties

Fully Hodge–Newton Decomposable Shimura Varieties

Some Results on Affine Deligne–lusztig Varieties

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