Mod $p$ points on Shimura varieties of abelian type

Kisin, Mark

doi:10.1090/jams/867

Cited by 80 publications

(218 citation statements)

References 37 publications

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“…is the (promoted) K-level structure coming from the K -level structure ε p x,K on A x , cf. [40]. By Corollary 1.3.11 of [40] the data (A x , λ x , ε p x,K , (s α,0,x )) uniquely determines the point x ∈ S K (G, X)(k).…”

Section: Rapoport-zink Uniformization For Shimura Varieties Of Abeliamentioning

confidence: 95%

“…It should be possible that the strategy of [40] 3.8 enables us to prove the following refinement of Proposition 6.4…”

Section: For Any Sufficiently Small Open Compact Subgroupmentioning

confidence: 99%

“…where X p (φ) and X p (φ) are certain sets canonically associated to φ, such that (cf. Lemma 3.3.4 of [40]) X p (φ) X G µ (b) M red (F p ) and X p (φ) is a G(A p f )-torsor. Take an unramified Shimura datum of Hodge type (G 1 , X 1 ), together with a central isogeny G der 1 → G der , such that it induces an isomorphism of the associated adjoint Shimura data (G ad 1 , X ad 1 ) (G ad , X ad ).…”

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confidence: 97%

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On Some Generalized Rapoport–Zink Spaces

Shen¹

2019

Can. J. Math.-J. Can. Math.

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We enlarge the class of Rapoport-Zink spaces of Hodge type by modifying the centers of the associated p-adic reductive groups. These such-obtained Rapoport-Zink spaces are called of abelian type. The class of Rapoport-Zink spaces of abelian type is strictly larger than the class of Rapoport-Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport-Zink spaces of abelian type can be viewed as moduli spaces of local G-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport-Zink spaces of abelian type. We construct and study the Ekedahl-Oort stratifications for the special fibers of Rapoport-Zink spaces of abelian type. As an application, we deduce a Rapoport-Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

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Section: Rapoport-zink Uniformization For Shimura Varieties Of Abeliamentioning

confidence: 95%

“…It should be possible that the strategy of [40] 3.8 enables us to prove the following refinement of Proposition 6.4…”

Section: For Any Sufficiently Small Open Compact Subgroupmentioning

confidence: 99%

mentioning

confidence: 97%

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On Some Generalized Rapoport–Zink Spaces

Shen¹

2019

Can. J. Math.-J. Can. Math.

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“…Note that the minuscule coweight case is especially important for applications in number theory. Kisin [38] proved the Langlands-Rapoport conjecture for modp points on Shimura varieties of abelian type with hyperspecial level structure. Compared to the function field analogous of Langlands-Rapoport conjecture [61], there are extra complication coming from algebraic geometry and the explicit description of the connected components of X(µ, b) in [5] is used in an essential way to overcome the complication.…”

Section: Connected Componentsmentioning

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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“…One such feature is the existence of smooth integral models at primes not dividing the level, and their subsequent utility for studying the action of Frobenius on the Galois representations arising from Shimura varieties crucial for the Langlands programme. Such results have been available in the PEL type case for a long time, thanks largely to the programme of Kottwitz, but have recently been extended to the much more general case of abelian type Shimura varieties in work culminating with the recent papers of Kisin [12], [13].…”

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confidence: 99%

Integral canonical models for automorphic vector bundles of abelian type

Lovering

2017

Alg. Number Th.

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We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type.More precisely, we first build on Kisin's work to construct integral canonical models over OE[1/N ] for Shimura varieties of abelian type with hyperspecial level at all primes not dividing N compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these in hand, one immediately also gets integral models for automorphic vector bundles.

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Mod $p$ points on Shimura varieties of abelian type

Abstract: Abstract. We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.

Cited by 80 publications

References 37 publications

On Some Generalized Rapoport–Zink Spaces

On Some Generalized Rapoport–Zink Spaces

Some Results on Affine Deligne–lusztig Varieties

Integral canonical models for automorphic vector bundles of abelian type

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