Stratifications in the reduction of Shimura varieties

He, Xuhua; Rapoport, Michael

doi:10.1007/s00229-016-0863-x

Cited by 48 publications

(106 citation statements)

References 37 publications

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“…Here part (1) was obtained in joint work with Zhou [33]. As an application, we verified the Axioms in [32] for certain PEL type Shimura varieties. In [33], the set of connected components of X(µ, b) was also studied for nonbasic b.…”

Section: Connected Componentssupporting

confidence: 55%

“…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 applications and current status of the axioms. It is shown in [32] that under those axioms, the Newton strata, the Ekedahl-Oort strata, the Kottwitz-Rapoport strata, and the Ekedahl-Kottwitz-Oort-Rapoport strata discussed in §7, are all nonempty in their natural range. Furthermore, under those axioms several relations between these various stratifications are also established in [32].…”

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

“…In joint work with Nie [31], based on the framework of [32], we studied the density problem of the µ-ordinary locus. Under the axioms of [32] we gave several explicit criteria on the density of the µ-ordinary locus.…”

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

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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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Section: Connected Componentssupporting

confidence: 55%

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

Section: Some Applications To Shimura Varietiesmentioning

confidence: 99%

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Some Results on Affine Deligne–lusztig Varieties

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Proof. For a special monodromy datum, every Newton polygon ν ∈ ν(B(µ m , f)) occurs for a point of Z(m, N, a) [15] where m = 8, N = 4, and a = (2, 4, 5, 5); the signature is f = (1, 1, 0, 0, 2, 0, 1). There are two degenerations of compact type of a:…”

Section: Pel-indecomposable Newton Polygonsmentioning

confidence: 99%

Newton polygons arising from special families of cyclic covers of the projective line

Mantovan

Pries

et al. 2019

Res. number theory

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By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the µ-ordinary Ekedahl-Oort type, occurring in the characteristic p reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl-Oort types of Jacobians of smooth curves in characteristic p. Under certain congruence conditions on p, these include: the supersingular Newton polygon for genus 5, 6, 7; fourteen new non-supersingular Newton polygons for genus 5 − 7; eleven new Ekedahl-Oort types for genus 4 − 7 and, for all g ≥ 6, the Newton polygon with p-rank g − 6 with slopes 1/6 and 5/6.

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