Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

Vasiu, Adrian

doi:10.1002/mana.201700415

Cited by 2 publications

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“…The following theorem was conjectured by Langlands, Milne (cf. [Mil92], Conjecture 4.25), and was established by Vasiu ([Vas99], [Vas07], [Vas08]) and Kisin ([Kis09], [Kis10]).…”

Section: 12mentioning

confidence: 96%

Nonemptiness of Newton strata of Shimura varieties of Hodge type

Lee¹

2018

Alg. Number Th.

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For a Shimura variety of Hodge type with hyperspecial level at a prime p, the Newton stratification on its special fiber at p is a stratification defined in terms of the isomorphism class of the Dieudonne module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline cycles ("F -isocrystal with G Qp -structure"). There has been a conjectural group-theoretic description of the F-isocrystals that are expected to show up in the special fiber. We confirm this conjecture by two different methods. More precisely, for any Fisocrystal with G Qp -structure that is expected to appear (in a precise sense), first we construct a special point which has good reduction and whose reduction has associated F -isocrystal equal to given one. Secondly, we produce a Kottwtiz triple (with trivial Kottwitz invariant) with the F -isocrystal component being the given one. According to a recent result of Kisin which establishes the Langlands-Rapoport conjecture, such Kottwitz triple arises from a point in the reduction. point of S K ⊗ O κ(℘) defined over k. By smoothness of S K and the moduli interpretation of S K , the Dieudonne module D(A z ) is supplied with a set of Frobnius-invariant tensors {t α,z } α∈J , and there exists an L(k)-isomorphism between the dual space W ∨ ⊗ L(k) and the Dieudonne module D(A z ) which matches the G-invariant tensors on W ∨ and the tensors {t α,z } α∈J on D(A z ), which is thus canonically determined up to the action of G L(k) on W ∨ L(k) (cf. §3.2). Choosing such an isomorphism, we transport the Frobenius operator Φ to W ∨ L(k) and get an element b ∈ G(L(k)) by

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“…The following theorem was conjectured by Langlands, Milne (cf. [Mil92], Conjecture 4.25), and was established by Vasiu ([Vas99], [Vas07], [Vas08]) and Kisin ([Kis09], [Kis10]).…”

Section: 12mentioning

confidence: 96%