A motivic conjecture of Milne

Vasiu, Adrian

doi:10.1515/crelle-2012-0009

Cited by 7 publications

(38 citation statements)

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“…For the µ-ordinary locus, this was first proved by Wedhorn ([Wed99]) in the PEL-type cases, using an equi-characteristic deformation argument (which is thus yet unavailable for Hodge type Shimura varieties) and by Wortmann [Wor13] for general Hodge-type cases, along the line of Viehmann-Wedhorn [VW13] comparing the Newton stratification with the Ekedahl-Oort stratification. As such, Wortmann's work makes an essential use of the Kisin's results on the integral p-adic comparison between the etale cohomology with Z p -coefficients and the crystalline cohomology with integral coefficients, endowed with tensors (which is also established by [Vas12]). There was also a group-theoretic approach of Bultel [Bul01].…”

Section: 27mentioning

confidence: 99%

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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Section: 27mentioning

confidence: 99%

Nonemptiness of Newton strata of Shimura varieties of Hodge type

Lee¹

2018

Alg. Number Th.

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“…The most serious problem is the presence of connections ∇ 0 , which cannot be removed in general (unless it is asserted in Theorem 9.4). In order to handle the connections, we use the theory of moduli of connections [Vas12,§3]. We review (and slightly generalise) the theory in §10.…”

Section: Classification Via Torsion Kisin Modulesmentioning

confidence: 99%

“…Finally, the semi-linear algebra objects appearing in Theorem 1 and Corollary 3 carry connections. In some cases when the (anti-)equivalences of categories were constructed without connection, we can remove connections from the statement using Vasiu's construction of moduli of connections [Vas12,§3]. (See Corollary 10.3.1 for more details).…”

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confidence: 99%

“…To prove Theorem 5, we generalise the strategy of [Kis06,§2.3]. Due to the presence of connections, the actual argument is quite elaborate using the theory of moduli of connections [Vas12,§3].…”

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“…In §9 we state our main results on finite locally free group schemes of p-power order. The proof uses the theory of moduli of connection (in a slightly generalised form than the original version in [Vas12,§3]), which will be studied in §10.…”

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confidence: 99%

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The Relative Breuil–Kisin Classification ofp-Divisible Groups and Finite Flat Group Schemes

Kim

2014

Int Math Res Notices

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Assume that p > 2, and let O K be a p-adic discrete valuation ring with residue field admitting a finite p-basis, and let R be a formally smooth formally finite-type O K -algebra. (Indeed, we allow slightly more general rings R.) We construct an anti-equivalence of categories between the categories of p-divisible groups over R and certain semi-linear algebra objects which generalise (ϕ, S)-modules of height 1 (or Kisin modules). A similar classification result for p-power order finite flat group schemes is deduced from the classification of p-divisible groups. We also show compatibility of various construction of (Zp-lattice or torsion) Galois representations, including the relative version of Faltings' integral comparison theorem for p-divisible groups. We obtain partial results when p = 2.Ki S (ϕ, ∇ 0 ) of certain "torsion Kisin S-modules" (Definition 9.2.1). Furthermore, we have a natural G R∞ -equivariant isomorphism H(R) ∼ = T * (M * (H)).Let us review previous results when p > 2. When R = O K with perfect residue field, all our main results are known already. Theorem 4(1) is proved by Faltings [Fal99, §6], while the first assertion was already proved by Fontaine [Fon82, 1 Indeed, we will work with slightly more general rings than Brinon [Bri06,Bri08]. See §4.4 for more details.2 In this paper, we only consider commutative finite locally free group schemes, so we will often suppress the adjective "commutative". Definition 6.1.6) as follows:Lemma 6.2.2. Assume that p = 2 and R satisfies the p-basis condition ( §2.2.1). Then the functors Mod S (ϕ) ϕ nilp → MF S (ϕ) ϕ nilp and Mod S (ϕ) ψ nilp → MF S (ϕ) ψ nilp , defined by S ⊗ ϕ,S (·), are fully faithful. The same statement holds for the full subcategories of ϕand ψnilpotent objects in Mod S (ϕ, ∇) and Mod KiThe same proof of Lemma 6.2 shows that it suffices to prove the lemma when R is a p-adic discrete valuation ring with perfect residue field, which follows from combining Proposition 1.1.9 and Theorem 1.2.8 in [Kis09]. Proposition 6.3. Assume that R satisfies the p-basis condition ( §2.2.1). Then the functors Mod, are essentially surjective. In particular, they are equivalences of categories if p > 2 or if they are restricted to ϕand ψnilpotent objects, and are equivalence of categories up to isogeny if p = 2.We prove the proposition later in §6.4. Let us record some interesting corollaries. The following is immediate from Corollary 6.3.1. Assume that R satisfies the p-basis condition ( §2.2.1). If p > 2 then there exists an exact contravariant functorsatisfies the normality assumption ( §2.2.4), and an anti-equivalence of categories if R satisfies the formally finite-type assumption ( §2.2.2). If p = 2 then we have an exact contravariant functor M * [ 1 p ] : G → M * (G)[ 1 p ] on the isogeny categories {p-divisible groups over R}[1/p] → Mod S (ϕ, ∇)[1/p], which is fully faithful if R satisfies the formally finite-type assumption ( §2.2.2).When p = 2, we have the following strengthening of Corollary 6.3.1 for ϕand ψnilpotent objects: Corollary 6.3.2. Let ...

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

Vasiu

2020

Mathematische Nachrichten

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We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0,). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0,) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

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A motivic conjecture of Milne

Cited by 7 publications

References 28 publications

Nonemptiness of Newton strata of Shimura varieties of Hodge type

Nonemptiness of Newton strata of Shimura varieties of Hodge type

The Relative Breuil–Kisin Classification ofp-Divisible Groups and Finite Flat Group Schemes

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

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