On the structure of some $p$-adic period domains

Chen, Miaofen; Fargues, Laurent; Shen, Xu

doi:10.4310/cjm.2021.v9.n1.a4

Cited by 13 publications

(25 citation statements)

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“…One also has the so called weakly admissible locus Q wa (K 0 ) which can be described as the set of isotropic lines in V Φ K 0 ⊗ K 0 that is not contained in any rational isotropic space contained in V Φ K 0 . In fact we have the equality Q a = Q wa which follows from that the local Shimura datum being fully Hodge-Newton decomposable [CFS18] and [Shen17]. The period map π dR is defined for a general Hodge-type Rapoport-Zink spaces is defined by [Kim18] and extended to the abelian case in [Shen17].…”

Section: Liftings the Superspecial Locusmentioning

confidence: 95%

On the superspecial loci of orthogonal type Shimura varieties

Wang

2019

Preprint

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In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n−2, 2) with n ≥ 3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type Shimura variety and its lift in the integral model by certain homogeneous spaces. As applications, we provide a mass formula for the superspecial locus. We also indicate how Gross's conjectures can be generalized to the Coxeter type Shimura varieties using the group theoretic method of Görtz and He.

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Section: Liftings the Superspecial Locusmentioning

confidence: 95%

On the superspecial loci of orthogonal type Shimura varieties

Wang

2019

Preprint

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“…In the quasi-split case, we have the following equivalent definition for the HN-decomposability. 1)-( 3) is the same as that of [6] Lemma 4.11. It is deduced from 1.7 (1).…”

Section: Hodge-newton-decomposabilitymentioning

confidence: 99%

“…Let A := Ker(π 1 (M ) Γ → π 1 (G) Γ ) which is torsion free by the proof of [6] Lemma 4.11. As the functor (−) Γ is right exact, there exists a natural surjection Q Γ → A.…”

Section: Kottwitz Setmentioning

confidence: 99%

“…We also need the following proposition which is a key ingredient of the proof of the main result. For the necessity, if b M is not basic in M or (M, µ) is not fully HN-indecomposable, it's equivalent to say that bM is not basic in M or ( M , w 0 µ) is not fully HN-indecomposable, then after applying Proposition 4.5 (in the first case) or [6] Proof of theorem 4.3. The reduction to the quasi-split case is similar to the proof of [6] Theorem 6.1.…”

Section: The Action Of Jb On the Modifications Of G-bundlesmentioning

confidence: 99%

“…For the necessity, if b M is not basic in M or (M, µ) is not fully HN-indecomposable, it's equivalent to say that bM is not basic in M or ( M , w 0 µ) is not fully HN-indecomposable, then after applying Proposition 4.5 (in the first case) or [6] Proof of theorem 4.3. The reduction to the quasi-split case is similar to the proof of [6] Theorem 6.1. We may assume that G is adjoint, then G = J b * is an extended pure inner form of H, where [b * ] ∈ B(H) basic .…”

Section: The Action Of Jb On the Modifications Of G-bundlesmentioning

confidence: 99%

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Fargues-Rapoport conjecture for p-adic period domains in the non-basic case

Chen¹

2020

Preprint

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We prove the Fargues-Rapoport conjecture for p-adic period domains in the nonbasic case with minuscule cocharacter. More precisely, we give a group theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide. ContentsIntroduction 1 Notations 2 1. Kottwitz set and G-bundles on the Fargues-Fontaine curve 3 2. Admissible locus and weakly admissible locus 9 3. Hodge-Newton-decomposability 13 4. The action of Jb on the modifications of G-bundles 18 5. Newton stratification and weakly admissible locus 23 References 24

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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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On the structure of some $p$-adic period domains

Cited by 13 publications

References 0 publications

On the superspecial loci of orthogonal type Shimura varieties

On the superspecial loci of orthogonal type Shimura varieties

Fargues-Rapoport conjecture for p-adic period domains in the non-basic case

Fully Hodge–Newton Decomposable Shimura Varieties

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