On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

Oki, Yasuhiro

doi:10.1007/s00229-020-01265-4

Cited by 3 publications

(4 citation statements)

References 27 publications

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“…The appearance of the two copies of O λ [Sh(B, K H )] m is related to the fact that the irreducible components of the supersingular locus of X Pa (B) are naturally parametrized by two copies of Sh(B, K H ). This fact along with other finer properties of the supersingular locus are obtained in author's work [Wang19a,Wang19b] and Oki's work [Oki22].…”

Section: Then We Have a Canonical Isomorphismsupporting

confidence: 56%

“…It would be interesting to generalize the results of this article to cuspidal automorphic representations π of GSp 4 (A F ) where F is a totally real field. The main reason we make this restriction in this article is that the results in [Wang19a,Wang19b] and [Oki22] are obtained only for the quaternionic unitary Shimura variety over Q. As far as the author's knowledge, the current Rapoport-Zink uniformization theorem does not apply to those Shimura varieties studied in this article if one considers totally real field.…”

Section: Then We Have a Canonical Isomorphismmentioning

confidence: 99%

“…For a definite choice of such alternating form, see [KR00] or [Oki22]. Let D be the quaternion division algebra over Q p .…”

Section: Then We Have a Canonical Isomorphismmentioning

confidence: 99%

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Arithmetic level raising for certain quaternionic unitary Shimura variety

Wang¹

2022

Preprint

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In this article we prove an arithmetic level raising theorem for the symplectic group of degree four in the ramified case. This result is a key step towards the Beilinson-Bloch-Kato conjecture for certain Rankin-Selberg motives associated to orthogonal groups within the framework of the Gan-Gross-Prasad conjecture. The theorem itself can be also viewed as an analogue of the Ihara's lemma or the Tate conjectures for special fibers of Shimura varieties at ramified characteristics. The proof relies heavily on the description of the supersingular locus of certain quaternionic unitary Shimura variety which is closely related to the classical Siegel threefold. Contents 1. Introduction 1.1. Main results 1.2. Applications and prospect 1.3. Notations and conventions Acknowledgements 2. Quaternionic unitary Shimura variety 2.1. Quaternionic unitary groups 2.2. The local quaternionic unitary Shimura variety 2.3. The quaternionic unitary Shimura variety 3. Level raising conditions 4. Monodromy on the cohomology of the quaternionic unitary Shimura variety 4.1. Picard-Lefschetz formula 4.2. Descriptions of α and β 4.3. Nearby cycles on certain Shimura varieties 5. The level raising matrix 6. Computing the level raising matrix Key words and phrases. Shimura varieties, Arithmetic level raising, Vanishing cycles.HAINING WANG 6.1. A Hecke operator identity 30 6.2. Computing the level raising matrix 32 7. Tate cycles on the quaternionic unitary Shimura variety 37 7.1. Computing the supersingular matrix 38 7.2. Tate classes in the quaternionic unitary Shimura variety 40 8. Galois deformation rings and cohomology of Shimura varieties 43 8.1. Galois representation for GSp 4 43 8.2. Galois deformation rings 45 9. Arithmetic level raising and applications 47 9.1. Proof of Theorem 9.5 50 9.2. Level raising for GSp 4 54 References 55

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Section: Then We Have a Canonical Isomorphismsupporting

confidence: 56%

Section: Then We Have a Canonical Isomorphismmentioning

confidence: 99%

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Arithmetic level raising for certain quaternionic unitary Shimura variety

Wang¹

2022

Preprint

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“…For Deligne-Lusztig varieties in orthogonal groups, cf. also Oki's paper [38], Section 6. Note that this theorem corrects a few omissions of smooth cases in [13] Section 7 and the erratum, and the incorrect claim regarding non-smoothness in type B in [13].…”

Section: N) Smoothmentioning

confidence: 96%

Basic loci of Coxeter type with arbitrary parahoric level

Görtz¹,

Nie³

2022

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

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Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their "group-theoretic models" -generalized affine Deligne-Lusztig varieties -in cases where they have a particularly nice description. Continuing the work of [13] and [14] we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new and open cases from the point of view of Shimura varieties/Rapoport-Zink spaces.

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Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure

Oki¹

2020

Preprint

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In this article, we give a concrete description of the underlying reduced subscheme of the Rapoport-Zink spaces for spinor similitude groups with special maximal parahoric (and nonhyperspecial) level structure. Moreover, we give two applications of the above result. One of which is describing the structure of the basic loci of mod p reductions of Kisin-Pappas integral models of Shimura varieties for spinor similitude groups with special maximal parahoric level structure at p. The other is constructing a variant of the result of He, Li and Zhu, which gives a formula on the intersection multiplicity of the GGP cycles associated codimension 1 embeddings of Rapoport-Zink spaces for spinor similitude groups.

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On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

Cited by 3 publications

References 27 publications

Arithmetic level raising for certain quaternionic unitary Shimura variety

Arithmetic level raising for certain quaternionic unitary Shimura variety

Basic loci of Coxeter type with arbitrary parahoric level

Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure

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