On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space

Wang, Haining

doi:10.1007/s00208-019-01938-w

Cited by 5 publications

(3 citation statements)

References 29 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: 55%

“…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%

“…Denote by M Pa the underlying reduced subscheme of N Pa (0). The main result of [Wang19a] concerns the structure of M Pa . The same result is also obtained by Oki in [Oki22] using a different method.…”

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: 55%

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

Oki

2021

manuscripta math.

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On the supersingular locus of the Shimura variety for GU(2,2) over a ramified prime

Oki

2023

Int. J. Math.

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We study the structure of the supersingular locus of the Rapoport–Zink integral model of the Shimura variety for GU(2,2) over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely 2-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the projective line.

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On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space

Cited by 5 publications

References 29 publications

Arithmetic level raising for certain quaternionic unitary Shimura variety

Arithmetic level raising for certain quaternionic unitary Shimura variety

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

On the supersingular locus of the Shimura variety for GU(2,2) over a ramified prime

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