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

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In this article we study the special fiber of the Rapoport–Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat–Tits stratification and is intimately related to the Bruhat–Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

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Acknowledgements

The author would like to thank Henri Darmon for supporting his postdoctoral studies. He would like to thank Liang Xiao for many helpful conversations regarding to Shimura varieties and beyond. He is grateful to Ulrich Görtz for all the help and his comments on this article. He is inspired by reading many works of Chia-Fu Yu on Siegel modular varieties. He also would like to thank Ben Howard, Eyal Goren, Yichao Tian, Xu Shen and Benedict Gross for valuable discussions related to this article. He would like to thank the referees for carefully reading the article and pointing out all the corrections. While this article is being reviewed, Yasuhiro Oki obtained similar results about the supersingular locus of the quaternionic unitary Shimura variety independently. His method is completely different. He exploited the exceptional isomorphisms between the group \(\mathrm {GU}_{B}(2)\) and the non-split \(\mathrm {GSpin}(3,2)\). Then he embedded the non-split \(\mathrm {GSpin}(3,2)\) in the split \(\mathrm {GSpin}(4,2)\). This allows him to use the results of Howard-Pappas [11] mentioned before. We would like to thank Yoichi Mieda for sending Oki’s work to us.

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  1. Department of Mathematics, McGill University, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada

    Haining Wang

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  1. Haining Wang
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Correspondence to Haining Wang.

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Communicated by Wei Zhang.

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Wang, H. On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space. Math. Ann. 376, 1107–1144 (2020). https://doi.org/10.1007/s00208-019-01938-w

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