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Cell decomposition of some unitary group Rapoport–Zink spaces

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In this paper we study the \(p\)-adic analytic geometry of the basic unitary group Rapoport–Zink spaces \(\mathcal {M}_K\) with signature \((1,n-1)\). Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber \(\mathcal {M}_{red}\) defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain \(\mathcal {D}_K\) in \(\mathcal {M}_K\) for the action of \(G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)\), the product of the associated \(p\)-adic reductive groups, and prove that \(\mathcal {M}_K\) admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).

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Acknowledgments

I would like to thank Prof. Laurent Fargues sincerely, since without his guide this paper would not be accomplished. I should thank Yoichi Mieda, who had proposed some useful questions after the first version of this article. I should also thank the referee for careful reading and suggestions.

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Shen, X. Cell decomposition of some unitary group Rapoport–Zink spaces. Math. Ann. 360, 825–899 (2014). https://doi.org/10.1007/s00208-014-1048-0

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