Cell decomposition of some unitary group Rapoport-Zink spaces

Xu Shen Universität Bonn


Distinguished Paper Award in 2017

Mathematische Annalen, 2014.5
In this paper we study the p-adic analytic geometry of the basic unitary group Rapoport–Zink spaces MK with signature (1, n − 1). Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandteMathematik 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 Mred defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain DK in MK for the action of G(Qp)×Jb(Qp), the product of the associated p-adic reductive groups, and prove that MK 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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  title={Cell decomposition of some unitary group Rapoport-Zink spaces},
  author={Xu Shen},
  booktitle={Mathematische Annalen},
Xu Shen. Cell decomposition of some unitary group Rapoport-Zink spaces. 2014. In Mathematische Annalen. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170719190511118267802.
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