Cell decomposition of some unitary group Rapoport-Zink spaces

TL;DR

This paper investigates the $p$-adic geometry of certain unitary Rapoport-Zink spaces, constructing a cell decomposition and establishing a Lefschetz trace formula using advanced stratification and filtration techniques.

Contribution

It introduces a cell decomposition for basic unitary Rapoport-Zink spaces and proves a Lefschetz trace formula via the action of elliptic elements.

Findings

Constructed a relatively compact fundamental domain in the space.

Proved the existence of a locally finite cell decomposition.

Established a Lefschetz trace formula for the space.

Abstract

In this paper we study the $p$-adic analytic geometry of the basic unitary group Rapoport-Zink spaces $\M_K$ with signature $(1,n-1)$. Using the theory of Harder-Narasimhan filtration of finite flat groups developed by Fargues in \cite{F2},\cite{F3}, and the Bruhat-Tits stratification of the reduced special fiber $\M_{red}$ defined by Vollaard-Wedhorn in \cite{VW}, we find some relatively compact fundamental domain $\D_K$ in $\M_K$ for the action of $G(\Q_p)\times J_b(\Q_p)$, the product of the associated $p$-adic reductive groups, and prove that $\M_K$ 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 \cite{Mi2}.