On the Lefschetz trace formula for Lubin-Tate spaces

TL;DR

This paper provides a new proof of the Lefschetz trace formula for Lubin-Tate spaces using cell decompositions and Mieda's theorem, with potential for generalization to other Rapoport-Zink spaces.

Contribution

It introduces a novel proof approach for the Lefschetz trace formula for Lubin-Tate spaces, different from previous methods, and suggests broader applicability to other Rapoport-Zink spaces.

Findings

Reproved the Lefschetz trace formula for Lubin-Tate spaces.

Developed a proof based on cell decompositions and Mieda's theorem.

Extended the approach to certain unitary Rapoport-Zink spaces.

Abstract

We reprove the Lefschetz trace formula for Lubin-Tate spaces, based on the locally finite cell decompositions of these spaces obtained by Fargues, and Mieda's theorem of Lefschetz trace formula for certain open adic spaces (\cite{Mi1} theorem 3.13). This proof is rather different from those of Strauch in \cite{St} (theorem 3.3.1) and of Mieda in \cite{Mi1} (example 4.21), and is quite hopeful to generalized to some other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, we proved a Lefschetz trace formula for some unitary Rapoport-Zink spaces in \cite{Sh} by using similar ideas here.