A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres

TL;DR

This paper establishes a Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibers within Arakelov geometry, extending Thomason's work and confirming a conjecture by Maillot and Rössler.

Contribution

It introduces a new fixed point formula for singular arithmetic schemes, generalizing previous results to the arithmetic setting and verifying a conjecture in the field.

Findings

Proved a Lefschetz fixed point formula for singular arithmetic schemes.

Extended Thomason's Lefschetz formula to the arithmetic context.

Confirmed a conjecture by Maillot and Rössler.

Abstract

In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. R\"{o}ssler.