Concentration theorem and relative fixed point formula of Lefschetz type in Arakelov geometry

TL;DR

This paper establishes a concentration theorem for arithmetic K-theory and uses it to derive a simpler proof of a Lefschetz-type fixed point formula in Arakelov geometry, confirming a conjecture by K"{o}hler and Roessler.

Contribution

It introduces a new, simpler proof of a Lefschetz fixed point formula in Arakelov geometry based on an arithmetic concentration theorem.

Findings

Proved an arithmetic concentration theorem for K_0-theory.

Derived a Lefschetz fixed point formula in Arakelov geometry.

Provided a more natural and less computational proof of the fixed point formula.

Abstract

In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. K\"{o}hler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.