The Maillot-R\"ossler current and the polylogarithm on abelian schemes

TL;DR

This paper offers a conceptual proof linking the polylogarithm on abelian schemes with higher analytic torsion forms, and characterizes the arithmetic Chern character of the Poincaré bundle through invariance properties.

Contribution

It provides a new axiomatic approach to the arithmetic Chern character and a decomposition of the arithmetic Chow group, advancing understanding of polylogarithms and arithmetic geometry.

Findings

Polylogarithm realization described via Bismut-Köhler torsion form

Axiomatic characterization of the arithmetic Chern character

Decomposition result for the arithmetic Chow group

Abstract

We give a conceptual proof of the fact that the realisation of the degree zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut-K\"ohler higher analytic torsion form of the Poincar\'e bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincar\'e bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.