The Maillot–Rössler current and the polylogarithm on abelian schemes

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.