A note on polylogarithms on curves and abelian schemes

TL;DR

This paper explores the relationship between polylogarithms on curves and abelian schemes, demonstrating how polylogarithms on abelian schemes relate to those on sub-curves through push-forward and cup-product operations.

Contribution

It establishes a new connection showing that polylogarithms on abelian schemes can be derived from those on sub-curves, especially in the case of Jacobians.

Findings

Polylogarithm on abelian schemes is obtained as a push-forward of the polylogarithm on a sub-curve.

When the abelian scheme is a Jacobian, the push-forward equals a cup-product with the curve's fundamental class.

The work links polylogarithms on curves to those on abelian schemes via geometric operations.

Abstract

In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable sub-curve. If the abelian scheme is the Jacobian of a smooth projective curve, this push-forward can also be written as a cup-product with the fundamental class of the curve.