The algebraic de Rham realization of the elliptic polylogarithm via the Poincar\'e bundle

TL;DR

This paper develops an algebraic de Rham framework for elliptic polylogarithms using the Poincaré bundle, generalizing prior results and enabling explicit computation of Eisenstein classes.

Contribution

It introduces a new algebraic de Rham realization of elliptic polylogarithms applicable to families of elliptic curves, extending previous work and providing explicit formulas for Eisenstein classes.

Findings

Explicit algebraic formulas for de Rham Eisenstein classes

Generalization of previous elliptic polylogarithm results

Application to families of elliptic curves

Abstract

In this paper, we describe the algebraic de Rham realization of the elliptic polylogarithm for arbitrary families of elliptic curves in terms of the Poincar\'e bundle. Our work builds on previous work of Scheider and generalizes results of Bannai-Kobayashi-Tsuji and Scheider. As an application, we compute the de Rham Eisenstein classes explicitly in terms of certain algebraic Eisenstein series.