Multiple Elliptic Polylogarithms

TL;DR

This paper introduces multiple elliptic polylogarithms as multivalued functions on elliptic curve configuration spaces, expressing iterated integrals and fundamental group periods in terms of these new functions.

Contribution

It defines multiple elliptic polylogarithms and demonstrates their ability to express all iterated integrals and periods on elliptic curve configuration spaces.

Findings

All iterated integrals on $E^{(n)}$ can be expressed using these functions.

Periods of the unipotent fundamental group of the punctured elliptic curve are represented by these polylogarithms.

The functions are constructed via a general averaging procedure.

Abstract

We study the de Rham fundamental group of the configuration space $E^{(n)}$ of $n+1$ marked points on an elliptic curve $E$, and define multiple elliptic polylogarithms. These are multivalued functions on $E^{(n)}$ with unipotent monodromy, and are constructed by a general averaging procedure. We show that all iterated integrals on $E^{(n)}$, and in particular the periods of the unipotent fundamental group of the punctured curve $E \backslash \{0\}$, can be expressed in terms of these functions.