Multiple Elliptic Polylogarithms

Understand

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.