A construction of single-valued elliptic polylogarithms

TL;DR

This paper develops a method to construct single-valued elliptic polylogarithms on punctured elliptic curves, extending Brown's genus-zero framework to elliptic curves with applications like the elliptic Bloch-Wigner dilogarithm.

Contribution

It introduces a new formalism for single-valued elliptic polylogarithms using elliptic associators, generalizing Brown's approach from genus-zero to elliptic curves.

Findings

Constructed elliptic single-valued polylogarithms on punctured elliptic curves.

Extended Brown's genus-zero framework to elliptic curves.

Provided explicit examples including the elliptic Bloch-Wigner dilogarithm.

Abstract

We establish a general construction of single-valued elliptic polylogarithms as functions on the once-punctured elliptic curve. Our formalism is an extension of Brown's construction of genus-zero single-valued polylogarithms to the elliptic curve: the condition of trivial monodromy for solutions to the Knizhnik-Zamolodchikov-Bernard equation is expressed in terms of elliptic associators and involves two representations of a two-letter alphabet. Our elliptic single-valued condition reduces to Brown's genus-zero condition upon degeneration of the torus. We provide several examples for our construction, including the elliptic Bloch-Wigner dilogarithm.