The meta-abelian elliptic KZB associator and periods of Eisenstein series

TL;DR

This paper explicitly computes the image of the elliptic KZB associator in a specific quotient, linking it to Eisenstein series, period polynomials, and elliptic polylogarithms, thus advancing understanding of elliptic associators and their periods.

Contribution

It provides an explicit formula for the image of the elliptic KZB associator in the meta-abelian quotient, connecting it to Eichler integrals and period polynomials.

Findings

Explicit formula for the elliptic KZB associator in the meta-abelian quotient

Retrieval of Zagier's extended period polynomials from the associator

Connection of the associator to elliptic polylogarithm values at zero

Abstract

We compute the image of Enriquez' elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne's computation of the depth one quotient of the Drinfeld associator. We also show how to retrieve Zagier's extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson--Levin's elliptic polylogarithms from the meta-abelian elliptic KZB associator.