The de Rham period map for punctured elliptic curves and the KZB equation

Abstract

We demonstrate that the algebraic KZB connection of Levin–Racinet and Luo on a once-punctured elliptic curve represents Kim’s universal unipotent connection, and we observe that the Hodge filtration on the KZB connection has a particularly simple form. This allows us to generalise previous work of Beacom by writing down explicitly the maximal metabelian quotient of Kim’s de Rham period map in terms of elliptic polylogarithms. As far as we are aware this is the first time that the de Rham period map has been written out for an infinite dimensional quotient of the de Rham fundamental group on any curve of positive genus.