Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism

Understand

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves.

• As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities.
• We study the properties of our iterated integrals and their relationship to the multiple elliptic polylogarithms from the mathematics literature.
• On the one hand, we find that our iterated integrals span essentially the same space of functions as the multiple elliptic polylogarithms.