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

TL;DR

This work generalizes polylogarithms to genus-one elliptic curves by introducing elliptic polylogarithms defined through iterated integrals with simple-pole kernels on E. It shows these E_3 (and E_4 for quartics) functions span (and are equivalent to) the established multiple elliptic polylogarithms under the torus-curve map, while offering a more directly usable kernel basis for physics computations. An explicit algorithm for computing primitives of elliptic polylogarithms multiplied by rational functions is provided, with detailed treatment of cubic and quartic curves and regularization to preserve shuffle structure. The authors demonstrate practical applications by expressing Laurent coefficients of hypergeometric and Appell functions in terms of elliptic polylogarithms and discuss the implications for high-energy physics calculations, including future work on sunrise integrals and differential equations in elliptic settings.

Abstract

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. On the other, our formulation allows for a more direct use to solve a large variety of problems in high-energy physics. We demonstrate the use of our functions in the evaluation of the Laurent expansion of some hypergeometric functions for values of the indices close to half integers.