Schottky-Kronecker forms and hyperelliptic polylogarithms

TL;DR

This paper introduces Schottky-Kronecker forms as higher-genus analogues of Kronecker's series, enabling the numerical evaluation of hyperelliptic polylogarithms through explicit constructions and Poincaré series.

Contribution

It constructs explicit higher-genus polylogarithm differentials using Schottky parametrization, extending Bernard's ideas, and provides a numerical approach for hyperelliptic cases.

Findings

Explicit construction of higher-genus polylogarithm differentials.

Numerical evaluation of genus-two hyperelliptic polylogarithms.

Verification of convergence assumptions for real hyperelliptic curves.

Abstract

Elliptic polylogarithms can be defined as iterated integrals on a genus-one Riemann surface of a set of integration kernels whose generating series was already considered by Kronecker in the 19th century. In this article, we employ the Schottky parametrization of a Riemann surface to construct higher-genus analogues of Kronecker's generating series, which we refer to as Schottky-Kronecker forms. Our explicit construction generalizes ideas from Bernard's higher-genus construction of the Knizhnik-Zamolodchikov connection. Integration kernels generated from the Schottky-Kronecker forms are defined as Poincar\'e series. Under technical assumptions, related to the convergence of these Poincar\'e series on the underlying Riemann surface, we argue that these integration kernels coincide with a set of differentials defined by Enriquez, whose iterated integrals constitute higher-genus analogues of polylogarithms. Enriquez' original definition is not well-suited for numerical evaluation of higher-genus polylogarithms. In contrast, the Poincar\'e series defining our integration kernels can be evaluated numerically for real hyperelliptic curves, for which the above-mentioned convergence assumptions can be verified. We numerically evaluate several examples of genus-two polylogarithms, thereby paving the way for numerical evaluation of hyperelliptic analogues of polylogarithms.