Translating auxiliary symmetries between Schottky uniformization and Jacobi parametrization

TL;DR

This paper compares the symmetries of Riemann surfaces in Jacobi and Schottky parametrizations, enabling translation of functional relations and improving numerical evaluations of special functions.

Contribution

It provides a systematic translation of symmetries between Jacobi and Schottky descriptions, facilitating cross-language functional relations and numerical computations.

Findings

Translated symplectic transformations to M"obius transformations

Enabled transfer of functional relations between languages

Improved numerical evaluation of special functions

Abstract

The explicit description and computation of functions defined on Riemann surfaces of various genera depends on the choice of language: while the Jacobi parametrization is widely known and used, the Schottky uniformization has been proven to provide an alternative approach, useful in particular for (but not limited to) numerical calculations. Despite capturing the geometry of the Riemann surface completely, the two languages are subject to rather different sets of auxiliary symmetries. In this article we translate and compare the symplectic transformations inherent in the Jacobi parametrization to the freedom in choosing M\"obius transformations generating the Schottky group for the Schottky uniformization. Our results are aimed at transferring functional relations expressed in the Schottky language to the Jacobi language and vice versa. An immediate application would be the efficient numerical evaluation of special functions in a physics context by favorably tuning the Schottky cover leading to quicker convergence.