Classical Schottky Uniformizations of Genus 2A Package for Mathematica

Abstract

The general theory of Riemann surfaces asserts that a closed Riemann surface \(S\) of genus \(g \geq 2\) be seen as (i) the quotient by a Kleinian group \(G\) or (ii) a plane algebraic curve \(C\) (possible with singularities) or (iii) a symmetric complex \(g × g\) matrix \(Z\) with positive imaginary part (a Riemann period matrix). Numerical uniformization problem ask for numerical relations between these objects for suitable choices of \(G, C\) and \(Z\). In this note we discuss the case of genus two for \(G\) a classical Schottky group. The algorithm has been implemented into a mathematica package for the case of M-real curves of genus 2, but it can easily be rewritten for the general case.