Numerical Schottky Uniformizations

Abstract

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.