Geometriae Dedicata

, Volume 111, Issue 1, pp 125–157 | Cite as

Numerical Schottky Uniformizations

  • Rubén A. Hidalgo
  • Jaime Figueroa


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.


Schottky groups real Riemann surfaces Riemann period matrices 

Mathematics Subject Classifications (2000)

30F40 30F10 


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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Departamento de MatemáticasUTFSMValparaísoChile

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