### 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. © Springer 2005.

Original language | English |
---|---|

Pages (from-to) | 125-157 |

Number of pages | 33 |

Journal | Geometriae Dedicata |

DOIs | |

Publication status | Published - 1 Mar 2005 |

## Fingerprint Dive into the research topics of 'Numerical Schottky uniformizations'. Together they form a unique fingerprint.

## Cite this

Hidalgo, R. A., & Figueroa, J. (2005). Numerical Schottky uniformizations.

*Geometriae Dedicata*, 125-157. https://doi.org/10.1007/s10711-004-7514-1