Computation of period matrices of real algebraic curves
type: Item
instance of: scholarly article
title: Computation of period matrices of real algebraic curves
zbMATH Open document ID: 0805.14029
DOI: 10.1007/BF02573995
published in: Discrete \& Computational Geometry
publication date: 1995-02-05T00:00:00Z
full work available at URL: https://eudml.org/doc/131289
review text: This very well written paper explains a way to compute the period matrix of a real algebraic curve when it is given via uniformization, and not via equations. It describes an algorithm which has as input Möbius transformations freely generating a Fuchsian group \(G\) of the second kind, and as output the period matrix of the real curve \(X = \Omega (G)/G\), where \(\Omega(G)\) is the domain of discontinuity of \(G\). This accounts for real curves of genus \(g\) which separate their complexification, the case where the real curve does not separate may be reduced to the former using a double covering. The main idea is to express the holomorphic abelian differentials on \(X\) via Poincaré series, where the summation is over \(G\). The needed facts (convergence of Poincaré series, basis for holomorphic differentials for \(X\) given by Poincaré series easily obtained from the generators of \(G)\) are explained following closely a paper by \textit{C. J. Earle} and \textit{A. Marden} [Ill. J. Math. 13, 202-219 (1969; Zbl 0169.102)]. -- The author gives an estimation of the error made when the Poincaré series is truncated, and indicates that there are in this estimation constants which may be difficult to calculate from the presentation of \(G\). The paper closes with an example for genus 2, for which the computation time is given.
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 14P05
zbMATH DE Number: 502908
zbMATH Keywords: compute the period matrix of a real algebraic curve
zbMATH Keywords: uniformization
zbMATH Keywords: Poincaré series
author: Mika Seppälä
reviewed by: Michel Coste
MaRDI profile type: Publication
cites work: Foundations of the theory of Klein surfaces
cites work: On Poincaré series with application to \(H^ P\) spaces on bordered Riemann surfaces
cites work: Real algebraic curves
cites work: Moduli spaces of stable real algebraic curves
cites work: Moduli spaces for real algebraic curves and real abelian varieties
cites work: Riemann surfaces, plane algebraic curves and their period matrices
cites work: Compactifications of moduli spaces in real algebraic geometry
OpenAlex ID: W1966060217
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description: scientific article; zbMATH DE number 502908