Geometriae Dedicata

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

Numerical Schottky Uniformizations



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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  1. 1.
    Aigon, A., Silhol, R. 2002JLondon Math. Soc.66671690Google Scholar
  2. 2.
    Bers, L. 1975Automorphic forms for Schottky groupsAdv. Math.16332361CrossRefGoogle Scholar
  3. 3.
    Burnside, W. 1892On a class of automorphic functionsProc. London Math. Soc.234988Google Scholar
  4. 4.
    Buser, P., Silhol, R. 2001Geodesics, periods and equations of real hyperelliptic curvesDuke Math. J.108211250CrossRefGoogle Scholar
  5. 5.
    Chuckrow, V. 1968On Schottky groups with application to Kleinian groupsAnn. Math.884761Google Scholar
  6. 6.
    Farkas, H. and Kra, I.: Riemann Surfaces. Springer-Verlag, Berlin.Google Scholar
  7. 7.
    Greenleaf, N., May, C.L. 1982Bordered Klein surfaces with maximal symmetryTrans. Amer. Math. Soc.274265283Google Scholar
  8. 8.
    Gianni, P., Seppälä, M., Silhol, R., Trager, B. 1998Riemann surfaces, plane algebraic curves and their period matricesJ. Symbolic Comput.26789803CrossRefGoogle Scholar
  9. 9.
    Hidalgo, R. A.: Real Surfaces, Riemann matrices and algebraic curves. In: Complex Manifolds and Hyperbolic Geometry (II Iberoamerican Congress on Geometry, January 4–9, 2001, CIMAT, Guanajuato, Mexico), Contemp Math. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 277–299,Google Scholar
  10. 10.
    Keen, L.: On hyperelliptic Schottky groups, Ann. Acad. Sci. Fenn. Series A.I. Mathematica 5 (1980).Google Scholar
  11. 11.
    Koebe, P. 1907Über die Uniformisierung reeller algebraischer KurvenNachr. Akad. Wiss. Goettingen 177190Google Scholar
  12. 12.
    Koebe, P. 1910Über die Uniformisierung der Algebraischen Kurven IIMath. Ann.69181CrossRefGoogle Scholar
  13. 13.
    Maskit, B. 1988Kleinian Groups. Grundlehren Math. Wiss. 287Springer-VerlagBerlinGoogle Scholar
  14. 14.
    Maskit, B. 1967A characterization of Schottky groupsJ. Anal. Math.19227230Google Scholar
  15. 15.
    May, C. L. 1975Automorphisms of compact Klein surfaces with boundaryPacific J. Math.59199210Google Scholar
  16. 16.
    May, C. L. 1977A bound for the number of automorphisms of a compact Klein surface with boundaryProc. Amer. Math. Soc.63273280Google Scholar
  17. 17.
    Seppälä M. (1994) Computation of period matrices of real algebraic curves. Discrete Comput. Geom. 11:65–81Google Scholar
  18. 18.
    Seppälä, M.: Myrberg’s numerical uniformization of hyperelliptic curves, Preprint.Google Scholar
  19. 19.
    Silhol, R.: Hyperbolic lego and algebraic curves in genus 2 and 3. In: Complex Manifolds and Hyperbolic Geometry, Contemp. Math. 311, Amer, Math. Soc. Providence, 2001, pp., 313–334.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Departamento de MatemáticasUTFSMValparaísoChile

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