Myrberg's numerical uniformization of hyperelliptic curves

 
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  1. Preliminaries
  2. Numerical uniformization problem
  3. History of the problem
  4. Present state of the problem
Poincaré (1908):
The true method of foreseeing the future of mathematics is to study its history and its actual state.
 
 
 

References

[1] W. Burnside. Note on the equation y2=x(x4-1). Proc. London Math. Soc. (1), 24:17-20, 1893.

[2] Peter Buser. Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser Verlag, Basel-Boston-New York, 1992.

[3] Peter Buser and Robert Silhol. Geodesics, Periods and Equations of Real Hyperelliptic Curves, 1998 Manuscript submitted for publication.

[4] Fernanda Esser. Die algebraische Uniformisierung mit numerischen Beispielen nach Myrberg. Manuscript probably written in the 1930's, exact date not known.

[5] César Gómez and Germán Sierra. A note on Liouville theory and the uniformization of Riemann surfaces. In Quantum field theory, statistical mechanics, quantum groups and topology (Coral Gables, FL, 1991), pages 115-122. World Sci. Publishing, River Edge, NJ, 1992.

[6] Dennis A. Hejhal. Sur les paramétres accessoires pour l'uniformisation de Schottky. C. R. Acad. Sci. Paris Sér. A, 279:713-716, 1974.

[7] Dennis A. Hejhal. Sur les paramétres accessoires pour l'uniformisation de Schottky. C. R. Acad. Sci. Paris Sér. A, 279:695-697, 1974.

[8] Dennis A. Hejhal. Sur les paramétres accessoires pour l'uniformisation fuchsienne. C. R. Acad. Sci. Paris Sér. A-B, 282(8):Ai, A403-A406, 1976.

[9] L. Keen, H. E. Rauch, and A. T. Vasquez. Moduli of punctured tori and the accessory parameter of Lamé's equation. Trans. Amer. Math. Soc., 255:201-230, 1979.

[10] Linda Keen. Collars on Riemann surfaces. In: Discontinuous Groups and Riemann Surfaces, volume 79 of Ann. of Math. Studies, pages 263-268. Princeton University Press, 1974.

[11] Linda Keen. Accessory parameters and the uniformization of punctured tori. In: Modular functions in analysis and number theory, pages 132-149. Univ. Pittsburgh, Pittsburgh, Pa., 1983.

[12] Linda Keen. Review on [13] by Irwin Kra. MR 89j:30062, 1989. Mathematics Review.

[13] Irwin Kra. Accessory parameters for punctured spheres. Trans. Amer. Math. Soc., 313(2):589-617, 1989.

[14] Ravi S. Kulkarni. Riemann surfaces admitting large automorphism groups. In: Extremal Riemann surfaces (San Francisco, CA, 1995), pages 63-79. Amer. Math. Soc., Providence, RI, 1997.

[15] O. Lehto and K. I. Virtanen. Quasiconformal mappings in the plane, volume 126 of Die Grundlehren der mathematischen Wissenschaften}. Springer-Verlag, Berlin-Heidelberg-New York, 1973. Translated from the {G}erman by K. W. Lucas. 2nd ed.

[16] P. J. Myrberg. Über die Numerische Ausführung der Uniformisierung. Acta Soc. Sci. Fenn., XLVIII(7):1 - 53, 1920.

[17] H. Poincaré. Sur les Groupes des équations Linéaires. Acta Math., IV:201 - 312, 1884.

[18] R. A. Rankin. Sir Edmund Whittaker's work on automorphic functions. Proc. Edinburgh Math. Soc., 11:25-30, 1958.

[19] R. A. Rankin. Burnside's uniformization. Acta Arith., 79(1):53-57, 1997.

[20] Rubí E. Rodríguez and Víctor González-Aguilera. Fermat's Quartic Curve, Klein's Curve and their Tetrahedron. Contemporary Matematics, 201:43-62, 1997.

[21] Simon J. Smith and Joachim A. Hempel. The accessory parameter problem for the uniformization of the twice-punctured disc. J. London Math. Soc. (2), 40(2):269-279, 1989.

[22] L. A. Takhtajan. Uniformization, local index theorem, and geometry of the moduli spaces of Riemann surfaces and vector bundles. In : Theta functions-Bowdoin 1987, Part 1 (Brunswick, ME, 1987), pages 581-596. Amer. Math. Soc., Providence, RI, 1989.

[23] E. T. Whittaker. On the connexion of algebraic functions with automorphic functions. Phil. Trans., 192A:1-32, 1898.

[24] J. M. Whittaker. The uniformisation of algebraic curves. J. London Math. Soc., 5:150-154, 1930.

[25] P. G. Zograf and L. A. Takhtajan. On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0. Mat. Sb. (N.S.)}, 132(174)(2):147-166, 1987.