Drawing Curves Over Number Fields

  • G. B. Shabat
  • V. A. Voevodsky
Part of the Modern Birkhäuser Classics book series (volume 88)


This paper develops some of the ideas outlined by Alexander Grothendieck in his unpublished Esquisse d’un programme [0] in 1984.


Modulus Space Riemann Surface Number Field Fuchsian Group Cell Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [0]
    Grothendieck A., Esquisse d’un programme, Preprint 1984.zbMATHGoogle Scholar
  2. [1]
    Belyi G.V., On Galots extensions of the maximal cyclotomic field (in Russian). Izvestiya An SSSR, ser. matem., 43:2 (1979), 269–276.Google Scholar
  3. [2]
    Mumford D., Curves and their Jacobians, Univ. of Michigan Press, 1975.zbMATHGoogle Scholar
  4. [3]
    Coxeter H.S.M. and Moser W.O.J., Generators and relations for discrete groups, Springer-Verlag, 1972.CrossRefGoogle Scholar
  5. [4]
    Penner R.C., The Teichmuller Space of a Punctured Surface, preprint.Google Scholar
  6. [5]
    Klein F., Ueber die Transformation siebenter Ordnung der elliptischen Funktionen, Math. Ann. 14 B.3 (1878), 428–471.MathSciNetCrossRefGoogle Scholar
  7. [6]
    Hartshorne, R.S., Algebraic Geometry, Springer-Verlag, 1977.CrossRefGoogle Scholar
  8. [7]
    Holzapfel, R.-P., Around Euler Partial Differential Equations, Berlin: Deutscher Verlag der Wissenschaften, 1986.zbMATHGoogle Scholar
  9. [8]
    Yui, N., Explicit form of modular equations, J. Reine und Angew Math, 1978, Bd. 299–300.zbMATHGoogle Scholar
  10. [9]
    Vladutz, S.G., Modular curves and the codes of polynomial complexity (in Russian). Preprint, 1978.Google Scholar
  11. [10]
    Voevodsky V.A., and Shabat G.B., Equilateral triangulations of Riemann surfaces and curves over algebraic number fields (in Russian). Doklady ANSSSR (1989), 204:2, 265–268.MathSciNetGoogle Scholar
  12. [11]
    Douady A. and Hubbard J., On the density of Strebel differentials. Invent. Math. (1975), 30 , N2, 175–179.MathSciNetCrossRefGoogle Scholar
  13. [12]
    Boulatov D.V., Kazakov V.A., Kostov I.K., Migdal A.A., Analytical and numerical study of dynamically triangulated surfaces. Nucl. Phys. B275 [FS17] (1986), 641–686.MathSciNetCrossRefGoogle Scholar
  14. [13]
    Manin Yu. I., Reflections on arithmetical physics, Talk at the Poiana-Brashov School on Strings and Conformai Field Theory, Sept. 1987, 1–14.Google Scholar
  15. [14]
    Itzyczon, C., Random Geometry, Lattices and Fields. “New perspectives in Quantum Field Theory”, Jaca (Spain), 1985.Google Scholar
  16. [15]
    Threlfall, W., Gruppenbilder, Abh. Sachs. Akad. Wiss. Math.-Phys. Kl.41, S. 1–59.Google Scholar
  17. [16]
    Abikoff, W., The Real-Analytic Theory of Teichmuller Space, Lecture Notes in Mathematics, Springer-Verlag, 820, 1980.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2007

Authors and Affiliations

  • G. B. Shabat
    • 1
  • V. A. Voevodsky
    • 2
  1. 1.MoscowUSSR
  2. 2.Mech.-Math facultyMoscow State UniversityMoscowUSSR

Personalised recommendations