Algebraic Values of Schwarz Triangle Functions

  • Hironori Shiga
  • Jürgen Wolfart
Part of the Progress in Mathematics book series (PM, volume 260)

Abstract

We consider Schwarz maps for triangles whose angles are rather general rational multiples of π. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.

Keywords

Schwarz triangle functions hypergeometric functions algebraic values transcendence complex multiplication 

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References

  1. [1]
    N. Archinard, Hypergeometric Abelian Varieties, Canad. J. Math. 55(5) (2003), 897–932.MATHMathSciNetGoogle Scholar
  2. [2]
    D. Bertrand, Endomorphismes de groupes algébriques; applications arithmétiques, pp. 1–45 in Approximations Diophantiennes et Nombres Transcendants, Progr. Math. 31, Birkhäuser, 1983.Google Scholar
  3. [3]
    Cl. Chevalley, A. Weil, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Hamburger Math. Sem. 10 (1934), 358–361.Google Scholar
  4. [4]
    P. Cohen, J. Wolfart, Modular embeddings for some non-arithmetic Fuchsian groups, Acta Arithmetica 56 (1990), 93–110.MATHMathSciNetGoogle Scholar
  5. [5]
    P.B. Cohen, J. Wolfart, Algebraic Appell-Lauricella Functions, Analysis 12 (1992), 359–376.MathSciNetMATHGoogle Scholar
  6. [6]
    P.B. Cohen, G. Wüstholz, Application of the André-Oort Conjecture to some Questions in Transcendence, pp. 89–106 in A Panorama in Number Theory or The View from Baker’s Garden, ed.: G. Wüstholz, Cambridge Univ. Press, 2002.Google Scholar
  7. [7]
    B. Edixhoven, A. Yafeev, Subvarieties of Shimura varieties, Ann. Math. 157 (2003), 621–645.MATHGoogle Scholar
  8. [8]
    A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, Vol. 1., McGraw-Hill, 1953.Google Scholar
  9. [9]
    Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, 1978.Google Scholar
  10. [10]
    F. Klein, Vorlesungen über die hypergeometrische Funktion, Springer, 1933.Google Scholar
  11. [11]
    N. Koblitz, A. Ogus, Algebraicity of some products of values of the Г function, Proc. Symp. Pure Math. 33 (1979), 343–346.Google Scholar
  12. [12]
    K. Koike, On the family of pentagonal curves of genus 6 and associated modular forms on the ball, J. Math. Soc. Japan, 55 (2003), 165–196.MATHMathSciNetGoogle Scholar
  13. [13]
    S. Lang, Complex Multiplication, Springer, 1983.Google Scholar
  14. [14]
    H.A. Schwarz, Über diejenigen Fälle, in welchen die Gaußische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math. 75 (1873), 292–335.Google Scholar
  15. [15]
    H. Shiga, T. Tsutsui, J. Wolfart, Fuchsian differential equations with apparent singularities, Osaka J. Math. 41 (2004), 625–658.MATHMathSciNetGoogle Scholar
  16. [16]
    H. Shiga, J. Wolfart, Criteria for complex multiplication and transcendence properties for automorphic functions, J. reine angew. Math. 463 (1995), 1–25.MATHMathSciNetGoogle Scholar
  17. [17]
    G. Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. Math. 78 (1963), 149–192.CrossRefMathSciNetGoogle Scholar
  18. [18]
    C.L. Siegel, Lectures on Riemann Matrices, Tata Inst., Bombay, 1963.MATHGoogle Scholar
  19. [19]
    K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), 91–106.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Waldschmidt, Transcendance de périodes: état de connaissances, to appear in the proceedings of a conference in Mahdia 2003.Google Scholar
  21. [21]
    J. Wolfart, Werte hypergeometrischer Funktionen, Invent. math. 92 (1988), 187–216.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    J. Wolfart, G. Wüstholz, Der Überlagerungsradius algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen, Math. Ann. 273 (1985), 1–15.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    G. Wüstholz, Algebraic Groups, Hodge Theory, and Transcendence, pp. 476–483 in Proc. of the ICM Berkeley 1986 (ed.: A.M. Gleason), AMS, 1987.Google Scholar
  24. [24]
    G. Wüstholz, Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Ann. of Math. 129 (1989), 501–517.CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Hironori Shiga
    • 1
  • Jürgen Wolfart
    • 2
  1. 1.Inst. of Math. and PhysicsChiba UniversityChibaJapan
  2. 2.Math. Sem. der Univ.Frankfurt a.M.Germany

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