Invariant Functions with Respect to the Whitehead-Link

Part of the Progress in Mathematics book series (PM, volume 260)


We survey our construction of invariant functions on the real 3-dimensional hyperbolic space ℍ3 for the Whitehead-link-complement group WGL 2(ℤ[i]) and for a few groups commensurable with W. We make use of theta functions on the bounded symmetric domain \( \mathbb{D} \) of type I 2,2 and an embedding i : ℍ3\( \mathbb{D} \). The quotient spaces of ℍ3 by these groups are realized by these invariant functions. We review classical results on the λ-function, the j-function and theta constants on the upper half space; our construction is based on them.


Whitehead link hyperbolic structure automorphic forms theta functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [F]
    E. Freitag, Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper, Sitzungsber. Heidelb. Akad. Wiss., 1 (1967), 1–49.Google Scholar
  2. [IKSY]
    K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé — A modern theory of special functions, Aspects of Mathematics, E16, Friedrich Vieweg & Sohn, Braunschweig, 1997.Google Scholar
  3. [I]
    J. Igusa, Theta Functions, Springer-Verlag, Berlin, Heidelberg, New York, 1972.zbMATHGoogle Scholar
  4. [K]
    T. Kimura, Hypergeometric Functions of Two Variables, Seminar Note in Math. Univ. of Tokyo, 1973.Google Scholar
  5. [M1]
    K. Matsumoto, Theta functions on the bounded symmetric domain of type I 2,2 and the period map of 4-parameter family of K3 surfaces, Math. Ann., 295 (1993), 383–408.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [M2]
    K. Matsumoto, Algebraic relations among some theta functions on the bounded symmetric domain of type I r,r, to appear in Kyushu J. Math..Google Scholar
  7. [M3]
    K. Matsumoto, Automorphic functions for the Borromean-rings-complement group, preprint, 2005.Google Scholar
  8. [MNY]
    K. Matsumoto, H. Nishi and M. Yoshida, Automorphic functions for the Whitehead-link-complement group, preprint, 2005.Google Scholar
  9. [MSY]
    K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a 4-parameter family of K3 surfaces and the Aomoto-Gel’fand hypergeometric function of type (3,6), Internat. J. of Math., 3 (1992), 1–164.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [MY]
    K. Matsumoto and M. Yoshida, Invariants for some real hyperbolic groups, Internat. J. of Math., 13 (2002), 415–443.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [T]
    W. Thurston, Geometry and Topology of 3-manifolds, Lecture Notes, Princeton Univ., 1977/78.Google Scholar
  12. [W]
    N. Wielenberg, The structure of certain subgroups of the Picard group. Math. Proc. Cambridge Philos. Soc., 84 (1978), no. 3, 427–436.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [Y]
    M. Yoshida, Hypergeometric Functions, My Love, Aspects of Mathematics, E32, Friedrich Vieweg & Sohn, Braunschweig, 1997.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations