Abstract
We survey our construction of invariant functions on the real 3-dimensional hyperbolic space ℍ3 for the Whitehead-link-complement group W ⊂ GL 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Freitag, Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper, Sitzungsber. Heidelb. Akad. Wiss., 1 (1967), 1–49.
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.
J. Igusa, Theta Functions, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
T. Kimura, Hypergeometric Functions of Two Variables, Seminar Note in Math. Univ. of Tokyo, 1973.
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.
K. Matsumoto, Algebraic relations among some theta functions on the bounded symmetric domain of type I r,r, to appear in Kyushu J. Math..
K. Matsumoto, Automorphic functions for the Borromean-rings-complement group, preprint, 2005.
K. Matsumoto, H. Nishi and M. Yoshida, Automorphic functions for the Whitehead-link-complement group, preprint, 2005.
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.
K. Matsumoto and M. Yoshida, Invariants for some real hyperbolic groups, Internat. J. of Math., 13 (2002), 415–443.
W. Thurston, Geometry and Topology of 3-manifolds, Lecture Notes, Princeton Univ., 1977/78.
N. Wielenberg, The structure of certain subgroups of the Picard group. Math. Proc. Cambridge Philos. Soc., 84 (1978), no. 3, 427–436.
M. Yoshida, Hypergeometric Functions, My Love, Aspects of Mathematics, E32, Friedrich Vieweg & Sohn, Braunschweig, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Matsumoto, K. (2007). Invariant Functions with Respect to the Whitehead-Link. In: Holzapfel, RP., Uludağ, A.M., Yoshida, M. (eds) Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol 260. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8284-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8284-1_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8283-4
Online ISBN: 978-3-7643-8284-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)