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On Schottky groups arising from the hypergeometric equation with imaginary exponents

Author(s): Takashi Ichikawa; Masaaki Yoshida
Journal: Proc. Amer. Math. Soc. 132 (2004), 447-454.
MSC (2000): Primary 33C05
Posted: June 12, 2003
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Abstract: In an article by Sasaki and Yoshida (2000), we encountered Schottky groups of genus 2 as monodromy groups of the hypergeometric equation with purely imaginary exponents. In this paper we study automorphic functions for these Schottky groups, and give a conjectural infinite product formula for the elliptic modular function $\lambda$.

References:

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E.D.BELOKOLOS, A.I.BOBENKO, V.Z.ENOL'SKII, A.R.ITS AND V.B.MATVEEV, Algebro-geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, New York, 1994.

[GP]
L.GERRITZEN AND M.VAN DER PUT, Schottky groups and Mumford curves, Lecture Notes in Math. 817(1980), Springer. MR 82j:10053

[SY]
T.SASAKI AND M.YOSHIDA, A geometric study of the hypergeometric function with imaginary exponents, Experimental Math. 10(2000), 321-330.

[S]
F.SCHOTTKY, Über eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments univerändert bleibt, J. Reine Angew. Math. 101(1887) 227-272.

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Additional Information:

Takashi Ichikawa
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502 Japan
Email: ichikawa@ms.saga-u.ac.jp

Masaaki Yoshida
Affiliation: Department of Mathematics, Kyushu University, Fukuoka 810-8560 Japan
Email: myoshida@math.kyushu-u.ac.jp

DOI: 10.1090/S0002-9939-03-07117-X
PII: S 0002-9939(03)07117-X
Keywords: Hypergeometric equation, monodromy group, Schwarzian derivative, reflection principle, Schottky group, automorphic functions, infinite product, $\lambda$ function.
Received by editor(s): October 1, 2001
Received by editor(s) in revised form: October 2, 2002
Posted: June 12, 2003
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2003, American Mathematical Society

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The following works have cited this article

T. Ichikawa and M. Yoshida, On Schottky groups arising from the hypergeometric equation with imaginary exponents, Proc. Amer. Math. Soc. (2) 132 (2003), 447-454.

T. Ichikawa and M. Yoshida, On Schottky groups arising from the hypergeometric equation with imaginary exponents, Proc. Amer. Math. Soc. 132 (2004), 447-454.

T. Ichikawa and M. Yoshida, On Schottky groups arising from the hypergeometric equation with imaginary exponents, Proc. Amer. Math. Soc. 132 (2004), 447-454.

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