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ISSN 1088-6826(e) ISSN 0002-9939(p)

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
MathSciNet review: 2022368
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Abstract | References | Similar articles | Additional information

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:

[BBEIM]: 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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