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Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb{Z})$

Author: Hiroyuki Tsutsumi
Journal: Proc. Amer. Math. Soc. 134 (2006), 931-941
MSC (2000): Primary 11F03, 11F11, 11F25
Published electronically: July 20, 2005
MathSciNet review: 2196023
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Abstract: We give a definition of the modular differential equations of weight $k$ for a discrete subgroup for $\Gamma \subset SL_2(\mathbb{R})$; in this paper we set $\Gamma = SL_2(\mathbb{Z})$. We solve such equations admitting regular singularities at elliptic points for $SL_2(\mathbb{Z})$ in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer $k$, and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight $k$.

References [Enhancements On Off] (What's this?)

  • 1. Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig, 1991, A modern theory of special functions. MR 1118604 (92j:33001)
  • 2. M Kaneko and M Koike, On modular forms arising from a differential equation of hypergeometric type, Ramanujan J. 7 (2003), 145-164. MR 2035798 (2005a:11050)
  • 3. M. Kaneko and D. Zagier, Supersingular $j$-invariants, hypergeometric series, and Atkin's orthogonal polynomials, Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 97-126. MR 1486833 (99b:11064)
  • 4. Masanobu Kaneko and Naoya Todaka, Hypergeometric modular forms and supersingular elliptic curves, Proceedings on Moonshine and related topics (Montréal, QC, 1999) (Providence, RI), CRM Proc. Lecture Notes, vol. 30, Amer. Math. Soc., 2001, pp. 79-83. MR 1877758 (2002m:11036)
  • 5. Masaaki Yoshida, Hypergeometric functions, my love, Aspects of Mathematics, E32, Friedr. Vieweg & Sohn, Braunschweig, 1997, Modular interpretations of configuration spaces. MR 1453580 (98k:33024)

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Additional Information

Hiroyuki Tsutsumi
Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504 Japan
Address at time of publication: Osaka University of Health and Sports Science, 1-1 Asashirodai, Kumatori-cho, Sennan-gun, Osaka 590-0496, Japan

Keywords: Modular form, hypergeometric series
Received by editor(s): June 3, 2004
Received by editor(s) in revised form: October 26, 2004
Published electronically: July 20, 2005
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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