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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
DOI: https://doi.org/10.1090/S0002-9939-05-08115-3
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 $k$.


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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
Email: tsutsumi@math.shimane-u.ac.jp, tsutsumi@ouhs.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-05-08115-3
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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