Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb {Z})$
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- by Hiroyuki Tsutsumi PDF
- Proc. Amer. Math. Soc. 134 (2006), 931-941 Request permission
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
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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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2196023