On modular solutions of certain modular differential equation and supersingular polynomials
Article
First Online:
- 62 Downloads
Abstract
We extend the results of Kaneko–Zagier and Baba–Granath on relations of supersingular polynomials and solutions of certain second-order modular differential equations.
Keywords
Modular form Supersingular polynomials Hypergeometric seriesMathematics Subject Classification
11F11 11F25Notes
Acknowledgements
The author would like to thank Professor Masanobu Kaneko for helpful advice. He also thanks the anonymous referee for valuable suggestions.
References
- 1.Kaneko, M., Zagier, D.: Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. In: Computational Perspectives on Number Theory (Chicago, IL, 1995). AMS/IP Stud. Adv. Math., 7, pp. 97–126. American Mathematical Society, Providence, RI (1998)Google Scholar
- 2.Baba, S., Granath, H.: Orthogonal systems of modular forms and supersingular polynomials. Int. J. Number Theory 7, 249–259 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14, 197–272 (1941)CrossRefzbMATHGoogle Scholar
- 4.Tsutsumi, H.: Modular differential equations of second order with regular singularities at elliptic points for \(SL_{2}(\mathbb{Z})\). Proc. Am. Math. Soc. 134, 931–941 (2006)CrossRefzbMATHGoogle Scholar
Copyright information
© Springer Science+Business Media, LLC, part of Springer Nature 2018