On modular solutions of certain modular differential equation and supersingular polynomials

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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 series 

Mathematics Subject Classification

11F11 11F25 

Notes

Acknowledgements

The author would like to thank Professor Masanobu Kaneko for helpful advice. He also thanks the anonymous referee for valuable suggestions.

References

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    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
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    Baba, S., Granath, H.: Orthogonal systems of modular forms and supersingular polynomials. Int. J. Number Theory 7, 249–259 (2011)MathSciNetCrossRefMATHGoogle Scholar
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    Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14, 197–272 (1941)MathSciNetCrossRefMATHGoogle Scholar
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    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)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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