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On Modular Forms Arising from a Differential Equation of Hypergeometric Type

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Abstract

Modular and quasimodular solutions of a specific second order differential equation in the upper-half plane, which originates from a study of supersingular j-invariants in the first author's work with Don Zagier, are given explicitly. Positivity of Fourier coefficients of some of the solutions as well as a characterization of the differential equation are also discussed.

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References

  1. M. Kaneko and D. Zagier, “Supersingular j-invariants, Hypergeometric series, and Atkin's orthogonal polynomials,” AMS/IP Studies in Advanced Mathematics 7 (1998), 97-126.

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  2. Ikuo Satake, “Flat structure for the simple elliptic singularity of type ?6 and Jacobi form,” in Proc. of the Japan Academy 69A(7) (1993), 247-251.

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  3. Ikuo Satake, “Flat structure and the prepotential for the elliptic root system of type D (1,1) 4,” in Topological Field Theory, Primitive Forms and Related Topics (Kashiwara, Matsuo, Saito, and Satake eds.), Progress in Math. 160 (1998), 427-452

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University 33, Fukuoka, 812-8581, Japan

    Masanobu Kaneko

  2. Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810-8560, Japan

    Masao Koike

Authors
  1. Masanobu Kaneko
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  2. Masao Koike
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Kaneko, M., Koike, M. On Modular Forms Arising from a Differential Equation of Hypergeometric Type. The Ramanujan Journal 7, 145–164 (2003). https://doi.org/10.1023/A:1026291027692

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  • DOI: https://doi.org/10.1023/A:1026291027692

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