The Ramanujan Journal

, Volume 7, Issue 1–3, pp 145–164 | Cite as

On Modular Forms Arising from a Differential Equation of Hypergeometric Type

  • Masanobu Kaneko
  • Masao Koike


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.

modular/quasimodular forms hypergeometric differential equation 


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  1. 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.Google Scholar
  2. 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.Google Scholar
  3. 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), 427-452Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Masanobu Kaneko
    • 1
  • Masao Koike
    • 2
  1. 1.Graduate School of MathematicsKyushu University 33FukuokaJapan
  2. 2.Graduate School of MathematicsKyushu University, RopponmatsuFukuokaJapan

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