On Modular Forms Arising from a Differential Equation of Hypergeometric Type

  • Masanobu Kaneko
  • Masao Koike
Part of the Developments in Mathematics book series (DEVM, volume 10)


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.

Key words

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.MathSciNetGoogle Scholar
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    Ikuo Satake, “Flat structure for the simple elliptic singularity of type E6 and Jacobi form,” in Proc. of the Japan Academy 69A(7) (1993),247–251.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ikuo Satake, “Flat structure and the prepotential for the elliptic root system of type (41’1),” 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

© Springer Science+Business Media New York 2003

Authors and Affiliations

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

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