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.
In memory of Robert A. Rankin
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References
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.
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.
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), 427–452
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© 2003 Springer Science+Business Media New York
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Kaneko, M., Koike, M. (2003). On Modular Forms Arising from a Differential Equation of Hypergeometric Type. In: Berndt, B., Ono, K. (eds) Number Theory and Modular Forms. Developments in Mathematics, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6044-6_12
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DOI: https://doi.org/10.1007/978-1-4757-6044-6_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5395-7
Online ISBN: 978-1-4757-6044-6
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