Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

Details Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

2008-07-07

arxiv.org/abs/0807.1081v5

Mathematics

Number Theory

40 pages, final version, accepted by Manuscripta Mathematica

Scientific paper

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.

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