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arXiv · 0807.1081

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

Robert S. Maier
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Abstract

A unified treatment is given of low-weight modular forms on Γ_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 Γ_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 Γ(1) is treated.

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Robert S. Maier. 2010-06-13. Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms. https://doi.org/10.1007/s00229-010-0378-9

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