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Nonlinear differential equations satisfied by certain classical modular forms

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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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  1. Departments of Mathematics and Physics, University of Arizona, Tucson, AZ, 85721, USA

    Robert S. Maier

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Maier, R.S. Nonlinear differential equations satisfied by certain classical modular forms. manuscripta math. 134, 1–42 (2011). https://doi.org/10.1007/s00229-010-0378-9

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