Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms
Abstract
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.
Cite
@article{arxiv.0807.1081,
title = {Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms},
author = {Robert S. Maier},
journal= {arXiv preprint arXiv:0807.1081},
year = {2014}
}
Comments
40 pages, final version, accepted by Manuscripta Mathematica