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 Γ0(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 Γ0(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.

Original languageEnglish (US)
Pages (from-to)1-42
Number of pages42
JournalManuscripta Mathematica
Volume134
Issue number1
DOIs
StatePublished - Jan 1 2011

ASJC Scopus subject areas

  • Mathematics(all)

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