Nonlinear differential equations satisfied by certain classical modular forms

Research output: Contribution to journalArticle

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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 2011

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Modular Forms
Nonlinear Differential Equations
Generalized Equation
Triangle Group
Hypergeometric Equation
Divisor Function
Elliptic integral
Eisenstein Series
System of Nonlinear Equations
Series Representation
Coupled System
Triangle
Table
Subgroup
Form

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Nonlinear differential equations satisfied by certain classical modular forms. / Maier, Robert S.

In: Manuscripta Mathematica, Vol. 134, No. 1, 01.2011, p. 1-42.

Research output: Contribution to journalArticle

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