Parameterizations of the Chazy equation

TL;DR

This paper explores the parameterizations of the Chazy equation derived from automorphic properties of Schwarz triangle functions, connecting solutions to Ramanujan's Eisenstein series and providing new insights into modular forms.

Contribution

It systematically constructs solutions to the Chazy equation linked to Schwarz triangle functions and relates them to Ramanujan's Eisenstein series, offering new parametrizations.

Findings

Solutions correspond to specific parameter values of triangle functions.

All known parametrizations of Eisenstein series are recovered.

One new parametrization of Eisenstein series is introduced.

Abstract

The Chazy equation $y''' = 2yy'' - 3y'^2$ is derived from the automorphic properties of Schwarz triangle functions $S(\alpha, \beta, \gamma; z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $\alpha, \beta, \gamma$. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization of the Eisenstein series $P, Q, R$ introduced by Ramanujan in his modular theories of signature 2, 3, 4 and 6.