Parameterizations of the Chazy equation

英文摘要：

The Chazy equation $y''' = 2yy'' - 3y'^2$ is derived from the automorphic properties of Schwarz triangle functions $S(α, β, γ; z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $α, β, γ$. 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.