Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

Abstract

We study the relations governing the ring of quasiautomorphic forms associated with triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated with these triangle groups are shown to satisfy Ramanujan-like identities. These identities, in turn, allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated with the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.