Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

Abstract

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an ${\mathfrak{sl}}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $\Delta (n,m,\infty )$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $\Delta (n,m,\infty )$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $\Delta (2,5,\infty )$ from solutions of systems of nonlinear ODEs.

arXiv commentThe discussion of Ramanujan systems associated to general hyperbolic triangles (Section 3) has been substantially revised. To appear in Forum Mathematicum