Ramanujan systems of Rankin–Cohen type and hyperbolic triangles

Abstract

Abstract In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ ( C ) \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 Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ ( n , m , ∞ ) \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 Δ ⁢ ( 2 , 5 , ∞ ) \Delta(2,5,\infty) from solutions of systems of nonlinear ODEs.