Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

TL;DR

This paper characterizes special nonlinear differential systems linked to modular forms, introduces new systems for triangle groups, and derives novel relations for hypergeometric functions, expanding the understanding of modular and hyperbolic structures.

Contribution

It identifies Ramanujan systems of Rankin-Cohen type, connects them to triangle groups via modular embeddings, and constructs twisted modular forms from nonlinear ODE solutions.

Findings

Characterization of Ramanujan systems of Rankin-Cohen type.

Association of nonlinear ODE systems to triangle groups.

New relations for hypergeometric functions.

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.