2022/07/15 by Gabriele Bogo, Bogo, Gabriele, Younes Nikdelan +1 · 1 citation
Mathematics · #11F03 #11F55 (secondary) #16W50 #34A34 (primary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2207.07686
openalex publication_date 2022/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an \mathfraksl2(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,∞). 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 Δ(n,m,∞) 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,∞) from solutions of systems of nonlinear ODEs.