Ramanujan-type systems of nonlinear ODEs for $\Gamma_0(2)$ and $\Gamma_0(3)$

TL;DR

This paper develops nonlinear ODE systems whose solutions generate quasi-modular forms on specific subgroups, revealing algebraic structures and introducing Ramanujan-type tau functions with new recurrence and congruence properties.

Contribution

It introduces novel nonlinear ODE systems for $ ext{Gamma}_0(2)$ and $ ext{Gamma}_0(3)$, linking them to quasi-modular forms and algebraic structures.

Findings

Generated graded algebras have an $ ext{sl}_2( ext{C})$-module structure

Defined Ramanujan-type tau functions for $ ext{Gamma}_0(2)$ and $ ext{Gamma}_0(3)$

Derived new recurrence and congruence relations

Abstract

This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups $\Gamma_0(2)$ and $\Gamma_0(3)$. Using these systems, we provide the generated graded algebras with an $\mathfrak{sl}_2(\mathbb{C})$-module structure. As applications, we introduce Ramanujan-type tau functions for $\Gamma_0(2)$ and $\Gamma_0(3)$, and obtain some interesting and non-trivial recurrence and congruence relations.