Ramanujan-type systems of nonlinear ODEs for Γ0(2) and Γ0(3)

Nikdelan, Younes

doi:10.1016/j.exmath.2022.04.001

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Ramanujan-type systems of nonlinear ODEs for Γ₀(2) and Γ₀(3)

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Nikdelan, Younes
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1016/j.exmath.2022.04.001
(Publisher version)

https://doi.org/10.48550/arXiv.2111.01844
(Preprint)

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Citation

Nikdelan, Y. (2022). Ramanujan-type systems of nonlinear ODEs for Γ₀(2) and Γ₀(3). Expositiones Mathematicae, 40(3), 409-431. doi:10.1016/j.exmath.2022.04.001.

Cite as: https://hdl.handle.net/21.11116/0000-000A-8146-5

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.