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arXiv · 2103.04890

Quasimodular forms and modular differential equations which are not apparent at cusps: I

Chang-Shou Lin·Yifan Yang
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Abstract

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^+(N)$, $N=1,2,3$.

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Chang-Shou Lin, Yifan Yang. 2021-03-08. Quasimodular forms and modular differential equations which are not apparent at cusps: I. https://arxiv.org/abs/2103.04890

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