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

TL;DR

This paper investigates the relationship between quasimodular forms of depth 1 and second-order modular differential equations with regular singularities, focusing on groups generated by Atkin-Lehner involutions for specific levels.

Contribution

It establishes the converse connection between quasimodular forms and modular differential equations for certain modular groups, extending previous understanding.

Findings

Quasimodular forms divided by modular forms solve specific differential equations.

Main results provide a converse to known relationships for groups +N with N=1,2,3.

Exploration of differential equations associated with quasimodular forms on these groups.

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 $\Gamma=\Gamma_0^+(N)$ generated by $\Gamma_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($\Gamma_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). 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 $\Gamma_0^+(N)$, $N=1,2,3$.