On the common zeros of quasi-modular forms for $\Gamma_0^+(N)$ of level $N=1,2,3$

TL;DR

This paper investigates the zeros of derivatives of Eisenstein series for specific modular groups, proving that all such zeros are simple for levels 2 and 3, extending previous results for the full modular group.

Contribution

It generalizes existing results on zeros of Eisenstein series to quasi-modular forms for $ ext{Gamma}_0^+(N)$ at levels 2 and 3, showing all zeros are simple.

Findings

All zeros of derivatives are simple for levels 2 and 3.

Extends previous results from SL_2(Z) to $ ext{Gamma}_0^+(N)$.

Zeros are characterized for derivatives of Eisenstein series.

Abstract

In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for $\Gamma_0^+(N)$ of level $N=1,2$ and $3$, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms, and prove that all the zeros of the iterated derivatives of the Eisenstein series $\frac{d^m E_k^{(N)}(\tau)}{d\tau^m}$ of weight $k=2,4,6$ for $\Gamma_0^+(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterl\'{e} \cite{SJ20} for SL$_2(\mathbb{Z})$.