A note on zeros of Eisenstein series for genus zero Fuchsian groups

TL;DR

This paper extends previous results on the zeros of Eisenstein series for genus zero Fuchsian groups, showing most zeros lie on the lower arcs of a fundamental domain under certain conditions.

Contribution

It generalizes Hahn's results by demonstrating that most zeros of Eisenstein series are located on the lower arcs of the fundamental domain.

Findings

Most zeros of $E_{2k}^ ext{Gamma}$ lie on the lower arcs of the fundamental domain.

The result holds under similar assumptions as Hahn's original theorem.

The zeros are confined to a specific subset related to the real axis of the $j_ ext{Gamma}$-invariant.

Abstract

Let $\Gamma \subseteq \text{SL}_2(\mathbb{R})$ be a genus zero Fuchsian group of the first kind having $\infty$ as a cusp, and let $E_{2 k}^{\Gamma}$ be the holomorphic Eisenstein series associated with $\Gamma$ for the $\infty$ cusp that does not vanish at $\infty$ but vanishes at all the other cusps. In the paper "On zeros of Eisenstein series for genus zero Fuchsian groups", under assumptions on $\Gamma$, and on a certain fundamental domain $\mathcal{F}$, H. Hahn proved that all but at most $c(\Gamma, \mathcal{F})$ (a constant) of the zeros of $E_{2 k}^{\Gamma}$ lie on a certain subset of $\{z \in \mathfrak{H} : j_{\Gamma}(z) \in \mathbb{R}\}$. In this note, we consider a small generalization of Hahn's result on the domain locating the zeros of $E_{2 k}^{\Gamma}$. We can prove most of the zeros of $E_{2 k}^{\Gamma}$ in $\mathcal{F}$ lie on its lower arcs under the same assumption.