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

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 $E_{2k}^{\Gamma }$ 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 $E_{2k}^{\Gamma }$ 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 $E_{2k}^{\Gamma }$. We can prove most of the zeros of $E_{2k}^{\Gamma }$ in $\mathcal{F}$ lie on its lower arcs under the same assumption.

5 pages, 6 figures. http://www2.math.kyushu-u.ac.jp/~j.shigezumi/

References in corpus (1)

• On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II