On zeros of Eisenstein series for genus zero Fuchsian groups

Abstract

Let $\GN\leq\SLR$ be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let $\Ek$ be the holomorphic Eisenstein series of weight $2k$ on $\GN$ that is nonvanishing at $\infty$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on $\GN,$ and on a choice of a fundamental domain $\F$, we prove that all but possibly $c(\GN,\F)$ of the non-trivial zeros of $\Ek$ lie on a certain subset of $\{z\in\mathfrak{H} : \JN(z)\in\mathbb{R}\}$. Here $c(\GN,\F)$ is a constant that does not depend on the weight $2k$ and $\JN$ is the canonical hauptmodul for $\GN.$