On zeros of Eisenstein series for genus zero Fuchsian groups

Hahn, Heekyoung

doi:10.1090/S0002-9939-07-08763-1

On zeros of Eisenstein series for genus zero Fuchsian groups
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Proc. Amer. Math. Soc. 135 (2007), 2391-2401 Request permission

Abstract:

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

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