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On zeros of Eisenstein series for genus zero Fuchsian groups

Author: Heekyoung Hahn
Journal: Proc. Amer. Math. Soc. 135 (2007), 2391-2401
MSC (2000): Primary 11F03, 11F11
Published electronically: March 29, 2007
MathSciNet review: 2302560
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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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Additional Information

Heekyoung Hahn
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
MR Author ID: 707443

Keywords: Eisenstein series, modular forms, divisor polynomials
Received by editor(s): March 21, 2006
Received by editor(s) in revised form: April 27, 2006
Published electronically: March 29, 2007
Additional Notes: This research was supported in part by a National Science Foundation FRG grant (DMS 0244660)
Communicated by: Ken Ono
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.