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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$SL$ _{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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Heekyoung Hahn
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

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.

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