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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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On zeros of Eisenstein series for genus zero Fuchsian groups
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by Heekyoung Hahn PDF
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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Additional Information
  • Heekyoung Hahn
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • MR Author ID: 707443
  • Email: hahn@math.rochester.edu
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2391-2401
  • MSC (2000): Primary 11F03, 11F11
  • DOI: https://doi.org/10.1090/S0002-9939-07-08763-1
  • MathSciNet review: 2302560