Proceedings of the American Mathematical Society

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



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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  • 1. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  • 2. Matthew Boylan, Swinnerton-Dyer type congruences for certain Eisenstein series, 𝑞-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 93–108. MR 1874523, 10.1090/conm/291/04894
  • 3. Fred Diamond and John Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR 1357209
  • 4. Jayce Getz, A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2221–2231. MR 2052397, 10.1090/S0002-9939-04-07478-7
  • 5. J. S. Milne, Modular functions and modular forms, Course note,, Univ. of Michigan, 1997.
  • 6. K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $ q$-series, CBMS, 102, American Math. Soc., Providence, Rhode Island, 2004.
  • 7. K. Ono and K. Bringmann, Identities for traces of singular moduli, Acta Arith. 119 (2005), 317-327.
  • 8. F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169-170.
  • 9. R. A. Rankin, The zeros of certain Poincaré series, Compositio Math. 46 (1982), no. 3, 255–272. MR 664646
  • 10. Zeév Rudnick, On the asymptotic distribution of zeros of modular forms, Int. Math. Res. Not. 34 (2005), 2059–2074. MR 2181743, 10.1155/IMRN.2005.2059
  • 11. Bruno Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German by J. R. Smart and E. A. Schwandt; Die Grundlehren der mathematischen Wissenschaften, Band 203. MR 0412107
  • 12. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
  • 13. H. A. Verrill, Fundamental domain drawer, Java, fundomain.

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Additional Information

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.