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

DOI:
https://doi.org/10.1090/S0002-9939-07-08763-1

Published electronically:
March 29, 2007

MathSciNet review:
2302560

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\Gamma .$

- 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** - 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. 93â€“108. MR**1874523**, DOI https://doi.org/10.1090/conm/291/04894 - Fred Diamond and John Im,
*Modular forms and modular curves*, Seminar on Fermatâ€™s Last Theorem (Toronto, ON, 1993â€“1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39â€“133. MR**1357209** - 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, 2221â€“2231. MR**2052397**, DOI https://doi.org/10.1090/S0002-9939-04-07478-7 - J. S. Milne,
*Modular functions and modular forms*, Course note, http://www.jmilne.org/math, Univ. of Michigan, 1997. - 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. - K. Ono and K. Bringmann,
*Identities for traces of singular moduli*, Acta Arith.**119**(2005), 317â€“327. - F. K. C. Rankin and H. P. F. Swinnerton-Dyer,
*On the zeros of Eisenstein series*, Bull. London Math. Soc.**2**(1970), 169â€“170. - R. A. Rankin,
*The zeros of certain PoincarĂ© series*, Compositio Math.**46**(1982), no. 3, 255â€“272. MR**664646** - ZeĂ©v Rudnick,
*On the asymptotic distribution of zeros of modular forms*, Int. Math. Res. Not.**34**(2005), 2059â€“2074. MR**2181743**, DOI https://doi.org/10.1155/IMRN.2005.2059 - 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** - 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. KanĂ´ Memorial Lectures, No. 1. MR**0314766** - H. A. Verrill,
*Fundamental domain drawer, Java*, http://www.math.lsu.edu/~verrill/ fundomain.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11F03,
11F11

Retrieve articles in all journals with MSC (2000): 11F03, 11F11

Additional Information

**Heekyoung Hahn**

Affiliation:
Department of Mathematics, University of Rochester, Rochester, New York 14627

MR Author ID:
707443

Email:
hahn@math.rochester.edu

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.