On zeros of Eisenstein series for genus zero Fuchsian groups
Author:
Heekyoung Hahn
Journal:
Proc. Amer. Math. Soc. 135 (2007), 23912401
MSC (2000):
Primary 11F03, 11F11
Published electronically:
March 29, 2007
MathSciNet review:
2302560
Fulltext PDF Free Access
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Abstract: Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper halfplane, and is the canonical hauptmodul for
 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
(2000g:33001)
 2.
Matthew
Boylan, SwinnertonDyer type congruences for certain Eisenstein
series, physics (Urbana, IL, 2000) Contemp. Math., vol. 291,
Amer. Math. Soc., Providence, RI, 2001, pp. 93–108. MR 1874523
(2002k:11063), http://dx.doi.org/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
(97g:11044)
 4.
Jayce
Getz, A generalization of a theorem of
Rankin and SwinnertonDyer on zeros of modular forms, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2221–2231. MR 2052397
(2005e:11047), http://dx.doi.org/10.1090/S0002993904074787
 5.
J. S. Milne, Modular functions and modular forms, Course note, http://www.jmilne.org/math, Univ. of Michigan, 1997.
 6.
K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and 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), 317327.
 8.
F. K. C. Rankin and H. P. F. SwinnertonDyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169170.
 9.
R.
A. Rankin, The zeros of certain Poincaré series,
Compositio Math. 46 (1982), no. 3, 255–272. MR 664646
(83m:10036)
 10.
Zeév
Rudnick, On the asymptotic distribution of zeros of modular
forms, Int. Math. Res. Not. 34 (2005),
2059–2074. MR 2181743
(2006k:11099), http://dx.doi.org/10.1155/IMRN.2005.2059
 11.
Bruno
Schoeneberg, Elliptic modular functions: an introduction,
SpringerVerlag, New York, 1974. Translated from the German by J. R. Smart
and E. A. Schwandt; Die Grundlehren der mathematischen Wissenschaften, Band
203. MR
0412107 (54 #236)
 12.
Goro
Shimura, Introduction to the arithmetic theory of automorphic
functions, Publications of the Mathematical Society of Japan, No. 11.
Iwanami Shoten, Publishers, Tokyo, 1971. Kan\cflex o Memorial Lectures, No.
1. MR
0314766 (47 #3318)
 13.
H. A. Verrill, Fundamental domain drawer, Java, http://www.math.lsu.edu/~verrill/ fundomain.
 1.
 G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics 71, Cambridge Univ. Press, Cambridge, 1999. MR 1688958 (2000g:33001)
 2.
 M. Boylan, SwinnertonDyer type congruences for certain Eisenstein series, Contemporary Mathematics 291 (2001), 93108. MR 1874523 (2002k:11063)
 3.
 F. Diamond and J. Im, Modular forms and modular curves, Canadian Math. Soc. Conference Proceedings 17 (1995), 39133. MR 1357209 (97g:11044)
 4.
 J. Getz, A generalization of a theorem of Rankin and SwinnertonDyer on zeros of modular forms, Proc. American Math. Soc. 132 (2004), 22212231. MR 2052397 (2005e:11047)
 5.
 J. S. Milne, Modular functions and modular forms, Course note, http://www.jmilne.org/math, Univ. of Michigan, 1997.
 6.
 K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and 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), 317327.
 8.
 F. K. C. Rankin and H. P. F. SwinnertonDyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169170.
 9.
 R. A. Rankin, The zeros of certain Poincaré series, Compositio Math. 46 (1982), 255272. MR 0664646 (83m:10036)
 10.
 Z. Rudnick, On the asymptotic distribution of zeros of modular forms, IMRN, No. 34 (2005), 20592076. MR 2181743
 11.
 B. Schoeneberg, Elliptic modular functions: An introduction, SpringerVerlag, New York, Heidelberg, Berlin, 1974. MR 0412107 (54:236)
 12.
 G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, 1971. MR 0314766 (47:3318)
 13.
 H. A. Verrill, Fundamental domain drawer, Java, http://www.math.lsu.edu/~verrill/ fundomain.
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Additional Information
Heekyoung Hahn
Affiliation:
Department of Mathematics, University of Rochester, Rochester, New York 14627
Email:
hahn@math.rochester.edu
DOI:
http://dx.doi.org/10.1090/S0002993907087631
PII:
S 00029939(07)087631
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.
