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Zeros of some level 2 Eisenstein series

Authors: Sharon Garthwaite, Ling Long, Holly Swisher and Stephanie Treneer
Journal: Proc. Amer. Math. Soc. 138 (2010), 467-480
MSC (2000): Primary 11F11; Secondary 11F03
Published electronically: October 6, 2009
MathSciNet review: 2557165
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Abstract: The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on $ \Gamma(2)$ because of its elegant connection to a classical Jacobi elliptic function $ cn(u)$ which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allows us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an $ L$-series.

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  • [DJ08] W. Duke and Paul Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (2008), no. 4, part 1, 1327-1340. MR 2441704
  • [Gek01] E. U. Gekeler, Some observations on the arithmetic of Eisenstein series for the modular group $ \mathrm{SL}(2,{\mathbb{Z}})$, Arch. Math. (Basel) 77 (2001), no. 1, 5-21. MR 1845671 (2002f:11050)
  • [Get04] J. 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 (electronic). MR 2052397 (2005e:11047)
  • [Gun06] S. Gun, On the zeros of certain cusp forms, Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 2, 191-195. MR 2265867 (2008d:11035)
  • [Hah07] H. Hahn, On zeros of Eisenstein series for genus zero Fuchsian groups, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2391-2401 (electronic). MR 2302560 (2008a:11047)
  • [Han58] H. Hancock, Lectures on the theory of elliptic functions: Analysis, Dover Publications Inc., New York, 1958. MR 0100106 (20:6540)
  • [Koh04] W. Kohnen, Zeros of Eisenstein series, Kyushu J. Math. 58 (2004), no. 2, 251-256. MR 2117246 (2005h:11084)
  • [LLY05] W. C. Li, L. Long, and Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (2005), no. 1, 117-148. MR 2141761 (2006c:11053)
  • [LY05] L. Long and Y. Yang, A short proof of Milne's formulas for sums of integer squares, Int. J. Number Theory 1 (2005), no. 4, 533-551. MR 2196794 (2006j:11050)
  • [MNS07] T. Miezaki, H. Nozaki, and J. Shigezumi, On the zeros of Eisenstein series for $ \Gamma\sp *\sb 0(2)$ and $ \Gamma\sp *\sb 0(3)$, J. Math. Soc. Japan 59 (2007), no. 3, 693-706. MR 2344823 (2008m:11094)
  • [Mil02] S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J. 6 (2002), no. 1, 7-149. MR 1906722 (2003m:11060)
  • [Noz08] H. Nozaki, A separation property of the zeros of Eisenstein series for $ \mathrm{SL}(2,\mathbb{Z})$, Bull. Lond. Math. Soc. 40 (2008), no. 1, 26-36. MR 2409175 (2009d:11070)
  • [Ono04] K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $ q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 2004. MR 2020489 (2005c:11053)
  • [OP04] K. Ono and M. A. Papanikolas, $ p$-adic properties of values of the modular $ j$-function, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 357-365. MR 2059773 (2005a:11048)
  • [RSD70] F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169-170. MR 0260674 (41:5298)
  • [Shi07] J. Shigezumi, On the zeros of the Eisenstein series for $ \Gamma\sp *\sb 0(5)$ and $ \Gamma\sp *\sb 0(7)$, Kyushu J. Math. 61 (2007), no. 2, 527-549. MR 2362897 (2008j:11052)

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Additional Information

Sharon Garthwaite
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Ling Long
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Holly Swisher
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97301

Stephanie Treneer
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Received by editor(s): June 4, 2009
Published electronically: October 6, 2009
Additional Notes: The second author was supported in part by the NSA grant no. H98230-08-1-0076.
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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