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Zeros of some level 2 Eisenstein series

Author(s): Sharon Garthwaite; Ling Long; Holly Swisher; Stephanie Treneer
Journal: Proc. Amer. Math. Soc. 138 (2010), 467-480.
MSC (2000): Primary 11F11; Secondary 11F03
Posted: October 6, 2009
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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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Additional Information:

Sharon Garthwaite
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
Email: sharon.garthwaite@bucknell.edu

Ling Long
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: linglong@iastate.edu

Holly Swisher
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97301
Email: swisherh@math.oregonstate.edu

Stephanie Treneer
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: stephanie.treneer@wwu.edu

DOI: 10.1090/S0002-9939-09-10175-2
PII: S 0002-9939(09)10175-2
Received by editor(s): June 4, 2009
Posted: 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
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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