Zeros of some level 2 Eisenstein series

Garthwaite, Sharon; Long, Ling; Swisher, Holly; Treneer, Stephanie

doi:10.1090/S0002-9939-09-10175-2

Zeros of some level 2 Eisenstein series
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by Sharon Garthwaite, Ling Long, Holly Swisher and Stephanie Treneer PDF

Proc. Amer. Math. Soc. 138 (2010), 467-480 Request permission

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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