Zeros of some level 2 Eisenstein series

Details Zeros of some level 2 Eisenstein series Zeros of some level 2 Eisenstein series

2009-08-24

arxiv.org/abs/0908.3511v1

Mathematics

Number Theory

14 pages, 1 figure

Scientific paper

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 allow 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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