Zeros of some level 2 Eisenstein series

TL;DR

This paper investigates the distribution of zeros of certain level 2 Eisenstein series, extending known properties and connecting them to elliptic functions and L-series.

Contribution

It extends the distribution properties of Eisenstein series zeros to level 2 series linked to Jacobi elliptic functions and introduces a recursive polynomial method for zero calculation.

Findings

Zeros lie on specific arcs of the fundamental domain

Recursive polynomials enable zero computation

Zeros are connected to an L-series

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 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.