On the zeros of period functions associated to the Eisenstein series for \(\Gamma_0^+(N)\)
type: Item
instance of: scholarly article
title: On the zeros of period functions associated to the Eisenstein series for \(\Gamma_0^+(N)\)
zbMATH Open document ID: 1491.11043
author: Bo-Hae Im
author: SoYoung Choi
published in: Journal of Number Theory
publication date: 2022-01-28T00:00:00Z
review text: For a positive integer \(N\), let \(\Gamma_0(N)^+\) be the modular group generated by the elements of \(\Gamma_0(N)\) and the Fricke involution \(W_N\). Let \(E_{k}^+(z)\) be the Eisenstein series of even weight \(k\) for \(\Gamma_0(N)^+\). For a modular form \(f(z)=\sum_{n=0}^\infty a(n)e^{2\pi inz}\) of weight \(k\), the Eichler integral of \(f\) is defined by \(\mathcal{E}_f(z)=\sum_{n=0}^\infty a(n)n^{1-k}e^{2\pi inz}\). As an analogy of the period function of Eisenstein series for \(\mathrm{SL}_2(\mathbb Z)\), the authors define the period function of \(E_{k}^+\) by \(r(E_{k}^+;z)=-\frac{\Gamma(k-1)}{(2\pi i)^{k-1}}(\mathcal{E}_{E_{k}^+}-\mathcal{E}_{E_{k}^+}|_{2-k}W_N(z))\). The function \(r(E_{k}^+;z)-((k-1)N^{k/2}z)^{-1}\) is a polynomial of \(z\) of degree \(k-1\). From \(r(E_{k}^+;z)\), they consider three polynomials \(R_k(z)=(k-1)zr(E_{k}^+;z), R_k^-(z)=(k-1)zr^-(E_{k}^+;z)\) and \(P_k(z)\), where \(r^-(E_{k}^+;z)\) is the odd part of \(r(E_{k}^+;z)\) and, up to a nonzero constant, \(P_k(z)\) is defined by \(r(E_{k}^+;z)-(z^{k-1}+\frac{1}{N^{k/2}z})/(k-1)\). The polynomials \(R_k^-\) and \(R_k\) are of degree \(k\) and \(P_k\) is of degree \(k-2\). The authors are interested in the location of zeros of these three polynomials and show that all zeros of \(P_k\) (resp. \(R_k^-\) and \(R_k\)) lie on the circle \(|z|=1/\sqrt{N}\), if \(N\ge 3, k\ge 4\) (resp. \(N\ge 5, k\ge 10 \) and \(N\ge 17, k\ge 10\)), and the zeros of each polynomial are simple and regularly distributed on this circle. Further, if \(N=1,k\ge 4\) and \(N=2, k\ge 1\), then all zeros of \(P_k\) lie on the circle \(|z|=1/\sqrt{N}\). If \(N=2,3,4\) and \(k\ge 4\), then \(R_k^-\) has exactly \(k-4\) zeros lying on the circle \(|z|=1/\sqrt N\) and four distinct real zeros lying outside the circle. The key point of the proof is the \(N\)-self-inversive property of \(P_k,R_k\) and \(R_k^-\). Here a polynomial \(P(z)\) of degree \(d\) is \(N\)-self-inversive if \(P\) satisfies \( P(z)=\epsilon (\…
reviewed by: Noburo Ishii
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 7466549
zbMATH Keywords: period functions
zbMATH Keywords: Eisenstein series
zbMATH Keywords: \(N\)-self-inversive polynomials
Wikidata QID: Q114156865
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.06.021
OpenAlex ID: W3184845666
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DOI: 10.1016/J.JNT.2021.06.021
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description: scientific article; zbMATH DE number 7466549