On the zeros of period functions associated to the Eisenstein series for \(\Gamma_0^+(N)\)
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 \( P(z)=\epsilon (\sqrt Nz)^dP(1/Nz)\) for some \(\epsilon\in\mathbb C\). For \(N\)-self-inversive polynomial \(P(z)=\sum_{j=0}^dA_jz^j\in\mathbb C[z]\), if \(|A_d|\ge \frac12 \sum_{j=1}^{d-1}|A_j|\sqrt N^{d-1}\), then all of zeros of \(P\) lie on the circle \(|z|=1/\sqrt N\). Therefore, they check the above inequality for the coefficients of the polynomials under consideration. Further to show that the zeros are simple and regularly distributed on the circle, they deal with many fine inequalities.
scientific article; zbMATH DE number 7466549
scientific article