Interlacing of zeros of weakly holomorphic modular forms (Q2814363)

Let \(F\) and \(G\) be modular forms with respect to \(\mathrm{SL}_2(\mathbb Z)\) having zeros only on the circular arc \({\mathcal A}=\{e^{i\theta}:\pi/2\leq \theta\leq 2\pi/3\}\) of the canonical fundamental domain \(\mathcal F\) of \(\mathrm{SL}_2(\mathbb Z)\). One says that the zeros of \(F\) and \(G\) interlace on \(\mathcal A\) if every zero of one function is contained in an open interval whose endpoints are zeros of the other function, and each interval contains exactly one zero. It has been proven that the zeros of Eisenstein series \(E_k(z)\) and \(E_{k+12}(z)\) interlace on \(\mathcal A\) and that the zeros of \(j_n(z)\) and \(j_{n+1}(z)\) interlace on \(\mathcal A\), where \(j_n(z)\) is the function obtained from \(j(z)-744\) by the action of the \(n\)-th Hecke operator. For an even integer \(k\), let \(\ell\) be the integer defined by \(\ell=(k-k')/12\) with \(k'\in\{0,4,6,8,10,14\}\). In [\textit{W. Duke} and \textit{P. Jenkins}, Pure Appl. Math. Q. 4, No. 4, 1327--1340 (2008; Zbl 1200.11027)], a canonical basis \(\{f_{k,m}\}_{m=\ell}^\infty\) of the space of weakly holomorphic modular forms of weight \(k\) with respect to \(\mathrm{SL}_2(\mathbb Z)\) is constructed such that \(f_{k,m}(z)=q^{-m}+(q^{\ell+1})\) and all the zeros of \(f_{k,m}(z)\) in \(\mathcal F\) lie in \(\mathcal A\).NEWLINENEWLINEAs principal results in this article, the authors prove that for an even integer \(k\) larger than \(3\), the zeros of \(f_{k,0}(z)\) interlace with the zeros of \(f_{k+12,0}\) on \(\mathcal A\) and generally prove that for a positive number \(\varepsilon\) and a fixed positive integer \(m\) (resp. \(k\in\mathbb Z\)), the zeros of \(f_{k,m}(z)\) interlace with the zeros of \(f_{k+12,m}(z)\) (resp.\(f_{k,m+1}(z)\)) on the arc \({\mathcal A}_\varepsilon=\{e^{i\theta}:\pi/2<\theta<2\pi/3-\varepsilon\}\) for \(k\) (resp. \(m\)) large enough. Let \(b_{k,m}(\theta)=k\theta/2-2\pi m \cos \theta\). The authors show that for \(k\) and \(m\) such that \(m\geq |\ell|-\ell\) the zeros of \(\cos(b_{k,m}(\theta))\) interlace on the interval \((\pi/2,2\pi/3)\) with the zeros of \(\cos(b_{k+12,m}(\theta))\) and with the zeros of \(\cos (b_{k,m+1}(\theta))\). Those principal results are obtained from computing the approximation to \(f_{k,m}(e^{i\theta})\) by \(\cos (b_{k,m}(\theta))\) in full details. However, in order to prove the result for \(f_{k,0}\) and \(f_{k+12,0}\) in the interval \([7\pi/12,2\pi/3)\), they additionally need a trigonometric function \(H_k(\theta)=2\cos(k\theta/2)+(2\cos(\theta/2))^{-k}\). The zeros of \(H_k(\theta)\) interlace with the zeros of \(H_{k+12}(\theta)\) on \([7\pi/12,2\pi/3)\).