Bounding the order of vanishing of cuspidal newforms via the \(n^{\text{th}}\) centered moments (Q6631468)

Let \(H_{k}^{*}(N)\) denote the set of holomorphic cusp forms of weight \(k\) that are newforms of level \(N\). For every \(f\in{H_{k}^{*}(N)}\), we have \(f(z)=\sum_{n=1}^{\infty}a_{f}(n)e(nz)\). We set the \(L\)-function associated to \(f\) \N\[\NL(s,f)=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}}, \ \lambda_{f}(n)=a_{f}(n)n^{-(k-1)/2}.\N\]\NThe completed \(L\)-function \(\Lambda(s, f )\) satisfies the functional equation \N\[\N\Lambda(s, f ) = \varepsilon_{f}\Lambda(1-s, f )\N\]\Nwith \(\varepsilon_{f} = \pm1\) and \(\Lambda(s,f)=(\sqrt{N}/2\pi)^{s}\Gamma(s+(k-1)/2)L(s,f)\). We note \(H_{k}^{+}(N)=\{f\in{H_{k}^{*}(N)}; \ \varepsilon_{f} = +1\}\) and \(H_{k}^{-}(N)=\{f\in{H_{k}^{*}(N)}; \ \varepsilon_{f} = -1\}\). The \(n\)-level density of an \(L\)-function \( L(s, f )\) is defined as \N\[\ND_{n}(f,\phi):=\sum_{j_,{1},\ldots,j_{n};\ j_{i}\neq\pm j_{k}}\phi\left(\frac{\log c_{f}}{2\pi}\gamma_{f}^{(j_{1})},\ldots,\frac{\log c_{f}}{2\pi}\gamma_{f}^{(j_{n})}\right),\N\]\Nwhere \(\phi:\mathbb{R}^{n}\to\mathbb{R}\) is a test function, \(c_f\) is the analytic conductor of \(f\) and \(\gamma_{f}^{(j)}\) represents the imaginary parts of the zeroes of an \(L\)-function associated with the modular form \(f\in{H_{k}^{+}(N)}\) with an additional zero \(\gamma_{f}^{(0)}= 0\) if \(f\in{H_{k}^{-}(N)}\).\N\NIn this paper under review, the authors use the \(n\)-level density to bound how often forms in the family \(F_N\) vanish with \(F_N\) being the basis of the set \(H_{k}^{*}(N)\) of cuspidal newforms of level \(N\) and some fixed weight \(k\), with \(N\to\infty\) through the primes.