Extending the support of 1- and 2-level densities for cusp form \(L\)-functions under square-root cancellation hypotheses (Q6595588)

This paper is a continuation of the seminal work of \textit{H. Iwaniec} et al. [Publ. Math., Inst. Hautes Étud. Sci. 91, 55--131 (2000; Zbl 1012.11041)]. More precisely, the authors extend their results for certain one-level densities to 2-level, and discuss how to generalize to arbitrary \(n\). Specifically, it is shown that a natural generalization of their Hypothesis S on cancellation in certain prime sums, which led to increasing the support of the Fourier transform of the test function in a larger open interval \(\supsetneq (-2,2)\) for the \(1\)-level density, leads to increased support where the \(2\)-level density of certain families of cuspidal newforms and random matrix theory agree. Such results have obviously applications in bounding the order of vanishing of \(L\)-functions at the central point.