On the moments of one-level densities in families of holomorphic cusp forms in the level aspect

Abstract

We study the $n^{\mathrm{th}}$ centered moments of the $1$-level density for the low-lying zeros of $L$-functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the Generalized Riemann Hypotheses, we compute this statistic for any $n\ge 1$ and for all test functions whose Fourier transforms are supported in $\left(-2/n,2/n\right)$. This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range $(-1/n,1/n)$ and surpasses the previous record of $(-1/(n-1),1/(n-1))$ whenever $n>2$. The Katz-Sarnak philosophy predicts that the aforementioned statistic can be modeled by the corresponding statistic for the eigenvalues of random orthogonal matrices. We prove that this is the case for test functions with Fourier support contained in $(-2/n,2/n)$. The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey-Snaith and Mason-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the $L$-functions in our family.

arXiv comment58 pages. Revised version, to appear in Algebra & Number Theory

References in corpus (13)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Low-lying zeros of families of elliptic curves
• The effect of convolving families of L-functions on the underlying group symmetries
• Low lying zeros of L-functions with orthogonal symmetry
• The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions
• An Orthogonal Test of the L-Functions Ratios Conjecture
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
• Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups
• Low-lying zeroes of Maass form $L$-functions
• Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions
• One-level density estimates for Dirichlet L-functions with extended support
• Low-lying zeros in families of holomorphic cusp forms: the weight aspect
• Determining optimal test functions for $2$-level densities