Low-lying zeros of L-functions with orthogonal symmetry

Abstract

We investigate the moments of a smooth counting function of the zeros near the central point of $L$-functions of weight $k$ cuspidal newforms of prime level $N$. We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in $(-1/n,1/n)$, as $N\to \infty$ the first $n$ centered moments are Gaussian. By extending the support to $(-1/(n-1),1/(n-1))$, we see non-Gaussian behavior; in particular, the odd-centered moments are nonzero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in $(-2/n,2/n)$ if $2k\ge n$. The $n$th-centered moments agree with random matrix theory in this extended range, providing additional support for the Katz-Sarnak conjectures. The proof requires calculating multidimensional integrals of the nondiagonal terms in the Bessel-Kloosterman expansion of the Petersson formula. We convert these multidimensional integrals to one-dimensional integrals already considered in the work of Iwaniec, Luo, and Sarnak [ILS] and derive a new and more tractable expression for the $n$th-centered moments for such test functions. This new formula facilitates comparisons between number theory and random matrix theory for test functions supported in $(-1/(n-1),1/(n-1))$ by simplifying the combinatorial arguments. As an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point