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 --> oo 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 non-zero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in (-2/n, 2/n) if 2k >= n. The nth 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 non-diagonal 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-Sarnak, and derive a new and more tractable expression for the nth 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.