Low-lying zeros of L -functions with orthogonal symmetry

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 → ∞ 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 2 k ≥ 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