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.

38 pages: revised draft: increased the support in the split cases from 1/(n - 2/3) to 1/(n-1), included applications of the centered moments to improved estimates on the order of vanishing at the central point. To apppear in the Duke Mathematical Journal

References in corpus (4)

• 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 Low Lying Zeros of a GL(4) and a GL(6) family of L-functions
• Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

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• One level density of low-lying zeros of families of $L$-functions
• Density Theorems for Exceptional Eigenvalues for Congruence Subgroups
• Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture
• Determining optimal test functions for $2$-level densities
• One level density of low-lying zeros of quadratic Hecke $L$-functions of Imaginary Quadratic Number Fields
• Lower order terms of the one level density of a family of quadratic Hecke $L$-functions
• On the moments of one-level densities in families of holomorphic cusp forms in the level aspect