Statistics for low-lying zeros of symmetric power L-functions in the level aspect

Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect

In a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2). %

Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are n…

低频零点与 $L$-函数中心值之间的联系

A connection between low-lying zeros and central values of $L$-functions

研究低频零点统计与 $L$-函数中心值分布的关系，通过部分结果推导出凯廷-萨尼思猜想的条件下界，并揭示对称性、傅里叶支持范围与下界质量之间的联系。

Low-lying Zeros of Number Field $L$-functions

One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithme…

Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves

The Katz-Sarnak philosophy predicts that statistics of zeros of families of L-functions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lower-order terms of the statistics of interest. In this paper we calculate lower-order terms of the 1-level density of some families of elliptic curves. We show that there are essentially two different effects on the distribution of low-lying zeros. First, low-lying zeros are more numerous in families of elliptic curves E with relatively large numbers of points (mod p…

Small first zeros of L-functions

From a family of L-functions with unitary symmetry, Hughes and Rudnick obtained results on the height of its lowest zero. We extend their results to other families of Lfunctions according to the type of symmetry coming from statistics for low-lying zeros.

Low-lying zeroes of Maass form $L$-functions

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the l…

Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory

The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n…

Dirichlet $L$-函数族临界线上低零点的分布

Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions

本文通过引入高维Mellin变换，克服了短区间内Selberg磨光器计算中的交叉项障碍，证明了Dirichlet $L$-函数族在临界线上低零点数量的新下界。

Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters

While Random Matrix Theory has successfully modeled many quantities of families of L-functions, it frequently cannot see the family's arithmetic. In some situations this requires an extended theory that inserts arithmetic factors depending on the family, while in other cases these factors result in contributions which vanish in the limit, and are thus not detected. We review the general theory associated to one of the most important statistics, the n-level density of zeros near the central point. According to the Katz-Sarnak density conjecture, to each family of L-functions there is a correspo…

Linnik's problem in fiber bundles over quadratic homogeneous varieties

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…

The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions

We investigate the large weight (k --> oo) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even Hecke-Maass cusp form and H_k(1) is a Hecke eigenbasis for the space H_k(1) of holomorphic cusp forms of weight k for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of…