论文标题
在整体尖缘家族中的一级密度的时刻，在水平方面形成
On the moments of one-level densities in families of holomorphic cusp forms in the level aspect
论文作者
论文摘要
我们研究了$ n^{\ rm th} $中心时刻的$ 1 $级别密度的$ 1级密度，用于$ l $ functions的低洼零，上面固定在大质量和固定重量的圆锥形尖端新形式。假设有概括的Riemann假设，我们为任何$ n \ ge 1 $计算此统计量，以及所有在$ \ left（-2/n，\，2/n \ right）$中支持其傅立叶变换的测试功能。据信这是当前技术的自然限制。我们的工作大大扩展到了琐碎的范围$（-1/n，\，1/n）$，并超过了$（-1/（N-1），\，1/（n-1））$的先前记录，只要$ n> 2 $。 Katz-Sarnak哲学预测，上述统计量可以由随机正交矩阵的特征值的相应统计量来建模。我们证明，在$（2/n，\，2/n）$中包含傅立叶支持的测试功能就是这种情况。主要的技术创新是评估术语组合动物园的一种可行性优势，类似于Conrey-Snaith和Mason-Snaith的工作。作为一个应用程序，我们的工作为我们家庭中$ l $ functions的中心点消失的顺序提供了更好的界限。
We study the $n^{\rm th}$ centered moments of the $1$-level density for the low-lying zeros of $L$-functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the Generalized Riemann Hypotheses, we compute this statistic for any $n\ge 1$ and for all test functions whose Fourier transforms are supported in $\left(-2/n, \, 2/n\right)$. This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range $(-1/n, \, 1/n)$ and surpasses the previous record of $(-1/(n-1),\, 1/(n-1))$ whenever $n>2$. The Katz-Sarnak philosophy predicts that the aforementioned statistic can be modeled by the corresponding statistic for the eigenvalues of random orthogonal matrices. We prove that this is the case for test functions with Fourier support contained in $(-2/n,\, 2/n)$. The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey-Snaith and Mason-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the $L$-functions in our family.
