On the orthogonal symmetry of L-functions of a family of Hecke Grössencharacters
On the orthogonal symmetry of L-functions of a family of Hecke Grössencharacters
复制标题
关于 Hecke Grössen 字符族 L 函数的正交对称性
DOI：
10.4064/aa157-4-2
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发表时间：
2012
期刊：
影响因子：
0.7
通讯作者：
N. Snaith
中科院分区：
文献类型：
--
作者：
J. Conrey;N. Snaith
复乘法椭圆曲线上的L-函数的对称幂族已经从代数、自守和p-adic的角度得到了广泛的关注。在这里，我们从经典解析数论和随机矩阵理论的角度来研究这个家族，特别是专注于家族对称类型的证据。特别是，我们调查的值在中心点，并给出证据表明，这个家庭可以模拟的正交矩阵的合奏。我们证明了这些L值的平均值，它再现，通过一个完全不同的方法，证明了格林伯格和Villegas-Zagier的渐近公式与功率节省的渐近公式。我们给出了二阶矩的一个上界，这个上界实际上只大了一个对数。我们也给出了一个明确的猜想，这个家庭的第二时刻，与功率节省。最后，我们用一个检验函数计算了这个族的一个能级密度，该检验函数的傅里叶变换具有有限的支持度。这是已知的工作维勒加斯-扎吉尔的子集，这些L-职能，甚至职能方程永远不会消失，我们表明在何种程度上这一结果是反映了我们的分析结果。
The family of symmetric powers of an L-function associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and p-adic points of view. Here we examine this family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by ensembles of orthogonal matrices. We prove an asymptotic formula with power savings for the average of these L-values, which reproduces, by a completely different method, an asymptotic formula proven by Greenberg and Villegas–Zagier. We give an upper bound for the second moment which is conjecturally too large by just one logarithm. We also give an explicit conjecture for the second moment of this family, with power savings. Finally, we compute the one level density for this family with a test function whose Fourier transform has limited support. It is known by the work of Villegas – Zagier that the subset of these L-functions which have even functional equations never vanish; we show to what extent this result is reflected by our analytic results.