The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions
The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions
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DOI：
10.1090/memo/1394
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发表时间：
2023-02
影响因子：
1.9
通讯作者：
V. Blomer;É. Fouvry;E. Kowalski;P. Michel;Djordje Milićević;W. Sawin
中科院分区：
文献类型：
--
作者：
V. Blomer;É. Fouvry;E. Kowalski;P. Michel;Djordje Milićević;W. Sawin
对于一个相当普遍的家庭L L -功能，我们调查的已知后果的存在的渐近公式与节能的误差项（扭曲）的第一和第二时刻的中心值在家庭。然后，我们详细考虑家庭的扭曲的一个固定的尖形原始Dirichlet字符模一个素数q q的重要特殊情况，并证明它满足这样的公式。我们得出算术结果：正比例的中心值L（f χ，1 / 2）L（f\otimes \chi，1/2）是非零的，并且确实是有界的;存在许多中心L L -值非常大的特征χ \chi;大解析秩的概率以指数速度衰减。最后，我们将展示如何建立一个特殊的情况下的Mazur和鲁宾猜想的分布模块化符号的二阶矩估计。
For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(f\otimes \chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ \chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.