The relative trace formula and central L-values

本研究的目的是建立涉及L函数中心值的新的显式恒等式和其他与自同构形式相关的数据。相对迹公式是用群上的轨道积分来表示给定群上自同构形的周期积分的一种方法。根据设置的不同，周期积分可以给出所需的频谱信息，如傅立叶系数或L值。其中一个例子是Kuznetsov的GL(2)傅立叶迹公式，它在解析数论中已变得普遍存在，并应用于Klosterman和的有界和、Riemann Zeta函数的矩、模L函数的次凸性等各种重要问题。Pi和他的合作者C.Li将发展其他的傅立叶迹公式，并使用相对迹公式直接研究L函数的各种平均。这个项目是数论的，研究整数及其代数性质。根据朗兰兹的一个猜想，有理数的代数结构被伽罗瓦群捕获，用模形式编码。这些是高等微积分中表现出特殊对称性的函数。拟议的研究将加深我们对模块化形式的理解。

The purpose of the proposed research is to establish new explicit identities involving the central values of L-functions and other data attached to automorphic forms. The relative trace formula is a technique which expresses period integrals of automorphic forms on a given group in terms of orbital integrals on the group. Depending on the setup, the period integrals can give desirable spectral information, like Fourier coefficients or L-values. One example is the GL(2) Fourier trace formula of Kuznetsov, which has become ubiquitous in analytic number theory with applications to such diverse important problems as bounding sums of Kloosterman sums, moments of the Riemann zeta function, and subconvexity for modular L-functions. The PI and his collaborator C. Li will develop other Fourier trace formulas, and also directly investigate various averages of L-functions using the relative trace formula.This project is in number theory, which is the study of the integers and their algebraic properties. According to a conjecture of Langlands, the algebraic structure of the rational numbers, as captured by the Galois group, is encoded by modular forms. These are functions in advanced calculus which exhibit special symmetries. The proposed research will deepen our understanding of modular forms.