Trace Formulas, L-Functions, and Automorphic and Arithmetic Periods

本项目的主要目的是研究L函数与不同时期的关系，包括自同构期和算术期。L函数是现代数论问题研究中的一个重要对象，特别是解决丢番图方程，如著名的Birch和Swinnerton-Dyer猜想及其推广。L函数在变换群表示的研究中也是至关重要的，它在物理学中具有重要的作用。这项工作有望对高维情况下的这些关系提供重要的新理解。研究人员计划组织与该项目相关的研讨会，为该领域的学生、博士后和其他研究人员提供大量的指导、讨论和合作机会。该项目的范围包括四个部分。在第一部分中，作者将利用迹公式研究中的一些新技巧来解决L函数与自同构周期之间的关系--那些通过对某些子群积分自同构形而得到的周期。第二部分旨在通过去掉某些通常难以检验的假设，将自同构周期的一些显式公式，即Ichino-Ikeda公式，推广到更一般的情况。第三部分旨在为关于L函数与Chow群关系的Beilinson-Bloch猜想提供无条件证据，从而将Birch和Swinnerton-Dyer猜想中的类似结果推广到高维情形。在最后部分，研究人员计划采用一种新的方法，通过相对迹公式来获得关于三个模椭圆曲线乘积的某些循环的高度的算术三重积公式的变体。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The main objective of this project is to study the relation between L-functions and various periods, both automorphic and arithmetic. The L-function is an important object in the modern study of questions in number theory, in particular, solving Diophantine equations like the famous Birch and Swinnerton-Dyer conjecture and its generalizations. L-functions are also crucial in the study of representations of transformation groups, which have an important role in physics. The work is expected to provide important new understanding of these relations in higher dimensional cases. The investigator plans to organize workshops related to the project, which will provide students, postdocs, and other researchers in the area substantial opportunities for instruction, discussion, and collaboration.The scope of this project consists of four parts. In the first part, the investigator will use some new techniques in the study of trace formulas to solve questions concerning the relation between L-functions and automorphic periods -- those periods obtained by integrating automorphic forms along certain subgroups. The second part aims to extend some explicit formulas of automorphic periods, known as the Ichino-Ikeda formula, to more general cases by removing certain assumptions that are usually hard to check. The third part aims to provide unconditional evidence toward the Beilinson-Bloch conjecture on the relation between L-functions and Chow groups, which will generalize parallel results in the Birch and Swinnerton-Dyer conjecture to higher dimensional cases. In the last part, the investigator plans to adopt a new approach via a relative trace formula to obtain variants of the arithmetic triple product formula concerning the height of certain cycles on a product of three modular elliptic curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.