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

迹公式、L 函数以及自守和算术周期

• 批准号: 2000533
• 负责人: Congling Qiu
• 金额: $ 30万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000533&HistoricalAwards=false
• 关键词: Trace; Formulas; Functions; Automorphic; Arithmetic

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.

该项目的主要目标是研究 L 函数与各个周期（自守和算术）之间的关系。 L 函数是现代数论问题研究的重要对象，特别是求解丢番图方程（如著名的 Birch 和 Swinnerton-Dyer 猜想及其推广）的重要对象。 L 函数在变换群表示的研究中也至关重要，变换群在物理学中具有重要作用。这项工作预计将为高维情况下这些关系提供重要的新理解。研究者计划组织与该项目相关的研讨会，这将为该领域的学生、博士后和其他研究人员提供大量的指导、讨论和合作的机会。该项目的范围由四个部分组成。在第一部分中，研究者将在迹公式研究中使用一些新技术来解决有关L函数和自同构周期（通过沿某些子群整合自同构形式获得的周期）之间的关系的问题。第二部分旨在通过删除某些通常难以检查的假设，将一些显式的自守周期公式（称为 Ichino-Ikeda 公式）扩展到更一般的情况。第三部分旨在为L函数与Chow群之间的关系的Beilinson-Bloch猜想提供无条件证据，这将Birch和Swinnerton-Dyer猜想的并行结果推广到更高维的情况。在最后一部分中，研究人员计划采用一种新方法，通过相对迹公式来获得关于三个模椭圆曲线乘积上某些周期的高度的算术三重乘积公式的变体。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。