Arithmetic and Geometry Around Relative Trace Formulae

该项目涉及数论研究，研究整数的性质，是现代密码学的核心。数论的一个中心主题是研究代数和几何对象之间的关系以及l函数的特殊值（黎曼zeta函数的推广，由欧拉在18世纪引入，黎曼在19世纪进行了广泛的研究）。一个激励的例子是Birch和Swinnerton-Dyer的猜想，它将椭圆曲线（多项式方程中最简单的一类）上的有理点与l函数的解析性质联系起来。本研究项目探讨了数论中的几个主题，旨在加深对这些关系的理解。本项目旨在研究Birch和Swinnerton-Dyer猜想及其高维推广，其中一个是酉Shimura变的算术Gan-Gross-Prasad猜想。研究者先前发现了一种相对迹公式的方法来研究某些自同构l函数的一阶导数和Shimura变种上的交数，研究者和合作者最近的工作将这一思想扩展到函数域上一般二阶线性群的l函数的高阶导数。本研究项目旨在将相对迹公式方法扩展到更一般的设置中，例如，扩展到一般高阶线性群的l函数。

This project concerns research in number theory, which studies properties of the whole numbers and is at the core of modern cryptography. One central theme in number theory is to study the relationship between algebraic and geometric objects and special values of L-functions (generalizations of the Riemann zeta function introduced by Euler in the eighteenth century and studied extensively by Riemann in the nineteenth century). A motivating example is the conjecture of Birch and Swinnerton-Dyer, which relates rational points on elliptic curves (one of the simplest classes of polynomial equations) to the analytic property of L-functions. This research project explores several topics in number theory and aims to deepen understanding of these relationships.The project aims to study the Birch and Swinnerton-Dyer conjecture and its high dimensional generalizations, one of which is the arithmetic Gan-Gross-Prasad conjecture for unitary Shimura varieties. The investigator previously discovered a relative trace formula approach to study the first derivative of certain automorphic L-functions and intersection numbers on Shimura varieties, and recent work of the investigator and collaborator extends the idea to higher derivatives for L-functions for the general linear group of rank two over function fields. This research project aims to extend the relative trace formula approach to more general settings, for instance, to L-functions for the general linear groups of higher rank.