Arithmetic and Geometry Around Relative Trace Formulae

这个项目涉及数论的研究，它研究整数的性质，是现代密码学的核心。数论的一个中心主题是研究代数对象、几何对象和L函数(18世纪由欧拉提出并在19世纪被黎曼广泛研究的黎曼Zeta函数的推广)的特定值之间的关系。一个令人振奋的例子是Birch和Swinnerton-Dyer的猜想，它将椭圆曲线(最简单的多项式方程之一)上的有理点与L函数的解析性质联系起来。本研究项目探索数论中的几个主题，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，其中之一是酉Shimura簇的算术Gan-Gross-Prasad猜想。研究人员以前发现了一种相对迹公式方法来研究某些自同构的L-函数的一阶导数和Shimura簇上的交数，最近的研究人员和合作者的工作将这一思想推广到函数域上秩二的一般线性群的L-函数的高阶导数。本研究的目的是将相对迹公式方法推广到更一般的情形，例如，推广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.