Arithmetic and Geometry Around Relative Trace Formulae

围绕相对迹公式的算术和几何

• 批准号: 1838118
• 负责人: Wei Zhang
• 金额: $ 15.94万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1838118&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Around; Relative; Trace

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.

该项目涉及数字理论的研究，该研究研究了整数的属性，并且是现代密码学的核心。数字理论的一个中心主题是研究代数和几何对象与L功能的特殊值之间的关系（Euler在18世纪引入的Riemann Zeta函数的概括，并由Riemann在19世纪进行了广泛的研究）。一个激励的例子是桦木和swinnerton-dyer的猜想，它将椭圆曲线（最简单的多项式方程类别之一）上的理性点与L功能的分析特性有关。该研究项目探讨了数量理论中的几个主题，并旨在加深对这些关系的理解。该项目旨在研究桦木和swinnerton-dyer的猜想及其高维概括，其中之一是算术Gan-gross-prossad猜想，用于单一的Shimura品种。 研究者先前发现了一种相对的痕量公式方法，用于研究Shimura品种上某些自动型L功能和相交数的第一个衍生物，研究者和合作者的最新工作将该概念扩展到高级范围的较高衍生物，用于级别的两种等级线性群体的L-Functions。该研究项目旨在将相对痕量公式的方法扩展到更一般的环境，例如，较高等级的一般线性群体的功能。