Algebraic Cycles and L-functions

代数圈和 L 函数

• 批准号: 2401337
• 负责人: Chao Li
• 金额: $ 23万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401337&HistoricalAwards=false
• 关键词: Algebraic; Cycles; functions

The research in this project concerns one of the basic questions in mathematics: solving algebraic equations. The information of the solutions are encoded in various mathematical objects: algebraic cycles, automorphic forms and L-functions. The research will deepen the understanding of these mathematical objects and the connection between them, especially in high dimensions, which requires solving many new problems, developing new tools and interactions in diverse areas, and appealing to new perspectives which may shed new light on old problems. It will also advance the techniques for understanding the arithmetic of elliptic curves, particularly the Birch and Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute. The PI will continue to mentor graduate students, organize conferences and workshops, and write expository articles. The PI will work on several projects relating arithmetic geometry with automorphic L-function, centered around the common theme of the generalization and applications of the Gross--Zagier formula. The PI will investigate the Kudla--Rapoport conjecture for parahoric levels. The PI will extend the arithmetic inner product formula to orthogonal groups, and study the Bloch--Kato conjecture of symmetric power motives of elliptic curves and endoscopic cases of the arithmetic Gan--Gross--Prasad conjectures. The PI will also investigate a new arithmetic relative trace formula approach towards a Gross--Zagier type formula for orthogonal Shimura varieties.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.

该项目的研究涉及数学中的基本问题之一：解决代数方程。解决方案的信息在各种数学对象中编码：代数周期，自动形式和L功能。这项研究将加深对这些数学对象的理解及其之间的联系，尤其是在高维度上，这需要解决许多新问题，在不同领域开发新的工具和互动，并吸引可能对旧问题的新观点。它还将推进理解椭圆曲线算术的技术，尤其是桦木和Swinnerton-Dyer猜想，这是粘土数学研究所的七千年奖项问题之一。 PI将继续指导研究生，组织会议和讲习班，并撰写说明性文章。 PI将在几个项目上工作，这些项​​目将算术几何形状与自动型L功能相关，围绕着概括 - Zagier公式的概括和应用的共同主题。 PI将针对脊骨水平调查库德拉（Kudla-Ropoport）的猜想。 PI将将算术内产物公式扩展到正交组，并研究椭圆形曲线的对称功率动力的Bloch-Kato猜想和算术gan- gross-gross-gross-- prasasad猜想的内窥镜案例。 PI还将研究一种新的算术相对痕量公式方法，用于正交Shimura品种的总体 - Zagier类型公式。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估标准通过评估来进行评估的。