Arithmetic Geometry and Automorphic L-Functions

算术几何和自同构 L 函数

• 批准号: 2101157
• 负责人: Chao Li
• 金额: $ 22.26万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101157&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Automorphic; Functions

The research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects, including arithmetic varieties and automorphic L-functions. This research aims to deepen the understanding of these mathematical objects, especially in higher dimension, which requires developing new tools and interactions in diverse areas and appealing to new perspectives that may shed new light on longstanding open questions. The project also aims to 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 research consists of several projects relating arithmetic geometry with automorphic L-functions, centered on the common theme of the generalization and applications of the Gross–Zagier formula. The PI will investigate the Kudla–Rapoport conjecture for ramified unitary groups and orthogonal groups. The PI plans to extend the arithmetic inner product formula and apply it to the Beilinson–Bloch and Bloch–Kato conjectures of symmetric power motives of elliptic curves. 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.

该研究项目涉及数学中的一个基本问题：求解代数方程。关于解的信息编码在各种数学对象中，包括算术变量和自同构l函数。本研究旨在加深对这些数学对象的理解，特别是在更高维度，这需要在不同领域开发新的工具和相互作用，并吸引新的视角，这可能会为长期存在的开放性问题提供新的线索。该项目还旨在推进理解椭圆曲线算法的技术，特别是Birch和Swinnerton-Dyer猜想，该猜想是克莱数学研究所的七个千年奖问题之一。本研究包括几个与算术几何和自同构l函数相关的项目，围绕Gross-Zagier公式的推广和应用这一共同主题。PI将研究分支酉群和正交群的Kudla-Rapoport猜想。PI计划扩展算术内积公式，并将其应用于椭圆曲线对称动力动机的Beilinson-Bloch和Bloch-Kato猜想。PI还将研究一种新的算术相对迹公式方法，用于正交Shimura变量的Gross-Zagier型公式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Kudla-Rapoport cycles and derivatives of local densities

库德拉-拉波波特循环和局部密度的导数

• DOI: 10.1090/jams/988
• 发表时间: 2021
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Li, Chao;Zhang, Wei
• 通讯作者: Zhang, Wei

On the arithmetic Siegel–Weil formula for GSpin Shimura varieties

GSpin Shimura 簇的算术 Siegel Weil 公式

• DOI: 10.1007/s00222-022-01106-z
• 发表时间: 2022
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Li, Chao;Zhang, Wei
• 通讯作者: Zhang, Wei

Chow groups and L-derivatives of automorphic motives for unitary groups

Chow 群和酉群自守动机的 L-导数

• DOI: 10.4007/annals.2021.194.3.6
• 发表时间: 2021
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Li, Chao;Liu, Yifeng
• 通讯作者: Liu, Yifeng

A note on Tate’s conjectures for abelian varieties

关于泰特关于阿贝尔簇的猜想的注释

• DOI: 10.2140/ent.2022.1.41
• 发表时间: 2022
• 期刊: Essential Number Theory
• 作者: Li, Chao;Zhang, Wei
• 通讯作者: Zhang, Wei

Chow groups and L -derivatives of automorphic motives for unitary groups, II.

Chow 群和酉群自守动机的 L 导数，II。

• DOI: 10.1017/fmp.2022.2
• 发表时间: 2022
• 期刊: Pi
• 作者: Li, Chao;Liu, Yifeng
• 通讯作者: Liu, Yifeng