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的基本问题之一：求解代数方程的信息被编码在各种数学对象中，包括算术簇和自同构 L 函数。这项研究旨在加深对这些数学对象的理解，特别是在高等数学对象中。该项目还旨在推进理解椭圆曲线算术的技术，特别是正在发展的 Birch 和椭圆曲线。斯温纳顿-戴尔猜想，克莱数学研究所七大千年奖问题之一。该研究由几个将算术几何与自守L函数相关的项目组成，围绕格罗斯-扎吉尔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应用。 PI 还将研究一种新的算术相对迹公式方法，以实现正交 Shimura 簇的 Gross-Zagier 型公式。该奖项反映了 NSF 的法定使命和使命。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，该项目被认为值得支持。

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: 2021-09
• 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: 2022-06
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Li, Chao;Zhang, Wei
• 通讯作者: Zhang, Wei

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: 2022-01
• 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: 2022-01
• 期刊: Pi
• 作者: Li, Chao;Liu, Yifeng
• 通讯作者: Liu, Yifeng

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: 2021-11
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Li, Chao;Liu, Yifeng
• 通讯作者: Liu, Yifeng