Heegner Points, L-Functions of Elliptic Curves, and Generalizations

海格纳点、椭圆曲线的 L 函数和概括

• 批准号: 1802269
• 负责人: Chao Li
• 金额: $ 14.36万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802269&HistoricalAwards=false
• 关键词: Heegner; Points; Functions; Elliptic; Curves

This research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects: algebraic cycles, automorphic forms, and L-functions. The project aims to deepen the understanding of these mathematical objects and the connection between them. 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. Elliptic curves and similar algebraic equations have wide application in other disciplines such as cryptography; results of the research are expected to advance understanding in these areas as well.The research comprises several projects in number theory: the arithmetic of Heegner points, L-functions of elliptic curves, and their higher dimensional generalizations. The work will investigate the congruences between Heegner points and their various applications, including Goldfeld's conjecture on elliptic curves in quadratic twists families and the rank of elliptic curves in Rubin-Silverberg families. The project will also investigate the Birch and Swinnerton-Dyer formula for elliptic curves at additive primes. For higher-dimensional generalizations, the investigator will explore arithmetic intersection problems on Rapoport-Zink spaces arising from arithmetic Gan-Gross-Prasad conjectures and the arithmetic fundamental lemma. It is also planned to initiate a new program for simultaneous generalization of the Waldspurger and Gross-Zagier formulas.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基本问题之一：解决代数方程。有关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。该项目旨在加深对这些数学对象及其之间的联系的理解。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。椭圆曲线和类似的代数方程在其他学科（例如密码学）中具有广泛的应用。该研究的结果也有望提高这些领域的理解。该研究包括数字理论的几个项目：Heegner点的算术，椭圆曲线的L功能及其更高的概括。这项工作将调查Heegner点与其各种应用之间的一致性，包括戈德菲尔德在二次曲折家庭中对椭圆曲线的猜想，以及鲁宾·塞尔弗伯格（Rubin-Silverberg）家庭中的椭圆形曲线等级。该项目还将调查在添加素数的椭圆形曲线的桦木和Swinnerton-Dyer配方。 对于高维概括，研究者将探索由算术Gan-gross-pros-prosad猜想和算术基本引理引起的Rapoport-Zink空间上的算术相交问题。还计划启动一项新的计划，以同时概括Waldspurger和Gross-Zagier公式。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。

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

Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

• DOI: 10.4310/cjm.2021.v9.n1.a1
• 发表时间: 2021-02
• 期刊: arXiv: Number Theory
• 作者: Yifeng Liu

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

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

FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

精细设计-LUSZTIG 品种和算术基本引理

• DOI: 10.1017/fms.2019.45
• 发表时间: 2019
• 期刊: Sigma
• 作者: HE, XUHUA;LI, CHAO;ZHU, YIHANG
• 通讯作者: ZHU, YIHANG