Algebraic Cycles and L-functions

本课题的研究涉及数学中的一个基本问题：求解代数方程。解的信息被编码成各种数学对象：代数循环、自同构形式和l函数。这项研究将加深对这些数学对象及其之间的联系的理解，特别是在高维中，这需要解决许多新问题，开发新的工具和在不同领域的相互作用，并吸引新的视角，这可能会对旧问题产生新的启示。它还将推进理解椭圆曲线算法的技术，特别是Birch和Swinnerton-Dyer猜想，它是克莱数学研究所的七个千年奖问题之一。PI将继续指导研究生，组织会议和研讨会，并撰写说明性文章。PI将在几个与自同构l函数相关的算术几何项目上工作，围绕Gross- Zagier公式的推广和应用这一共同主题。PI将研究库德拉-拉波波特猜想的副水平。PI将算术内积公式推广到正交群，并研究椭圆曲线对称幂动机的Bloch—Kato猜想和算术Gan—Gross—Prasad猜想的内镜情况。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。

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.