Algebraic Cycles and L-functions

这个项目的研究涉及到数学中的一个基本问题：代数方程的求解。解的信息被编码成各种数学对象：代数圈、自同构形和L函数。这项研究将加深对这些数学对象以及它们之间的联系的理解，特别是在高维方面，这需要解决许多新问题，在不同领域开发新工具和相互作用，并吸引新的视角，可能会对旧问题有新的启发。它还将提高理解椭圆曲线算术的技术，特别是伯奇和斯温纳顿-戴尔猜想，这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将致力于几个与算术几何和自同构L函数相关的项目，围绕着格罗斯-扎吉尔公式的推广和应用这一共同主题。PI将调查Kudla-Rapoport猜想的意指水平。PI将把算术内积公式推广到正交群，并研究椭圆曲线的对称幂动机的Bloch-Kato猜想和算术Gan-Gross-Prasad猜想的内窥镜情况。PI还将研究一种新的算术相对跟踪公式方法，以获得针对正交Shimura品种的Gross-Zagier类型公式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.