Algebraic Cycles and L-functions

调查员建议继续他的项目的几个主题在数论，自守形式和算术几何。PI将与各种合作者一起研究Gan-Gross-Prasad周期，Ichino-Ikeda和N. Harris等提出的Gros-Zagier公式对高维Shimura簇的表示理论。张某PI还将追求算术基本引理，即Rapoport-Zink空间上某个轨道积分与某个相交数之间的相关数学恒等式。 这项研究涉及一种由代数方程定义的特殊类型的数学对象，称为代数圈，其中包含有关几何和算术的重要信息。他们的应用算术的椭圆曲线，特别是伯奇-斯温纳顿-戴尔猜想，七个千年奖的问题之一，克莱数学研究所。椭圆曲线的研究在密码学、信息安全等领域有着重要的意义。

The investigator proposes to continue his project on several topics in number theory, automorphic forms and arithmetic geometry. With various collaborators, the PI will investigate the Gan-Gross-Prasad periods, the generalized Waldspurger formula for unitary groups conjectured by Ichino--Ikeda and N. Harris, as well as the representation theoretical formulation of Gross--Zagier formula to higher dimensional Shimura varieties by S. Zhang. The PI will also pursue the arithmetic fundamental lemma, a relevant conjectural identity between a certain orbital integral and a certain intersection number on Rapoport--Zink space. This research concerns a special type of mathematical object defined by algebraic equations, known as algebraic cycles, which contain important information about geometry and arithmetic. They have applications to the arithmetic of elliptic curves, particularly the Birch--Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute. The study of elliptic curve is crucial in many areas such as cryptography and information security.