Algebraic Cycles and L-functions

研究者建议在数论、自同构形式和算术几何方面继续他的项目。与不同的合作者，PI将研究Gan-Gross-Prasad周期，由Ichino- Ikeda和N. Harris推测的单位群的广义Waldspurger公式，以及S. Zhang的Gross- Zagier公式对高维Shimura变量的表示理论公式。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交点数之间的相关推测恒等式。本研究涉及由代数方程定义的一种特殊类型的数学对象，称为代数循环，它包含有关几何和算术的重要信息。它们应用于椭圆曲线的计算，特别是伯奇-斯温纳顿-戴尔猜想，它是克莱数学研究所的七个千禧年奖问题之一。椭圆曲线的研究在密码学和信息安全等许多领域都具有重要意义。

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.