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.