This award supports the principal investigator's research on number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool, with diverse applications in areas such as data transmission and processing, communication systems, and internet security. The PI is one of the organizers of a workshop, the Arizona Winter School. He is also training graduate students in topics related to this award. In more detail, the project centers around the theory of automorphic forms, a branch of number theory which studies natural numbers through symmetries. The theory of L-functions and its special values is of particular interest in this subject. The main goal is to apply various techniques from representation theory and harmonic analysis to prove identities linking special values of L-functions to other objects, e.g. period integrals, or heights of special cycles on Shimura varieties. More concretely the PI will study the following conjectures: (1) Gan--Gross--Prasad conjectures and its arithmetic variant; (2)Guo--Jacquet conjecture on linear periods. 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.