Periods and special values of L-functions for unitary groups

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The study of L-functions or zeta functions has a long history in mathematics, dating back to Gauss, Riemann, Dirichlet, Hecke, Artin, et al. In the last fifty years, lots of progress has been made after Langlands introduced the deep philosophy between number theory and automorphic forms. Among these, the relation between special values of L-functions and periods, which the PI proposes to study, is one of the most important problems. Precisely, the first part of the proposal is about the global restriction problems of automorphic representations in the framework of the Gan-Gross-Prasad conjecture. With various collaborators, the PI will formulate an explicit conjecture for those periods, generalizing the work of Ichino-Ikeda, as well as study the case of unitary groups using the tool of relative trace formula, proposed by Jacquet-Rallis in a special case and the PI in general. The second part of the proposal is to understand the relation between central derivatives of L-functions and certain arithmetic periods, namely, height of cycles on Shimura varieties. The PI proposes to study such relation via arithmetic theta lifting after Kudla and the PI himself, in particular, in the next unknown case of U(2,2).The ultimate goal of this research is to better understand one of the central questions in mathematics: how to solve equations, particularly, algebraic equations in number fields. These equations hide themselves in various kinds of mathematical objects, such as algebraic cycles, representations, period relations, etc. The Birch and Swinnerton-Dyer conjecture is one of the most famous questions toward this direction. Outside pure mathematics, algebraic equations like elliptic curves, have important application in cryptography.