Periods and special values of L-functions for unitary groups

对L函数或Zeta函数的研究在数学上有着悠久的历史，最早可以追溯到Gauss、Riemann、Dirichlet、Hecke、Artin等人。在过去的五十年里，自从朗兰兹介绍了数论和自同构形式之间的深刻哲学之后，取得了许多进展。其中，派所主张研究的L函数的特殊价值与时期的关系是最重要的问题之一。确切地说，该提案的第一部分是关于Gan-Gross-Prasad猜想框架下自同构表示的全局限制问题。与不同的合作者一起，PI将对这些周期形成一个显式猜想，推广了Ichino-Ikeda的工作，并利用Jacquet-Rallis在特殊情况下提出的相对迹公式和PI的一般形式来研究酉群的情况。建议的第二部分是为了理解L函数的中心导数与某些算术周期，即Shimura簇的圈的高度之间的关系。PI建议在Kudla和PI本人之后，特别是在U(2，2)的下一个未知情况下，通过算术theta提升来研究这种关系。这项研究的最终目的是更好地理解数学中的核心问题之一：如何求解方程，特别是数域中的代数方程。这些方程隐藏在各种数学对象中，如代数环、表示法、周期关系等。Birch和Swinnerton-Dyer猜想是这方面最著名的问题之一。在纯数学之外，像椭圆曲线这样的代数方程在密码学中有重要的应用。

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.