Periods and special values of L-functions for unitary groups

酉群 L 函数的周期和特殊值

• 批准号: 1302000
• 负责人: Yifeng Liu
• 金额: $ 13万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302000&HistoricalAwards=false
• 关键词: Periods; special; values; functions; unitary

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.

l函数或ζ函数的研究在数学中有着悠久的历史，可以追溯到高斯、黎曼、狄利克雷、赫克、马丁等人。在朗兰兹引入数论与自同构形式之间的深层哲学之后，近50年来取得了很大的进展。其中，PI所研究的l函数的特殊值与周期之间的关系是最重要的问题之一。确切地说，本文的第一部分是关于Gan-Gross-Prasad猜想框架下自同构表示的全局约束问题。PI将与不同的合作者一起，为这些时期制定一个明确的猜想，推广Ichino-Ikeda的工作，并使用Jacquet-Rallis在特殊情况下和PI一般情况下提出的相对迹公式工具研究酉群的情况。第二部分是了解l -函数的中心导数与某些算术周期，即Shimura变异上的环高度之间的关系。PI建议在Kudla和PI自己之后，通过算术提升来研究这种关系，特别是在下一个未知的U（2,2）的情况下。本研究的最终目的是为了更好地理解数学中的一个核心问题：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，如代数循环、表示、周期关系等。Birch和Swinnerton-Dyer猜想是这个方向上最著名的问题之一。在纯数学之外，椭圆曲线等代数方程在密码学中也有重要的应用。