P-adic Aspects of L-Values, Congruences Between Automorphic Forms, and Arithmetic Applications

L 值的 P 进数方面、自守形式之间的同余以及算术应用

• 批准号: 2001527
• 负责人: Zheng Liu
• 金额: $ 12.12万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001527&HistoricalAwards=false
• 关键词: adic; Aspects; Values; Congruences; Between

In the study of many fundamental problems in number theory, for example distribution of prime numbers and rational solutions to some polynomial equations, a function on the complex plane, called the L-function, has been associated with the arithmetic object under study. Various properties of the L-functions are believed and have been proved in many special cases to encode deep arithmetic information. This research project is mainly concerned with congruences between special values of L-functions, as well as congruences between automorphic forms, which are multi-variable functions with many symmetries. This project also aims at applying these congruences to study the connection between the special values of L-functions and the sizes of arithmetic cohomology groups in some special cases. In this project, one aspect is the study of the congruences modulo powers of p between automorphic forms related to theta correspondence. This includes studying the p-adic properties of certain theta lifts by using suitable integral representations of L-functions and L-values for automorphic representations, constructing certain p-adic L-functions and constructing Hida families of theta lifts. The Iwasawa--Greenberg Main Conjecture predicts a connection between the p-adic L-functions and the Selmer groups attached to p-adic deformations of Galois representations. Another aspect of this project is to exploit the results on the congruences to study this conjecture in the special case of the Rankin--Selberg product of a pair of modular forms whose weights are of the same parity.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.

在数论中的许多基本问题的研究中，例如素数的分布和一些多项式方程的有理解，复平面上的一个函数称为L函数，它与所研究的算术对象有联系。L函数的各种性质被认为是正确的，并已在许多特殊情况下被证明可以用来编码深层算术信息。本课题主要研究L函数的特殊值之间的同余，以及具有多种对称性的多元函数的自同构型之间的同余。本文还利用这些同余关系研究了在某些特殊情况下，L函数的特殊值与算术上同调群的大小之间的关系。在这个项目中，一个方面是研究与theta对应有关的自同构形之间的p的模幂同余。这包括利用L-函数和L-值的适当的积分表示作为自同构表示来研究某些theta提升族的p-进的性质，构造某些p进的L函数和构造theta提升族的Hida族。岩泽-格林伯格猜想预言了P-进L函数与伴随于伽罗瓦表示的P-进变形的塞尔默群之间的联系。这个项目的另一个方面是利用同余的结果在Rankin-Selberg积的特殊情况下研究这一猜想，该猜想是一对权重相同的模形式的乘积。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Iwasawa–Greenberg main conjecture for nonordinary modular forms and Eisenstein congruences on GU(3,1)

岩泽 — 格林伯格关于 GU(3,1) 上的非普通模形式和爱森斯坦同余的主要猜想

• DOI: 10.1017/fms.2022.95
• 发表时间: 2022
• 期刊: Sigma
• 作者: Castella, Francesc;Liu, Zheng;Wan, Xin
• 通讯作者: Wan, Xin