Automorphic Forms and L-functions: P-adic Aspects and Applications

自守形式和 L 函数：P 进数方面和应用

• 批准号: 1501083
• 负责人: Ellen Eischen
• 金额: $ 13.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501083&HistoricalAwards=false
• 关键词: Automorphic; Forms; functions; adic; Aspects

This research project primarily concerns topics in number theory, one of the oldest branches of mathematics. Number theory is concerned with properties of whole numbers, such as factorization and related topics. In this project, the investigator will combine geometry, another classical branch of mathematics, with number theory, using deep geometric techniques together with properties of prime numbers to prove new results in number theory, in particular by studying certain families of number theoretic data. The techniques in this research project build on ones that have played a key role in the investigator's prior research, as well as ones that have been essential in some of the largest recent breakthroughs in mathematics, such as the proof of Fermat's Last Theorem. Other components of the project delve into exciting new areas in the field. This project focuses on four closely connected topics: L-functions, automorphic forms, differential operators, and related geometric problems. Most of the research concerns p-adic aspects, but the investigator will also work on some purely Archimedean problems that have arisen in the context of her previous work on p-adic interpolation of values of Eisenstein series and L-functions. The planned work has consequences primarily in number theory but also conjecturally in representation theory. The p-adic L-functions that form part of this research program conjecturally play a key role in Iwasawa theory, a p-adic theory for studying arithmetic data, such as class groups and Galois representations. The geometric problems in this research program also will contribute to the understanding of p-adic L-functions, Galois representations, and their role in Iwasawa theory.

该研究项目主要涉及数字理论的主题，这是数学最古老的分支之一。 数字理论与整数的属性有关，例如分解和相关主题。 在该项目中，研究者将使用深度几何技术以及质数的属性结合数学，这是数学的另一个古典分支，并将数字理论结合在一起，以证明数字理论的新结果，特别是通过研究数量理论数据的某些家庭。 该研究项目中的技术建立在研究者先前研究中发挥关键作用的技术，以及在一些最近最大的数学突破中至关重要的研究，例如Fermat的最后一个定理证明。该项目的其他组成部分深入到该领域令人兴奋的新领域。 该项目着重于四个紧密连接的主题：L功能，自动形式，差异操作员和相关的几何问题。 大多数研究都涉及P-ADIC方面，但是研究人员还将处理一些纯粹的Archimedean问题，这些问题是在她先前关于Eisenstein系列和L功能值的P-Adic插值工作的背景下出现的。 计划的工作主要在数量理论中具有后果，但在表示理论中也有猜想。 构成该研究计划的一部分的P-ADIC L功能在伊瓦沙瓦理论中起着关键作用，这是一种用于研究算术数据的P-ADIC理论，例如课堂组和GALOIS表示。 该研究计划中的几何问题还将有助于理解P-ADIC L功能，GALOIS表示及其在Iwasawa理论中的作用。