L-Functions, Elliptic Curves and Siegel Zeros

L 函数、椭圆曲线和西格尔零点

• 批准号: 9801642
• 负责人: Henryk Iwaniec
• 金额: $ 50.68万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801642&HistoricalAwards=false
• 关键词: Functions; Elliptic; Curves; Siegel; Zeros

9801642 Iwaniec This award supports a joint project by W. Duke and H. Iwaniec. Some of the most important problems in number theory concern families of L-functions associated to automorphic forms and elliptic curves. The study of analytic aspects of these topics has progressed over the last twenty years and now constitutes one of the most rapidly expanding areas of modern analytic number theory. This project will address central questions of this subject. In particular, issues surrounding the old controversy about the existence of Siegel zeros form part of the proposal. A new line of investigation on this difficult unsolved problem is proposed using automorphic theory. Another aim is to broaden the scope of analytic number theory by implementing its techniques in novel ways in the arithmetic theory of elliptic curves, number fields, and modular forms. This will be accomplished through a series of new interdisciplinary problems about L-functions, division fields, and exponential sums associated to modular curves. The project also aims to have educational impact at several levels. It is a significant benefit for a young student to gain experience conducting research and presenting mathematics before his or her dissertation begins. When done in a collaborative and supportive environment it provides a needed bridge between the phase of learning foundational mathematics to that of thinking independently and creatively, as is required to write a good dissertation. The investigators plan to provide several students with an opportunity to engage in such research before their thesis. This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

小行星9801642 该奖项支持W.杜克和H.伊万尼克数论中一些最重要的问题涉及与自守形式和椭圆曲线相关的L-函数族。这些主题的分析方面的研究在过去的二十年中取得了进展，现在构成了现代解析数论最迅速扩展的领域之一。本项目将解决这一主题的核心问题。特别是，围绕西格尔零存在的老争议的问题构成了提案的一部分。利用自守理论对这一难题提出了一条新的研究思路。另一个目标是通过在椭圆曲线、数域和模形式的算术理论中以新颖的方式实现解析数论的技术来拓宽解析数论的范围。 这将通过一系列新的跨学科的L-函数，除法领域，和指数和相关的模曲线的问题。 该项目还旨在在多个层面产生教育影响。这是一个显着的好处，为年轻的学生获得经验进行研究，并提出数学之前，他或她的论文开始。当在一个协作和支持的环境中完成时，它提供了基础数学学习阶段与独立和创造性思维阶段之间的必要桥梁，这是写一篇好论文所必需的。研究人员计划为几名学生提供一个机会，在他们的论文之前从事这样的研究。 本研究属于一般数学领域的数论研究福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学最古老的分支之一，几个世纪以来出于纯粹的美学原因而受到人们的追求。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。