L-Functions, Elliptic Curves and Siegel Zeros

9801642伊万涅克该奖项支持W.Duke和H.Iwaniec的一个联合项目。数论中一些最重要的问题涉及到L函数族--与自同构型和椭圆曲线有关的函数族。对这些主题的解析方面的研究在过去的二十年里取得了进展，现在构成了现代解析数论中发展最快的领域之一。该项目将解决这一主题的核心问题。特别是，围绕西格尔零星存在的旧争议的问题构成了该提案的一部分。利用自同构理论对这一悬而未决的难题提出了新的研究思路。另一个目的是通过在椭圆曲线、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扩大解析数论的范围。这将通过关于L的一系列新的跨学科问题来实现-函数、除域和与模曲线相关的指数和。该项目还旨在在几个层面上产生教育影响。对于一个年轻的学生来说，在他或她的论文开始之前获得进行研究和展示数学的经验是一个重大的好处。当在协作和支持的环境中完成时，它在学习基础数学的阶段到独立思考和创造性思维的阶段之间提供了必要的桥梁，这是写一篇好论文所必需的。研究人员计划在他们的论文之前为几个学生提供一个从事这样的研究的机会。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

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.