Applications of Double Dirichlet Series to Automorphic Forms and Number Theory

双狄利克雷级数在自守形式和数论中的应用

基本信息

批准号：
0088921
负责人：
Jeffrey Hoffstein
金额：
$ 8.7万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2000
资助国家：
美国
起止时间：
2000-07-01 至 2003-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0088921&HistoricalAwards=false
关键词：
Applications Double Dirichlet Series Automorphic

项目摘要

The investigator continues his investigation of applications of double Dirichlet series to the study of automorphic L-series and number theory. He and various collaborators have been developing the theory of double Dirichlet series for several years. It offers a simple replacement for the Rankin-Selberg method in many instances. For example, it provides a very short proof of the entirety of the symmetric square L-series of a generic automorphic form on GL(2). The investigator hopes to extend this technique as far as possible to understand higher symmetric power L-functions and the collective behavior of higher order twists of standard L-functions. As an additional source of input for trying to guess potential applications of the double Dirichlet series method, the investigator studies the Mellin transforms of certain generalizations of metaplectic forms. These are constructed from other non congruence subgroups. The theory of these forms is correspondingly rich and should contribute further insights into applications of double Dirichlet series.The study of L-series has been an essential part of the development of mathematics over the last 100 years. For example, certain L-series provided vital links in the chain that led to the recent proof of Fermat's last theorem. The study of double Dirichlet series is leading to an improved understanding of L-series which in turn leads to further knowledge of some very deep areas of mathematics. It is impossible to predict the ultimate impact of this investigation, but potential areas of application include cryptography.

研究人员继续研究双迪里奇系列在自增长L系列和数字理论研究中的应用。他和各种合作者多年来一直在发展双迪里奇系列的理论。在许多情况下，它为Rankin-Selberg方法提供了一个简单的替代品。例如，它提供了一个非常简短的证明，证明了GL（2）上通用自动形式的整体正方形L系列。研究人员希望尽可能地扩展该技术，以了解更高的对称功率l功能和高阶标准L功能的集体行为。作为试图猜测双Dirichlet系列方法的潜在应用的其他输入来源，研究者研究了Mellin的某些元容器形式的概括。这些是由其他非一致亚组构建的。这些形式的理论相应丰富，应进一步洞察双迪里奇系列的应用。在过去的100年中，对L系列的研究一直是数学发展的重要组成部分。例如，某些L系列在链条中提供了重要的联系，从而导致了Fermat的最后一个定理的证据。对Double Dirichlet系列的研究导致人们对L系列的了解得到了改进，这反过来又导致了一些非常深入的数学领域的了解。无法预测这项调查的最终影响，但是潜在的应用领域包括密码学。