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

• 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

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.

作者继续研究二重狄里克莱级数在自同构L级数和数论研究中的应用。几年来，他和多位合作者一直在发展双狄里克莱级数理论。它在许多情况下为Rankin-Selberg方法提供了一种简单的替代方法。例如，它给出了GL(2)上一般自同构型的对称平方L级数的整性的一个非常简短的证明。作者希望将这一方法推广到理解更高对称的幂L函数和标准L函数的高阶扭转的集体行为上。作为猜测双Dirichlet级数方法潜在应用的另一个输入源，研究者研究了某些亚可解形式的推广的Mellin变换。这些都是由其他非同余子群构成的。这些形式的理论相应地丰富，应该有助于进一步洞察二重狄里克莱级数的应用。L级数的研究是近百年来数学发展的重要组成部分。例如，某些L级数提供了链中的重要环节，导致了最近费马大定理的证明。对二重狄里克莱级数的研究有助于加深对L级数的理解，进而加深对数学中一些很深的领域的认识。无法预测这项调查的最终影响，但潜在的应用领域包括密码学。