Multiple Dirichlet Series with Applications to Automorphic Representation Theory

多重狄利克雷级数及其在自守表示理论中的应用

• 批准号: 0702438
• 负责人: Benjamin Brubaker
• 金额: $ 17.04万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0702438&HistoricalAwards=false
• 关键词: Multiple; Dirichlet; Series; Applications; Automorphic

In this project, Principal Investigator Brubaker and his collaborators seek to understand metaplectic forms, a generalization of automorphic forms to certain covers of split, reductive algebraic groups. Specifically, the proposed research focuses on constructing Dirichlet series in several complex variables, termed ``Multiple Dirichlet Series'' (MDS), with good analytic properties (e.g. functional equations and meromorphic continuation) and connecting these series to the Fourier-Whittaker coefficients of Eisenstein series on metaplectic groups. Since the construction of the metaplectic cover is intimately tied to reciprocity laws in number fields, the Dirichlet series and its polar residues contain families of automorphic forms twisted by characters built from power residue symbols. Analytic properties of the Dirichlet series then translate to arithmetic applications for automorphic forms, including non-vanishing results for automorphic L-functions. The construction and subsequent proof of analytic properties of these MDS uses new techniques in combinatorial representation theory, and another primary objective of this work is a deeper understanding of the connections between this representation theory and metaplectic forms. In the process, this structure is expected to illuminate further the relationship between combinatorial representation theory and the special case of Fourier-Whittaker coefficients of automorphic forms.Historically, problems in number theory have centered around integer solutions to polynomial equations, so called ``Diophantine equations,'' which could be simply stated, but often extraordinarily hard to prove. It once appeared that these questions were the stuff of pure thought experiments, since the integers are discrete and should therefore have no bearing on the world and its continuous phenomena. But in the late 1960's, Robert Langlands developed a series of far-reaching conjectures known today as the Langlands' Program to investigate connections among number theory, arithmetic geometry and harmonic analysis; in fact, his conjectures were based on calculations involving a special case of the highly symmetric functions known as Eisenstein series, which are the principal objects of study in this proposal. The reach of Langlands' conjectures has been greatly expanded in the last several decades and now extends from methods for solving Diophantine equations to geometric versions with intimate connections to quantum field theory and string theory, which attempt to explain the origins and expansion of our universe via a uniform treatment of fundamental forces including gravity and electromagnetism. That is, motivated by natural questions about solutions of polynomial equations in the integers, one obtains a new interpretation for profoundly important physical phenomena; in trying to answer discrete problems, one finds explanations of the continuous world. This project attempts to bring together a previously disparate community of researchers and students in number theory, automorphic forms, Lie groups, and combinatorics to further investigate analogous connections by studying large classes of more general Eisenstein series and their relations to the aforementioned disciplines.

在这个项目中，首席研究员 Brubaker 和他的合作者试图理解元逻辑形式，即自同构形式对分裂、还原代数群的某些覆盖的概括。 具体来说，所提出的研究重点是在多个复变量中构造狄利克雷级数，称为“多重狄利克雷级数”（MDS），具有良好的分析性质（例如函数方程和亚纯延拓），并将这些级数连接到爱森斯坦关于超群群的系列。由于超触覆盖的构造与数域中的互反律密切相关，狄利克雷级数及其极性留数包含由幂留数符号构建的字符扭曲的自守形式族。然后，狄利克雷级数的解析性质转化为自同构形式的算术应用，包括自同构 L 函数的非零结果。这些 MDS 的分析属性的构造和随后的证明使用了组合表示理论中的新技术，这项工作的另一个主要目标是更深入地理解这种表示理论和元逻辑形式之间的联系。在此过程中，这种结构有望进一步阐明组合表示论与自守形式的傅立叶-惠特克系数的特殊情况之间的关系。历史上，数论中的问题都集中在多项式方程的整数解上，即所谓的“丢番图”方程”可以简单地表述，但通常很难证明。曾经看来，这些问题是纯粹思想实验的内容，因为整数是离散的，因此应该与世界及其连续现象无关。但在 1960 年代末，罗伯特·朗兰兹提出了一系列影响深远的猜想，即今天的朗兰兹纲领，以研究数论、算术几何和调和分析之间的联系；事实上，他的猜想是基于涉及称为爱森斯坦级数的高度对称函数的特殊情况的计算，这是该提案中的主要研究对象。朗兰兹猜想的范围在过去几十年里得到了极大的扩展，现在从解决丢番图方程的方法扩展到与量子场论和弦理论密切相关的几何版本，这些猜想试图通过以下方式解释我们宇宙的起源和膨胀：对包括重力和电磁力在内的基本力的统一处理。也就是说，在整数多项式方程解的自然问题的推动下，人们获得了对极其重要的物理现象的新解释；在试图回答离散问题的过程中，人们找到了对连续世界的解释。该项目试图将数论、自同构形式、李群和组合学方面以前不同的研究人员和学生聚集在一起，通过研究更一般的爱森斯坦级数及其与上述学科的关系来进一步研究类似的联系。