FRG: Collaborative Research: Combinatorial representation theory, multiple Dirichlet series, and moments of L-functions

FRG：协作研究：组合表示理论、多重狄利克雷级数和 L 函数矩

• 批准号: 0652529
• 负责人: Benjamin Brubaker
• 金额: $ 9.84万
• 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=0652529&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Combinatorial; representation

Great progress has been made in recent years in the theory of multiple Dirichlet series. A variety of previously studied examples have been organized into a coherent framework. The emergent structures serve to both suggest natural generalizations---often with applications to analytic number theory---and point towards unexpected connections with such diverse areas of mathematics as the spectral theory of automorphic forms, arithmetic of function fields, the geometry of affine root systems and combinatorial representation theory. Many applications in analytic number theory have been found and many more are expected. These include moment estimates and convexity breaking for L-functions over an arbitrary number field, nonvanishing results for L-functions over number fields and function fields and results on the nature of the mysterious Whittaker coefficients of metaplectic Eisenstein series and higher order theta functions. Moreover, during the past several years the combined efforts of the investigators have demonstrated that Weyl group multiple Dirichlet series have a beautiful structure that was previously unknown, and by elucidating this structure, new connections with other areas of mathematics are rapidly emerging. The grant will fund continued investigation of these rapidly developing areas. In addition, two workshops are planned for the dissemination of these results and new techniques to research mathematicians and graduate students.Number theory began thousands of years ago and was initially inspired by questions about prime numbers. Dirichlet series are infinite series, such as the Riemann zeta function, and are a primary tool in the study of prime numbers. More recently they have come to fore by providing interconnections between many diverse areas of pure mathematics and physics. Multiple Dirichlet series are simply Dirichlet series in several variables -- they have the merit that the number theoretic quantities they measure can themselves be Dirichlet series, in particular L-functions, which are fundamental objects that can be associated with many classes of number-theoretic data, such as elliptic curves, representations of Galois groups, or modular forms.

近年来多重狄利克雷级数理论取得了很大进展。先前研究的各种示例已被组织成一个连贯的框架。这些新出现的结构既表明了自然的概括——通常应用于解析数论——也指出了与自守形式的谱理论、函数域算术、仿射几何等不同数学领域的意想不到的联系。根系统和组合表示理论。解析数论中的许多应用已经被发现，并且预计还会有更多应用。其中包括任意数域上 L 函数的矩估计和凸性破缺、数域和函数域上 L 函数的非零结果，以及有关超波爱森斯坦级数和高阶 theta 函数的神秘 Whittaker 系数性质的结果。此外，在过去几年中，研究人员的共同努力证明，Weyl群多重狄利克雷级数具有以前未知的美丽结构，并且通过阐明这种结构，与其他数学领域的新联系正在迅速出现。这笔赠款将资助对这些快速发展领域的持续调查。此外，还计划举办两个研讨会，向研究数学家和研究生传播这些成果和新技术。数论始于数千年前，最初受到素数问题的启发。狄利克雷级数是无限级数，例如黎曼 zeta 函数，是研究素数的主要工具。最近，它们通过提供纯数学和物理的许多不同领域之间的互连而脱颖而出。 多重狄利克雷级数只是多个变量的狄利克雷级数 - 它们的优点是它们测量的数论量本身可以是狄利克雷级数，特别是 L 函数，它们是可以与许多类数论相关联的基本对象数据，例如椭圆曲线、伽罗瓦群的表示或模形式。