Problems in Analytic Theory of L-functions and Automorphic Forms

L-函数和自同构形式的解析理论问题

• 批准号: 9700488
• 负责人: Jeffrey Stopple
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700488&HistoricalAwards=false
• 关键词: Problems; Analytic; Theory; functions; Automorphic

9700488 Stopple/Imamoglu This award funds research by the second named principal investigator on the following topics. First, she proposes to investigate a series of conjectural liftings from classical modular forms to Siegel modular forms. Second, she proposes to establish a Kronecker limit formula for the symplectic group of degree 4. This involves finding the Laurent series expansion of Siegel Eisenstein series about its pole. Such a result would have applications to arithmetic and geometry. Third, with J. Hoffstein, she proposes to develop a generalization of metaplectic forms. These new forms, as in the case of metaplectic forms, are expected to lead to striking results in the theory of L-functions. Last, jointly with L. Walling, she proposes to work on generalized Jacobi theta functions. These theta functions can also be used to make an appropriate Saito-Kurokawa conjecture since they provide a constructive example of such a lift. 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.

9700488 Stopple/Imamoglu该奖项资助第二位被提名的首席研究员就以下主题进行的研究。首先，她建议研究从经典模形式到Siegel模形式的一系列猜想提升。其次，建立了四次辛群的Kronecker极限公式。这涉及到求Siegel Eisenstein级数关于极点的Laurent级数展开。这样的结果将适用于算术和几何。第三，与J·霍夫斯坦一起，她提出了一种泛化形式。这些新形式，就像亚可解形式一样，有望在L函数理论中产生显著的结果。最后，她与L·沃林共同提出了广义Jacobi theta函数的研究。这些theta函数也可以用来做出适当的斋藤-黑川猜想，因为它们提供了这样一个提升的建设性例子。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。