Problems in Analytic Theory of L-functions and Automorphic Forms

该奖项资助第二名首席研究员对以下主题的研究。首先，她建议研究从经典模形式到西格尔模形式的一系列猜想提升。其次，她提出建立4次辛群的Kronecker极限公式。这包括找到西格尔·爱森斯坦级数关于其极点的劳伦级数展开。这样的结果可以应用于算术和几何。第三，与J. Hoffstein一起，她提出发展形而上学形式的概括。这些新形式，就像形而上学形式的情况一样，有望在l函数理论中产生惊人的结果。最后，她与L. Walling共同提出了对广义雅可比函数的研究。这些函数也可以用来做一个适当的Saito-Kurokawa猜想，因为它们提供了这样一个提升的建设性例子。本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，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。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各种应用领域不可或缺的工具。

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.