Mathematical Sciences: Analytic Number Theory on Groups

这个项目涉及对离散群的研究，具体应用于解析数论。主要研究人员证明了L函数的导数的特殊值可以表示为某些模积分的推广的单余循环的有限线性组合。这个项目的一个目标是发展一种推广著名的模积分理论的单上循环理论，并发展与模形式相关的L函数的导数的应用。另一个目标是找到定义在函数域上的椭圆曲线的Shafarevich-Tate群的锐界。最近，首席研究员与Hoffstein和Lieman一起证明了从GL(2)到GL(3)的伴随平方提升的Zeta函数不存在Siegel零点。我们将继续使用证明方法来研究GL(N)上的Siegel零点。最终目标是开发基于Zeta函数的新密码系统。这项研究属于数论的一般领域。数论的历史根源在于研究整数，它解决的问题是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。

This project involves the study of discrete groups with specific applications to analytic number theory. The principal investigator has shown that special values of derivatives of L-functions are expressible as finite linear combinations of certain one-cocycles which are generalizations of modular integrals. One goal of this project is to develop a theory of these one-cocycles which generalizes the well known theory of modular integrals, and to develop applications of derivatives of L-functions associated to modular forms. Another goal is to find sharp bounds for the Shafarevich-Tate group for elliptic curves defined over function fields. Recently, the principal investigator in conjunction with Hoffstein and Lieman has shown that there are no Siegel zeros for zeta functions associated to adjoint square lifts from GL(2) to GL(3). The methods used for this proof will be pursued to study Siegel zeros on GL(n). A final goal is to develop new crypto-systems based on zeta functions. This research falls into the general area of number theory. Number theory has its historical roots in the study of the whole numbers, addressing suah 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.