Mathematical Sciences: Analytic Number Theory on Groups

这个项目涉及离散群的研究与具体应用解析数论。主要研究者证明了L-函数的导数的特殊值可以表示为某些单上循环的有限线性组合，这些单上循环是模积分的推广。这个项目的一个目标是发展一个理论，这些一个cocycles推广了著名的理论，模积分，并开发应用程序的衍生物的L-功能相关的模块形式。另一个目标是找到尖锐的边界的Shafarevich-Tate群的椭圆曲线定义在功能领域。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函数没有Siegel零点。将继续研究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。 这项研究属于一般的数论领域福尔斯。数论有它的历史根源在研究整个数字，解决suah问题的那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.