The Rankin-Selberg Method, Zeros of Special Functions and Models of Representations Over Finite Fields

该奖项为研究者继续研究Rankin-Selberg方法提供了资金。这项工作的大部分可能是联合工作。研究可能会从寻找新的Rankin-Selberg积分转向从现有的Rankin-Selberg结构中提取更多信息。一个初始问题是GL(2)上对称立方l函数的阿基米德局部积分的不消失。Friedberg， Hoffstein和Bump一直在探索来自GL(n)和二次字符上具有自同构l函数的双Dirichlet级数的反射群。我们将研究GL(2)和立方字符的相应问题。关于自同构形式的其他问题也将讨论。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。

This award provides funding for the investigator's continued research into the Rankin-Selberg method. Much of this work will likely be joint work. The research will likely shift from the search for new Rankin-Selberg integrals to extracting more information from existing Rankin-Selberg constructions. An initial problem will be the nonvanishing of the archimedean local integrals for the symmetric cube L-functions on GL(2). Friedberg, Hoffstein, and Bump have been exploring reflection groups coming from double Dirichlet series with automorphic L-functions on GL(n) and quadratic characters. The corresponding question for GL(2) and cubic characters will be studied. Other problems on automorphic forms will be investigated. 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.