Double Dirichlet Series and Generalized Metaplectic Forms

本文提出了研究自同构的两种新方法：Dirichlet和Hecke l -级数。第一种技术涉及使用二重狄利克雷级数、相关的泛函方程群和两个复变量函数的凸性原理。我们的目的是研究这些二重狄利克雷级数作为系数在另一个变量w的狄利克雷级数展开中的一变量s的数论兴趣l级数的性质。这类二重级数经常出现在高阶群上的元爱森斯坦级数理论中。在过去的几年里，我们做了大量的工作来获取这些双级数中出现的l系列集合的信息。主要的工具是将Rankin-Selberg方法推广到元塑性爱森斯坦级数和尖形。然而，进展的一个主要障碍是分析与这些由Rankin-Selberg过程产生的l系列相关的阿基米德因素的困难。我们相信，上述技术应该适用于所有在理论上可以用Rankin-Selberg方法获得的情况，但应该避免在无限处工作的困难。我们希望用这个方法证明GL(2)上自同构形式的对称四幂l级数的全面性。这将导致最著名的傅立叶系数指数在质量形式估计中的改进从5/28到1/6。其他应用还包括对称立方l序列的全集性、GL(2)自同构l序列在任意点上的无限次三次扭曲的不消失性、Hecke l序列任意阶扭曲的均值估计。第二种技巧与形而上学形式的某些概括有关。在全模群的同余子群作用下，GL(2)上的元形变换。在这个项目中，研究者和他的合作者将用同余子群的表示取代久保田符号来构建一种新的元群。这可以用来构造爱森斯坦级数。研究者希望与G的伽罗瓦群a子群的基域扩展相关的Artin l级数能与这个爱森斯坦级数相连。如果成立，对爱森斯坦级数的积分变换的研究可以得到关于某些类别的Artin l级数的新信息。至少，对形而上学形式的这种概括的研究应该是有趣的，因为它们似乎是全新的对象。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。

9700757 Hoffstein In this proposal we put forward two new techniques for the study of automorphic, Dirichlet and Hecke L-series. The first technique involves the use of double Dirichlet series, associated groups of functional equations and a convexity principle for functions of two complex variables. Our purpose is to study the properties of L-series of number theoretic interest in one variable s that arise in these double Dirichlet series as coefficients in the expansion as a Dirichlet series in another variable w. Such double series often arise in the theory of metaplectic Eisenstein series on higher rank groups. A good deal of our efforts in past years have been devoted to obtaining information about collections of L-series that appear in these double series. The main tool has been generalizations of the Rankin-Selberg method applied to metaplectic Eisenstein series and cusp forms. A major obstacle to progress, however, has been the difficulty of analyzing the archimedian factors associated to these L-series that are created by the Rankin-Selberg process. It is our belief that the technique described above should be applicable in all the cases that are theoretically accessible by the Rankin-Selberg method, but the difficulties of working at the infinite place should be avoided. We hope to use this method to prove the entirety of the symmetric fourth power L-series associated to an automorphic form on GL(2) . This would have as a consequence an improvement of the best known exponent in Fourier coefficients estimates for Maass forms from 5/28 to 1/6 . Other applications should include the entirety of the symmetric cube L-series, non-vanishing of infinitely many cubic twists of a GL(2) automorphic L-series at arbitrary points, and mean value estimates for arbitrary order twists of Hecke L-series. The second technique is connected to certain generalizations of metaplectic forms. Metaplectic forms on GL(2) transform under the action of a congruence subgroup of the fu ll modular group. In this project, the investigator and his collaborators will construct a new kind of metaplectic group with the Kubota symbol replaced by a representation of the congruence subgroup. This could then be used to construct an Eisenstein series. The investigator hopes that the Artin L-series associated to extensions of the base field with Galois group a subgroup of G will be connected to this Eisenstein series. If true, an investigation of integral transforms of the Eisenstein series could lead to new information about certain classes of Artin L-series. At the very least, an investigation of such generalizations of metaplectic forms should prove interesting, as they appear to be completely new objects. 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.