Double Dirichlet Series and Generalized Metaplectic Forms

9700757霍夫斯坦在这个建议中，我们提出了两种研究自同构的新方法，狄利克莱特和赫克L级数。第一种方法涉及到双重狄里克莱级数、相关的函数方程组和两个复变量函数的凸性原理的使用。我们的目的是研究一元数论中的L级数的性质，这些重Dirichlet级数作为展开式中的系数出现在这些重Dirichlet级数中，作为另一个变量w的Dirichlet级数，这种重级数经常出现在高阶群上的亚可解Eisenstein级数理论中。在过去的几年里，我们致力于获取有关出现在这两个系列中的L系列收藏的信息。主要的工具是将Rankin-Selberg方法推广到亚可解的Eisenstein级数和尖点形式。然而，进展的一个主要障碍是很难分析与这些由兰金-塞尔伯格进程创造的L系列相关的阿基米德因素。我们认为，上述技术应该适用于理论上可用Rankin-Selberg方法获得的所有情况，但应避免在无穷大处工作的困难。我们希望用这个方法证明GL(2)上与自同构型相关的对称四次L级数的整性。这将导致Maass形式的傅立叶系数估计中最著名的指数从5/28提高到1/6。其他的应用还包括对称立方L级数的整体性，GL(2)自同构的L级数在任意点的无穷多个三次扭转不为零，以及Hecke L级数的任意阶扭转的均值估计。第二种技巧与某些亚可解形式的推广有关。在全模群的同余子群作用下，GL(2)变换的亚辛形式。在这个项目中，研究者和他的合作者将用同余子群的表示代替久保田符号来构造一种新的亚普勒群。然后，这可以用来构建爱森斯坦系列。作者希望G的子群为伽罗华群的与基域扩张有关的Artin L级数能与这个Eisenstein级数相连。如果这是真的，对艾森斯坦级数的积分变换的研究可能会得到有关L级数某些类的新信息。至少，研究这种亚普勒式的推广应该是有趣的，因为它们看起来是全新的对象。这项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，它解决了一些问题，比如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。

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.