Double Dirichlet Series and Generalized Metaplectic Forms

小行星9700757 在这个建议中，我们提出了两个新的技术研究自守，Dirichlet和Hecke L-级数。 第一种技术涉及使用双重狄利克雷级数，相关的功能方程组和凸性原则的功能，两个复杂的变量。我们的目的是研究数论上感兴趣的一个变量s的L-级数的性质，这些二重Dirichlet级数作为系数在另一个变量w的Dirichlet级数的展开中出现。这类二重级数经常出现在高阶群上的亚群Eisenstein级数理论中。 在过去的几年里，我们的大量努力都致力于获得关于出现在这些双重级数中的L-级数集合的信息。 主要工具已被推广的Rankin-Selberg方法应用到metaplectic爱森斯坦系列和尖点形式。 然而，进展的一个主要障碍是分析与这些由Rankin-Selberg过程创建的L系列相关的阿基米德因子的困难。我们相信，上述技术应该适用于所有的情况下，理论上可以访问的Rankin-Selberg方法，但在无限的地方工作的困难，应避免。 我们希望用这种方法来证明与GL（2）上的自守形式相关联的对称四次幂L-级数的整体性。 这将作为一个结果的改善最知名的指数傅立叶系数估计的马斯形式从5/28到1/6。 其他的应用应该包括对称立方L-级数的整体性，GL（2）自守L-级数在任意点的无穷多立方扭转的非零性，以及Hecke L-级数任意阶扭转的均值估计。第二种技术是连接到某些概括的metaplectic形式。 GL（2）变换在全模群的同余子群作用下的亚代数形式在这个项目中，研究者和他的合作者将构造一种新的元群，其中Kubota符号被同余子群的表示所取代。这可以用来构造一个爱森斯坦级数。研究者希望，与基域的伽罗瓦群的扩张相关的阿廷L-级数G的子群将连接到这个爱森斯坦级数。 如果是真的，对爱森斯坦级数的积分变换的研究可能会导致关于某些类别的阿廷L-级数的新信息。 至少，对元形式的这种推广的研究应该是有趣的，因为它们似乎是全新的对象。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如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。

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.