Mathematical Sciences: Sums of L-functions, the Metaplectic Group, and Non-Generic Representations

弗里德伯格9531957这项调查将涉及四个主要研究方向。首先，主要研究人员建议继续他与D.Bump和J.Hoffstein的合作，通过对两个复变量中的某些自然产生的Dirichlet级数的系统研究，获得关于自同构L函数的信息。这些L函数本身不是欧拉乘积，但它们的各个系数是欧拉的。这些级数作为Rankin-Selberg型积分产生，具有亚纯延拓和函数方程。积分可以用局部表示论的方法来分析，狄里克莱级数系数与L函数有关。然后利用积分的性质得到L函数的性质。主要研究某些正交、辛群及其亚辛覆盖上的积分，它应该给出关于对象的分析信息，如GL(2)的三次特征标自同构形式的扭曲，二次扩张的理想类特征标的扭曲的均方，以及与GL(3)上的自同构表示相关的标准L函数的二次扭曲的非零性和平均大小。还将研究更一般的涉及两个以上复变量的L函数的和。第二，主要研究者建议研究与高次亚可积自同构型相关的欧拉乘积。这种欧拉乘积的存在是通过亚可解形式与非拟可解形式之间的假设对应来预测的；然而，它们只在少数情况下被展示出来。在普惠制(4)的封面上已知有两个这样的欧拉产品，一个是首席调查员和Wong在双层封面上的产品，另一个是主要调查员的学生T.Goetze在三重封面上的产品。主要研究人员建议首先通过将这些工作与Rallis和Piatetski-Shapiro提出的非唯一模型理论相联系来给予这些工作较少的计算基础，然后使用这种新方法将它们推广到3级以上的覆盖和GSP(4)以外的群。这项工作可能会与D.Bump合作。第三，首席调查员将继续研究相对迹线公式。这个公式结合了从$L$-函数的积分表达式中得到的周期考虑和朗兰兹函数性，在许多情况下最终应该允许人们建立L-包包含类属成员。在最近完成的一个大规模项目中，首席研究员和Jacquet证明了一个这样的公式的基本引理。第四，在最近与D.Goldberg合作的工作中，主要研究者已经开始使用某些不是Whittaker模型的模型来直接处理正交和酉群上的非一般表示。我们建议利用这些模型来建立局部结果(例如，关于Plancerel测度的朗兰兹猜想)和许多源于艾森斯坦级数的L函数的整体延拓，甚至对于这些群的非一般表示。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。

Friedberg 9531957 This investigation will deal with four main directions of research. First, the principal investigator proposes to continue his work, joint with D. Bump and J. Hoffstein, to obtain information about automorphic L-functions through the systematic study of certain naturally occurring Dirichlet series in two complex variables. These L-functions are not themselves Euler products, but their individual coefficients are Eulerian. These series arise as integrals of Rankin-Selberg type, possessing meromorphic continuation and functional equation. The integrals may be analyzed by local representation- theoretic methods, and the Dirichlet series coefficients related to L-functions. The properties of the integral are then used to obtain properties of the L-functions. The principal investigator will investigate integrals on certain orthogonal and symplectic groups and on their metaplectic covers, which should give analytic information concerning such objects as twists of GL(2) automorphic forms by cubic characters, mean squares of twists by ideal class characters of quadratic extensions, and the nonvanishing and mean size of quadratic twists of the standard L-function associated to an automorphic representation on GL(3).The study of more general sums of L-functions, involving more than two complex variables, is also anticipated. Second, the principal investigator proposes to investigate the Euler products associated with higher degree metaplectic automorphic forms. The existence of such Euler products is predicted by the hypothetical correspondence between metaplectic forms and non-metaplectic ones; however, they have only been exhibited in a few cases. There are two such Euler products known on covers of GSp(4), one on the double cover due to the principal investigator and Wong, and the second on the triple cover due to the principal investigator's student T. Goetze. The principal investigator proposes to first give these works less computational foundations by connecting them to the theory of non-unique models presented by Rallis and Piatetski-Shapiro, and then to use this new approach to generalize them to covers of degree higher than 3, and to groups other than GSp(4). This work will probably be joint with D. Bump. Third, the principal investigator will continue to work on the relative trace formula. This formula, which combines period considerations arising from integral expressions of $L$-functions with Langlands functoriality, should ultimately allow one to establish in many cases that L-packets contain generic members.In a recently completed massive project, the principal investigator and Jacquet have proved the fundamental lemma for one such formula. Fourth, in recent work with D. Goldberg, the principal investigator has begun to use certain models which are not Whittaker models to deal directly with non-generic representations on orthogonal and unitary groups. It is proposed to use these models to establish both local results (e.g. Langlands conjecture on Plancherel measure) and the global continuation of many L-functions arising from Eisenstein series a la Langlands-Shahidi, even for non-generic representations of these groups. 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.