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

Friedberg 9531957这项调查将涉及四个主要的研究方向。首先，主要研究者建议继续他的工作，与D。Bump和J. Hoffstein，通过系统地研究某些自然发生的两个复变量的Dirichlet级数来获得关于自守L-函数的信息。这些L-函数本身不是欧拉乘积，但它们的系数是欧拉的。这些级数作为Rankin-Selberg型积分出现，具有亚纯延拓和函数方程。 积分可以用局部表示论方法分析，Dirichlet级数系数与L-函数有关。 积分的性质然后被用来获得L-函数的性质。主要研究者将研究某些正交和辛群及其亚复盖上的积分，这些积分应提供有关对象的分析信息，例如GL（2）自守形式的三次特征的扭曲，二次扩张的理想类特征的扭曲的均方，以及与GL（3）上的自守表示相联系的标准L-函数的二次扭的非零大小和平均大小.更一般的L-函数和的研究，涉及两个以上的复杂变量，也是预料之中的。第二，主要研究者建议研究与更高次的元自守形式相关的欧拉乘积。 这种欧拉乘积的存在是由亚代数形式和非亚代数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GSp（4）的覆盖上已知有两个这样的欧拉积，一个是由主要研究者和Wong在双重覆盖上得到的，另一个是由主要研究者的学生T在三重覆盖上得到的。格茨 首席研究员建议首先让这些作品更少的计算基础， 将它们与Rallis和Piatetski-Shapiro提出的非唯一模型理论联系起来，然后使用这种新方法将它们推广到度大于3的覆盖和GSp（4）以外的群。这项工作可能与D.砰第三，主要研究者将继续研究相对痕量公式。这个公式，它结合了期间的考虑所产生的积分表达式的$L$-功能与朗兰兹函，最终应允许一个建立在许多情况下，L-包包含通用members.In最近完成的一个大规模的项目，首席研究员和雅凯已经证明了基本引理的一个这样的公式。第四，在最近的工作与D。Goldberg，主要研究者已经开始使用某些模型，而不是惠特克模型，直接处理正交和酉群上的非一般表示。 建议使用这些模型来建立本地的结果（例如朗兰兹猜想Plancherel措施）和全球的许多L-功能所产生的爱森斯坦系列一拉Langlands-Shahidi，即使非通用表示这些群体。 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。

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.