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

这个调查将涉及四个主要的研究方向。首先，首席研究员提议继续他的工作，与D. Bump和J. Hoffstein合作，通过系统研究某些自然发生的Dirichlet级数在两个复变量中获得关于自同态l函数的信息。这些l函数本身不是欧拉积，但是它们的系数是欧拉积。这些级数是Rankin-Selberg型的积分，具有亚纯延拓和泛函方程。积分可以用局部表示理论的方法来分析，并且可以用与l函数相关的狄利克雷级数系数来分析。然后利用积分的性质来得到l函数的性质。主要研究者将研究某些正交群和辛群上的积分以及它们的元复盖上的积分，这些积分将给出关于GL(2)的三次自同构形式的扭曲、二次扩展的理想类特征的扭曲的均方，以及与GL(3)上的自同构表示相关的标准l函数的二次扭曲的不消失和平均大小等对象的解析信息。对涉及两个以上复变量的l函数的更一般的和的研究也被期待。其次，主要研究者建议研究与高次元塑自同构形式相关的欧拉积。这类欧拉积的存在是通过假设的变形形式与非变形形式的对应关系来预测的；然而，它们只在少数情况下被展示出来。在GSp(4)的封面上有两个这样的欧拉积，一个在双封面上，这是主要研究者和Wong的功劳，另一个在三封面上，这是主要研究者的学生T. Goetze的功劳。首席研究员建议首先通过将这些作品与Rallis和Piatetski-Shapiro提出的非唯一模型理论联系起来，从而减少这些作品的计算基础，然后使用这种新方法将它们推广到高于3度的覆盖范围，以及GSp(4)以外的群体。这项工作可能会与D. Bump合作。第三，首席研究员将继续研究相对痕量公式。这个公式结合了由L函数的积分表达式和Langlands泛函性引起的周期考虑，最终应该允许人们在许多情况下建立L包包含泛型成员。在最近完成的一个大型项目中，首席研究员和Jacquet已经证明了一个这样的公式的基本引理。第四，在最近与D. Goldberg的工作中，首席研究员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泛型表示。我们建议使用这些模型来建立局部结果（如Plancherel测度上的Langlands猜想）和许多l -函数的全局延拓，这些l -函数是由爱森斯坦级数a la Langlands- shahidi引起的，甚至对于这些群的非一般表示也是如此。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。

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.