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

数学科学：L 函数之和、元波群和非泛型表示

• 批准号: 9531957
• 负责人: Solomon Friedberg
• 金额: $ 4.4万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-09-01 至 1999-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531957&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Sums; functions; Metaplectic

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.

弗里德伯格9531957这项调查将处理四个主要研究方向。首先，主要研究人员建议继续与D. Bump和J. Hoffstein的联合工作，以通过对两个复杂变量中某些天然发生的Dirichlet系列进行系统研究来获取有关自动型L功能的信息。这些L功能本身不是Euler产品，但它们的个体系数是Eulerian。这些系列作为Rankin-Selberg类型的积分出现，具有Meromororphic延续和功能方程。 可以通过局部表示方法和与L功能相关的DIRICHLET系列系数分析积分。 然后使用积分的特性来获得L功能的属性。 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）。还期望对涉及两个以上复杂变量的更通用的L功能总和进行研究。其次，首席研究人员建议研究与高度替代自动形式相关的欧拉产品。 这种欧拉产物的存在是通过元容器形式与非属于非偏度形式之间的假设对应关系来预测的。但是，它们仅在少数情况下才展出。 GSP（4）的封面上有两种此类Euler产品，其中一个是由于首席研究员和Wong而在双重盖上，而第二个则是由于主要研究人员的学生T. Goetze而在三重封面上。 首席研究者建议首先通过将这些作品与Rallis和Piatetski-Shapiro提出的非唯一模型的理论联系起来，从而使这些作品较少，然后使用这种新方法将其推广以覆盖高度高于3的学位，以及除GSP以外的其他基团（4）。这项工作可能与D. Bump联合。第三，主要研究人员将继续处理相对痕量公式。该公式结合了由$ l $ functions与Langlands功能性的整体表达式引起的时期考虑，最终应允许在许多情况下建立LAPCACKET包含通用成员的一个元素。在最近完成的大规模项目中，首席调查员和Jacquet证明了这样一种公式的基本障碍。第四，在与D. Goldberg的最新工作中，主要研究人员已经开始使用某些模型，这些模型不是Whittaker模型直接处理正交和统一群体的非列加性表示。 建议使用这些模型来建立局部结果（例如，兰格兰对plancherel措施的猜想），也可以建立由Eisenstein系列A La Langlands-Shahidi引起的许多L功能的全球延续，甚至对于这些群体的非总体表示。 这项研究属于数量理论的一般数学领域。 数字理论在整个数字的研究中具有历史根源，解决了诸如处理一个整数的问题的问题。它是数学最古老的分支之一，由于纯粹的审美原因而被追捕了许多世纪。但是，在过去的半个世纪中，它已成为数据传输和处理以及通信系统等领域的不同应用中必不可少的工具。