Langlands Reciprocity and Automorphic Forms

朗兰兹互易和自守形式

• 批准号: 1500759
• 负责人: Freydoon Shahidi
• 金额: $ 27万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500759&HistoricalAwards=false
• 关键词: Langlands; Reciprocity; Automorphic; Forms

This research project concerns reciprocity laws, which are correspondences between different sets of objects preserving certain quantities, each defined by separate means. Reciprocity laws are found in abundance in many disciplines, ranging from mathematics and physics to engineering, network theory, and social sciences. The two sets of objects may a priori have no way of seeing each other and that makes such laws truly fascinating. One of the deepest examples of reciprocity are those appearing in number theory, a rather general form of which is due to Artin and Langlands, for which the famous "Quadratic Reciprocity Law" is just a first example. Such reciprocity laws suggest an indexing of certain presentations of "Galois groups" by complex matrices, objects of arithmetic nature, with infinite dimensional presentations of general linear groups over local fields, objects of analytic nature, preserving certain complex functions (root numbers and L-functions) attached to them by totally separate means. An important part of this project is to show that this reciprocity is robust by developing an approach to establishing this equality for all such factors. More precisely, this project suggests an approach to establishing the equality of certain Artin factors (Artin root numbers and L-functions) with those obtained from analytic methods, e.g., those coming from Langlands-Shahidi method. These will carry information from one side to the other including equality of conductors, root numbers and possibly R-groups. There will be consequences in representation theory and automorphic forms such as many cases of tempered L-packet and its converse, generic A-packet conjectures, as well as normalization of intertwining operators by means of Artin factors as demanded by Arthur and Langlands and the conjecture of Lapid and Mao as well as others. As another project, one hopes to obtain results on p-adic L-functions by means of Fourier coefficients of Eisenstein series where their complex versions show up. Certain intertwining relations for covering groups will also be established, as well as study of Weyl's law by means of twisted trace formula as part of a student doctorate thesis. The project suggests training of graduate students through teaching courses, mentoring, and advising.

该研究项目涉及互易律，互易律是保留特定量的不同对象组之间的对应关系，每个对象都通过不同的方式定义。互易定律广泛存在于许多学科中，从数学、物理学到工程学、网络理论和社会科学。这两组物体可能先验地无法看到彼此，这使得这些定律真正令人着迷。互易性最深刻的例子之一是数论中出现的例子，其相当普遍的形式是阿廷和朗兰兹提出的，著名的“二次互易律”只是第一个例子。这种互易律建议通过复数矩阵、算术性质的对象、具有局部域上的一般线性群的无限维表示、分析性质的对象来对“伽罗瓦群”的某些表示进行索引，并通过完全独立的方式保留某些复数函数（根数和 L 函数）附加到它们。该项目的一个重要部分是通过开发一种方法来为所有这些因素建立这种平等，以表明这种互惠性是强大的。 更准确地说，该项目提出了一种方法，用于建立某些 Artin 因子（Artin 根数和 L 函数）与通过分析方法获得的因子（例如来自 Langlands-Shahidi 方法的因子）的相等性。这些将把信息从一侧传送到另一侧，包括导体的相等性、根号和可能的 R 基团。这将在表示论和自同构形式中产生后果，例如调节 L 包及其逆向通用 A 包猜想的许多情况，以及通过 Arthur 和 Langlands 所要求的 Artin 因子以及 Lapid 和 Mao 等人的猜想对交织算子进行归一化。作为另一个项目，人们希望通过爱森斯坦级数的傅里叶系数获得 p 进 L 函数的结果，其中出现了复数版本。还将建立覆盖群的某些交织关系，以及通过扭曲迹公式研究韦尔定律，作为学生博士论文的一部分。该项目建议通过教学课程、指导和建议来培训研究生。