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.

该研究项目涉及互惠定律，这些定律是保留一定数量的不同对象集之间的对应关系，每个对象由单独的均值定义。在许多学科中发现了互惠法，从数学和物理学到工程，网络理论和社会科学。两组对象可能先验地看不到彼此，这使得这样的法律确实令人着迷。互惠的最深例子之一是那些出现在数字理论中的人，这是一种相当普遍的形式，是由于Artin和Langlands造成的，著名的“二次互惠法”只是第一个例子。这种互惠定律表明，复杂矩阵，算术性质的对象对“ Galois群体”的某些介绍，并在本地领域，分析性质的对象，保留某些复杂功能（根数和l- - l--功能）通过完全独立的手段附加到它们上。该项目的一个重要部分是表明，通过开发一种为所有这些因素建立这种平等的方法，这种互惠是可靠的。 更确切地说，该项目提出了一种方法，可以通过从Langlands-Shahidi方法（例如分析方法获得的方法）建立某些ARTIN因素（Artin Root数和L功能）的平等。这些将从一侧到另一侧的信息，包括导体平等，根号和可能的R组。代表理论和自动形式将会产生后果，例如许多钢化L-actet的病例及其相反的通用A包装猜想，以及通过Arthur和Langlands和猜想所要求的Artin因素对交织操作员的归一化。 Lapid和Mao以及其他人。作为另一个项目，人们希望通过Eisenstein系列的傅立叶系数来获得P-ADIC L功能的结果，其中它们的复杂版本显示出来。还将建立某些覆盖组的交织关系，并通过扭曲的痕量公式作为学生博士学位论文的一部分来研究Weyl的定律。该项目建议通过教学课程，指导和建议对研究生进行培训。