Langlands Reciprocity and Automorphic Forms

这个研究项目涉及互易定律，这是不同的对象之间保持一定数量的对应关系，每个对象都由单独的方法定义。互易定律在许多学科中都有大量的发现，从数学和物理学到工程学、网络理论和社会科学。这两组物体可能先验地没有办法看到对方，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出现在数论中的那些，其中一个相当普遍的形式是由于阿廷和朗兰兹，著名的“二次互易定律”只是第一个例子。这样的互反律建议索引的某些介绍“伽罗瓦群”的复杂矩阵，对象的算术性质，与无限维介绍一般线性群在当地的领域，对象的分析性质，保持某些复杂的功能（根号码和L-功能）附加到他们完全不同的手段。该项目的一个重要部分是通过制定一种方法来为所有这些因素建立这种平等来表明这种互惠性是强大的。 更确切地说，该项目提出了一种方法来建立某些Artin因子（Artin根数和L函数）与从分析方法获得的因子的相等性，例如，Langlands-Shahidi方法。这些将从一侧到另一侧携带信息，包括导体的相等性，根数和可能的R群。在表示论和自守形式中会有一些结果，例如回火L-包和它的匡威、一般A-包的许多情况，以及通过亚瑟和朗兰兹要求的阿廷因子对交织算子进行归一化，以及拉皮德和毛泽东等人的猜想。作为另一个项目，人们希望通过Eisenstein级数的傅立叶系数得到关于p-adic L-函数的结果，其中它们的复数形式出现。也将建立覆盖群的某些缠绕关系，以及作为学生博士论文的一部分，通过扭曲迹公式研究外尔定律。该项目建议通过教授课程、指导和咨询来培训研究生。

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.