Langlands Reciprocity and Automorphic Forms

这个研究项目涉及互惠定律，这是保持一定数量的不同对象之间的对应关系，每个对象都用不同的方法定义。从数学、物理到工程、网络理论和社会科学，互惠定律在许多学科中都有大量的应用。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出现在数论中的互易性，它的一种相当普遍的形式是由Artin和Langlands提出的，其中著名的“二次互易性定律”只是第一个例子。这样的互易律建议用复矩阵来索引“伽罗瓦群”的某些表示，具有算术性质的对象，具有局部域上一般线性群的无限维表示，具有解析性质的对象，以完全不同的方式保留附加在它们上的某些复函数（根数和l函数）。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，从而表明这种互惠是稳健的。更准确地说，本项目提出了一种方法来建立某些Artin因子（Artin根数和l -函数）与解析方法（例如来自Langlands-Shahidi方法）的相等性。它们将把信息从一边传递到另一边，包括导体的相等性、根数和可能的r群。这将在表征理论和自同构形式中产生影响，例如许多情况下的回调l包及其逆，一般a包猜想，以及Arthur和Langlands要求的通过Artin因子和Lapid和Mao等人的猜想对交织算子进行规范化。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得到p进l函数的结果，在那里它们的复数版本出现了。还将建立覆盖群体的一定的缠结关系，以及通过扭曲迹公式研究Weyl定律作为学生博士论文的一部分。该项目建议通过授课、指导和咨询等方式培养研究生。

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.