Langlands Reciprocity and Automorphic Forms

这个研究项目涉及互易律，它是保持一定数量的不同对象集合之间的对应关系，每个对象都以不同的方式定义。互惠定律在许多学科中都有大量的发现，从数学和物理到工程学、网络理论和社会科学。这两组物体可能先验地无法看到对方，这使得这样的定律真正令人着迷。互易性最深刻的例子之一是出现在数论中的那些例子，它的一个相当普遍的形式是由于Artin和朗兰兹，著名的“二次互易定律”只是第一个例子。这样的互易定律表明，“伽罗华群”的某些表示是通过复矩阵来索引的，它是算术性质的对象，具有局部域上一般线性群的无限维表示，是解析性质的对象，通过完全独立的方法保持某些复函数(根数和L函数)。这个项目的一个重要部分是通过开发一种方法来建立对所有这些因素的这种平等，从而表明这种互惠是强大的。更确切地说，这个项目提出了一种方法来建立某些Artin因子(Artin根数和L函数)与那些从分析方法获得的相等，例如来自朗兰兹-沙希迪方法的那些。它们将把信息从一边传送到另一边，包括导体、根数和可能的R-群的相等。在表示理论和自同构形式中会有一些结果，如许多情况下调和的L包及其逆，一般的A包猜想，以及像Arthur和Langland所要求的利用Artin因子和Lapid和Mao等人的猜想使交织算子的正规化。作为另一项工作，我们希望通过Eisenstein级数的傅里叶系数得到关于p-进L函数的结果，其中给出了它们的复数形式。还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Weyl定律。该项目建议通过教学课程、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.