Automorphic Forms and L-Functions

自守形式和 L 函数

基本信息

批准号：
1801497
负责人：
Solomon Friedberg
金额：
$ 17.5万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801497&HistoricalAwards=false
关键词：
Automorphic Forms Functions

项目摘要

This research project concerns number theory. It focuses on automorphic forms---functions that are invariant under a large discrete group of symmetries. When the order of applying the symmetries does not matter, such functions may be studied by Fourier analysis, discovered in the nineteenth century; however, for more complicated symmetries new ideas are needed and our knowledge is far from complete. The fundamental Langlands Functoriality Conjectures predict that highly symmetric functions are the keys to understanding solutions to polynomial equations, making a bridge between the continuous (functions) and the discrete (solutions). A specific case of the conjectures played a key role in Wiles's proof of Fermat's Conjecture, but most cases of the Langlands Conjectures are still unproved. This project will provide new information about automorphic forms, including functoriality, and about quantities in number theory and in other areas of mathematics related to automorphic forms.In more detail, this project includes problems related to functoriality, L-functions, and covering groups. In one series of problems, the principal investigator proposes to give applications of, and to generalize, a new recent construction, the twisted doubling integral, which makes it possible to analyze L-functions for non-generic automorphic forms. A second series of problems concerns automorphic forms on general covering groups, the metaplectic groups. The principal investigator proposes to give broad examples of the principle that constructions involving automorphic forms on linear groups (or the double covers described by Weil) can be extended to general covering groups. A third set of problems concerns new constructions involving automorphic forms, representation theory, and number theory, including the study of functionals on p-adic groups related to Iwahori-Hecke algebras and quantum groups. This research will advance our knowledge of automorphic forms on reductive groups and their covers, with consequences for number theory, representation theory, and string theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目涉及数字理论。它重点介绍了大型离散对称组下不变的自动形式。当应用对称性的顺序无关紧要时，可以通过19世纪发现的傅立叶分析来研究此类功能。但是，对于更复杂的对称性，需要新想法，我们的知识远非完整。基本的Langlands功能性猜想预测，高度对称函数是理解多项式方程的解决方案，在连续（函数）和离散（解决方案）之间建立桥梁的关键。猜想的特定案例在威尔斯的fermat猜想证明中起着关键作用，但大多数兰兰人猜想的案例仍然没有证实。该项目将提供有关自动形式的新信息，包括功能性，以及数量理论的数量以及与自动形式相关的数学领域的数量。更详细地说，该项目包括与功能性，L功能和覆盖组有关的问题。在一系列问题中，首席研究人员提议为新的新结构提供应用，并概括扭曲的加倍积分，这使得可以分析非遗传自动形式的L功能。第二系列问题涉及一般覆盖组，元容器组的自动形式。首席研究者建议将涉及线性组上涉及自动形式的构造（或Weil所描述的双重覆盖物）的构造可以扩展到一般覆盖组。第三组问题涉及涉及自动形式，表示理论和数字理论的新结构，包括研究与伊瓦霍里·赫克（Iwahori-Hecke）代数和量子组相关的P-ADIC组的功能。这项研究将促进我们对还原群体及其封面上的自态形式的了解，以及对数字理论，代表理论和弦理论的影响。该奖项反映了NSF的法定使命，并被认为是值得通过基金会的智力优点和更广泛的影响来通过评估来支持的。