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.

该研究项目涉及数论。它关注自同构形式——在大的离散对称群下不变的函数。当应用对称性的顺序并不重要时，可以通过 19 世纪发现的傅里叶分析来研究此类函数。然而，对于更复杂的对称性，需要新的想法，而我们的知识还远未完成。基本的朗兰兹函数性猜想预测，高度对称的函数是理解多项式方程解的关键，在连续（函数）和离散（解）之间架起了一座桥梁。猜想的一个具体案例在怀尔斯证明费马猜想中发挥了关键作用，但朗兰兹猜想的大多数案例仍未得到证明。该项目将提供有关自同构形式（包括函子性）以及数论和与自同构形式相关的其他数学领域中的数量的新信息。更详细地说，该项目包括与函子性、L 函数和覆盖群相关的问题。在一系列问题中，主要研究者提出应用并推广一种新的最近构造，即扭曲重倍积分，它使得分析非泛自同构形式的 L 函数成为可能。第二系列问题涉及一般覆盖群（元波群）上的自同构形式。主要研究者建议给出有关线性群（或韦尔描述的双覆盖）上的自同构形式的构造可以扩展到一般覆盖群的原理的广泛示例。第三组问题涉及涉及自守形式、表示论和数论的新构造，包括与 Iwahori-Hecke 代数和量子群相关的 p 进群泛函的研究。这项研究将增进我们对约简群及其覆盖层的自同构形式的了解，并对数论、表示论和弦理论产生影响。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和知识进行评估，被认为值得支持。更广泛的影响审查标准。