Automorphic L-Functions and Langlands Functoriality

自同构 L 函数和朗兰兹函数性

• 批准号: 0200325
• 负责人: Freydoon Shahidi
• 金额: $ 46.6万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200325&HistoricalAwards=false
• 关键词: Automorphic; Functions; Langlands; Functoriality

Recent striking results establishing the existence of the functorial symmetric powers of degree 3 and 4 for cusp forms on GL(2) as automorphic forms on GL(4) and GL(5), as well as transfer of generic cusp forms from odd special orthogonal groups to general linear groups, by the investigator and his collaborator, has opened a new front in automorphic forms and number theory. They have resulted in surprising new estimates towards Ramanujan--Selberg and Sato--Tate conjectures, together with a large number of impressive, definitive and new results in number theory, automorphic forms and geometry obtained by other mathematicians. The investigator explores extensions of these to higher powers of forms on GL(2) as well as transfers of generic forms on other classical groups and their simply connected similitude coverings. The well-known and still open case of transfer from GSp(4) to GL(4) then becomes a special case of this. Beside exploring new ideas of Langlands on transferring beyond endoscopy, a situation which is already present in the third and fourth symmetric powers, he plans to investigate any possible extension of his method to infinite dimensional groups. He has a solid approach to establishing the stability of root numbers necessary for these transfers, by means of his method, and studies Bessel functions and the full generality of all the root numbers defined from his method. Stability of a subclass of these root numbers is the last serious problem in establishing these transfers. The investigator is also working on a number of problems concerning poles of local intertwining operators as well as those of automorphic L-functions coming from his method. Much of this work is carried out in collaboration with collaborators.The theory of automorphic forms is a very powerful, exciting and promising part of modern mathematics. Through a number of deep conjectures, mainly due to Robert Langlands of the Institute for Advanced Study (Langlands program), it tries to unify objects from different parts of mathematics such as number theory, analysis and geometry. Wiles' proof of Fermat's Last Theorem, which is a consequence of relating plane curves defined by equations of degree three with rational coefficients to functions on complex upper half plane, provides an excellent example of this vast program. The investigator's recent work with his collaborators has led to new, striking and surprising correspondences of this sort with many consequences in number theory and geometry. While this has resolved some very long standing and significant problems, many more important questions need to be answered. In this project, the investigator uses methods of analysis, i.e., the study of continuous objects, that he has developed over his career and have been fundamental in the recent progress, to establish new correspondences of this kind between objects of a discrete nature with many applications to different parts of number theory and geometry. The project involves many collaborations and training for graduate students and postdocs.

最近的引人注目的结果确定了GL（2）上牙尖和4度的功能对称能力的存在，作为GL（4）和GL（4）和GL（5）上的自动形式，以及将通用尖端形式从奇特的特殊正交组转移到研究者的一般线性组中，由研究者及其合作者开放，是由自动形式和新的形式进行了自动理论和数字编号。 他们为Ramanujan-塞尔伯格和萨托特猜想提供了令人惊讶的新估计，以及其他数学家获得的数字理论，自动形式和几何形式的大量令人印象深刻的，确定的和新的结果。 研究者探索了这些扩展到GL（2）上更高的形式的扩展，以及对其他古典组的通用形式的转移及其简单相关的相似覆盖物。 然后，从GSP（4）转移到GL（4）的众所周知且仍开放的案例成为了特殊情况。 除了探索兰兰兹（Langlands）的新想法外，关于第三和第四对称力量已经存在的情况，他计划调查其方法可能将其方法扩展到无限的维度群体。 他采用坚实的方法来通过他的方法来确定这些转移所需的根数稳定性，并研究贝塞尔功能以及从他的方法定义的所有根号的全部一般性。 这些根数的子类的稳定性是建立这些转移的最后一个严重问题。 研究人员还在研究许多有关当地交织操作员以及他方法的自动形态L功能的问题。 这项工作的大部分是与合作者合作进行的。自动形式的理论是现代数学的非常强大，令人兴奋和有前途的一部分。 通过许多深层猜想，主要是由于高级研究所的罗伯特·兰兰兹（Robert Langlands）（Langlands计划），它试图从数学的不同部分（例如数字理论，分析和几何学）统一对象。 威尔斯的Fermat的最后一个定理证明，这是由第三学位方程与复杂上半平面上的功能定义的平面曲线的结果，提供了这个庞大的程序的一个很好的例子。 研究者与他的合作者的最新工作导致了这种新的，惊人的和令人惊讶的对应关系，并在数字理论和几何学上产生了许多后果。 尽管这已经解决了一些很长时间的问题，但需要回答许多更重要的问题。 在这个项目中，研究者使用分析方法，即对连续对象的研究，他已经在职业生涯中发展了，并且在最近的进步中一直是基础，以在离散性质的对象之间建立新的对应关系，这些对象在数量理论的不同部分和几何学的不同部分中都有许多应用程序。 该项目涉及许多合作和研究生和博士后的培训。