Automorphic L-Functions and Langlands Functoriality

研究者和合作者最近的惊人成果证明GL(2)上的尖形作为GL(4)和GL(5)上的自同构形式存在3次和4次的泛函对称幂，以及一般尖形从奇特殊正交群向一般线性群的迁移，开辟了自同构形式和数论的新前沿。他们对Ramanujan- Selberg和Sato- Tate猜想产生了令人惊讶的新估计，以及其他数学家在数论、自同构形式和几何方面获得的大量令人印象深刻的、确定的新结果。研究者探索了这些形式在GL(2)上的高幂形式的扩展，以及一般形式在其他经典群上的转移及其单连通的相似覆盖。从GSp(4)到GL(4)的转移，这一众所周知且仍未解决的情况，成为了这一问题的一个特例。除了探索Langlands关于超越内窥镜的转移的新想法（这种情况已经存在于第三和第四对称幂中），他还计划研究将他的方法扩展到无限维群的任何可能。他有一个坚实的方法来建立这些转移所必需的根数的稳定性，通过他的方法，并研究贝塞尔函数和由他的方法定义的所有根数的完全一般性。这些根数的子类的稳定性是建立这些传输的最后一个严重问题。研究者还研究了一些关于局部缠结算子的极点的问题，以及来自他的方法的自同构l函数的极点问题。这项工作的大部分是在合作者的合作下进行的。自同构形式理论是现代数学中一个非常强大、令人兴奋和有前途的部分。通过一些深刻的猜想，主要是由高级研究所（Langlands program）的罗伯特·朗兰兹（Robert Langlands）提出的，它试图统一数学中不同部分的对象，如数论、分析和几何。怀尔斯对费马大定理的证明，是将带有有理系数的三次方程定义的平面曲线与复上半平面上的函数联系起来的结果，为这个庞大的程序提供了一个很好的例子。这位研究者最近与他的合作者的工作已经导致了这种新的、惊人的、令人惊讶的对应关系，在数论和几何领域产生了许多影响。虽然这解决了一些长期存在的重大问题，但还有许多更重要的问题需要回答。在这个项目中，研究者使用分析方法，即对连续对象的研究，这是他在职业生涯中发展起来的，在最近的进展中是基础性的，在离散性质的对象之间建立了这种新的对应关系，并在数论和几何的不同部分中有许多应用。该项目涉及许多合作以及对研究生和博士后的培训。

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.