Automorphic L-Functions and Langlands Functoriality

最近，研究者和他的合作者证明了GL(2)上的尖形的3次和4次函数式对称幂作为GL(4)和GL(5)上的自同构形式的存在性，以及一般尖形从奇特殊正交群到一般线性群的转换，开辟了自同构形式和数论的一个新的前沿。它们导致了对Ramanujan-Selberg和Sato-Tate猜想的惊人的新估计，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、确凿的和新的结果。研究了GL(2)上形式的高次方的推广，以及一般形式在其他经典群和它们的单连通相似覆盖上的转移。从GSP(4)转移到GL(4)的众所周知且仍未解决的情况则成为这方面的一个特例。除了探索朗兰兹关于超越内窥镜检查的新想法外，他还计划研究将他的方法扩展到无限维群的任何可能。他有一个可靠的方法来建立这些转移所必需的根数的稳定性，并用他的方法研究了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转移的最后一个严重问题。这位研究者还研究了一些与局部交织算子的极点有关的问题，以及由他的方法得到的自同构L函数的极点问题。这项工作的大部分是与合作者合作完成的。自同构形式理论是现代数学中非常强大、令人兴奋和有前途的部分。通过许多深刻的猜想，主要是由于高级研究所的罗伯特·朗兰兹(朗兰兹计划)，它试图统一数学中不同部分的对象，如数论、分析和几何。Wiles对费马大定理的证明，是将具有有理系数的三次方程所定义的平面曲线与复上半平面上的函数联系起来的结果，为这一庞大的程序提供了一个极好的例子。这位研究人员最近与他的合作者的工作导致了这种新的、引人注目的和令人惊讶的通信，在数论和几何学中产生了许多结果。虽然这已经解决了一些非常长期和重大的问题，但还有许多更重要的问题需要回答。在这个项目中，研究人员使用分析方法，即对连续对象的研究，这是他在职业生涯中发展起来的，在最近的进展中一直是基础的，以建立离散性质的对象之间的这种新的对应关系，并在数论和几何的不同部分有许多应用。该项目涉及对研究生和博士后的许多合作和培训。

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.