Automorphic L-Functions and Langlands Functoriality

最近的惊人结果确定了 GL(2) 上的尖点形式的 3 次和 4 次函数对称幂作为 GL(4) 和 GL(5) 上的自同构形式的存在，以及研究者和他的合作者将通用尖点形式从奇数特殊正交群转移到一般线性群，开辟了自同构形式和数论的新前沿。 他们对拉马努金-塞尔伯格和佐藤-泰特猜想做出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、明确的和新的结果。 研究人员探索了这些在 GL(2) 上形式的更高幂的扩展，以及在其他经典群上的通用形式的迁移及其简单连接的相似覆盖。 从 GSp(4) 到 GL(4) 转移的众所周知且仍然悬而未决的案例就成为了这种情况的一个特例。 除了探索朗兰兹关于超越内窥镜检查的新想法（这种情况已经存在于三次和四次对称幂中）之外，他还计划研究将他的方法扩展到无限维群的任何可能的扩展。 他有一个可靠的方法来通过他的方法建立这些传递所需的根数的稳定性，并研究贝塞尔函数和从他的方法定义的所有根数的全面通用性。 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。 研究人员还正在研究一些与局部交织算子的极点以及来自他的方法的自同构 L 函数的极点有关的问题。 这项工作的大部分都是与合作者合作完成的。自守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。 通过一些深刻的猜想，主要来自高等研究院的罗伯特·朗兰兹（朗兰兹计划），它试图统一来自数学不同部分（如数论、分析和几何）的对象。 怀尔斯对费马大定理的证明，是将由三阶方程和有理系数定义的平面曲线与复数上半平面上的函数联系起来的结果，为这个庞大的计划提供了一个很好的例子。 研究人员最近与他的合作者的合作产生了这种新的、引人注目的和令人惊讶的对应关系，在数论和几何中产生了许多后果。 虽然这解决了一些长期存在的重大问题，但许多更重要的问题需要得到解答。 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，这种方法是他在职业生涯中发展起来的，并且在最近的进展中发挥了基础作用，在离散性质的对象之间建立这种新的对应关系，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.