Some Problems on Fourier Coefficients of Automorphic Forms and L-functions

这个项目的重点是自同构形式的现代理论和朗兰兹纲领。自同构形式是具有丰富对称性的函数。这些对称性是我们理解宇宙中物体内在结构的指南。在数学中，这些对称性是许多不同理论的共同基础，如几何、数论、数学物理、代数和分析。因此，自同构形式的现代理论，本质上是朗兰兹纲领，为这些领域的进一步研究提供了组织原则。PI的研究目标是建立自同构形式的基本结构。PI将培养研究生和博士后，并在各种场合和会议上进行公开讲座、初级讲座和研究讲座等，向更广泛的社区发表研究报告。PI蒋迪华将继续他对平方可积自同构形式的离散谱、l函数和朗兰泛函猜想的研究。PI研究的基本问题是经典群上自同构形式离散谱的精细结构，自同构l函数的解析和算术性质，以及通过自同构积分变换的平方可积自同构形式的显式Langlands泛函转移。内窥镜理论是朗兰兹在20世纪80年代发现的，并通过b.c.的基础工作得到证实。Ngo和J. Arthur等人通过轨迹公式方法。一方面，PI打算在内窥镜存在的基础上研究精细结构。另一方面，PI打算通过具有自同构核函数的积分变换来构造倒牙自同构形式的显式模，从而使内窥镜转换可以通过积分变换来实现。此外，PI将发展扭曲自同构下降理论，该理论可用于证明全局Gan-Gross-Prasad猜想的大量新情况。建立了张量积函数中心临界值不消失的新高阶情形。同时，PI还计划发展局部理论，将局部场上群的谐波分析的基本问题与局部朗兰兹猜想给出的算术数据联系起来。PI的长期研究目标是理解自同构形式理论中的一般局部-全局自同构原理，它反映了算术和数论中的基本原理之一。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on the modern theory of automorphic forms and the Langlands Program. Automorphic Forms are functions with abundant symmetries. These symmetries are the guidelines to understanding the intrinsic structures of objects in our universe. In Mathematics, these symmetries are common grounds for many different theories such as Geometry, Number Theory, Mathematical Physics, Algebra and Analysis. Hence the modern theory of automorphic forms, essentially the Langlands program, provides the organizing principle for further research in these areas. The research of the PI has a goal of establishing basic structures for automorphic forms. The PI will train graduate students and postodcs, and give lectures on his research to broader community, including public lectures, primary lectures and research talks in various occasions and conferences. The PI, Dihua Jiang, will continue his research on the discrete spectrum of square-integrable automorphic forms, L-functions and the Langlands functoriality conjectures. The basic problems that the PI has been investigating are refined structures of the discrete spectrum of automorphic forms on classical groups, analytic and arithmetic properties of automorphic L-functions, and explicit Langlands functorial transfers for square-integrable automorphic forms via automorphic integral transforms. The theory of endoscopy, the existence of which was discovered by R. Langlands in 1980's and confirmed through the fundamental work of B.-C. Ngo and J. Arthur and others via the trace formula approach. On the one hand, the PI intends to study refined structure based on the existence of endoscopy. On the other hand, the PI intends to construct explicit modules for the cuspidal automorphic forms via integral transform with automorphic kernel functions, so that the endoscopic transfers can be realized via integral transforms, Moreover, the PI will develop the theory of twisted automorphic descents that can be used to prove substantially new cases of the global Gan-Gross-Prasad conjecture, establish new higher rank cases of non-vanishing of the central critical value of tensor product $L$-functions. Meanwhile, the PI also plans to develop the local theory, relating basic problems in harmonic analysis of groups over a local filed to the arithmetic data that are given by the local Langlands conjecture. The long term research goal of the PI is to understand the general local-global-automorphic principles in the theory of automorphic forms, which reflects one of the basic principles in the arithmetic and number 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.