L-functions, Fourier Transforms, and Gamma Factors

任何数学理论中的一个重要目标都是通过另一组对象来理解另一组对象。当这两个集合是一对一对应时，人们可以将这种对应称为互易律。互易定律的一个深层例子是Artin和朗兰兹定律，它是高斯发现的二次互易定律的一个巨大推广，对于求解模为素数的整数上的方程是重要的，这也是更广泛的朗兰兹程序的起源。虽然朗兰兹计划内的一般互易定律仍远未得到适当的制定，即使是对于超过有理数的对象，人们也可以考虑“朗兰兹函数原理”，这是Artin-朗兰兹互易性的结果，也是目前朗兰兹计划的核心猜想。这个项目同时涉及到倒易性和函数性，并将建立新的技术和工具来研究它们。更详细地，PI将比较由Braverman-Kazhdan定义的傅里叶变换和由Ngo定义的Hankel变换，以寻找经典群的标准L函数。从长远来看，这将导致任何可约群上尖型的L函数及其L群的任何不可约表示的完整理论。这不仅将通过逆定理得到相当一般的函数性情况，而且还将提供超越内窥镜方法中所需的适当的泊松求和公式。在互易性方面，按照前面证明的GL(N)的外部因子和对称平方因子的技巧，将建立一些情况，其中Artin因子与通过GL(N)的局部朗兰兹对应(局部互易)由朗兰兹-沙希迪方法定义的因子相等。此外，还将继续开展涉及p元L函数、覆盖群的局部系数矩阵以及L的Rankin积的GSpin群函数的项目。该奖项反映了美国国家科学基金会的法定使命，并已通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

An important goal in any mathematical theory is to understand one set of objects by means of another one. When the two sets are in a one-to-one correspondence, then one may call the correspondence a reciprocity law. One deep example of a reciprocity law is that of Artin and Langlands, which is a vast generalization of Quadratic Reciprocity Law discovered by Gauss that is important for solving equations over the integers modulo a prime, and which is the genesis of the more general Langlands Program. While a general reciprocity law within the Langlands program is still far from properly formulated, even for objects over rational numbers, one can consider the "Langlands Functoriality Principle," which is a consequence of Artin-Langlands reciprocity and is a conjecture that is currently at the core of Langlands program. This project deals with both reciprocity and functoriality and will establish new techniques and tools to study them.In more detail, the PI will compare the Fourier transform defined by Braverman-Kazhdan and the Hankel transform defined by Ngo, for the standard L-functions for classical groups. In the long run this will lead to a full theory of L-functions for cusp forms on any reductive group and any irreducible representation of its L-group. This will not only lead to fairly general cases of functoriality through converse theorems, but will also provide suitable Poisson summation formulas that are needed in the Beyond Endoscopy approach to functoriality. In terms of reciprocity, a number of cases where the equality of Artin factors with the factors defined by the Langlands-Shahidi method through the local Langlands correspondence (local reciprocity) for GL(n) will be established, following the techniques used in the cases of exterior and symmetric square factors for GL(n) proved earlier. Projects involving p-adic L-functions, local coefficients matrices for covering groups, and Rankin products of L-functions for GSpin groups will also be pursued.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.