L-functions, Fourier Transforms, and Gamma Factors

任何数学理论的一个重要目标是通过另一组对象来理解一组对象。当两个集合是一对一的对应关系时，我们可以称这种对应关系为互易律。一个关于互易律的深刻例子是阿廷和朗兰兹的，它是对高斯发现的二次互易律的广泛推广，这对于求解整数模a素数上的方程很重要，它是更普遍的朗兰兹纲领的起源。虽然朗兰兹纲领中的一般互易律还远远没有得到适当的表述，即使对于有理数上的物体，人们也可以考虑“朗兰兹泛函原理”，这是阿廷-朗兰兹互易的结果，也是朗兰兹纲领目前核心的一个猜想。该项目涉及互惠性和功能性，并将建立新的技术和工具来研究它们。更详细地说，PI将比较Braverman-Kazhdan定义的傅里叶变换和Ngo定义的Hankel变换，用于经典群的标准l函数。从长远来看，这将导致在任何可约群及其l群的任何不可约表示上的尖形的l函数的完整理论。这不仅会通过逆定理得出功能性的一般情况，而且还会提供超出内窥镜方法所需的合适的泊松求和公式。在互反方面，在GL(n)的外部因子和对称平方因子的情况下所使用的技术之后，通过局部朗兰兹对应（局部互反），将建立一些情况，其中Artin因子与Langlands- shahidi方法定义的因子相等。涉及p进l函数、覆盖群的局部系数矩阵和GSpin群的l函数的Rankin积的项目也将继续进行。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。

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.