L-functions, Fourier Transforms, and Gamma Factors

任何数学理论的一个重要目标是通过一组对象来理解另一组对象。当两个集合一一对应时，我们可以称之为互反律。互反律的一个深层次的例子是阿廷和朗兰兹的互反律，它是高斯发现的二次互反律的一个广泛推广，对于求解以素数为模的整数上的方程很重要，并且是更一般的朗兰兹纲领的起源。虽然朗兰兹纲领中的一般互反定律仍然远未被正确地表述，即使对于有理数上的对象，人们也可以考虑“朗兰兹功能性原理”，这是阿廷-朗兰兹互反的结果，也是目前朗兰兹纲领的核心猜想。该项目涉及互易性和函性，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，这将导致一个完整的理论L-函数的尖形式的任何约化群和任何不可约表示的L-群。这不仅将通过匡威定理得到相当一般的函性情况，而且还将提供在超越内窥镜方法中所需的合适的泊松求和公式。在互易性方面，将建立一些情况下，Artin因子与Langlands-Shahidi方法通过GL（n）的局部Langlands对应（局部互易性）定义的因子的相等性，遵循先前证明的GL（n）的外部和对称平方因子的情况下使用的技术。涉及p-adic L-函数、覆盖群的局部系数矩阵和GSpin群的L-函数的兰金产品的项目也将被追求。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.