L-functions, Fourier Transforms, and Gamma Factors

L 函数、傅立叶变换和伽玛因子

• 批准号: 1801273
• 负责人: Freydoon Shahidi
• 金额: $ 27.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801273&HistoricalAwards=false
• 关键词: functions; Fourier; Transforms; Gamma; Factors

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.

任何数学理论的一个重要目标都是通过一组对象来理解另一组对象。当这两个集合一一对应时，我们可以将这种对应关系称为互易律。互易律的一个深刻的例子是 Artin 和 Langlands 的例子，它是高斯发现的二次互易律的广泛推广，对于求解以素数为模的整数方程非常重要，也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易律还远没有得到正确的表述，即使对于有理数以上的对象，人们也可以考虑“朗兰兹函数性原理”，它是阿廷-朗兰兹互易性的结果，也是目前朗兰兹纲领核心的一种猜想。该项目涉及互易性和函子性，并将建立新的技术和工具来研究它们。更详细地讲，PI 将比较 Braverman-Kazhdan 定义的傅里叶变换和 Ngo 定义的 Hankel 变换，作为经典群的标准 L 函数。从长远来看，这将导致任何还原群及其 L 群的任何不可约表示的尖点形式的 L 函数的完整理论。这不仅会通过逆定理产生相当一般的函数性案例，而且还将提供超越内窥镜函数性方法所需的合适的泊松求和公式。在互易性方面，遵循先前证明的 GL(n) 的外部和对称平方因子情况中使用的技术，将建立一些 Artin 因子与通过 GL(n) 的局部朗兰兹对应（局部互易）由 Langlands-Shahidi 方法定义的因子相等的情况。还将开展涉及 p-adic L 函数、覆盖群体的局部系数矩阵以及 GSpin 群体的 L 函数的 Rankin 产品的项目。该奖项反映了 NSF 的法定使命，并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。