Whittaker Models and p-Adic Deformation in the Langlands Program

朗兰兹纲领中的 Whittaker 模型和 p-Adic 变形

• 批准号: 2234339
• 负责人: Gilbert Moss
• 金额: $ 9.54万
• 依托单位: University of Maine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2234339&HistoricalAwards=false
• 关键词: Whittaker; Models; Adic; Deformation; Langlands

Primes numbers are integers, or whole numbers, which cannot be factored into a product of smaller integers. When considered as parts of larger and more abstract number systems, for example those containing imaginary numbers, there are several ways primes may factor. The splitting behavior of prime numbers is a fascinating and mysterious part of nature, and number theory is concerned with studying it systematically, especially by connecting it with the arithmetic of polynomials and the geometry of their solution sets. Over the past several decades, the Langlands program has revealed striking connections between these solution sets of polynomials and sets of symmetries that naturally occur in infinite-dimensional spaces. It has been especially fruitful in the Langlands program to consider the way these symmetries deform within families, and this project approaches two outstanding problems in this area by using the framework of Whittaker models to make them more concrete, or to turn them into explicit computational problems.Spaces of Whittaker functions of automorphic forms are called Whittaker models. Symmetries in Whittaker models capture what is needed to construct L-functions in the Langlands program, in many cases. This project will study the deformation theory of Whittaker models in order to understand the behavior of Langlands reciprocity with respect to congruences modulo prime numbers, and with respect to deformation in families. The first goal is to generalize Ihara’s lemma by controlling congruences between automorphic forms in terms of their local Whittaker models. The approach is to use a computer to test, in many examples, the congruences that can occur in the action of the local Iwahori-Hecke algebra on a non-Eisenstein integral eigenform. The second goal is to formulate, and prove in the banal case, a local Langlands conjecture in families for reductive groups other than the general linear group, especially quasi-split groups that split over a tame extension. The approach is to construct a moduli space of Langlands parameters, study its geometry, and use converse theorems to connect it to the integral Bernstein center.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.

素数是整数或整数，不能被分解成更小整数的乘积。当被视为更大更抽象的数字系统的一部分时，例如那些包含虚数的系统，质数可以通过几种方式分解。素数的分裂行为是自然界中一个迷人而神秘的部分，数论关注的是系统地研究它，特别是将它与多项式的算术及其解集的几何联系起来。在过去的几十年里，朗兰兹程序揭示了这些多项式解集与无限维空间中自然出现的对称集之间的惊人联系。在朗兰兹计划中，考虑这些对称在家庭中变形的方式尤其富有成果，这个项目通过使用惠特克模型的框架使它们更具体，或者把它们变成明确的计算问题，来解决这个领域的两个突出问题。自同构形式的Whittaker函数的空间称为Whittaker模型。在许多情况下，惠特克模型中的对称性捕捉到了在朗兰兹程序中构造l函数所需要的东西。本项目将研究惠特克模型的变形理论，以了解朗兰兹互易对同余模素数的行为，以及对家庭变形的行为。第一个目标是通过控制自同构形式在局部惠特克模型中的同余来推广伊哈拉引理。该方法是使用计算机来测试，在许多例子中，局部Iwahori-Hecke代数在非爱森斯坦积分特征形式上可能发生的同余。第二个目标是在平凡的情况下，对除一般线性群以外的约化群，特别是在温和扩展上分裂的拟分裂群，在族中形成并证明一个局部朗兰兹猜想。该方法是构造一个朗兰兹参数的模空间，研究其几何形状，并利用逆定理将其与积分伯恩斯坦中心联系起来。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。