Whittaker Models and p-Adic Deformation in the Langlands Program

素数是整数或整数，不能分解为较小整数的乘积。当被认为是更大和更抽象的数字系统的一部分时，例如那些包含虚数的系统，素数可以有几种方式分解。素数的分裂行为是自然界中迷人而神秘的一部分，数论致力于系统地研究它，特别是通过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，朗兰兹计划揭示了这些多项式的解集和自然发生在无限维空间中的对称集之间的惊人联系。朗兰兹计划在考虑这些对称性在族内变形的方式方面取得了特别丰硕的成果，该计划通过使用惠特克模型的框架来处理这一领域的两个突出问题，使它们更加具体，或者将它们转化为显式计算问题。自守形式的惠特克函数空间称为惠特克模型。在许多情况下，惠特克模型中的对称性捕获了朗兰兹纲领中构造L函数所需的内容。本项目将研究惠特克模型的变形理论，以了解朗兰兹互易性对模素数同余的行为，以及对族中变形的行为。第一个目标是通过控制自守形式之间的局部Whittaker模型的同余来推广Ihara引理。该方法是使用计算机来测试，在许多例子中，可以发生在当地的岩堀-赫克代数的非爱森斯坦积分本征形的行动的同余。第二个目标是制定，并证明在平凡的情况下，一个本地朗兰兹猜想在家庭的还原群以外的一般线性群，特别是准分裂群分裂超过驯服的扩展。该方法是构建一个朗兰兹参数的模空间，研究其几何形状，并使用匡威定理将其连接到积分伯恩斯坦中心。该奖项反映了NSF的法定使命，并已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的支持。

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.