Whittaker Models and p-Adic Deformation in the Langlands Program

素数是整数，也就是不能分解成较小整数乘积的整数。当被认为是更大和更抽象的数系的一部分时，例如那些包含虚数的数系，素数可能有几种方式。素数的分裂行为是自然界中一个迷人而神秘的部分，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出现在无限维空间中的对称集之间的惊人联系。在朗兰兹计划中，考虑这些对称性在族中的变形方式是特别有成效的，这个项目通过使用Whittaker模型的框架来处理这一领域中的两个突出问题，使它们更具体，或者把它们变成显式的计算问题。惠特克模型中的对称性捕捉到了构造L所需要的东西--在许多情况下，朗兰兹程序中的函数。这个项目将研究惠特克模型的形变理论，以了解朗兰兹互易关于模素数同余的行为，以及关于族中的形变。第一个目标是通过控制自同构形之间的同余来推广Ihara引理。在许多例子中，方法是使用计算机来测试局部Iwahori-Hecke代数在非Eisenstein积分本征形式上的作用中可能出现的同余。第二个目标是在一般情况下，对一般线性群以外的约化群，特别是在驯服扩张上分裂的拟分裂群，在族中阐明并证明一个局部朗兰兹猜想。方法是构建一个由朗兰兹参数组成的模空间，研究它的几何结构，并使用逆定理将其与完整的伯恩斯坦中心联系起来。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.