Whittaker Models and p-Adic Deformation in the Langlands Program

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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.