Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole number solutions of some equation of interest. The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. (Number theorists refer to the conjectured relationship between L-functions and automorphic forms as a "reciprocity law.") Langlands developed an array of powerful representation theoretic methods to study the conjectures. These are methods that exploit the many symmetries of automorphic forms and L-functions to analyze their mathematical properties; these methods have been incorporated into a body of mathematics known as "the Langlands program." A more recent approach to the study of automorphic forms and L-functions is the use of p-adic methods. These are methods that involve using divisibility properties with respect to some fixed prime number p to study the Taylor series coefficients of the automorphic forms and L-functions. Recently, the representation theoretic methods and p-adic methods have begun to be unified into a so-called "p-adic Langlands program." This project aims to develop new results and methods in the p-adic Langlands program, and to use them to establish new reciprocity laws.
The goal of the research project is to investigate the p-adic aspects of the Langlands program. At the heart of the Langlands program is a conjectured reciprocity law relating automorphic representations to p-adic Galois representations. The precise description of the manner in which automorphic forms and Galois representations are supposed to correspond involves local reciprocity laws, that is, reciprocity laws that relate the behavior of the automorphic representation at a prime q to the behavior of the Galois representation at that same prime. These local laws are most subtle when q is taken to be the same prime p that governs the coefficients of the Galois representation; indeed, in this case such a local reciprocity law would constitute a p-adic local Langlands correspondence, and its existence remains conjectural other than in the abelian case, and the case of GL_2(Q_p). Together with various collaborators, the principal investigator aims to investigate this hoped-for but mysterious p-adic local reciprocity law in various ways. The PI's study of moduli stacks of p-adic Galois representations will provide new geometric insight into p-adic local Langlands and related problems such as the Breuil--Mezard conjecture. The PI's proposed trace formula for p-adically completed cohomology should yield new insight into local-global compatibility in the p-adic context. The PI's research on growth of cohomology will increase the range of applicability of stable trace formula, one of the most powerful tools available for studying reciprocity laws associated to automorphic forms.
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.
