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 arising from the etale cohomology of algebraic varieties over number fields. The description of this reciprocity is in terms of local laws, that is, reciprocity laws that relate the behavior of the automorphic representation at a prime k to the behavior of the Galois representation at that same prime. These local laws are most subtle when k 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). With collaborators, the principal investigator aims to investigate this conjectural p-adic local Langlands correspondence in various ways. One part of the work aims to construct moduli stacks of p-adic Galois representations. This will allow for the introduction of new geometric methods into the study of the Galois-theoretic side of the correspondence. Another part of the project will attempt to construct the p-adic local Langlands correspondence in various new contexts. More precisely, previous work of the principal investigator and collaborators used global methods to construct a candidate for the p-adic local Langlands correspondence in some generality; this project aims to establish that the correspondence so constructed is truly local.
