The goal of the proposed 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 of a certain type (namely those whose Langlands parameter at the archimedean place is ``integral'', or ``algebraic'') to Galois representations arising from the etale cohomology of algebraic varieties over number fields (i.e. Galois representations that are ``motivic''). A basic fact in the theory of Galois representations is that such representations are naturally parametrized by certain p-adic analytic spaces (Galois deformation spaces), in which the motivic Galois representations float in a much larger sea of non-motivic representations. Studying this entire sea of Galois representations is an interesting and important problem, which turns out to be of fundamental importance even if one is ultimately interested only in the motivic points of this space, since most successful approaches to establishing reciprocity have involved p-adically interpolating information across the sea of all Galois representations. In the case of two-dimensional representations of the absolute Galois group of Q, the proposer has proved the desired reciprocity in most cases, by establishing a local-global compatibility result relating the p-adically completed cohomology of modular curves to the p-adic local Langlands correspondence of Breuil, Colmez, and Paskunas. The proposed project involves generalizing these results to other contexts, such as to representations of the/absolute Galois group of a totally real field. Since there isno construction of a p-adic local Langlands correspondence in this case, a major part of the project will involve constructing such a correspondence.
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 Langlands conjectured relationship between L-functions and automorphic forms as a ``reciprocity law''.) Langlands developed an array of powerful representation theoretic methods to study his 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''. The proposer aims to develop new results and methods in the p-adic Langlands program, and to use them to establish new results about L-functions, and, in particular, to establish new reciprocity laws.
