The goal of the proposed project is to investigate some of the p-adic aspects of the Langlands program. The Langlands program studies the interrelations between several subjects, including the representation theory of reductive groups (both over local fields and over the adeles of a global field), automorphic forms, global Galois representations, and L-functions. Each of these subjects has a p-adic aspect: one may consider p-adic or adelic groups acting on topological p-adic vector spaces, p-adic automorphic forms (or their close cousins, cohomology classes with p-adic coefficients on arithmetic quotients of symmetric spaces), p-adic representations of Galois groups and their deformation theory, and p-adic L-functions. The proposed investigation will encompass all these. Some of the particular results expected to follow from this investigation are: new results on the modularity of two-dimensional p-adic representations of the absolute Galois group of the rational numbers, and the construction of several variable p-adic L-functions attached to p-adic families of modular forms.
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-adicmethods 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, to use them to establish new results about L-functions, and, in particular, to establish new reciprocity laws.
