The goal of the proposed project is to make progress on two fundamental problems in the arithmetic of automorphic forms: the construction and study of p-adic analytic families of Hecke eigenforms, and the construction and study of p-adic L-functions; and to do so by employing the methods of group representation theory. Developments in number theory over the last few decades have led to a broadening of the concept of automorphic form, to include notions such as p-adic automorphic forms, which can be interpolated in p-adic analytic families, and the related p-adic L-functions to which they give rise. These objects play an increasingly important role in number theory, but they can be difficult to study, because the powerful representation theoretic methods that plays such a crucial role in the classical theory of automorphic forms do not apply to them. Work of the proposer shows that they can be studied representation theoretically, however, by using methods from the recently introduced locally analytic representation theory of p-adic groups. The proposed project will employ these methods to construct p-adic analytic families of automorphic Hecke eigenforms for arbitrary reductive groups, and to construct and study p-adic L-functions attached to these automorphic forms. An integral part of the project will be the development of the representation-theoretic tools that underlie these constructions.
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. Automorphic forms and L-functions are two kinds of functions that arise in this way, and that play a particularly important role in number theory. One traditional approach to studying these functions is to use representation theoretic methods. These are methods that exploit the many symmetries of automorphic forms and L-functions to analyse their mathematical properties. A more recent approach to their study, that is playing an ever more important role, is to use 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. A large part of their usefulness comes from that fact that they allow one to group automorphic forms and the related L-functions into families called ``p-adic families'', and study the members of the families simultaneously. Until recently, the representation theoretic approach and the p-adic approaches have remained quite distinct. The goal of the proposed project is to unite the two approaches using methods of so-called ``locally analytic representation theory'' (an emerging branch of representation theory). The proposer will develop new tools in this theory, and apply them to construct and study new examples of p-adic families of automorphic forms and L-functions, as well as to improve our understanding of those families that are already known to exist.
