Locally analytic representation theory and p-adic interpolation

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The goal of the proposed project is to make progress on twofundamental problems in the arithmetic of automorphic forms:the construction and study of p-adic analytic families ofHecke 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 ledto a broadening of the concept of automorphic form, to include notionssuch as p-adic automorphic forms, which can be interpolated inp-adic analytic families, and the related p-adic L-functions to whichthey give rise. These objects play an increasingly importantrole in number theory, but they can be difficult to study, becausethe powerful representation theoretic methods that plays such a crucialrole in the classical theory of automorphic forms do not apply to them.Work of the proposer shows that they can be studied representationtheoretically, however, by using methods from the recently introducedlocally analytic representation theory of p-adic groups. The proposedproject will employ these methods to construct p-adic analytic familiesof automorphic Hecke eigenforms for arbitrary reductive groups, and toconstruct and study p-adic L-functions attached to these automorphicforms. An integral part of the project will be the development ofthe representation-theoretic tools that underlie these constructions.Number theory is the branch of mathematics that studies phenomenarelated to properties of whole numbers. A typical number theoreticquestion is to determine the number of whole number solutions of someequation of interest. The answers to such questions can often beencoded in certain mathematical functions. Automorphic forms andL-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 userepresentation theoretic methods. These are methods that exploitthe many symmetries of automorphic forms and L-functions to analysetheir 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 respectto some fixed prime number p to study the Taylor series coefficientsof the automorphic forms and L-functions. A large part of theirusefulness comes from that fact that they allow one to groupautomorphic forms and the related L-functions into familiescalled ``p-adic families'', and study the members of the familiessimultaneously. Until recently, the representation theoretic approachand the p-adic approaches have remained quite distinct. The goal of theproposed project is to unite the two approaches using methods of so-called``locally analytic representation theory'' (an emerging branch ofrepresentation theory). The proposer will develop new tools in this theory,and apply them to construct and study new examples of p-adic families ofautomorphic forms and L-functions, as well as to improve our understandingof those families that are already known to exist.