Functoriality and L-functions

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

The investigator is carrying out a program in the theory of automorphic representations. The program is to find different methods to produce automorphic funtoriality and then relate such methods. The point of finding such connections is to determine automorphic-type formulae( such as periods or theta integrals) for special values of automorphic L functions. Specifically the techniques of the regular trace formula (as well as the relative trace formula) and the use of the converse theorem produce examples of such functorial lifting. The investigator relates these techniques to theta correspondence and the descent method. Such new techniques make it possible to determine the nonvanishing of the new liftings in terms of special values. The motivation of the above work is to have an analytic formalism to study the special values of certain automorphic L functions. It turns out that some of the most famous problems in number theory (such as Fermat' s last theorem and more generally the determination of integral solutions to polynomial identities) have as a goal the concrete analysis of such L functions as above. The information given by the investigator's approach may be useful in the qualitative study of these L functions.