Converse theorems, functoriality, and applications to number theory

Abstract

There has been a recent coming together of the Converse Theorem for $\gln$ and the Langlands-Shahidi method of controlling the analytic properties of automorphic $L$-functions which has allowed us to establish a number of new cases of functoriality, or the lifting of automorphic forms. In this article we would like to present the current state of the Converse Theorem and outline the method one uses to apply the Converse Theorem to obtain liftings. We will then turn to an exposition of the new liftings and some of their applications.