Converse theorems, functoriality, and applications to number theory

TL;DR

This paper reviews recent advances in Converse Theorems and the Langlands-Shahidi method, highlighting their role in establishing new cases of functoriality and automorphic form liftings in number theory.

Contribution

It presents the current state of Converse Theorems, explains their application to automorphic liftings, and discusses new results and their number-theoretic applications.

Findings

New cases of functoriality established

Automorphic liftings constructed using Converse Theorems

Applications to number theory demonstrated

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.