Cyclic base change of cuspidal automorphic representations over function fields

Let G be a split semisimple group over a global function field K. Given a cuspidal automorphic representation of G satisfying a technical hypothesis, we prove that for almost all primes, there is a cyclic base change lifting of along any Z/ Z -extension of K. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group G over a local function field F, and almost all primes, any irreducible admissible representation of G (F) admits a base change along any Z/ Z -extension of F. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.