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 $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$ , there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$ -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 $\ell$ , any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$ -extension of $F$ . Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.