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Cyclic base change of cuspidal automorphic representations over function fields

https://doi.org/10.1112/S0010437X24007243

Böckle, Gebhard; Feng, Tony; Harris, Michael; Khare, Chandrashekhar B; Thorne, Jack A (September 2024, Compositio Mathematica)

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.

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