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Cyclic base change of cuspidal automorphic representations over function fields
Published online by Cambridge University Press: 11 September 2024
Abstract
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
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