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Cyclic base change of cuspidal automorphic representations over function fields
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 9 / September 2024
- Published online by Cambridge University Press:
- 11 September 2024, pp. 1959-2004
- Print publication:
- September 2024
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- Article
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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.
