Abstract
Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\widehat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$) should be associated with an everywhere unramified automorphic representation of the group $G$.
We show that for any homomorphism $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.
Citation
Gebhard Böckle. Michael Harris. Chandrashekhar Khare. Jack A. Thorne. "$\hat{G}$-local systems on smooth projective curves are potentially automorphic." Acta Math. 223 (1) 1 - 111, September 2019. https://doi.org/10.4310/ACTA.2019.v223.n1.a1