Ĝ-local systems on smooth projective curves are potentially automorphic

Abstract

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\Bbb F_q$, and let $G$ be a split semisimple algebraic group over $\Bbb F_q$. Its dual group $\widehat{G}$ is a split reductive group over $\Bbb Z$. Conjecturally, any $l$-adic $\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \widehat{G}(\overline{\Bbb Q}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $\pi_1(X) \to \widehat{G}(\overline{\Bbb Q}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.