  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 $\hat{G}$ is a split reductive group over
$\mathbb{Z}$. Conjecturally, any $l$-adic $\hat{G}$-local system on $X$
(equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to
\hat{G}(\bar{\mathbb{Q}}_l)$) should be associated to an everywhere unramified
automorphic representation of the group $G$.
  We show that for any homomorphism $\pi_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$
of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which
the associated local system becomes automorphic.
