Let $X$ be a smooth, projective, geometrically connected curve over a finite field $Fq$, and let $G$ be a split semisimple algebraic group over $Fq$. Its dual group $G^$ is a split reductive group over $Z$. Conjecturally, any $l$-adic $G^$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi 1(X)\to G^(Q―l)$) should be associated to an everywhere unramified automorphic representation of the group $G$.

We show that for any homomorphism $\pi 1(X)\to G^(Q―l)$ of Zariski dense image, there exists a finite Galois cover $Y\to X$ over which the associated local system becomes automorphic.