$\hat{G}$-local systems on smooth projective curves are potentially automorphic

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 $\hat{G}$ 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.

Accepted manuscript. To appear in Acta Mathematica. With two appendices by Dennis Gaitsgory