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# Acta Mathematica

## Volume 223 (2019)

### Number 1

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

Pages: 1 – 111

DOI: https://dx.doi.org/10.4310/ACTA.2019.v223.n1.a1

#### Authors

#### 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.

Received 2 October 2016

Received revised 4 January 2019

Accepted 5 May 2019

Published 30 September 2019