Acta Mathematica

Volume 223 (2019)

Number 1

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

Pages: 1 – 111



Gebhard Böckle (Interdisciplinary Center, for Scientific Computing, Universität Heidelberg, Germany)

Michael Harris (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Chandrashekhar Khare (Department of Mathematics, University of California Los Angeles)

Jack A. Thorne (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


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