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September 2019 $\hat{G}$-local systems on smooth projective curves are potentially automorphic
Gebhard Böckle, Michael Harris, Chandrashekhar Khare, Jack A. Thorne
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Acta Math. 223(1): 1-111 (September 2019). DOI: 10.4310/ACTA.2019.v223.n1.a1


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.


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Gebhard Böckle. Michael Harris. Chandrashekhar Khare. Jack A. Thorne. "$\hat{G}$-local systems on smooth projective curves are potentially automorphic." Acta Math. 223 (1) 1 - 111, September 2019.


Received: 2 October 2016; Revised: 4 January 2019; Published: September 2019
First available in Project Euclid: 16 April 2020

zbMATH: 07128508
MathSciNet: MR4018263
Digital Object Identifier: 10.4310/ACTA.2019.v223.n1.a1

Rights: Copyright © 2019 Institut Mittag-Leffler


Vol.223 • No. 1 • September 2019
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