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G^-local systems on smooth projective curves are potentially automorphic

Gebhard Böckle Universität Heidelberg Michael Harris Columbia University Chandrashekhar Khare University of California Los Angeles Jack A. Thorne University of Cambridge

Number Theory mathscidoc:1911.43042

Acta Mathematica, 223, (1), 1-111, 2019
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 Gˆ is a split reductive group over Z. Conjecturally, any l-adic Gˆ-local system on X (equivalently, any conjugacy class of continuous homomorphisms π1(X)→Gˆ(Q¯¯¯¯l)) should be associated with an everywhere unramified automorphic representation of the group G. We show that for any homomorphism π1(X)→Gˆ(Q¯¯¯¯l) of Zariski dense image, there exists a finite Galois cover Y→X over which the associated local system becomes automorphic.
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@inproceedings{gebhard2019g^-local,
  title={G^-local systems on smooth projective curves are potentially automorphic},
  author={Gebhard Böckle, Michael Harris, Chandrashekhar Khare, and Jack A. Thorne},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128153619390541553},
  booktitle={Acta Mathematica},
  volume={223},
  number={1},
  pages={1-111},
  year={2019},
}
Gebhard Böckle, Michael Harris, Chandrashekhar Khare, and Jack A. Thorne. G^-local systems on smooth projective curves are potentially automorphic. 2019. Vol. 223. In Acta Mathematica. pp.1-111. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128153619390541553.
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