Israel Journal of Mathematics

, Volume 157, Issue 1, pp 155–191 | Cite as

On De Jong’s conjecture

  • D. Gaitsgory


Let X be a smooth projective curve over a finite field F q . Let ρ be a continuous representation π(X) → GL n (F), where F = F l ((t)) with F l being another finite field of order prime to q.

Assume that \( \rho \left| {_{\pi (\bar X)} } \right. \) is irreducible. De Jong’s conjecture says that in this case \( \rho (\pi (\bar X)) \) is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture.

In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.


Local System Canonical Isomorphism Abelian Category Symmetric Power Koszul Complex 
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Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • D. Gaitsgory
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

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