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Israel Journal of Mathematics

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

On De Jong’s conjecture

  • D. Gaitsgory
Article

Abstract

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.

Keywords

Local System Canonical Isomorphism Abelian Category Symmetric Power Koszul Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque 100 (1982).Google Scholar
  2. [2]
    A. Beilinson and V. Drinfeld, Chiral algebras, AMS Colloquium Publications 51, American Mathematical Society, Providence, RI, 2004.MATHGoogle Scholar
  3. [3]
    W. Borho and R. MacPherson, Small resolutions of nilpotent varieties, Astérisque 101–102 (1982), 23–74.Google Scholar
  4. [4]
    A J. de Jong, A conjecture on arithmetic fundamental groups, Israel Journal of Mathematics 121 (2001), 61–64.MATHMathSciNetGoogle Scholar
  5. [5]
    E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, Journal of the American Mathematical Society 15 (2002), 367–417.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    D. Gaitsgory, Automorphic sheaves and Eisenstein series, PhD Thesis, Tel Aviv University, 1997.Google Scholar
  7. [7]
    D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Inventiones Mathematicae 144 (2001), 253–280.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, math.AG/0204081.Google Scholar
  9. [9]
    L. Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), Astérisque 82–83 (1981), 161–172.MathSciNetGoogle Scholar
  10. [10]
    G. Laumon, Faisceaux automorphes pour GL n : la première construction de Drinfeld, alg-geom/9511004.Google Scholar
  11. [11]
    I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, math.RT/0401222.Google Scholar

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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