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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

On the geometric Langlands conjecture


Authors: E. Frenkel, D. Gaitsgory and K. Vilonen
Journal: J. Amer. Math. Soc. 15 (2002), 367-417
MSC (2000): Primary 11R39, 11F70; Secondary 14H60, 22E55
Published electronically: December 31, 2001
MathSciNet review: 1887638
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Abstract: Let $X$ be a smooth, complete, geometrically connected curve over a field of characteristic $p$. The geometric Langlands conjecture states that to each irreducible rank $n$ local system $E$ on $X$ one can attach a perverse sheaf on the moduli stack of rank $n$ bundles on $X$ (irreducible on each connected component), which is a Hecke eigensheaf with respect to $E$. In this paper we derive the geometric Langlands conjecture from a certain vanishing conjecture. Furthermore, using recent results of Lafforgue, we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the ground field is finite.


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Additional Information

E. Frenkel
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

D. Gaitsgory
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

K. Vilonen
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208

DOI: http://dx.doi.org/10.1090/S0894-0347-01-00388-5
PII: S 0894-0347(01)00388-5
Received by editor(s): February 14, 2001
Published electronically: December 31, 2001
Article copyright: © Copyright 2001 by E. Frenkel, D. Gaitsgory, K. Vilonen