On the Geometric Langlands Conjecture

Let X X be a smooth, complete, geometrically connected curve over a field of characteristic p p . The geometric Langlands conjecture states that to each irreducible rank n n local system E E on X X one can attach a perverse sheaf on the moduli stack of rank n n bundles on X X (irreducible on each co...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2002-04, Vol.15 (2), p.367-417
Main Authors: Frenkel, E., Gaitsgory, D., Vilonen, K.
Format: Article
Language:English
Subjects:
Algebra
Bunds
Cartesianism
Eigenfunctions
Functors
Integers
Mathematical vectors
Morphisms
Whittaker functions
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Summary:Let X X be a smooth, complete, geometrically connected curve over a field of characteristic p p . The geometric Langlands conjecture states that to each irreducible rank n n local system E E on X X one can attach a perverse sheaf on the moduli stack of rank n n bundles on X X (irreducible on each connected component), which is a Hecke eigensheaf with respect to E 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.
ISSN:0894-0347
1088-6834
DOI:10.1090/S0894-0347-01-00388-5