  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.
