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On the geometric Langlands conjecture

E. Frenkel·D. Gaitsgory·K. Vilonen
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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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E. Frenkel, D. Gaitsgory, K. Vilonen. 2001-10-17. On the geometric Langlands conjecture. https://arxiv.org/abs/math/0012255

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