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On the geometric Langlands conjecture. (English) Zbl 1071.11039

Summary: 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 \(E\). We derive the geometric Langlands conjecture from certain vanishing conjecture. Furthermore, using recent results of Lafforgue [cf. L. Lafforgue, Invent. Math. 147, 1–241 (2002; Zbl 1038.11075)], we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the ground field is finite.

MSC:

11G45 Geometric class field theory
11R39 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14H60 Vector bundles on curves and their moduli
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1038.11075
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References:

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