## 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

Zbl 1038.11075
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