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The Langlands correspondence for function fields (after Laurent Lafforgue). (La correspondance de Langlands sur les corps de fonctions (d’après Laurent Lafforgue).) (French) Zbl 1016.11052
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 207-265, Exp. No. 873 (2002).
This paper describes the strategy adopted by L. Lafforgue [Invent. Math. 147, 1–241 (2002; Zbl 1038.11075)] in order to prove the global Langlands conjecture for \(\text{GL}_r\) over function fields in positive characteristic. The main tool, invented by V. G. Drinfeld [Funct. Anal. Appl. 11, No. 3, 223–225 (1977); translation from Funkts. Anal. Prilozh. 11, No. 3, 74–75 (1977; Zbl 0392.12007)] to prove this conjecture (for \(\text{GL}_2\)), is the moduli stack of shtukas. This stack has very complicated structure (especially for \(r>2\)), and it was the main technical difficulty in the program initiated by Drinfeld and completed by Lafforgue.
Let \(X\) be a smooth projective curve over a finite field \(\mathbb{F}_q\), \(F\) its function field, and \(\mathbb{A}_F\) the corresponding ring of adèles. On the one hand, denote \(\mathcal{A}_r\) the set of isomorphism classes of irreducible cuspidal automorphic representations of \(\text{GL}_r(\mathbb{A}_F)\). On the other hand, denote \(\mathcal{G}_r\) the set of isomorphism classes of irreducible \(\ell\)-adic representations of rank \(r\) of \(\text{Gal}(F^{\text{sep}}/F)\) whose determinants are of finite order. The global Langlands conjecture states that there exists a natural bijection \(\sigma:\mathcal{A}_r\rightarrow\mathcal{G}_r\) preserving (almost all) local \(L\)-factors.
In the beginning of the paper, the author describes the moduli stack of Drinfeld’s shtukas, its compactification and related technical questions. The Langlands conjecture follows from the theorem of section 6.2 (a partial description of \(\ell\)-adic cohomology of the compactified moduli stack of shtukas) and an inductive procedure attributed to Deligne. The rest of the paper contains a sketch of Lafforgue’s proof of this auxiliary theorem.
For the entire collection see [Zbl 0981.00011].

11R39 Langlands-Weil conjectures, nonabelian class field theory
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
11F52 Modular forms associated to Drinfel’d modules
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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