Lafforgue's theorem

The Langlands conjectures for GL₁(K) follow from (and are essentially equivalent to) class field theory. More precisely the Artin map gives a map from the idele class group to the abelianization of the Weil group.

The representations of GL_n(F) appearing in the Langlands correspondence are automorphic representations.

Here F is a global field of some positive characteristic p, and ℓ is some prime not equal to p.

Lafforgue's theorem states that there is a bijection σ between:

• Equivalence classes of cuspidal representations π of GL_n(F), and
• Equivalence classes of irreducible ℓ-adic representations σ(π) of dimension n of the absolute Galois group of F

that preserves the L-function at every place of F.

The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π. The idea of doing this is to look in the ℓ-adic cohomology of the moduli stack of shtukas of rank n that have compatible level N structures for all N. The cohomology contains subquotients of the form

π⊗σ(π)⊗σ(π)^∨

which can be used to construct σ(π) from π. A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.

Lafforgue's theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GL_n(F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1.

Lafforgue's theorem implies the conjecture of Deligne (1980, 1.2.10) that an irreducible finite-dimensional l-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.

• Local Langlands conjectures

• Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR 0546608
• Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS, 52 (52): 137–252, doi:10.1007/BF02684780, ISSN 1618-1913, MR 0601520, S2CID 189769469
• Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions, Generalized functions, vol. 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673
• Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105
• Lafforgue, Laurent (2002), "Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands." (Drinfeld shtukas, Arthur-Selberg trace formula and Langlands correspondence) Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 383–400, Higher Ed. Press, Beijing, 2002.
• Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654
• Langlands, Robert (1967), Letter to Prof. Weil
• Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, vol. 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614
• Gérard Laumon (2002), "The work of Laurent Lafforgue", Proceedings of the ICM, Beijing 2002, vol. 1, 91–97,
• G. Laumon (2000), "La correspondance de Langlands sur les corps de fonctions (d'après Laurent Lafforgue)" (The Langlands correspondence over function fields (according to Laurent Lafforgue)), Séminaire Bourbaki, 52e année, 1999–2000, no. 873.

• Lafforgue's publications
• The work of Robert Langlands
• Rapoport, The work of Laurent Lafforgue (PDF)