In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups.
The Langlands conjectures were introduced by Langlands (1967, 1970) and describe a correspondence between representations of the Weil group of an algebraic function field and representations of algebraic groups over the function field, generalizing class field theory of function fields from abelian Galois groups to non-abelian Galois groups.
Langlands conjectures for GL1
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.
Automorphic representations of GLn(F)
The representations of GLn(F) appearing in the Langlands correspondence are automorphic representations.
Lafforgue's theorem for GLn(F)
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 GLn(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
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 GLn(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.
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- Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS (52): 137–252, ISSN 1618-1913, MR 0601520
- Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) , Representation theory and automorphic functions, Generalized functions, 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673
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- 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.
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- 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, 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.