abstract duality: opposite category,
certain representations – called automorphic representations – of the -dimensional general linear group with coefficients in the ring of adeles of , arising within the representations given by functions on the double coset space (where is the ring of integers of all formal completions of ).
This is motivated from the abelian case (), which is fully understood: For then an -dimensional representation of the Galois group factors through and hence through an abelian group. Therefore, by adjunction, it is equivalently a representation of the abelianization of the Galois group. The Kronecker-Weber theorem says that for , then the abelianized Galois group is the idele class group , and hence 1-dimensional representations of the Galois group are equivalently representations of this. Moreover, one finds that for any prime number , then those representations which are “unramified at ” are invariant under the subring of p-adic integers, hence are representations of the double quotient group . More generally, the Artin reciprocity law says that for number fields there is an isomorphism between and the abelianized Galois group.
Conjecture 1 To each such automorphic representation is associated an L-function – the automorphic L-function – and in generalization of Artin reciprocity the conjecture of Langlands is that the Artin L-function associated with the given Galois representation is equal to this: (Gelbhart 84, conjecture 1 (page 27 (204))).
More generally, analogous statements are supposed to hold for general reductive algebraic groups other than . Here now a L-function is assigned to data which in addtion to the Galois representation consists of a linear representation of the Langlands dual group of .
First of all:
Conjecture 2 This more general L-function is conjectured to indeed behave like a decent L-function in that it has meromorphic analytic continuation to the complex plane and satisfies the “functional equation”-invariance under sending its parameter to (Gelbhart 84, conjecture 2’ (page 29 (205))).
Conjecture 3 This construction is supposed to behave well with respect to an analytic homomorphism in that when changing the representation of by precomposition with this homomorphism one may find an accompanying change of Galois representation/automorphic representation from to such that the associated L-function remains invariant under these joint changes. This statement is what Robert Langlands calls functoriality (Gelbhart 84, conjecture 3 (page 31 (207)))
In fact this last conjecture implies the previous two (Gelbhart 84, (page 32 (208))).
Various versions and refinements of this conjecture have since been considered, for some perspective see (Taylor 02, Langlands 14, Harris 14). On the one hand the “localization” of the program to local fields leads to the conjecture of local Langlands correspondences (Gelbhart 84, (page 34 (210))). On the other hand, the interpretation of the above story dually in arithmetic geometry in view of the function field analogy motivates the conjectural geometric Langlands correspondence, based on the following analogy:
|arithmetic Langlands correspondence||geometric Langlands correspondence|
|ring of integers of global field||structure sheaf on complex curve|
|Galois group||fundamental group of|
|Galois representation||flat connection/local system on|
|idele class group mod integral adeles||moduli stack of line bundles on|
|nonabelian “||moduli stack of vector bundles on|
|automorphic representation||Hitchin connection D-module on bundle of conformal blocks over the moduli stack|
From this arithmetic geometry point of view the Langlands conjecture seems to speak of a correspondence that sends Dirac distributions on the moduli space of flat connections over an algebraic curve to certain “automorphic” functions on the moduli stack of bundles on the same curve. This suggests that the Langlands correspondence should be understood as a nonabelian version of a Fourier-Mukai-type integral transform. This version of the conjecture is known as the geometric Langlands correspondence. See there for more details.
The original conjecture is due to
Surveys of the state of the program include
Richard Taylor, Galois Representations, Proceedings of the ICM 2002 (long version pdf).
Michael Harris, Automorphic Galois representations and Shimura varieties, Proceedings of the ICM 2014 (pdf).
Introductions and expository surveys include
Mark Goresky, Langlands’ conjectures for physicists (pdf)
Peter Toth, Geometric abelian class field theory, 2011 (web)
More resources are at
The Work of Robert Langlands at the Institute of Advanced Study.