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 conjecture is motivated from the following special 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.
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
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219; Edward Frenkel, Commentary on “An elementary introduction to the Langlands Program” by Steven Gelbart, Bull. Amer. Math. Soc. 48 (2011), 513-515, abstract, pdf
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.