Current Status
The Langlands conjectures for GL(1, K) follow from (and are essentially equivalent to) class field theory.
Langlands proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible representations.
Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.
Andrew Wiles' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures. Unfortunately, his method cannot be extended to arbitrary number fields.
The Langlands conjecture for GL(2, Q) still remains unproved.
Laurent Lafforgue proved Lafforgue's theorem verifying the Langlands conjectures for the general linear group GL(n, K) for function fields K. This work continued earlier investigations by Vladimir Drinfel'd, who proved the case GL(2, K)
Read more about this topic: Langlands Program
Famous quotes containing the word current:
“A man is a little thing whilst he works by and for himself, but, when he gives voice to the rules of love and justice, is godlike, his word is current in all countries; and all men, though his enemies are made his friends and obey it as their own.”
—Ralph Waldo Emerson (1803–1882)