4The Langlands correspondence
IV Topics in Number Theory
4 The Langlands correspondence
Local class field theory says we have an isomorphism
W
ab
F
∼
=
F
×
.
If we want to state this in terms of the full Weil group, we can talk about the
one-dimensional representations of
W
F
, and write local class field theory as a
correspondence
characters of GL
1
(F )
←→
1-dimensional representations
of W
F
The Langlands correspondence aims to understand the representations of
GL
n
(
F
), and it turns out this corresponds to
n
-dimensional representations of
L
F
. That is, if we put enough adjectives in front of these words.