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.

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