1Class field theory
IV Topics in Number Theory
1.2 Local class field theory
Local class field theory is a (collection of) theorems that describe abelian exten-
sions of a local field. The key takeaway is that finite abelian extensions of
F
correspond to open finite index subgroups of
F
×
, but the theorem says a bit
more than that:
Theorem (Local class field theory).
(i)
Let
F
be a local field. Then there is a continuous homomorphism, the
local Artin map
Art
F
: F
×
→ Γ
ab
F
with dense image characterized by the properties
(a) The following diagram commutes:
F
×
Γ
ab
F
Γ
F
/I
F
Z
ˆ
Z
v
F
Art
F
∼
(b) If F
0
/F is finite, then the following diagram commutes:
(F
0
)
×
Γ
ab
F
0
= Gal(F
0ab
/F
0
)
F
×
Γ
ab
F
= Gal(F
ab
/F )
Art
F
0
N
F
0
/F
restriction
Art
F
(ii) Moreover, the existence theorem says Art
−1
F
induces a bijection
open finite index
subgroups of F
×
←→
open subgroups of Γ
ab
F
Of course, open subgroups of Γ
ab
F
further corresponds to finite abelian
extensions of F .
(iii) Further, Art
F
induces an isomorphism
O
×
F
∼
→ im(I
F
→ Γ
ab
F
)
and this maps (1 +
πO
F
)
×
to the image of
P
F
. Of course, the quotient
O
×
F
/(1 + πO
F
)
×
∼
=
k
×
= µ
∞
(k).
(iv)
Finally, this is functorial, namely if we have an isomorphism
α
:
F
∼
→ F
0
and
extend it to
¯α
:
¯
F
∼
→
¯
F
0
, then this induces isomorphisms between the Galois
groups
α
∗
: Γ
F
∼
→
Γ
F
0
(up to conjugacy), and
α
ab
∗
◦Art
F
=
Art
F
0
◦α
ab
∗
.
On the level of finite Galois extensions
E/F
, we can rephrase the first part
of the theorem as giving a map
Art
E/F
:
F
×
N
E/F
(E
×
)
→ Gal(E/F )
ab
which is now an isomorphism (since a dense subgroup of a discrete group is the
whole thing!).
We can write down these maps explicitly in certain special cases. We will
not justify the following example:
Example. If F = Q
p
, then
F
ab
= Q
p
(µ
∞
) =
[
Q
p
(µ
n
) = Q
ur
p
(µ
p
∞
).
Moreover, if we write x ∈ Q
×
p
as p
n
y with y ∈ Z
×
p
, then
Art
Q
(x)|
Q
ur
p
= Frob
n
p
, Art
Q
(x)|
Q
p
(µ
p
∞
)
= (ζ
p
n
7→ ζ
y mod p
n
p
n
).
If we had the arithmetic Frobenius instead, then we would have a
−y
in the
power there, which is less pleasant.
The cases of the Archimedean local fields are easy to write down and prove
directly!
Example.
If
F
=
C
, then Γ
F
= Γ
ab
F
= 1 is trivial, and the Artin map is similarly
trivial. There are no non-trivial open finite index subgroups of
C
×
, just as there
are no non-trivial open subgroups of the trivial group.
Example.
If
F
=
R
, then
¯
R
=
C
and Γ
F
= Γ
ab
F
=
Z/
2
Z
=
{±
1
}
. The Artin
map is given by the sign map. The unique open finite index subgroup of R
×
is
R
×
>0
, and this corresponds to the finite Galois extension C/R.
As stated in the theorem, the Artin map has dense image, but is not surjective
(in general). To fix this problem, it is convenient to introduce the Weil group.
Definition
(Weil group)
.
Let
F
be a non-Archimedean local field. Then the
Weil group of F is the topological group W
F
defined as follows:
– As a group, it is
W
F
= {γ ∈ Γ
F
| γ|
F
ur
= Frob
n
q
for some n ∈ Z}.
Recall that
Gal
(
F
ur
/F
) =
ˆ
Z
, and we are requiring
γ|
F
ur
to be in
Z
. In
particular, I
F
⊆ W
F
.
–
The topology is defined by the property that
I
F
is an open subgroup with
the profinite topology. Equivalently,
W
F
is a fiber product of topological
groups
W
F
Γ
F
Z
ˆ
Z
where Z has the discrete topology.
Note that W
F
is not profinite. It is totally disconnected but not compact.
This seems like a slightly artificial definition, but this is cooked up precisely
so that
Proposition. Art
F
induces an isomorphism of topological groups
Art
W
F
: F
×
→ W
ab
F
.
This maps O
×
F
isomorphically onto the inertia subgroup of Γ
ab
F
.
In the case of Archimedean local fields, we make the following definitions.
They will seem rather ad hoc, but we will provide some justification later.
–
The Weil group of
C
is defined to be
W
C
=
C
×
, and the Artin map
Art
W
R
is defined to be the identity.
– The Weil group of R is defined the non-abelian group
W
R
= hC
×
, σ | σ
2
= −1 ∈ C
×
, σzσ
−1
= ¯z for all z ∈ C
×
i.
This is a (non)-split extension of C
×
by Γ
R
,
1 → C
×
→ W
R
→ Γ
R
→ 1,
where the last map sends
z 7→
0 and
σ 7→
1. This is in fact the unique
non-split extension of Γ
R
by C
×
where Γ
R
acts on C
×
in a natural way.
The map Art
W
R
is better described by its inverse, which maps
(Art
W
R
)
−1
: W
ab
R
∼
−→ R
×
z 7−→ z¯z
σ 7−→ −1
To understand these definitions, we need the notion of the relative Weil
group.
Definition
(Relative Weil group)
.
Let
F
be a non-Archimedean local field, and
E/F Galois but not necessarily finite. We define
W
E/F
= {γ ∈ Gal(E
ab
/F ) : γ|
F
ur
= Frob
n
q
, n ∈ Z} =
W
F
[W
E
, W
E
]
.
with the quotient topology.
The W
¯
F /F
= W
F
, while W
F/F
= W
ab
F
= F
×
by local class field theory.
Now if
E/F
is a finite extension, then we have an exact sequence of Galois
groups
1 Gal(E
ab
/E) Gal(E
ab
/F ) Gal(E/F ) 1
1 W
ab
E
W
E/F
Gal(E/F ) 1.
By the Artin map,
W
ab
E
=
E
×
. So the relative Weil group is an extension of
Gal(E/F ) by E
×
. In the case of non-Archimedean fields, we have
lim
E
E
×
= {1},
where the field extensions are joined by the norm map. So
¯
F
×
is invisible in
W
F
=
lim W
E/F
. The weirdness above comes from the fact that the separable
closures of R and C are finite extensions.
We are, of course, not going to prove local class field theory in this course.
However, we can say something about the proofs. There are a few ways of
proving it:
–
The cohomological method (see Artin–Tate, Cassels–Fr¨ohlich), which only
treats the first part, namely the existence of
Art
K
. We start off with a
finite Galois extension E/F , and we want to construct an isomorphism
Art
E/F
: F
×
/N
E/F
(E
×
) → Gal(E/F )
ab
.
Writing G = Gal(E/F ), this uses the cohomological interpretation
F
×
/N
E/F
(E
×
) =
ˆ
H
0
(G, E
×
),
where
ˆ
H
is the Tate cohomology of finite groups. On the other hand, we
have
G
ab
= H
1
(G, Z) =
ˆ
H
−2
(G, Z).
The main step is to compute
H
2
(G, E
×
) =
ˆ
H
2
(G, E
×
)
∼
=
1
n
Z/Z ⊆ Q/Z = H
2
(Γ
F
,
¯
F
×
).
where
n
= [
E
:
F
]. The final group
H
2
(Γ
F
,
¯
F
×
) is the Brauer group
Br
(
F
),
and the subgroup is just the kernel of Br(F ) → Br(E).
Once we have done this, we then define
Art
E/F
to be the cup product
with the generator of
ˆ
H
2
(
G, E
×
), and this maps
ˆ
H
−2
(
G, Z
)
→
ˆ
H
0
(
G, E
×
).
The fact that this map is an isomorphism is rather formal.
The advantage of this method is that it generalizes to duality theorems
about
H
∗
(
G, M
) for arbitrary
M
, but this map is not at all explicit, and
is very much tied to abelian extensions.
–
Formal group methods: We know that the maximal abelian extension of
Q
p
is obtained in two steps — we can write
Q
ab
p
= Q
ur
p
(µ
p
∞
) = Q
ur
p
(torsion points in
ˆ
G
m
),
where
ˆ
G
m
is the formal multiplication group, which we can think of as
(1 + m
¯
Q
p
)
×
. This generalizes to any F/Q
p
— we have
F
ab
= F
ur
(torsion points in
ˆ
G
π
),
where
ˆ
G
π
is the “Lubin–Tate formal group”. This is described in Iwasawa’s
book, and also in a paper of Yoshida’s. The original paper by Lubin and
Tate is also very readable.
The advantage of this is that it is very explicit, and when done correctly,
gives both the existence of the Artin map and the existence theorem. This
also has a natural generalization to non-abelian extensions. However, it
does not give duality theorems.
–
Neukrich’s method: Suppose
E/F
is abelian and finite. If
g ∈ Gal
(
E/F
),
we want to construct
Art
−1
E/F
(
g
)
∈ F
×
/N
E/F
(
E
×
). The point is that there
is only one possibility, because
hgi
is a cyclic subgroup of
Gal
(
E/F
), and
corresponds to some cyclic extension
Gal
(
E/F
0
). We have the following
lemma:
Lemma.
There is a finite
K/F
0
such that
K ∩E
=
F
0
, so
Gal
(
KE/K
)
∼
=
Gal(E/F
0
) = hgi. Moreover, KE/K is unramified.
Let
g
0
|
E
=
g
, and suppose
g
0
=
Frob
a
KE/K
. If local class field theory is
true, then we have to have
Art
−1
KE/K
(g
0
) = π
a
K
(mod N
KE/K
(KE
×
)).
Then by our compatibility conditions, this implies
Art
−1
E/F
(g) = N
K/F
(π
a
K
) (mod N
E/F
(E
×
)).
The problem is then to show that this does not depend on the choices,
and then show that it is a homomorphism. These are in fact extremely
complicated. Note that everything so far is just Galois theory. Solving
these two problems is then where all the number theory goes in.
–
When class field theory was first done, we first did global class field theory,
and deduced the local case from that. No one does that anymore nowadays,
since we now have purely local proofs. However, when we try to generalize
to the Langlands programme, what we have so far all start with global
theorems and then proceed to deduce local results.