1Class field theory
IV Topics in Number Theory
1.1 Preliminaries
Class field theory is the study of abelian extensions of local or global fields.
Before we can do class field theory, we must first know Galois theory.
Notation.
Let
K
be a field. We will write
¯
K
for a separable closure of
K
, and
Γ
K
= Gal(
¯
K/K). We have
Γ
K
= lim
L/K finite separable
Gal(L/K),
which is a profinite group. The associated topology is the Krull topology.
Galois theory tells us
Theorem (Galois theory). There are bijections
closed subgroups of
Γ
K
←→
subfields
K ⊆ L ⊆
¯
K
open subgroups of
Γ
K
←→
finite subfields
K ⊆ L ⊆
¯
K
Notation.
We write
K
ab
for the maximal abelian subextension of
¯
K
, and then
Gal(K
ab
/K) = Γ
ab
K
=
Γ
K
[Γ
K
, Γ
K
]
.
It is crucial to note that while
¯
K
is unique, it is only unique up to non-
canonical isomorphism. Indeed, it has many automorphisms, given by elements
of
Gal
(
¯
K/K
). Thus, Γ
K
is well-defined up to conjugation only. On the other
hand, the abelianization Γ
ab
K
is well-defined. This will be important in later
naturality statements.
Definition
(Non-Archimedean local field)
.
A non-Archimedean local field is a
finite extension of Q
p
or F
p
((t)).
We can also define Archimedean local fields, but they are slightly less inter-
esting.
Definition
(Archimedean local field)
.
An Archimedean local field is a field that
is R or C.
If
F
is a non-Archimedean local field, then it has a canonical normalized
valuation
v = v
F
: F
×
Z.
Definition
(Valuation ring)
.
The valuation ring of a non-Archimedean local
field F is
O = O
F
= {x ∈ F : v(x) ≥ 0}.
Any element
π
=
π
f
∈ O
F
with
v
(
π
) = 1 is called a uniformizer. This generates
the maximal ideal
m = m
F
= {x ∈ O
F
: v(x) ≥ 1}.
Definition
(Residue field)
.
The residue field of a non-Archimedean local field
F is
k = k
F
= O
F
/m
F
.
This is a finite field of order q = p
r
.
A particularly well-understood subfield of
F
ab
is the maximal unramified
extension F
ur
. We have
Gal(F
ur
/F ) = Gal(
¯
k/k) =
ˆ
Z = lim
n≥1
Z/nZ.
and this is completely determined by the behaviour of the residue field. The rest
of Γ
F
is called the inertia group.
Definition (Inertia group). The inertia group I
F
is defined to be
I
F
= Gal(
¯
F /F
ur
) ⊆ Γ
F
.
We also define
Definition
(Wild inertia group)
.
The wild inertia group
P
F
is the maximal
pro-p-subgroup of I
F
.
Returning to the maximal unramified extension, note that saying
Gal
(
¯
k/k
)
∼
=
ˆ
Z
requires picking an isomorphism, and this is equivalent to picking an element
of
ˆ
Z to be the “1”. Naively, we might pick the following:
Definition
(Arithmetic Frobenius)
.
The arithmetic Frobenius
ϕ
q
∈ Gal
(
¯
k/k
)
(where |k| = q) is defined to be
ϕ
q
(x) = x
q
.
Identifying this with 1
∈
ˆ
Z
leads to infinite confusion, and we shall not do so.
Instead, we define
Definition (Geometric Frobenius). The geometric Frobenius is
Frob
q
= ϕ
−1
q
∈ Gal(
¯
k/k).
We shall identify Gal(
¯
k/k)
∼
=
ˆ
Z by setting the geometric Frobenius to be 1.
The reason this is called the geometric Frobenius is that if we have a scheme
over a finite field
k
, then there are two ways the Frobenius can act on it — either
as a Galois action, or as a pullback along the morphism (
−
)
q
:
k → k
. The latter
corresponds to the geometric Frobenius.
We now turn to understand the inertia groups. The point of introducing the
wild inertia group is to single out the “
p
-phenomena”, which we would like to
avoid. To understand
I
F
better, let
n
be a natural number prime to
p
. As usual,
we write
µ
n
(
¯
k) = {ζ ∈
¯
k : ζ
n
= 1}.
We also pick an nth root of π in
¯
F , say π
n
. By definition, this has π
n
n
= π.
Definition
(Tame mod
n
character)
.
The tame mod
n
character is the map
t(n) : I
F
= Gal(
¯
F /F
ur
) → µ
n
(
¯
k) given by
γ 7→ γ(π
n
)/π
n
(mod π).
Note that since γ fixes π = π
n
n
, we indeed have
γ(π
n
)
π
n
n
=
γ(π
n
n
)
π
n
n
= 1.
Moreover, this doesn’t depend on the choice of
π
n
. Any other choice differs by
an
n
th root of unity, but the
n
th root of unity lies in
F
ur
since
n
is prime to
p
. So
γ
fixes it and so it cancels out in the fraction. For the same reason, if
γ
moves
π
n
at all, then this is visible down in
¯
k
, since
γ
would have multiplied
π
n
by an nth root of unity, and these nth roots are present in
¯
k.
Now that everything is canonically well-defined, we can take the limit over
all n to obtain a map
ˆ
t : I
F
→ lim
(n,p)=1
µ
n
(
¯
k) =
Y
`6=p
lim
m≥1
µ
`
m
(
¯
k) ≡
Y
`6=p
Z
`
(1)(
¯
k).
This
Z
`
(1)(
¯
k
) is the Tate module of
¯
k
×
. This is isomorphic to
Z
`
, but not
canonically.
Theorem. ker
ˆ
t = P
F
.
Thus, it follows that maximal tamely ramified extension of
F
, i.e. the fixed
field of P
F
is
[
(n,p)=1
F
ur
(
n
√
π).
Note that
t
(
n
) extends to a map Γ
F
→ µ
n
given by the same formula, but
this now depends on the choice of
π
n
, and further, it is not a homomorphism,
because
t(n)(γδ) =
γδ(π
n
)
π
n
=
γ(π
n
)
π
n
γ
δ(π
n
)
π
n
= t(n)(γ) · γ(t(n)(δ)).
So this formula just says that
t
(
n
) is a 1-cocycle. Of course, picking another
π
n
will modify t(n) by a coboundary.