1Class field theory
IV Topics in Number Theory
1.4 Ideal-theoretic description of global class field theory
Originally, class field theory was discovered using ideals, and the ideal-theoretic
formulation is at times more convenient.
Let
m
be a modulus, and let
S
be the set of finite
v
such that
m
v
>
0. Let
I
S
be the group of fractional ideals prime to S. Consider
P
m
= {(x) ∈ I
S
: x ≡ 1 mod m}.
To be precise, we require that for all v ∈ S, we have v(x − 1) ≥ m
v
, and for all
infinite
v
real with
m
v
>
0, then
τ
(
x
)
>
0 for
τ
:
K → R
the corresponding to
v
.
In other words, x ∈ K
×
∩ U
m
.
Note that if
m
is trivial, then
I
S
/P
m
is the ideal class group. Thus, it makes
sense to define
Definition
(Ray class group)
.
Let
m
be a modulus. The generalized ideal class
group, or ray class group modulo m is
Cl
m
(K) = I
S
/P
m
.
One can show that this is always a finite group.
Proposition. There is a canonical isomorphism
J
K
K
×
U
m
∼
→ Cl
m
(K)
such that for v 6∈ S ∪ Σ
K,∞
, the composition
K
×
v
→ J
K
→ Cl
m
(K)
sends x 7→ p
−v(x)
v
.
Thus, in particular, the Galois group
Gal
(
L/K
) of the ray class field modulo
m
is
Cl
m
(
K
). Concretely, if
p 6∈ S
is an ideal, then [
p
]
∈ Cl
m
(
K
) corresponds to
σ
p
∈ Gal
(
L/K
), the arithmetic Frobenius. This was Artin’s original reciprocity
law.
When
m
= 0, then this map is the inverse of the map given by content.
However, in general, it is not simply (the inverse of) the prime-to-
S
content map,
even for ideles whose content is prime to
S
. According to Fr¨olich, this is the
“fundamental mistake of class field theory”.
Proof sketch. Let J
K
(S) ⊆ J
K
be given by
J
K
(S) =
Y
v6∈S∪Σ
K,∞
K
×
v
.
Here we do have the inverse of the content map
c
−1
: J
K
(S) I
S
(x
v
) 7→
Y
p
−v(x
v
)
v
We want to extend it to an isomorphism. Observe that
J
K
(S) ∩ U
m
=
Y
v6∈S∪Σ
K,∞
O
×
v
,
which is precisely the kernel of the map
c
−1
. So
c
−1
extends uniquely to a
homomorphism
J
K
(S)U
m
U
m
∼
=
J
K
(S)
J
K
(S) ∩ U
m
→ I
S
.
We then use that K
×
J
K
(S)U
m
= J
K
(weak approximation), and
K
×
∩ V
m
= {x ≡ 1 mod m, x ∈ K
∗
},
where
V
m
= J
K
(S)U
m
= {(x
v
) ∈ J
K
| for all v with m
v
> 0, x
v
∈ U
m
}.