In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.
Contents
Local conductor
Let
L
/
K
{\displaystyle L/K}
be a finite abelian extension of non-Archimedean local fields. The conductor of
L
/
K
{\displaystyle L/K}
, denoted
f
(
L
/
K
)
{\displaystyle {\mathfrak {f}}(L/K)}
, is the smallest non-negative integer
n
{\displaystyle n}
such that the higher unit group
U
(
n
)
=
1
+
m
K
n
=
{
u
∈
O
×
:
u
≡
1
(
mod
m
K
n
)
}
{\displaystyle U^{(n)}=1+{\mathfrak {m}}_{K}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,\left(\operatorname {mod} {\mathfrak {m}}_{K}^{n}\right)\right\}}
is contained in
N
L
/
K
(
L
×
)
{\displaystyle N_{L/K}(L^{\times })}
, where
N
L
/
K
{\displaystyle N_{L/K}}
is field norm map and
m
K
{\displaystyle {\mathfrak {m}}_{K}}
is the maximal ideal of
K
{\displaystyle K}
. Equivalently,
n
{\displaystyle n}
is the smallest integer such that the local Artin map is trivial on
U
K
(
n
)
{\displaystyle U_{K}^{(n)}}
. Sometimes, the conductor is defined as
m
K
n
{\displaystyle {\mathfrak {m}}_{K}^{n}}
where
n
{\displaystyle n}
is as above.
The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if and only if the conductor is zero, and it is tamely ramified if and only if the conductor is 1. More precisely, the conductor computes the non-triviality of higher ramification groups: if
s
{\displaystyle s}
is the largest integer for which the "lower numbering" higher ramification group Gs is non-trivial, then
f
(
L
/
K
)
=
η
L
/
K
(
s
)
+
1
{\displaystyle {\mathfrak {f}}(L/K)=\eta _{L/K}(s)+1}
, where
η
L
/
K
{\displaystyle \eta _{L/K}}
is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.
The conductor of
L
/
K
{\displaystyle L/K}
is also related to the Artin conductors of characters of the Galois group
Gal
(
L
/
K
)
{\displaystyle \operatorname {Gal} (L/K)}
. Specifically,
m
K
f
(
L
/
K
)
=
lcm
χ
m
K
f
χ
{\displaystyle {\mathfrak {m}}_{K}^{{\mathfrak {f}}(L/K)}=\operatorname {lcm} \limits _{\chi }{\mathfrak {m}}_{K}^{{\mathfrak {f}}_{\chi }}}
where
χ
{\displaystyle \chi }
varies over all multiplicative complex characters of
Gal
(
L
/
K
)
{\displaystyle \operatorname {Gal} (L/K)}
,
f
χ
{\displaystyle {\mathfrak {f}}_{\chi }}
is the Artin conductor of
χ
{\displaystyle \chi }
, and lcm is the least common multiple.
More general fields
The conductor can be defined in the same way for
L
/
K
{\displaystyle L/K}
a not necessarily abelian finite Galois extension of local fields. However, it only depends on
L
a
b
/
K
{\displaystyle L^{\mathrm {ab} }/K}
, the maximal abelian extension of
K
{\displaystyle K}
in
L
{\displaystyle L}
, because of the "norm limitation theorem", which states that, in this situation,
N
L
/
K
(
L
×
)
=
N
L
ab
Archimedean fields
Mostly for the sake of global conductors, the conductor of the trivial extension
R
/
R
{\displaystyle \mathbb {R} /\mathbb {R} }
is defined to be 0, and the conductor of the extension
C
/
R
{\displaystyle \mathbb {C} /\mathbb {R} }
is defined to be 1.
Global conductor
Algebraic number fields
The conductor of an abelian extension
L
/
K
{\displaystyle L/K}
of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let
θ
:
l
m
→
Gal
(
L
/
K
)
{\displaystyle \theta :l^{m}\to \operatorname {Gal} (L/K)}
be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m. We define the conductor of
L
/
K
{\displaystyle L/K}
, denoted
f
(
L
/
K