Conductor (class field theory)

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.

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)}

.