Artin conductor

In number theory, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin as an expression appearing in the functional equation of an Artin L-function.

Local Artin conductors

Suppose that

L

{\displaystyle L}

is a finite Galois extension of the local field

K

{\displaystyle K}

, with Galois group

G

{\displaystyle G}

. If

χ

{\displaystyle \chi }

is a character of

G

{\displaystyle G}

, then the Artin conductor of

χ

{\displaystyle \chi }

is the number

f

(

χ

)

=

∑

i

≥

0

g

i

g

0

(

χ

(

1

)

−

χ

(

G

i

)

)

,

{\displaystyle f(\chi )=\sum _{i\geq 0}{\frac {g_{i}}{g_{0}}}(\chi (1)-\chi (G_{i})),}

where

G

i

{\displaystyle G_{i}}

is the

i

{\displaystyle i}

-th ramification group (in lower numbering), of order

g

i

{\displaystyle g_{i}}

, and

χ

(

G

i

)

{\displaystyle \chi (G_{i})}

is the average value of

χ

{\displaystyle \chi }

on

G

i

{\displaystyle G_{i}}

. The local conductor is an integer. Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if

χ

{\displaystyle \chi }

is unramified, then its Artin conductor is zero. Thus, if

L

{\displaystyle L}

is unramified over

K

{\displaystyle K}

, then the Artin conductors of all

χ

{\displaystyle \chi }

are zero.

The wild invariant or Swan conductor of the character is

f

(

χ

)

−

(

χ

(

1

)

−

χ

(

G

0

)

)

,

{\displaystyle f(\chi )-(\chi (1)-\chi (G_{0})),}

in other words, the sum of the higher order terms with

i

>

0

{\displaystyle i>0}

.

Global Artin conductors

The global Artin conductor of a representation

χ

{\displaystyle \chi }

of the Galois group

G

{\displaystyle G}

of a finite extension

L

/

K

{\displaystyle L/K}

of global fields is an ideal of

K

{\displaystyle K}

, defined to be

f

(

χ

)

=

∏

p

p

f

(

χ

,

p

)

{\displaystyle {\mathfrak {f}}(\chi )=\prod _{p}p^{f(\chi ,p)}}

where the product is over the primes

p

{\displaystyle p}

of

K

{\displaystyle K}

, and

f

(

χ

,

p

)

{\displaystyle f(\chi ,p)}

is the local Artin conductor of the restriction of

χ

{\displaystyle \chi }

to the decomposition group of some prime of

L

{\displaystyle L}

lying over

p

{\displaystyle p}

. Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in

L

/

K

{\displaystyle L/K}

.