Weil group

In mathematics, a Weil group, introduced by André Weil, is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field

F

{\displaystyle F}

, its Weil group is generally denoted

W

F

{\displaystyle W_{F}}

. There also exists "finite level" modifications of the Galois groups: if

E

/

F

{\displaystyle E/F}

is a finite extension, then the relative Weil group of

E

/

F

{\displaystyle E/F}

is

W

E

/

F

=

W

F

/

W

E

c

{\displaystyle W_{E/F}=W_{F}/W_{E}^{c}}

(where the superscript

c

{\displaystyle c}

denotes the commutator subgroup).

Class formation

The Weil group of a class formation with fundamental classes

u

E

/

F

=

H

2

(

E

/

F

,

A

F

)

{\displaystyle u_{E/F}=H^{2}(E/F,A^{F})}

is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program.

If

E

/

F

{\displaystyle E/F}

is a normal layer, then the (relative) Weil group

W

E

/

F

{\displaystyle W_{E/F}}

of

E

/

F

{\displaystyle E/F}

is the extension

1

→

A

F

→

W

E

/

F

→

Gal

⁡

(

E

/

F

)

→

1

{\displaystyle 1\to A^{F}\to W_{E/F}\to \operatorname {Gal} (E/F)\to 1}

corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class

u

E

/

F

{\displaystyle u_{E/F}}

in

H

2

(

Gal

⁡

(

E

/

F

)

,

A

F

)

{\displaystyle H^{2}(\operatorname {Gal} (E/F),A^{F})}

. The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers

G

/

F

{\displaystyle G/F}

, for

F

{\displaystyle F}

an open subgroup of

G

{\displaystyle G}

.

The reciprocity map of the class formation

(

G

,

A

)

{\displaystyle (G,A)}

induces an isomorphism from

A

G

{\displaystyle A^{G}}

to the abelianization of the Weil group.

Archimedean local field

For archimedean local fields the Weil group is easy to describe: for

C

{\displaystyle \mathbb {C} }

it is the group

C

×

{\displaystyle \mathbb {C} ^{\times }}

of non-zero complex numbers, and for

R

{\displaystyle \mathbb {R} }

it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup

C

×

∪

j

C

×

{\displaystyle \mathbb {C} ^{\times }\cup j\mathbb {C} ^{\times }}

of the non-zero quaternions.

Finite field

For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism.