Galois representation

Galois representation

In general, let

K

be any field. Write

K

¯

for a

separable closure

of

K

, and

G

K

for the

absolute Galois group

Gal

⁢

(

K

¯

/

K

)

of

K

. Let

A

be a (

Hausdorff

)

Abelian

topological group

. Then an (

A

-valued)

Galois representation

for

K

is a

continuous

homomorphism

ρ

:

G

K

→

Aut

⁢

(

A

)

,

where we endow

G

K

with the

Krull topology

, and where

Aut

⁢

(

A

)

is the group of continuous

automorphisms

of

A

, endowed with the

compact-open topology

. One calls

A

the

representation

space for

ρ

.

The simplest case is where

A

=

ℂ

n

, the group of

n

×

1

column vectors with

complex

entries. Then

Aut

⁢

(

ℂ

n

)

=

GL

n

⁢

(

ℂ

)

, and we have what is usually called a complex representation of degree

n

. In the same manner, letting

A

=

F

n

, with

F

any field (such as

ℝ

or a

finite field

𝔽

q

) we obtain the usual definition of a degree

n

representation over

F

.

There is an alternate definition which we should also mention. Write

ℤ

⁢

[

G

K

]

for the

group ring

of

G

K

with

coefficients

in

ℤ

. Then a Galois representation for

K

is simply a continuous

ℤ

⁢

[

G

K

]

-module

A

(i.e. the action of

G

K

on

A

is given by a continuous homomorphism

ρ

). In other words, all the information in a representation

ρ

is preserved in considering the representation space

A

as a continuous

ℤ

⁢

[

G

K

]

-module. The equivalence of these two definitions is as described in the entry for the

group algebra

.

When

A

is

complete

, the continuity requirement is

equivalent

to the action of

ℤ

⁢

[

G

K

]

on

A

naturally extending to a

ℤ

⁢

[

[

G

K

]

]

-module

structure

on

A

. The notation

ℤ

⁢

[

[

G

K

]

]

denotes the completed group ring:

ℤ

⁢

[

[

G

]

]

=

lim

⟵

ℤ

⁢

[

G

/

H

]

,

where

G

is any

profinite group

, and

H

ranges over all

normal subgroups

of finite index.

A notation we will be using often is the following. Suppose

G

is a group,

ρ

:

G

→

Aut

⁢

(

A

)

is a representation and

H

⊆

G

a

subgroup

. Then we let

A

H

=

{

a

∈

A

∣

ρ

⁢

(

h

)

⁢

a

=

a

,

for

⁢

all

⁢

h

∈

H

}

,

the subgroup of

A

fixed

pointwise

by

H

.

Given a Galois representation

ρ

, let

G

0

=

ker

⁡

ρ

. By the fundamental theorem of infinite Galois theory, since

G

0

is a closed normal subgroup of

G

K

, it corresponds to a certain normal

subfield

of

K

¯

. Naturally, this is the

fixed field

of

G

0

, and we denote it by

K

⁢

(

ρ

)

. (The notation becomes better justified after we view some examples.) Notice that since

ρ

is

trivial

on

G

0

=

Gal

⁢

(

K

¯

/

K

⁢

(

ρ

)

)

, it factors through a representation

ρ

~

:

Gal

⁢

(

K

⁢

(

ρ

)

/

K

)

→

Aut

⁢

(

A

)

,

which is

faithful

. This property characterizes

K

⁢

(

ρ

)

.

In the case

A

=

ℝ

n

or

A

=

ℂ

n

, the so-called “no small subgroups”

argument

implies that the image of

G

K

is finite.

For a first application of definition, we say that

ρ

is

discrete

if for all

a

∈

A

, the

stabilizer

of

a

in

G

K

is open in

G

K

. This is the case when

A

is given the

discrete topology

, such as when

A

is finite and Hausdorff. The stabilizer of any

a

∈

A

fixes a

finite

extension

of

K

, which we denote by

K

⁢

(

a

)

. One has that

K

⁢

(

ρ

)

is the union of all the

K

⁢

(

a

)

.

As a second application, suppose that the image

ρ

⁢

(

G

K

)

is Abelian. Then the

quotient

G

K

/

G

0

is Abelian, so

G

0

contains the

commutator subgroup

of

G

K

, which means that

K

⁢

(

ρ

)

is

contained

in

K

ab

, the

maximal

Abelian extension

of

K

. This is the case when

ρ

is a

character

, i.e. a

1

-dimensional representation over some (

commutative

unital) ring,

ρ

:

G

K

→

GL

1

⁢

(

A

)

=

A

×

.

Associated to any field

K

are two basic Galois representations, namely those with representation spaces

A

=

L

and

A

=

L

×

, for any normal intermediate field

K

⊆

L

⊆

K

¯

, with the usual action of the

Galois group

on them. Both of these representations are discrete. The

additive

representation is rather simple if

L

/

K

is finite: by the normal basis theorem, it is merely a

permutation representation

on the normal basis. Also, if

L

=

K

¯

and

x

∈

K

¯

, then

K

⁢

(

x

)

, the field obtained by adjoining

x

to

K

, agrees with the fixed field of the stabilizer of

x

in

G

K

. This motivates the notation “

K

⁢

(

a

)

” introduced above.

By contrast, in general,

L

×

can become a rather complicated object. To look at just a piece of the representation

L

×

, assume that

L

contains the group

μ

m

of

m

-th

roots of unity

, where

m

is prime to the

characteristic

of

K

. Then we let

A

=

μ

m

. It is possible to choose an

isomorphism

of

Abelian groups

μ

m

≅

ℤ

/

m

, and it follows that our representation is

ρ

:

G

K

→

(

ℤ

/

m

)

×

. Now assume that

m

has the form

p

n

, where

p

is a prime not equal to the characteristic, and set

A

n

=

μ

p

n

. This gives a

sequence

of representations

ρ

n

:

G

K

→

(

ℤ

/

p

n

)

×

, which are

compatible

with the natural maps

(

ℤ

/

p

n

+

1

)

×

→

(

ℤ

/

p

n

)

×

. This compatibility allows us to glue them together into a big representation

ρ

:

G

K

→

Aut

⁢

(

T

p

⁢

𝔾

m

)

≅

ℤ

p

×

,

called the

p

-adic cyclotomic representation of

K

. This representation is often not discrete. The notation

T

p

⁢

𝔾

m

will be explained below.

This example may be generalized as follows. Let

B

be an Abelian

algebraic group

defined over

K

. For each

integer

n

, let

B

n

=

B

⁢

(

K

¯

)

⁢

[

p

n

]

be the set of

K

¯

-

rational points

whose order divides

p

n

. Then we define the

p

-adic Tate module of

B

via

T

p

⁢

B

=

lim

⟵

B

n

.

It acquires a natural Galois action from the ones on the

B

n

. The two most commonly treated examples of this are the cases

B

=

𝔾

m

(the

multiplicative group

, giving the cyclotomic representation above) and

B

=

E

, an

elliptic curve

defined over

K

.

The last thing which we shall mention about generalities is that to any Galois representation

ρ

:

G

K

→

Aut

⁢

(

A

)

, one may

associate

the Galois cohomology groups

H

n

⁢

(

K

,

ρ

)

, more commonly written

H

n

⁢

(

K

,

A

)

, which are defined to be the

group cohomology

of

G

K

(computed with continuous cochains) with coefficients in

A

.

Galois representations play a fundamental role in

algebraic number theory

, as many objects and properties related to

global fields

and

local fields

may be determined by certain Galois representations and their properties. We shall describe the local case first, and then the global case.

Let

K

be a local field, by which we mean the

fraction field

of a complete DVR with finite

residue field

. We write

v

K

for the normalized

valuation

,

𝒪

K

for the associated DVR,

𝔪

K

for the

maximal ideal

of

𝒪

K

,

k

K

=

𝒪

K

/

𝔪

K

for the residue field, and

ℓ

for the characteristic of

k

K

.

Let

L

/

K

be a finite

Galois extension

, and define

v

L

,

𝒪

L

,

𝔪

L

, and

k

L

accordingly. There is a natural

surjection

Gal

⁢

(

L

/

K

)

→

Gal

⁢

(

k

L

/

k

K

)

. We call the kernel of this map the

inertia group

, and write it

I

⁢

(

L

/

K

)

=

ker

⁡

(

Gal

⁢

(

L

/

K

)

→

Gal

⁢

(

k

L

/

k

K

)

)

. Further, the

p

-

Sylow subgroup

of

I

⁢

(

L

/

K

)

is normal, and we call it the

wild ramification

group, and denote it by

W

⁢

(

L

/

K

)

. One calls

I

/

W

the tame

ramification group

.

It happens that the formation of these group is compatible with extensions

L

′

/

L

/

K

, in that we have surjections

I

⁢

(

L

′

/

K

)

→

I

⁢

(

L

/

K

)

and

W

⁢

(

L

′

/

K

)

→

W

⁢

(

L

/

K

)

. This lets us define

W

K

⊂

I

K

⊂

G

K

to be the

inverse limits

of the subgroups

W

⁢

(

L

/

K

)

⊆

I

⁢

(

L

/

K

)

⊆

Gal

⁢

(

L

/

K

)

,

L

as usual ranging over all finite Galois extensions of

K

in

K

¯

.

Let

ρ

be a Galois representation for

K

with representation space

A

. We say that

ρ

is

unramified

if the inertia group

I

K

acts trivially on

A

, or in other words

I

K

⊆

ker

⁡

ρ

or

A

I

K

=

A

. Otherwise we say it is ramified. Similarly, we say that

ρ

is (at most) tamely ramified if the wild ramification group acts trivially, or

W

K

⊆

ker

⁡

ρ

, or

A

W

K

=

A

; and if not we say it is wildly ramified.

We let

K

ur

=

K

¯

I

K

be the maximal unramified extension of

K

, and

K

tame

=

K

¯

W

K

be the maximal tamely ramified extension of

K

.

Unramified or tamely ramified extensions are usually much easier to study than wildly ramified extensions. In the unramified case, it results from the fact that

G

K

/

I

K

≅

G

k

⁢

(

K

)

≅

ℤ

^

is

pro-cyclic

. Thus an unramified representation is completely determined by the action of

ρ

⁢

(

σ

)

for a topological

generator

σ

of

G

K

/

I

K

. (Such a

σ

is often called a Frobenius

element

.)

Given a

finite extension

L

/

K

, one defines the inertia degree

f

L

/

K

=

[

k

L

:

k

K

]

and the ramification degree

e

L

/

K

=

[

v

L

(

L

×

)

:

v

L

(

K

×

)

]

as usual. Then in the Galois case one may recover them as

f

L

/

K

=

[

Gal

(

L

/

K

)

:

I

(

L

/

K

)

]

and

e

L

/

K

=

#

⁢

I

⁢

(

L

/

K

)

. The tame inertia degree, which is the non-

p

-part of

e

L

/

K

, is equal to

[

I

⁢

(

L

/

K

)

:

W

⁢

(

L

/

K

)

]

, while the wild inertia degree, which is the

p

-part of

e

L

/

K

, is equal to

#

⁢

W

⁢

(

L

/

K

)

.

One finds that the inertia and ramification properties of

L

/

K

may be computed from the ramification properties of the Galois representation

𝒪

L

.

We now turn to global fields. We shall only treat the

number field

case. Thus we let

K

be a finite extension of

ℚ

, and write

𝒪

K

for its

ring of integers

. For each place

v

of

K

, write

K

v

for the

completion

of

K

with respect to

v

. When

v

is a

finite place

, we write simply

v

for its associated normalized valuation,

𝒪

v

for

𝒪

K

v

,

𝔪

v

for

𝔪

K

v

,

k

v

for

k

K

v

, and

ℓ

⁢

(

v

)

for the characteristic of

k

v

.

For each place

v

, fix an

algebraic closure

K

¯

v

of

K

v

. Furthermore, choose an

embedding

K

¯

↪

K

¯

v

. This choice is equivalent to choosing an extension of

v

to all of

K

¯

×

, and to choosing an embedding

G

K

v

↪

G

K

. We denote the image of this last embedding by

G

v

⊂

G

K

; it is called a

decomposition group

at

v

. Sitting inside

G

v

are two groups,

I

v

and

W

v

, corresponding to the inertia and wild ramification subgroups

I

K

v

and

W

K

v

of

G

K

v

; we call the images

I

v

and

W

v

the inertia group at

v

and the wild ramification group at

v

, respectively.

For a Galois representation

ρ

:

G

K

→

Aut

⁢

(

A

)

and a place

v

, it is profitable to consider the restricted representation

ρ

v

=

ρ

|

G

v

. One calls

ρ

a global representation, and

ρ

v

a local representation. We say that

ρ

is ramified or tamely ramified (or not) at

v

if

ρ

v

is (or isn’t). The

Tchebotarev density theorem

implies that the corresponding Frobenius elements

σ

v

∈

G

v

are dense in

G

K

, so that the union of the

G

v

is dense in

G

K

. Therefore, it is reasonable to try to reduce questions about

ρ

to questions about all the

ρ

v

independently. This is a manifestation of Hasse’s local-to-global principle.

Given a global Galois representation with representation space

ℤ

p

n

which is unramified at all but finitely many places

v

, it is a goal of

number theory

to prove that it arises naturally in

arithmetic geometry

(namely, as a subrepresentation of an étale cohomology group of a motive), and also to prove that it arises from an automorphic form. This can only be shown in certain special cases.

Title

Galois representation

Canonical name

GaloisRepresentation

Date of creation

2013-03-22 13:28:21

Last modified on

2013-03-22 13:28:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

18

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R32

Classification

msc 11R04

Classification

msc 11R34

Related topic

InverseLimit

Defines

tame inertia group

Defines

wild inertia group

Defines

tame inertia degree

Defines

wild inertia degree

Defines

discrete module

Defines

Tate module

Defines

cyclotomic representation

Defines

Galois cohomology

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