fundamental character of level n for the inertia group at p

fundamental character of level

n

for the inertia group at

p

Let

p

>

2

be a prime, fix

algebraic closures

ℚ

¯

and

ℚ

p

¯

, and fix an

embedding

of

ℚ

¯

↪

ℚ

p

¯

. This embedding corresponds with an

inclusion

of the

absolute Galois groups

:

Gal

⁡

(

ℚ

p

¯

/

ℚ

p

)

↪

Gal

⁡

(

ℚ

¯

/

ℚ

)

,

σ

↦

σ

|

ℚ

¯

.

Let

I

p

be the inertia

subgroup

of

Gal

⁡

(

ℚ

p

¯

/

ℚ

p

)

which we regard as a subgroup of

Gal

⁡

(

ℚ

¯

/

ℚ

)

via the injection above (for more information on the inertia subgroup at

p

,

I

p

, see the entry on

Galois representations

). Let

𝔽

p

n

be the

finite field

of

p

n

elements

. The purpose of this entry is to define

𝔽

p

n

-valued

characters

Ψ

n

, for every

n

≥

1

:

Ψ

n

:

I

p

⟶

𝔽

p

n

×

≅

ℤ

/

(

p

n

-

1

)

⁢

ℤ

which we will refer to as the fundamental character of level

n

of

I

p

.

Definition 1

.

Let

χ

p

:

Gal

⁡

(

Q

¯

/

Q

)

→

Z

p

×

be the

p

-adic

cyclotomic character

and let

χ

p

¯

be the

reduction

of

χ

p

modulo

p

. The fundamental character of level

1

is

Ψ

1

=

χ

p

¯

|

I

p

, i.e.

Ψ

1

is the

restriction

of the

p

-adic cyclotomic character

χ

p

to

I

p

, composed with reduction modulo

p

.

Next, we define the fundamental characters in more generality. Let

K

n

/

ℚ

p

be the unique

unramified

field extension

of degree

n

(it is unique by

local field

theory). The

residue field

of

K

n

is the field

k

n

=

𝔽

p

n

(because

k

must be an

extension

of degree

n

of

𝔽

p

).

Lemma 1

.

The field

K

n

contains all

(

p

n

-

1

)

th

roots of unity

.

Proof.

Clearly, the

polynomial

x

p

n

-

1

-

1

=

0

has

p

n

-

1

distinct roots in

k

n

=

𝔽

p

n

. Using Hensel’s lemma, one can check that each root in

k

n

lifts to an element of

K

n

.
∎

Let

K

n

′

=

K

n

⁢

(

(

-

p

)

1

p

n

-

1

)

. By the lemma, the

(

p

n

-

1

)

th roots of unity are

contained

in

K

n

. Therefore, the extension

K

n

′

/

K

n

is Galois. Moreover, by

Kummer theory

one has:

Gal

⁡

(

K

n

′

/

K

n

)

=

k

n

×

=

𝔽

p

n

×

.

Notice that the fact that

K

n

/

ℚ

p

is unramified implies that the

inertia group

I

p

injects into

Gal

⁡

(

ℚ

p

¯

/

K

n

)

↪

Gal

⁡

(

ℚ

p

¯

/

ℚ

p

)

. Therefore there is a map:

I

p

↪

Gal

⁡

(

ℚ

p

¯

/

K

n

)

→

Gal

⁡

(

K

n

′

/

K

n

)

→

𝔽

p

n

×

(1)

where the second map is simply given by restriction to

K

n

′

.

Definition 2

.

The fundamental character of level

n

≥

1

is the map

Ψ

n

:

I

p

→

F

p

n

×

given by Eq. (

1

).

Note from the author: I would like to thank Eknath Ghate for explaining this to me.

Title

fundamental character of level

n

for the inertia group at

p

Canonical name

FundamentalCharacterOfLevelNForTheInertiaGroupAtP

Date of creation

2013-03-22 15:36:26

Last modified on

2013-03-22 15:36:26

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R04

Classification

msc 11R32

Classification

msc 11R34

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