8Lubin–Tate theory

III Local Fields

8.1 Motivating example

We will work out the details of local Artin reciprocity in the case of

Q

p

as a

motivating example for the proof we are going to come up with later. Here we

will need the results of local class field theory to justify our claims, but this is

not circular since this is not really part of the proof.

Lemma. Let L/K be a finite abelian extension. Then we have

e

L/K

= (O

×

K

: N

L/K

(O

×

L

)).

Proof.

Pick

x ∈ L

×

, and

w

the valuation on

L

extending

v

K

, and

n

= [

L

:

K

].

Then by construction of w, we know

v

K

(N

L/K

(x)) = nw(x) = f

L/K

v

L

(x).

So we have a surjection

K

×

N(L/K)

Z

f

L/K

Z

v

K

.

The kernel of this map is equal to

O

×

K

N(L/K)

N(L/K)

∼

=

O

×

K

O

×

K

∩ N(L/K)

=

O

×

K

N

L/K

(O

×

L

)

.

So by local class field theory, we know

n = (K

×

: N(L/K)) = f

L/K

(O

×

K

: N

L/K

(O

×

L

)),

and this implies what we want.

Corollary.

Let

L/K

be a finite abelian extension. Then

L/K

is unramified if

and only if N

L/K

(O

×

L

) = O

×

K

.

Now we fix a uniformizer

π

K

. Then we have a topological group isomorphism

K

×

∼

=

hπ

K

i × O

×

K

.

Since we know that the finite abelian extensions correspond exactly to finite

index subgroups of

K

×

by taking the norm groups, we want to understand

subgroups of K

×

. Now consider the subgroups of K

×

of the form

hπ

m

K

i × U

(n)

K

.

We know these form a basis of the topology of

K

×

, so it follows that finite-index

open subgroups must contain one of these guys. So we can find the maximal

abelian extension as the union of all fields corresponding to these guys.

Since we know that

N

(

LM/K

) =

N

(

L/K

)

∩ N

(

M/K

), it suffices to further

specialize to the cases

hπ

K

i × U

(n)

K

and

hπ

m

K

i × O

K

separately. The second case is easy, because this corresponds to an unramified

extension by the above corollary, and unramified extensions are completely

characterized by the extension of the residue field. Note that the norm group

and the extension are both independent of the choice of uniformizer. The

extensions corresponding to the first case are much more difficult to construct,

and they depend on the choice of

π

K

. We will get them from Lubin–Tate theory.

Lemma.

Let

K

be a local field, and let

L

m

/K

be the extension corresponding

to hπ

m

K

i × O

K

. Let

L =

[

m

L

m

.

Then we have

K

ab

= K

ur

L,

Lemma. We have isomorphisms

W (K

ab

/K)

∼

=

W (K

ur

L/K)

∼

=

W (K

ur

/K) × Gal(L/K)

∼

=

Frob

Z

K

× Gal(L/K)

Proof.

The first isomorphism follows from the previous lemma. The second

follows from the fact that

K

ur

∩ L

=

K

as

L

is totally ramified. The last

isomorphism follows from the fact that

T

K

ur

/K

=

K

ur

trivially, and then by

definition W (K

ur/K

)

∼

=

Frob

Z

K

.

Example. We consider the special case of K = Q

p

and π

K

= p. We let

L

n

= Q

p

(ζ

p

n

),

where

ζ

p

n

is the primitive

p

n

th root of unity. Then by question 6 on example

sheet 2, we know this is a field with norm group

N(Q

p

(ζ

p

n

)/Q

p

) = hpi × (1 + p

n

Z

p

) = hpi × U

(n)

Q

p

,

and thus this is a totally ramified extension of Q

p

.

We put

Q

p

(ζ

p

∞

) =

∞

[

n=1

Q

p

(ζ

p

n

).

Then again this is totally ramified extension, since it is the nested union of

totally ramified extensions.

Then we have

Gal(Q

p

(ζ

p

∞

)/Q

p

)

∼

=

lim

←−

n

Gal(Q

p

(ζ

p

n

)/Q

p

)

= lim

←−

n

(Z/p

n

Z)

×

= Z

×

p

.

Note that we are a bit sloppy in this deduction. While we know that it is true

that

Z

×

p

∼

=

lim

←−

n

(

Z/p

n

Z

)

×

, the inverse limit depends not only on the groups

(

Z/p

n

Z

)

×

themselves, but also on the maps we use to connect the groups together.

Fortunately, from the discussion below, we will see that the maps

Gal(Q

p

(ζ

p

n

)/Q

p

) → Gal(Q

p

(ζ

p

n−1

)/Q

p

)

indeed correspond to the usual restriction maps

(Z/p

n

Z)

×

→ (Z/p

n−1

Z)

×

.

It is a fact that this is the inverse of the Artin map of

Q

p

restricted to

Z

×

p

.

Note that we have

W

(

Q

p

(

ζ

p

∞

)

/Q

p

) =

Gal

(

Q

p

(

ζ

p

∞

)

/Q

p

) because its maximal

unramified subextension is trivial.

We can trace through the above chains of isomorphisms to figure out what

the Artin map does. Let m = Z

×

p

. Then we can write

m = a

0

+ a

1

p + ··· ,

where a

i

∈ {0, ··· , p − 1} and a

0

6= 0. Now for each n, we know

m ≡ a

0

+ a

1

p + ··· + a

n−1

p

n−1

mod p

n

.

By the usual isomorphism Gal(Q

p

(ζ

p

n

)/Q

p

)

∼

=

Z/p

n

Z, we know m acts as

ζ

p

n

7→ ζ

a

0

+a

1

p+...+a

n−1

p

n−1

p

n

“=” ζ

m

p

n

on

Q

p

(

ζ

p

n

), where we abuse notation because taking

ζ

p

n

to powers of

p

greater

than

n

gives 1. It can also be interpreted as (1 +

λ

p

n

)

m

, where

λ

p

n

=

ζ

p

n

−

1 is

a uniformizer, which makes sense using binomial expansion.

So the above isomorphisms tells us that

Art

Q

p

restricted to

Z

×

p

acts on

Q

p

(ζ

p

∞

) as

Art

Q

p

(m)(ζ

p

n

) ≡ σ

m

−1

(ζ

p

n

) = ζ

m

−1

p

n

.

The full Artin map can then be read off from the following diagram:

Q

×

p

W (Q

ab

p

/Q

p

)

hpi × Z

×

p

W (Q

ur

p

/Q

p

) × Gal(Q

p

(ζ

p

∞

)/Q

p

)

∼

=

Art

Q

p

restriction

∼

where the bottom map sends

hp

n

, mi 7→ (Frob

n

Q

p

, σ

m

−1

).

In fact, we have

Theorem (Local Kronecker-Weber theorem).

Q

ab

p

=

[

n∈Z

≥1

Q

p

(ζ

n

),

Q

ur

p

=

[

n∈Z

≥1

(n,p)=1

Q

p

(ζ

n

).

Not a proof.

We will comment on the proof of the generalized version later.

Remark.

There is another normalization of the Artin map which sends a

uniformizer to the geometric Frobenius, defined to be the inverse of the arithmetic

Frobenius. With this convention, Art

Q

p

(m)|

Q

p

(ζ

p

∞

)

is σ

m

.

We can define higher ramification groups for general Galois extensions.

Definition

(Higher ramification groups)

.

Let

K

be a local field and

L/K

Galois.

We define, for s ∈ R

≥−1

G

s

(M/K) = {σ ∈ Gal(M/K) : σ|

L

∈ G

s

(L/K) for all finite

Galois subextension M/K}.

This definition makes sense, because the upper number behaves well when

we take quotients. This is one of the advantages of upper numbering. Note that

we can write the ramification group as the inverse limit

G

s

(M/K)

∼

=

lim

←−

L/K

G

s

(L/K),

as in the case of the Galois group.

Example.

Going back to the case of

K

=

Q

p

. We write

Q

p

n

for the unramified

extension of degree n of Q

p

. By question 11 of example sheet 3, we know that

G

s

(Q

p

n

(ζ

p

m

)/Q

p

) =

Gal(Q

p

n

(ζ

p

m

)/Q

p

) s = −1

Gal(Q

p

n

(ζ

p

m

)/Q

p

n

) −1 < s ≤ 0

Gal(Q

p

n

(ζ

p

m

)/ζ

p

k

) k − 1 < s ≤ k ≤ m − 1

1 s > m − 1

,

which corresponds to

hpi × U

(0)

hp

n

i × U

(m)

s = −1

hp

n

i × U

(0)

hp

n

i × U

(m)

−1 < s ≤ 0

hp

n

i × U

(k)

hp

n

i × U

(m)

k − 1 < s ≤ k ≤ m − 1

1 s > m − 1

under the Artin map.

By taking the limit as n, m → ∞, we get

Theorem. We have

G

s

(Q

ab

p

/Q

p

) = Art

Q

p

(1 + p

k

Z

p

) = Art

Q

p

(U

(k)

),

where k is chosen such that k − 1 < s ≤ k, k ∈ Z

≥0

.

Corollary. If L/Q

p

is a finite abelian extension, then

G

s

(L/Q

p

) = Art

Q

p

N(L/Q

p

)(1 + p

n

Z

p

)

N(L/Q

p

)

,

where n − 1 < s ≤ n.

Here Art

Q

p

induces an isomorphism

Q

×

p

N(L/Q

p

)

→ Gal(L/Q

p

).

So it follows that

L ⊆ Q

p

(

ζ

p

m

) for some

n

if and only if

G

s

(

L/Q

p

) = 1 for all

s > m − 1.