8Lubin–Tate theory

III Local Fields

8.2 Formal groups

The proof of local Artin reciprocity will be done by constructing the analogous

versions of

L

n

for an arbitrary local field, and then proving that it works. To

do so, we will need the notion of a formal group. The idea of a formal group is

that a formal group is a rule that specifies how we should multiply two elements

via a power series over a ring

R

. Then if we have a complete

R

-module, then

the formal group will turn the

R

-module into an actual group. There is then a

natural notion of a formal module, which is a formal group

F

with an

R

-action.

At the end, we will pick

R

=

O

K

. The idea is then that we can fix an

algebraic closure

¯

K

, and then a formal

O

K

-module will turn

m

¯

K

into an actual

O

K

-module. Then if we adjoin the right elements of

m

¯

K

to

K

, then we obtain

an extension of

K

with a natural

O

K

action, and we can hope that this restricts

to field automorphisms when we restrict to the unit group.

Notation. Let R be a ring. We write

R[[x

1

, ··· , x

n

]] =

X

k

1

,...,k

n

∈Z

≥0

a

k

1

,...,k

n

x

k

1

1

···x

k

n

n

: a

k

1

,...,k

n

∈ R

for the ring of formal power series in n variables over R.

Definition

(Formal group)

.

A (one-dimensional, commutative) formal group

over R is a power series F (X, Y ) ∈ R[X, Y ] such that

(i) F (X, Y ) ≡ X + Y mod (X

2

, XY, Y

2

)

(ii) Commutativity: F (X, Y ) = F (Y, X)

(iii) Associativity: F (X, F (Y, Z)) = F (F (X, Y ), Z).

This is most naturally understood from the point of view of algebraic geometry,

as a generalization of the Lie algebra over a Lie group. Instead of talking about

the tangent space of a group (the “first-order neighbourhood”), we talk about its

infinitesimal (formal) neighbourhood, which contains all higher-order information.

A lot of the seemingly-arbitrary compatibility conditions we later impose have

such geometric motivation that we unfortunately cannot go into.

Example.

If

F

is a formal group over

O

K

, where

K

is a complete valued field,

then

F

(

x, y

) converges for all

x, y ∈ m

K

. So

m

K

becomes a (semi)group under

the multiplication

(x, y) 7→ F (x, y) ∈ m

k

Example. We can define

ˆ

G

a

(X, Y ) = X + Y.

This is called the formal additive group.

Similarly, we can have

ˆ

G

m

(X, Y ) = X + Y + XY.

This is called the formal multiplicative group. Note that

X + Y + XY = (1 + X)(1 + Y ) −1.

So if

K

is a complete valued field, then

m

K

bijects with 1 +

m

k

by sending

x 7→

1 +

x

, and the rule sending (

x, y

)

∈ m

2

K

7→ x

+

y

+

xy ∈ m

K

is just the

usual multiplication in 1 + m

K

transported to m

K

via the bijection above.

We can think of this as looking at the group in a neighbourhood of the

identity 1.

Note that we called this a formal group, rather than a formal semi-group. It

turns out that the existence of identity and inverses is automatic.

Lemma. Let R be a ring and F a formal group over R. Then

F (X, 0) = X.

Also, there exists a power series i(X) ∈ X · R[[X]] such that

F (X, i(X)) = 0.

Proof. See example sheet 4.

The next thing to do is to define homomorphisms of formal groups.

Definition

(Homomorphism of formal groups)

.

Let

R

be a ring, and

F, G

be

formal groups over

R

. A homomorphism

f

:

F → G

is an element

f ∈ R

[[

X

]]

such that f(X) ≡ 0 mod X and

f(F (X, Y )) = G(f (X), f(Y )).

The endomorphisms

f

:

F → F

form a ring

End

R

(

F

) with addition +

F

given by

(f +

F

g)(x) = F (f(x), g(x)).

and multiplication is given by composition.

We can now define a formal module in the usual way, plus some compatibility

conditions.

Definition

(Formal module)

.

Let

R

be a ring. A formal

R

-module is a formal

group

F

over

R

with a ring homomorphism

R → End

R

(

F

), written,

a 7→

[

a

]

F

,

such that

[a]

F

(X) = aX mod X

2

.

Those were all general definitions. We now restrict to the case we really care

about. Let K be a local field, and q = |k

K

|. We let π ∈ O

K

be a uniformizer.

Definition

(Lubin–Tate module)

.

A Lubin–Tate module over

O

K

with respect

to π is a formal O

K

-module F such that

[π]

F

(X) ≡ X

q

mod π.

We can think of this condition of saying “uniformizer corresponds to the

Frobenius”.

Example.

The formal group

ˆ

G

m

is a Lubin–Tate

Z

p

module with respect to

p

given by the following formula: if a ∈ Z

p

, then we define

[a]

ˆ

G

m

(X) = (1 + X)

a

− 1 =

∞

X

n=1

a

n

X

n

.

The conditions

(1 + X)

a

− 1 ≡ aX mod X

2

and

(1 + X)

p

− 1 ≡ X

p

mod p

are clear.

We also have to check that

a 7→

[

a

]

F

is a ring homomorphism. This follows

from the identities

((1 + X)

a

)

b

= (1 + X)

ab

, (1 + X)

a

(1 + X)

b

= (1 + X)

ab

,

which are on the second example sheet.

The objective of the remainder of the section is to show that all Lubin–Tate

modules are isomorphic.

Definition

(Lubin–Tate series)

.

A Lubin–Tate series for

π

is a power series

e(X) ∈ O

K

[[X]] such that

e(X) ≡ πX mod X

2

, e(X) ≡ X

q

mod π.

We denote the set of Lubin–Tate series for π by E

π

.

Now by definition, if

F

is a Lubin–Tate

O

K

module for

π

, then [

π

]

F

is a

Lubin–Tate series for π.

Definition

(Lubin–Tate polynomial)

.

A Lubin–Tate polynomial is a polynomial

of the form

uX

q

+ π(a

q−1

X

q−1

+ ··· + a

2

X

2

) + πX

with u ∈ U

(1)

K

, and a

q−1

, ··· , a

2

∈ O

K

.

In particular, these are Lubin–Tate series.

Example. X

q

+ πX is a Lubin–Tate polynomial.

Example.

If

K

=

Q

p

and

π

=

p

, then (1 +

X

)

p

−

1 is a Lubin–Tate polynomial.

The result that allows us to prove that all Lubin–Tate modules are isomorphic

is the following general result:

Lemma. Let e

1

, e

2

∈ E

π

and take a linear form

L(x

1

, ··· , x

n

) =

n

X

i=1

a

i

X

i

, a

i

∈ O

K

.

Then there is a unique power series

F

(

x

1

, ··· , x

n

)

∈ O

K

[[

x

1

, ··· , x

n

]] such that

F (x

1

, ··· , x

n

) ≡ L(x

1

, ··· , x

n

) mod (x

1

, ··· , x

n

)

2

,

and

e

1

(F (x

1

, ··· , x

n

)) = F (e

2

(x

1

), e

2

(x

2

), ··· , e

2

(x

n

)).

For reasons of time, we will not prove this. We just build

F

by successive

approximation, which is not terribly enlightening.

Corollary.

Let

e ∈ E

π

be a Lubin–Tate series. Then there are unique power

series F

e

(X, Y ) ∈ O

K

[[X, Y ]] such that

F

e

(X, Y ) ≡ X + Y mod (X + Y )

2

e(F

e

(X, Y )) = F

e

(e(X), e(Y ))

Corollary.

Let

e

1

, e

2

∈ E

π

be Lubin–Tate series and

a ∈ O

K

. Then there exists

a unique power series [a]

e

1

,e

2

∈ O

K

[[X]] such that

[a]

e

1

,e

2

(X) ≡ aX mod X

2

e

1

([a]

e

1

,e

2

(X)) = [a]

e

1

,e

2

(e

2

(X)).

To simplify notation, if e

1

= e

2

= e, we just write [a]

e

= [a]

e,e

.

We now state the theorem that classifies all Lubin–Tate modules in terms of

Lubin–Tate series.

Theorem.

The Lubin–Tate

O

K

modules for

π

are precisely the series

F

e

for

e ∈ E

π

with formal O

K

-module structure given by

a 7→ [a]

e

.

Moreover, if

e

1

, e

2

∈ E

π

and

a ∈ O

K

, then [

a

]

e

1

,e

2

is a homomorphism from

F

e

2

→ F

e

1

.

If a ∈ O

×

K

, then it is an isomorphism with inverse [a

−1

]

e

2

,e

1

.

So in some sense, there is only one Lubin–Tate module.

Proof sketch.

If

F

is a Lubin–Tate

O

K

-module for

π

, then

e

= [

π

]

F

∈ E

π

by

definition, and

F

satisfies the properties that characterize the series

F

e

. So

F = F

e

by uniqueness.

For the remaining parts, one has to verify the following for all

e, e

1

, e

2

, e

3

∈ E

π

and a, b ∈ O

K

.

(i) F

e

(X, Y ) = F

e

(Y, X).

(ii) F

e

(X, F

e

(Y, Z)) = F

e

(F

e

(X, Y ), Z).

(iii) [a]

e

1

,e

2

(F

e

(X, Y )) = F

e

1

([a]

e

1

,e

2

(X), [a]

e

1

,e

2

(Y )).

(iv) [ab]

e

1

,e

3

(X) = [a]

e

1

,e

2

([b]

e

2

,e

3

(X)).

(v) [a + b]

e

1

,e

2

(X) = [a]

e

1

,e

2

(X) + [b]

e

1

,e

2

(X).

(vi) [π]

e

(X) = e(X).

The proof is just repeating the word “uniqueness” ten times.