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.