Abelian extensions via the Lubin-Tate construction

Section 4.6 Abelian extensions via the Lubin-Tate construction

Reference.

[37], I.2 and I.3; [4], IV.

Subsection Formal group laws

The idea here is to imitate Kronecker-Weber by identifying some kind of group object whose torsion points generate abelian extensions of a local field. We define the group object first.

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Definition 4.6.1.

A one-parameter commutative formal group law over a commutative ring

A

is a formal power series

F \in A [[X, Y]]

such that:

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$F \equiv X + Y (\mod (X, Y)^{2});$
$F (X, F (Y, Z)) = F (F (X, Y), Z)$ (associativity);
there exists a unique series $i_{F} \in X A [[X]]$ such that $F (X, i_{F} (X)) = 0$ (existence of inverses);
$F (X, Y) = F (Y, X)$ (commutativity).

Note that throughout this definition, we are using the fact that we can substitute a power series with constant term 0 (in any number of variables) into another power series to get a meaningful result.

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Suppose now that

A = o_{K}

where

K

is a local field. Then we can also substitute elements of the maximal ideal

m_{K}

into

F,

and this will define an exotic group structure on this set.

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Example 4.6.2.

Take

F = X + Y

(the formal additive group). When

A = o_{K},

this recovers the usual group structure on

m_{K} .

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Example 4.6.3.

Take

F = X + Y + X Y

(the formal multiplicative group). When

A = o_{K},

this corresponds to the group structure on

1 + m_{K}

via the bijection

x \mapsto x + 1

from

m_{K}

1 + m_{K} .

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Subsection Lubin-Tate formal groups

We next introduce a condition on a formal group law analogous to complex multiplication for an elliptic curve.

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Definition 4.6.4.

Let

F_{1}, F_{2}

be a formal group law over a ring

A .

A homomorphism from

F_{1}

F_{2}

is a power series

h \in T A [[T]]

such that

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h (F_{1} (X, Y)) = F_{2} (h (X), h (Y)) .

When

F_{1} = F_{2} = F

we call this an endomorphism of

F .

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There is a natural notion of addition of endomorphisms: if

f_{1}

and

f_{2}

are endomorphisms of

F,

then so is

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f_{1} +_{F} f_{2} = F (f_{1}, f_{2}) .

There is also a notion of multiplication of endomorphisms, which coincides with composition of formal power series. In particular this is in general not commutative.

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Example 4.6.5.

For any formal group law

F,

there exist unique endomorphisms

[n]

for

n \in Z

such that:

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$[0] = 0$ and $[1] = T;$
$[n_{1}] + [n_{2}] = [n_{1} + n_{2}];$
$[n_{1}] \circ [n_{2}] = [n_{1} n_{2}] .$

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Definition 4.6.6.

Let

K

be a local field with residue field of order

q

and fix a generator

π

m_{K} .

Let

F_{π}

be the subset of

f \in T o_{K} [[T]]

satisfying the following conditions:

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$f \equiv π T (\mod T^{2});$
$f \equiv T^{q} (\mod π) .$

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Proposition 4.6.7.

Let

K

be a local field with residue field of order

q

and fix a generator

π

m_{K} .

For any two

f, g \in F_{π},

for any homogeneous linear polynomial

ϕ_{1}

in the variables

X_{1}, \dots, X_{n}

over

o_{K},

there exists a unique power series

ϕ \in o_{K} [[X_{1}, \dots, X_{n}]]

such that:

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$ϕ \equiv ϕ_{1} (\mod (X_{1}, \dots, X_{n})^{2});$
$f (ϕ (X_{1}, \dots, X_{n})) = ϕ (g (X_{1}), \dots, g (X_{n})) .$

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Proof.

Writing

ϕ = \sum_{i = 1}^{\infty} ϕ_{i}

where

ϕ_{i}

is homogeneous of degree

i,

we construct the

ϕ_{i}

inductively. Suppose that

ϕ_{1}, \dots, ϕ_{i}

are already specified (initially with

i = 1

), and set

ϕ_{\leq i} = ϕ_{1} + \dots + ϕ_{i} .

By looking at Taylor expansions, we obtain

\begin{aligned} f ((ϕ_{\leq_{i}} + ϕ_{i + 1}) (X_{1}, \dots, X_{N})) & \equiv f (ϕ_{\leq i} (X_{1}, \dots, X_{n})) + π ϕ_{i + 1} (\mod (X_{1}, \dots, X_{n})^{i + 2}) \\ (ϕ_{\leq_{i}} + ϕ_{i + 1}) (g (X_{1}), \dots, g (X_{n})) & \equiv ϕ_{\leq i} (g (X_{1}), \dots, g (X_{n})) + π^{i + 1} ϕ_{i + 1} (\mod (X_{1}, \dots, X_{n})^{i + 2}) . \end{aligned}

This means that we must ensure that

ϕ_{i + 1} \equiv \frac{f (ϕ_{\leq i} (X_{1}, \dots, X_{n})) - ϕ_{\leq i} (g (X_{1}), \dots, g (X_{n}))}{(π^{i} - 1) π} (\mod (X_{1}, \dots, X_{n})^{i + 2}) .

But in fact the numerator of the fraction is divisible by

π :

modulo

π

both

f

and

g

are congruent to

T^{q}

and this map is a ring homomorphism. We thus obtain a unique choice of

ϕ_{i + 1}

that allows the construction to continue.

Corollary 4.6.8.

Let

K

be a local field with residue field of order

q

and fix a generator

π

m_{K} .

For any

f \in F_{π},

there exists a unique formal group

F

over

o_{K}

admitting

f

as an endomorphism. Moreover, for every

a \in o_{K},

F

admits a unique endomorphism

[a] \equiv α T (\mod T^{2})

which commutes with

f

(in particular,

[π] = f

), and these satisfy

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[a + b] = [a] +_{F} [b], [a b] = [a] \circ [b] .

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Proof.

We get

F

by applying Proposition 4.6.7 with

g = f

and

ϕ_{1} (X, Y) = X + Y .

Since

ϕ_{1} (X, Y)

also equals

Y + X,

the uniqueness aspect of Proposition 4.6.7 implies commutativity. We deduce existence of inverses by similarly taking

ϕ_{1} (X) = - X .

We deduce associativity by similarly taking

ϕ_{1} (X, Y, Z) = X + Y + Z

and invoking the uniqueness aspect. We produce

[a]

by similarly taking

ϕ_{1} (X) = a X;

we check that

[a + b] = [a] +_{F} [b]

and

[a b] = [a] \circ [b]

by invoking the uniqueness aspect. (Compare [37], Proposition I.2.12 and I.2.14.)

Remark 4.6.9.

For each

f \in F_{π},

we can find a unit

u \in o_{K} [[T]]^{\times}

with

u \equiv 1 (\mod π)

such that

u f

(which is also in

F_{π})

is a monic polynomial of degree

q .

This follows for example from the Weierstrass preparation theorem.

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Similarly, if we were to relax the second condition in Definition 4.6.6 to require only that

f \equiv T^{q} (\mod π, T^{q + 1}),

then we could again always find a unit

u \in o_{K} [[T]]^{\times}

with

u \equiv 1 (\mod π, T)

such that

u f

is a monic polynomial of degree

q .

(This relaxation would correspond to the definition of the height of a formal group.)

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Subsection Abelian extensions from a Lubin-Tate formal group

Definition 4.6.10.

Choose

K, f, F

as in Corollary 4.6.8. Let

Λ_{f}

be the

o_{K}

-module whose underlying set consists of all

α \in \overset{―}{K}

of absolute value less than 1, with addition given by

α + β := F (α, β)

and the action of

a \in o_{K}

β

given by

[a] (β) .

Let

K_{π}

be the subfield of

\overset{―}{K}

generated by the

[π^{k}]

-torsion elements of

Λ_{f}

k

runs over all positive integers. Note that this notation was used previously in Remark 4.1.9, and we will eventually show that these usages coincide (Theorem 4.6.16).

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Remark 4.6.11.

When applying Definition 4.6.10, we may use the fact that by Proposition 4.6.7, for any other series

g

satisfying

g \equiv π T (\mod T^{2})

and

g \equiv T^{q} (\mod π),

we get a canonical isomorphism

Λ_{f} ≅ Λ_{g}

o_{K}

-modules; in particular, this isomorphism is compatible with the action of

Gal (\overset{―}{K} / K) .

This lets us simplify some arguments by reducing to the case where

f

is a polynomial, or even to the case

f = π T + T^{q} .

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Lemma 4.6.12.

With notation as in Definition 4.6.10, the

[π^{k}]

-torsion submodule of

Λ_{f}

is isomorphic to

o_{K} / π^{k} .

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Proof.

For

k = 1

we see that the

[π]

-torsion submodule consists of

q

elements (by Remark 4.6.11 we need only check this for

f = π T + T^{q}

for which it is obvious), and hence is a one-dimensional

o_{K} / (π)

-vector space. Since

[π] : Λ_{f} \to Λ_{f}

is surjective, the general case now follows using Nakayama’s lemma (or see [37], lemma I.3.3).

Corollary 4.6.13.

The extension

K_{π} / K

is abelian and totally ramified; more precisely, there is a canonical isomorphism

Gal (K_{π} / K) \to o_{K}^{\times} .

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Proof.

The inclusion

Gal (K_{π} / K) \to o_{K}^{\times}

is immediate from Lemma 4.6.12. To check surjectivity and the totally ramified property, we check that the torsion points of exact order

π^{k}

are the roots of an Eisenstein polynomial over

o_{K} .

Again by Remark 4.6.11 we need only check this for

f = π T + T^{q},

in which case we can write

[π]^{k} = \sum_{i = 0}^{k} π^{k - i} T^{q^{i}} + \dots

and the omitted monomials do not contribute to the Newton polygon (each one is strictly divisible by one of the summands). By factoring according to the slopes of the Newton polygons (using Hensel’s lemma), we see that the roots of

[π^{k}]

which are not roots of

[π^{k - 1}]

are the roots of an Eisenstein polynomial over

o_{K} .

Example 4.6.14.

For

K = Q_{p},

we can take

π = p

and

f = (1 + T)^{p} - 1 .

In this case

K_{π} = ⋃_{k} Q (ζ_{p^{k}})

is the

p

-part of the cyclotomic tower.

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Subsection Local reciprocity from formal groups

We now fulfill a promise made for

K = Q_{p}

in Example 4.1.4, and use the Lubin-Tate construction to completely pin down the local reciprocity map. This can be done without even assuming the existence of a local reciprocity map, but we will not bother with that here.

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Lemma 4.6.15.

Let

K

be a local field. Let

π_{1}, π_{2}

be two uniformizers of

K

and set

a := π_{2} / π_{1} \in o_{K}^{\times} .

For

i = 1, 2,

pick

f_{i} \in F_{π_{i}}

and let

F_{i}

be the formal group associated to

f_{i}

via Corollary 4.6.8. Let

L / K

be the maximal unramified extension, let

R

be the completion of

o_{L},

let

σ : R \to R

be the Frobenius automorphism, and choose

ϵ \in R^{\times}

such that

ϵ^{σ} = a ϵ

(see Exercise 4). Then as formal groups over

R,

there is a unique isomorphism

θ

from

F_{1}

F_{2}

such that

θ \equiv ϵ T (\mod T^{2})

and

θ^{σ} = θ \circ [a]_{f_{1}}

(where

θ^{σ}

means apply

σ

to each coefficient of

θ

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Proof.

See Exercise 3.

Theorem 4.6.16. Lubin–Tate.

With notation as in Definition 4.6.10,

K^{ab} = K_{π} K^{unr} .

Moreover, the local reciprocity map may be written in the form

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ϕ_{K} : o_{K}^{\times} \times π^{Z} \to Gal (K_{π} / K) \times Gal (K^{unr} / K)

with the first factor mapping via the inverse of the isomorphism

o_{K}^{\times} \to Gal (K_{π} / K)

from Corollary 4.6.13 and the second factor mapping

π

to the Frobenius in

Gal (K^{unr} / K) .

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Proof.

By Corollary 4.6.13,

K_{π}

is a totally ramified abelian extension of

K

which is linearly disjoint from

K^{unr}

with Galois group

o_{K}^{\times} .

Hence

K_{π} K^{unr}

is an abelian extension of

K

with Galois group

Z \times o_{K}^{\times};

using local reciprocity (Theorem 4.1.2) we see that this extension must be all of

K^{ab} .

Now note that if

α \in Λ_{f}

is a

π^{k}

-torsion point, then

{Norm}_{K (α) / K} (α) = π .

Consequently, our proposed formula for

ϕ_{K}

has the property that

ϕ_{K} (π)

fixes

K_{π} .

It will thus suffice to show that

ϕ_{K} (ϖ)

fixes

K_{ϖ}

for any other uniformizer

ϖ

K,

as then we can conclude as in Remark 4.1.9. (In other words, we need to show that our formula does not depend on the choice of

π .

)

Set

a := ϖ / π \in o_{K}^{\times} .

For compatibility with the notation in Lemma 4.6.15, set

π_{1} := π,

π_{2} := ϖ .

Then

K_{ϖ}

is generated by

θ (α)

for

α

running over

K_{π},

and

\begin{aligned} ϕ_{K, 2} (π_{1}) (θ (α)) & = [a^{- 1}]_{f_{1}} (θ (α))^{σ} \\ = [a^{- 1}]_{f_{1}} (θ^{σ} (α)) \\ = (θ^{σ}) ([a^{- 1}]_{f_{1}} (α)) \\ = (θ \circ [a]_{f_{1}}) ([a^{- 1}]_{f_{1}} (α)) = θ (α) . \end{aligned}

(Compare [37], I.3.)

Remark 4.6.17.

We recall Remark 2.2.10: there is no analogue of the Lubin-Tate construction over a number field, except when

K = Q

for which the multiplicative group can be used (recovering the Kronecker-Weber theorem) or when

K

is imaginary quadratic for which an elliptic curve with complex multiplication by

K

can be used (see [10]). When

K

is a function field then there does exist such an analogue: for

K = F_{p} (t)

this was discovered by Carlitz, and this was (rediscovered and) generalized to arbitrary

K

by Drinfeld (see [19]).

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Exercises Exercises

1.

Prove that in Definition 4.6.1, the existence of inverses is a consequence of the other conditions.

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2.

Prove that in Definition 4.6.1, if the ring

A

has characteristic 0, then commutativity is a consequence of the other conditions.

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Hint.

First note that we can check commutativity after replacing

A

A,

so we may assume that

A

is a

Q

-algebra. In that case, show that

F

is isomorphic to the formal additive group.

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3.

Prove Lemma 4.6.15.

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Hint.

Use an inductive argument in the manner of Proposition 4.6.7, but this time making repeated use of Exercise 4 to match up increasingly many coefficients of

θ^{σ}

and

θ \circ [a]_{f_{1}} .

For more details, see [37], Proposition I.3.10.

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