An "abstract" reciprocity map

Section 7.3 An “abstract” reciprocity map

Reference.

[37] VII.5; [38] VI.4, but only loosely.

We next manufacture a canonical isomorphism

Gal (L / K)^{ab} \to C_{K} / {Norm}_{L / K} C_{L}

for any finite extension

L / K

of number fields, in which

C_{K}

and

C_{L}

are the corresponding idèle class groups (Theorem 7.3.8). However, we won’t yet know it agrees with our proposed reciprocity map, which is the product of the local reciprocity maps (Definition 6.4.6). We’ll come back to this point after we establish the existence theorem (see Section 7.5).

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Subsection Abstract unit groups and the class field axiom

We will be applying abstract class field theory with

k = Q

and

\overset{―}{k} = \overset{―}{Q},

an algebraic closure of

Q .

We first specify the

Gal (\overset{―}{Q} / Q)

-module

A

which will give rise to abstract unit groups.

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Definition 7.3.1.

Set

A := ⋃_{K} C_{K};

by Corollary 6.3.7,

A_{K} = C_{K}

for every number field

K .

Our earlier calculations (Theorem 7.1.2, Theorem 7.2.9) imply that the class field axiom is satisfied: for

L / K

a cyclic extension of number fields,

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# H_{T}^{0} (Gal (L / K), C_{L}) = [L : K], # H_{T}^{1} (Gal (L / K), C_{L}) = 1 .

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Remark 7.3.2.

In Definition 7.3.1, it will follow from the reciprocity law (Theorem 6.4.3) that the group

H_{T}^{0} (Gal (L / K), C_{L})

is cyclic. However, the class field axiom does not require advance knowledge of this.

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Subsection Cyclotomic extensions and abstract ramification theory

The cyclotomic extensions of a number field play a role in class field theory analogous to the role played by the unramified extensions in local class field theory. This makes it essential to make an explicit study of them for use in proving the main results. However, we will not need the Kronecker-Weber theorem (Theorem 1.1.2); instead, we will recover it as part of the reciprocity law.

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We first make a distinction which is of marginal significance in the totality of number theory, but is critical for our use of the machinery of abstract class field theory.

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Definition 7.3.3.

The extension

⋃_{n} Q (ζ_{n})

Q

obtained by adjoining all roots of unity has Galois group

{\hat{Z}}^{*} = \prod_{p} Z_{p}^{*} .

That group has a lot of torsion, since each

Z_{p}^{*}

contains a torsion subgroup of order

p - 1

(or 2, if

p = 2

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If we take the fixed field for the torsion subgroup of

Z^{*},

we get a slightly smaller extension, which I’ll call the small cyclotomic extension of

Q

and denote

Q^{smcy} .

Its Galois group is isomorphic to

\prod_{p} Z_{p} = \hat{Z},

but not canonically so.

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For

K

a number field, define

K^{smcy} := K Q^{smcy} .

Then

Gal (K^{smcy} / K) ≅ \hat{Z}

as well, even if

K

contains some extra roots of unity: the point is that any open subgroup of

\hat{Z}

is also isomorphic to

\hat{Z}

(but again not canonically so).

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With this definition in hand, we can set up the homomorphism

d

needed to define abstract ramification theory for the base field

k = Q .

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Definition 7.3.4.

Choose an isomorphism of

Gal (Q^{smcy} / Q)

with

\hat{Z};

our results are not going to depend on the choice (see Remark 7.3.11). That gives a continuous surjection

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d : Gal (\overset{―}{Q} / Q) \to Gal (Q^{smcy} / Q) ≅ \hat{Z};

recall that this means we are going to regard

Q^{smcy} / Q

as the “maximal unramified extension” of

Q .

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As in the general setup, for any finite extension

L / K

of number fields, we define the abstract ramification index

e_{L / K}

and the abstract inertia degree

f_{L / K}

by setting

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f_{L / K} := [L \cap Q^{smcy} : K \cap Q^{smcy}], e_{L / K} := \frac{[L : K]}{f_{L / K}} .

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Subsection An abstract henselian valuation

To complete the setup of abstract class field theory, we need an abstract henselian valuation

v : C_{Q} \to \hat{Z}

with respect to

d .

Recall from Definition 5.1.8 that this means:

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$v (C_{Q})$ is a subgroup $Z$ of $\hat{Z}$ containing $Z$ with $Z / n Z ≅ Z / n Z$ for all positive integers $n;$
$v ({Norm}_{K / Q} C_{K}) = f_{K / Q} Z$ for all finite extensions $K / Q .$

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Definition 7.3.5.

To define the map

v,

we write

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I_{Q} = Q^{*} \times R^{+} \times {\hat{Z}}^{*}

as in Remark 6.2.12. We then define

v

as the projection onto the third factor followed by the projection

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{\hat{Z}}^{*} ≅ Gal (Q^{cyc} / Q) \to Gal (Q^{smcy} / Q) ≅ \hat{Z} .

The first condition of Definition 5.1.8 holds by construction. We will check the second condition using Artin reciprocity for cyclotomic extensions.

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Lemma 7.3.6.

The map

v

defined in Definition 7.3.5 is a henselian valuation in the sense of Definition 5.1.8 (with respect to the map

d

from Definition 7.3.4).

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Proof.

Since we already know from Definition 7.3.5 that

v

factors through

C_{Q}

and surjects onto

\hat{Z},

it suffices to check that for every number field

K,

v ({Norm}_{K / Q} I_{K}) = f_{K / Q} \hat{Z} .

We may establish this by checking that the map

I_{K} \overset{{Norm}_{K / Q}}{\to} I_{Q} \to Gal (Q^{cyc} / Q)

has image

Gal (K^{cyc} / K) \subseteq Gal (Q^{cyc} / Q),

as then we get the desired condition by projecting from

Gal (Q^{cyc} / Q)

Gal (Q^{smcy} / Q) .

Note that for

K = Q,

this follows from Artin reciprocity for cyclotomic extensions (Definition 1.1.7).

In one direction, the fact that

I_{K}

maps into

Gal (K^{cyc} / K)

is a corollary of local reciprocity (Theorem 4.1.2) plus Artin reciprocity for cyclotomic extensions as used above.

In the other direction, the same logic shows that for each positive integer

n,

the image of

I_{K}

Gal (K (ζ_{n}) / K)

equals the image of the classical Artin map for

K (ζ_{n}) / K;

it will thus suffice to check that these maps are surjective. It is convenient to deduce this from the First Inequality; see Proposition 7.3.7 below.

Here is the consequence of the First Inequality used in the proof of Lemma 7.3.6.

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Proposition 7.3.7.

For

L / K

an abelian extension of number fields, the Artin map always surjects onto

Gal (L / K) .

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Proof.

Let

H

be the image of the Artin map; the fixed field

M

H

has the property that all but finitely many primes of

K

split completely in

M .

We’ve already seen that this contradicts the First Inequality unless

M = K

(Corollary 7.1.15, or if you prefer Proposition 7.2.3).

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Subsection Consequences of abstract CFT

We now apply abstract class field theory to obtain an “abstract adelic reciprocity law”.

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Theorem 7.3.8. Abstract adelic reciprocity law.

For every Galois extension

L / K

of number fields, we obtain an isomorphism

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r_{L / K}^{'} : C_{K} / {Norm}_{L / K} C_{L} \tilde{\to} Gal (L / K)^{ab} .

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Proof.

The hypotheses of abstract class field theory are satisfied by taking

k = Q

\overset{―}{k} = \overset{―}{Q};

A = ⋃_{K} C_{K}

as in Definition 7.3.1 (using Theorem 7.1.2 and Theorem 7.2.9 to verify the class field axiom);

d : Gal (\overset{―}{Q} / Q) \to \hat{Z}

as in Definition 7.3.4; and

v : C_{Q} \to \hat{Z}

as in Definition 7.3.5 (using Lemma 7.3.6). We may thus apply Theorem 5.3.9 to conclude.

Definition 7.3.9.

By Proposition 5.2.7, the maps

r_{L / K}^{'}

from Theorem 7.3.8 fit together to give a map

r_{K}^{'} : C_{K} \to Gal (K^{ab} / K);

but we do not yet know that this coincides with the product of the local reciprocity maps, so we cannot yet recover Artin reciprocity. However, we can at least deduce the norm limitation theorem (Theorem 6.4.5). See also Remark 7.3.11 below.

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Theorem 7.3.10. Norm limitation theorem.

L / K

is a finite extension of number fields and

M = L \cap K^{ab},

then

{Norm}_{L / K} C_{L} = {Norm}_{M / K} C_{M} .

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Proof.

Apply Corollary 5.3.11.

Remark 7.3.11.

Although we do not have a complete description of the isomorphism

r_{L / K}^{'}

coming from abstract class field theory, we do know one specific fact about this map: for “unramified” extensions

L / K

(i.e.,

L \subseteq K^{smcy}

), the “Frobenius” in

Gal (L / K)

maps to a “uniformizer” in

C_{K} .

That is, the element of

Gal (L / K)

coming from the element of

Gal (K^{smcy} / K)

which maps to 1 under

d_{K}

corresponds via reciprocity to the element of

C_{K}

which maps to 1 under

v_{K} .

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The broader point here is that the definitions of both

d

and

v

involve the same artificial choice of an isomorphism

Gal (Q^{smcy} / Q) ≅ \hat{Z},

which thus does not affect the reciprocity map. Compare Remark 5.1.11 and Exercise 4.

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