Adèles and idèles in field extensions

Section 6.3 Adèles and idèles in field extensions

Reference.

[38], VI.1 and VI.2.

Up to now, we have considered the ring of adèles associated to a single number field. We now turn to the effect of a field extension on this construction.

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Subsection Adèles in field extensions

Definition 6.3.1.

L / K

is an extension of number fields, we get an embedding

A_{K} ↪ A_{L}

as follows: given

α \in A_{K},

each place

w

L

restricts to a place

v

K,

so it makes sense to declare that the

w

-component of the image of

α

shall equal

α_{v} .

This embedding induces an inclusion

I_{K} ↪ I_{L}

of idèle groups. These embeddings are compatible with the topologies in both cases.

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All automorphisms of

L / K

act naturally (and continuously) on

A_{L}

and

I_{L} .

More generally, if

g \in Gal (\overset{―}{K} / K),

then

g

maps

L

to some other extension

L^{g}

K,

and

g

induces isomorphisms

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A_{L} \to A_{L^{g}}, I_{L} ≅ I_{L^{g}}, C_{L} \to C_{L^{g}} .

Explicitly, if

(α_{w})_{w} \in A_{L}

and

g \in G,

then

g

maps the completion

L_{w}

L

to a completion

L_{w^{g}}

L^{g} .

(Remember that a place

w

L

corresponds to an absolute value

| \cdot |_{w}

L;

the absolute value

| \cdot |_{w^{g}}

L^{g}

is given by

| a^{g} |_{w^{g}} = | a |_{w} .

Compare Definition 6.1.1 which was a special case of this.)

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Remark 6.3.2.

A more conceptual interpretation of the previous discussion is to identify

A_{L}

with the tensor product

A_{K} \otimes_{K} L .

In particular, this is a good way to see the Galois action on

A_{L} .

See Exercise 1.

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Remark 6.3.3.

When

K

is totally real, it is possible to show that every automorphism of

A_{K}

is induced by an automorphism of

K

over

Q,

even if we ignore topology and consider automorphisms of the underlying ring which need not be continuous. See Exercise 4. This breaks down when

K

has complex places because

C

has many automorphisms as a field without topology: the automorphism group acts transitively on

C ∖ \overset{―}{Q} .

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Subsection Trace and norm

Definition 6.3.4.

For

L / K

an extension of number fields, we define the trace map

{Trace}_{L / K} : A_{L} \to A_{K}

and the norm map

{Norm}_{L / K} : I_{L} \to I_{K}

by the formulas

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{Trace}_{L / K} (x) = \sum_{g} x^{g}, {Norm}_{L / K} (x) = \prod_{g} x^{g}

where

g

runs over coset representatives of

Gal (\overset{―}{K} / L)

Gal (\overset{―}{K} / K) .

Here the sum and product take place in the adèle and idèle rings of the Galois closure of

L

over

K;

in particular, if

L / K

is Galois,

g

simply runs over

Gal (L / K)

and the arithmetic takes place in

A_{L} .

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In terms of components, these definitions translate as

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\begin{aligned} ({Trace}_{L / K} (α))_{v} & = \sum_{w | v} {Trace}_{L_{w} / K_{v}} (α_{w}) \\ ({Norm}_{L / K} (α))_{v} & = \prod_{w | v} {Norm}_{L_{w} / K_{v}} (α_{w}) . \end{aligned}

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The trace and norm as defined here are compatible with the usual definitions for principal adèles/idèles. In particular, the norm induces a map

{Norm}_{L / K} : C_{L} \to C_{K} .

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Remark 6.3.5.

You can also define the trace of an adèle

α \in A_{L}

as the trace of addition by

α

as an endomorphism of the finite free

A_{K}

-module

A_{L},

and the norm of an idèle

α \in I_{L}

as the determinant of multiplication by

α

as an automorphism of the finite free

A_{K}

-module

A_{L} .

(Yes, the action is on the adèles in both cases. Remember from Remark 6.2.3 that idèles should be thought of as automorphisms of the adèles, not as elements of the adèle ring, in order to topologize them correctly.)

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Subsection Idèle groups and class groups

Proposition 6.3.6.

L / K

is a Galois extension with Galois group

G,

then

A_{L}^{G} = A_{K}

and

I_{L}^{G} = I_{K} .

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Proof.

v

is a place of

K,

then for each place

w

K

above

v,

the decomposition group

G_{w}

w

is isomorphic to

Gal (L_{w} / K_{v}) .

So if

(α)

is an adèle or idèle which is

G

-invariant, then

α_{w}

Gal (L_{w} / K_{v})

-invariant for each

w,

so belongs to

K_{v} .

Moreover,

G

acts transitively on the places

w

above

v,

α_{w} = α_{w^{'}}

for any two places

w, w^{'}

above

v .

Thus

(α)

is an adèle or idèle over

K .

This has the following consequence for idèle class groups. Note that for

L / K

any extension of number fields, we can see that

C_{K} \to C_{L}

is injective from the fact that

L^{*} \cap I_{K} = K^{*}

within

I_{L},

which follows from the fact that

L \cap A_{K} = K

within

A_{L}

(e.g., by Exercise 2).

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Corollary 6.3.7. Galois descent.

L / K

is a Galois extension with Galois group

G,

then

G

acts on

C_{L},

and the

G

-invariant elements are precisely

C_{K} .

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Proof.

We start with an exact sequence

1 \to L^{*} \to I_{L} \to C_{L} \to 1

G

-modules. Taking

G

-invariants, we get a long exact sequence

1 \to (L^{*})^{G} = K^{*} \to (I_{L})^{G} = I_{K} \to C_{L}^{G} \to H^{1} (G, L^{*}),

and the last term is 1 by Theorem 90 (Lemma 1.2.3). So we again have a short exact sequence, and

C_{L}^{G} ≅ I_{K} / K^{*} = C_{K} .

Remark 6.3.8.

There is no analogue of Corollary 6.3.7 for ideal class groups: the map

Cl (K) \to Cl (L)^{G}

is neither injective nor surjective in general (Exercise 5). This is our first hint of why the idèle class group will be a more convenient target for a reciprocity map than the ideal class group.

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The group

Cl (L)^{G}

is classically known as the group of ambiguous classes of

L / K .

This is related to the concept of a Pólya field from Remark 2.3.13; see [5].

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Exercises Exercises

1.

Let

L / K

be a finite extension of number fields. Prove that the natural map

A_{K} \otimes_{K} L \to A_{L}

is an isomorphism. In other words, if

α_{1}, \dots, α_{n}

form a basis of

L

as a

K

-vector space, then they also form a basis of

A_{L}

as an

A_{K}

-module. (Bonus observation: if

L / K

is Galois with Galois group

G,

then

A_{L}

is an induced

G

-module.)

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Hint.

Show first that for any place

v

K,

any basis of

L

as a

K

-vector space also forms a basis of

\prod_{w | v} L_{w}

as a

K_{v}

-vector space.

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2.

Let

K

be a number field. Prove that the integral closure of

Q

A_{K}

is equal to

K .

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Hint.

Suppose to the contrary that the integral closure contains some larger number field

L .

By Corollary 2.4.12, there are infinitely many primes of

K

which do not split in

L;

use one of these to obtain a contradiction.

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3.

Prove the following converse to Exercise 2: if

L / K

is an extension of number fields such that

K + A_{L, S} = A_{L}

for some finite set of places

S

L,

prove that

K = L .

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Hint.

Use the fact that there are infinitely many primes of

K

that do not split completely in

L

(Corollary 2.4.12).

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4.

Let

K

be a totally real Galois number field. Prove that the automorphism group of

A_{K}

as a bare ring (ignoring its topology) equals

Gal (K / Q) .

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Hint.

By Exercise 2, any automorphism of

A_{K}

acts on

K .

We thus have homomorphisms

Gal (K / Q) \to Aut (A_{K}) \to Gal (K / Q)

whose composition is the identity; it thus remains to check that

Aut (A_{K}) \to Gal (K / Q)

is injective. For this, apply Exercise 5 as in Exercise 6.

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5.

Let

L / K

be a Galois extension of number fields with Galois group

G .

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Give an example for which $Cl (K) \to Cl (L)^{G}$ fails to be injective.
Give an example for which $Cl (K) \to Cl (L)^{G}$ fails to be surjective.

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Hint.

One way to produce failures of injectivity is via the principal ideal theorem (Theorem 2.3.1). One way to produce failures of surjectivity is to find quadratic fields with class group

Z / 4 Z .

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