Idèles and class groups

Section 6.2 Idèles and class groups

Reference.

[37]; [38], VI.1 and VI.2; [34], VII.

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We now shift from additive to multiplicative considerations.

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Subsection Idèles

Definition 6.2.1.

Let

K

be a number field and let

A_{K}

be the ring of adèles associated to

K

(Definition 6.1.10). We define the group of idèles

I_{K}

associated to

K

as the group of units of the ring

A_{K} .

In other words, an element of

I_{K}

is a tuple

(α_{v}),

one element of

K_{v}^{*}

for each place

v

K,

such that

α_{v} \in o_{K_{v}}^{*}

for all but finitely many finite places

v .

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As a set,

I_{K}

is the restricted product of the pairs

(K_{v}^{*}, {1})

for infinite places

v

and

(K_{v}, o_{K_{v}}^{*})

for finite places

v .

We use this interpretation to give

I_{K}

the structure of a locally compact topological group.

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Definition 6.2.2.

For

S

a finite set, let

I_{K, S}

be the set of

x \in I_{K}

for which

x_{v} \in o_{K_{v}}^{*}

for each finite place

v \notin S;

then

I_{K} = ⋃_{S} I_{K, S} .

By analogy with Definition 6.1.14, the elements of

I_{K, S}

can be thought of as “adelic

S

-units”.

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Remark 6.2.3. Warning.

While the embedding of the idèle group

I_{K}

into the adèle group

A_{K}

is continuous, the restricted product topology on

I_{K}

does not coincide with the subspace topology for the embedding! For example, each set

I_{K, S}

is open in

I_{K}

but is not the intersection with an open subset of

A_{K} .

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This is a more serious version of the same issue that came up in Exercise 6, and the same fix applies: namely, identify

I_{K}

with

{GL}_{1} (A_{K})

and topologize it accordingly (Exercise 1).

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Subsection The idèle class group

Definition 6.2.4.

For each

α \in K^{*},

the principal adèle

α \in A_{K}

is an idèle, so we have an embedding

K^{*} ↪ I_{K} .

We refer to elements of the image of this embedding as principal idèles. Define the idèle class group of

K

as the quotient

C_{K} := I_{K} / K^{*}

of the idèles by the principal idèles.

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The terminology idèle class group is justified on account of the following construction.

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Definition 6.2.5.

There is a homomorphism from

I_{K}

to the group of fractional ideals of

K :

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(α_{ν})_{ν} \mapsto \prod_{p} p^{v_{p} (α_{p})},

which is continuous for the discrete topology on the group of fractional ideals. (Note that we are ignoring the infinite places; that is, this map factors through

I_{K}^{fin}

viewed as a quotient of

I_{K} .

) By unique factorization of fractional ideals, this homomorphism is surjective; its kernel is precisely

I_{K, S}

for

S

the set of infinite places.

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Under this map, the principal idèle corresponding to

α \in K

maps to the principal ideal generated by

α .

Thus we have a surjection

C_{K} \to Cl (K)

with kernel

I_{K, S} K^{*} / K^{*} .

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Remark 6.2.6.

What are the open subgroups of

I_{K} ?

For each formal product

m

of places, one gets an open subgroup of idèles

(α_{v})_{v}

such that:

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if $v$ is a real place occurring in $m,$ then $α_{v} > 0;$
if $v$ is a finite place corresponding to the prime $p,$ occurring to the power $e,$ then $α_{v} \equiv 1 (\mod p^{e}) .$

This then projects to an open subgroup of

C_{K},

the quotient by which is the ray class group of modulus

m!

(Here we are using the fact that any element of

C_{K}

can be represented by an element of

I_{K}

which has trivial valuation at any finite place dividing

m .

See Definition 7.2.1 for an elaboration of this point.)

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Consequently, the quotients of

C_{K}

by open subgroups are isomorphic to (and in bijection with) the generalized ideal class groups, with the added convenience that they are all quotients of one group (not a group that depends on

m

). This correspondence is what will allow us to translate between the classical and adelic versions of Artin reciprocity.

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Subsection Compactness and consequences

Definition 6.2.7.

By the product formula (Proposition 6.1.12), we get a well-defined norm map

| \cdot | : C_{K} \to R_{+}^{*} .

Let

C_{K}^{0}

be the kernel of the norm map; then

C_{K}^{0}

also surjects onto

Cl (K) .

(The surjection onto

Cl (K)

ignores the infinite places, so you can adjust there to force norm

1 .

)

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Proposition 6.2.8.

The group

C_{K}^{0}

is compact.

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Proof.

We first show (see Exercise 2) that there exists a real number

c > 1

with the following property: every idèle of norm

1

is (multiplicatively) congruent modulo

K^{*}

to an idèle whose components all have norms in

[c^{- 1}, c] .

The set of idèles with each component having norm in

[c^{- 1}, c]

is the product of “annuli” in the archimedean places and finitely many of the nonarchimedean places, and the group of units in the rest. (Most of the nonarchimedean places don’t have any absolute values strictly between

1

and

c .

) This is a compact set, the set of idèles therein of norm

1

is a closed subset and so is also compact, and the latter set surjects onto

C_{K}^{0},

so that’s compact too.

While Proposition 6.2.8 may look innocuous, it actually implies two key theorems of algebraic number theory which are traditionally proved using the Minkowski lattice construction. (In fact we are really doing the same arguments in slightly different language.)

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Corollary 6.2.9.

The class group

Cl (K)

K

is finite.

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Proof.

The group

C_{K}^{0}

is compact by Proposition 6.2.8 and it surjects onto

Cl (K),

so the latter must also be compact for the discrete topology, and hence finite.

Corollary 6.2.10.

There exists a finite set

S

of places of

K

such that

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I_{K} = I_{K, S} K^{*} .

In particular, for any such

S,

C_{K}

is a quotient of

I_{K, S} .

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Proof.

Since

Cl (K) = I_{L} / K^{*}

is finite, it is generated by some finite set of primes. By taking

S

to include the corresponding places, we achieve the desired effect.

Corollary 6.2.11. Dirichlet’s units theorem.

The group of units of

o_{K}

fits into an exact sequence

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0 \to μ_{K} \to o_{K}^{*} \to Z^{r + s - 1} \to 0

in which

μ_{K}

is the (finite cyclic) group of roots of unity of

K

and

r

and

s

are the number of real and complex places, respectively. More generally, for any finite set

S

of places containing all infinite places, the group of units of the ring

o_{K, S}

S

-integers fits into an exact sequence

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0 \to μ_{K} \to o_{K, S}^{*} \to Z^{# S - 1} \to 0 .

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Proof.

Define the map

\log : I_{K, S} \to R^{# S}

by taking log of the absolute value of the norm of each component in

S

(normalizing as in Definition 6.1.11). By the product formula (Proposition 6.1.12), this map carries

o_{K, S}^{*}

into the trace-zero hyperplane

H

R^{# S} .

By Kronecker’s theorem, any algebraic number with trivial valuation at all finite and infinite places must be a root of unity, so the kernel of

o_{K, S}^{*} \to H

equals

μ_{K} .

Restricting an element of

o_{K, S}^{*}

to a bounded subset of

H

bounds all of its absolute values. Hence the discreteness of

K

A_{K}

(Corollary 6.1.13) implies that the image of the group

o_{K, S}^{*}

is discrete in

H .

Let

W

be the span in

H

of the image of

o_{K, S}^{*};

it remains to check that

W = H,

as this will imply that

o_{K, S}^{*}

is a lattice in

H

and hence has rank

\dim H = # S - 1 .

We may check this after enlarging

S;

by Corollary 6.2.10, we can assume that

I_{K, S} K^{*} = I_{K}

and hence

C_{K} = I_{K} / K^{*} ≅ I_{K, S} / o_{K, S}^{*} .

Using this isomorphism, we obtain a continuous homomorphism

C_{K}^{0} \to H / W

whose image generates

H / W .

Since

C_{K}^{0}

is compact (Proposition 6.2.8), so is its image; this is a contradiction unless

H / W

is the zero vector space.

Remark 6.2.12.

One corollary of the proof of Corollary 6.2.9 is that the component group of

C_{K}

surjects onto

Cl (K),

and hence is nontrivial in general.

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This of course does not say anything in the case

K = Q,

and in this case one can give a more direct description of

C_{K} .

Namely, given an arbitrary idèle in

I_{Q},

there is a unique positive rational with the same norms at the finite places. Thus

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C_{Q} ≅ R^{+} \times \prod_{p} Z_{p}^{*} .

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Returning to the case of general

K,

there is a natural way to define a volume measure on

C_{K}

in such a way that the volume of the kernel of

C_{K}^{0} \to Cl (K)

is exactly the unit regulator of

K .

Consequently, the total volume of

C_{K}^{0}

equals the product of the class number and the unit regulator, and it is this product which shows up in the analytic class number formula (based on the residue of the Dedekind zeta function of

K

at the point

s = 1;

see Theorem 6.6.9).

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Subsection Aside: beyond class field theory

Remark 6.2.13.

Via the identification

I_{K} ≅ {GL}_{1} (A_{K})

from Remark 6.2.3, class field theory can be viewed as a correspondence between one-dimensional representations of

Gal (\overset{―}{K} / K)

and certain representations of

{GL}_{1} (A_{K}) .

This is the form in which class field theory generalizes to the nonabelian case: the Langlands program predicts a correspondence between

n

-dimensional representations of

Gal (\overset{―}{K} / K)

and certain representations of

{GL}_{n} (A_{K}) .

See Appendix A for further discussion.

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Exercises Exercises

1.

Show that the restricted direct product topology on

I_{K}

is the subspace topology for the embedding into

A_{K} \times A_{K}

given by the map

x \mapsto (x, x^{- 1}),

and moreover this embedding has closed image.

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2.

Complete the proof of Proposition 6.2.8 by establishing the existence of the constant

c .

This can be done using the finiteness of the class group (Corollary 6.2.9) or the units theorem (Corollary 6.2.11); alternatively, with more work one can give a direct proof via which both of the aforementioned results become corollaries of Proposition 6.2.8.

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Hint.

For the direct approach, see for example [34], Section V.1, Theorem 0.

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