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Notes on class field theory

Kiran S. Kedlaya

Section 6.2 Idèles and class groups

Reference.

[36]; [37], VI.1 and VI.2; [33], VII.
We now shift from additive to multiplicative considerations.

Subsection Idèles

Definition 6.2.1.

Let \(K\) be a number field and let \(\AA_K\) be the ring of adèles associated to \(K\) (Definition 6.1.9). We define the group of idèles \(I_K\) associated to \(K\) as the group of units of the ring \(\AA_K\text{.}\) In other words, an element of \(I_K\) is a tuple \((\alpha_v)\text{,}\) one element of \(K_v^*\) for each place \(v\) of \(K\text{,}\) such that \(\alpha_v \in \gotho_{K_v}^*\) for all but finitely many finite places \(v\text{.}\)
As a set, \(I_K\) is the restricted product of the pairs \((K_v^*, \{1\})\) for infinite places \(v\) and \((K_v, \gotho_{K_v}^*)\) for finite places \(v\text{.}\) We use this interpretation to give \(I_K\) the structure of a locally compact topological group.

Definition 6.2.2.

For \(S\) a finite set, let \(I_{K,S}\) be the set of \(x \in I_K\) for which \(x_v \in \gotho_{K_v}^*\) for each finite place \(v\notin S\text{;}\) then \(I_K = \bigcup_S I_{K,S}\text{.}\) By analogy with Definition 6.1.13, the elements of \(I_{K,S}\) can be thought of as “adelic \(S\)-units”.

Remark 6.2.3. Warning.

While the embedding of the idèle group \(I_K\) into the adèle group \(\AA_K\) 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 \(\AA_K\text{.}\)
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 \(\GL_1(\AA_K)\) and topologize it accordingly (Exercise 1).

Subsection The idèle class group

Definition 6.2.4.

For each \(\alpha \in K^*\text{,}\) the principal adèle \(\alpha \in \AA_K\) is an idèle, so we have an embedding \(K^* \hookrightarrow I_K\text{.}\) 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.
The terminology idèle class group is justified on account of the following construction.

Definition 6.2.5.

There is a homomorphism from \(I_K\) to the group of fractional ideals of \(K\text{:}\)
\begin{equation*} (\alpha_\nu)_\nu \mapsto \prod_{\gothp} \gothp^{v_{\gothp}(\alpha_\gothp)}\text{,} \end{equation*}
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 \(I_K\text{.}\)) By unique factorization of fractional ideals, this homomorphism is surjective; its kernel is precisely \(I_{K,S}\) for \(S\) the set of infinite places.
Under this map, the principal idèle corresponding to \(\alpha \in K\) maps to the principal ideal generated by \(\alpha\text{.}\) Thus we have a surjection \(C_K \to \Cl(K)\) with kernel \(I_{K,S} K^*/K^*\text{.}\)

Remark 6.2.6.

What are the open subgroups of \(I_K\text{?}\) For each formal product \(\gothm\) of places, one gets an open subgroup of idèles \((\alpha_v)_v\) such that:
  1. if \(v\) is a real place occurring in \(\gothm\text{,}\) then \(\alpha_v > 0\text{;}\)
  2. if \(v\) is a finite place corresponding to the prime \(\gothp\text{,}\) occurring to the power \(e\text{,}\) then \(\alpha_v \equiv 1 \pmod{\gothp^e}\text{.}\)
This then projects to an open subgroup of \(C_K\text{,}\) the quotient by which is the ray class group of modulus \(\gothm\text{!}\) (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 \(\gothm\text{.}\) See Definition 7.2.1 for an elaboration of this point.)
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 \(\gothm\)). This correspondence is what will allow us to translate between the classical and adelic versions of Artin reciprocity.

Subsection Compactness and consequences

Definition 6.2.7.

By the product formula (Proposition 6.1.11), we get a well-defined norm map \(|\cdot|: C_K \to \RR^*_+\text{.}\) Let \(C_K^0\) be the kernel of the norm map; then \(C_K^0\) also surjects onto \(\Cl(K)\text{.}\) (The surjection onto \(\Cl(K)\) ignores the infinite places, so you can adjust there to force norm 1.)

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 \([c^{-1}, c]\text{.}\)
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 \(c\text{.}\)) 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 \(C_K^0\text{,}\) 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.)

Proof.

The group \(C_K^0\) is compact by Proposition 6.2.8 and it surjects onto \(\Cl(K)\text{,}\) so the latter must also be compact for the discrete topology, and hence finite.

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.

Proof.

Define the map \(\log: I_{K,S} \to \RR^{\#S}\) by taking log of the absolute value of the norm of each component in \(S\) (normalizing as in Definition 6.1.10). By the product formula (Proposition 6.1.11), this map carries \(\gotho_{K,S}^*\) into the trace-zero hyperplane \(H\) in \(\RR^{\#S}\text{.}\) 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 \(\gotho_{K,S}^* \to H\) equals \(\mu_K\text{.}\)
Restricting an element of \(\gotho_{K,S}^*\) to a bounded subset of \(H\) bounds all of its absolute values. Hence the discreteness of \(K\) in \(\AA_K\) (Corollary 6.1.12) implies that the image of the group \(\gotho_{K,S}^*\) is discrete in \(H\text{.}\)
Let \(W\) be the span in \(H\) of the image of \(\gotho_{K,S}^*\text{;}\) it remains to check that \(W = H\text{,}\) as this will imply that \(\gotho_{K,S}^*\) is a lattice in \(H\) and hence has rank \(\dim H = \#S - 1\text{.}\) We may check this after enlarging \(S\text{;}\) by Corollary 6.2.10, we can assume that \(I_{K,S} K^* = I_K\) and hence
\begin{equation*} C_K = I_K/K^* \cong I_{K,S} / \gotho_{K,S}^*. \end{equation*}
Using this isomorphism, we obtain a continuous homomorphism \(C_K^0 \to H/W\) whose image generates \(H/W\text{.}\) 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. Thus

Remark 6.2.12.

One corollary of the proof of Corollary 6.2.9 is that the component group of \(C_K\) surjects onto \(\Cl(K)\text{,}\) and hence is nontrivial in general.
This of course does not say anything in the case \(K=\QQ\text{,}\) and in this case one can give a more direct description of \(C_K\text{.}\) Namely, given an arbitrary idèle in \(I_\QQ\text{,}\) there is a unique positive rational with the same norms at the finite places. Thus
\begin{equation*} C_\QQ \cong \RR^+ \times \prod_p \ZZ_p^*\text{.} \end{equation*}
Returning to the case of general \(K\text{,}\) there is a natural way to define a volume measure on \(C_K\) in such a way that the volume of the kernel of \(C^0_K \to \Cl(K)\) is exactly the unit regulator of \(K\text{.}\) Consequently, the total volume of \(C^0_K\) 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 \(s=1\text{;}\) see Theorem 6.6.9).

Subsection Aside: beyond class field theory

Remark 6.2.13.

Via the identification \(I_K \cong \GL_1(\AA_K)\) from Remark 6.2.3, class field theory can be viewed as a correspondence between one-dimensional representations of \(\Gal(\overline{K}/K)\) and certain representations of \(\GL_1(\AA_K)\text{.}\) 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 \(\Gal(\overline{K}/K)\) and certain representations of \(\GL_n(\AA_K)\text{.}\) See Appendix A for further discussion.

Exercises Exercises

1.

Show that the restricted direct product topology on \(I_K\) is the subspace topology for the embedding into \(\AA_K \times \AA_K\) given by the map \(x \mapsto (x,x^{-1})\text{,}\) and moreover this embedding has closed image.

2.

Complete the proof of Proposition 6.2.8 by establishing the existence of the constant \(c\text{.}\) 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.
Hint.
For the direct approach, see for example [33], Section V.1, Theorem 0.
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