Lemma 6.4.1.
For \(L/K\) an extension of number fields, \(\Norm_{L/K} C_L\) is an open subgroup of \(C_K\) of finite index.
Section 6.4 The adelic reciprocity law and Artin reciprocity
We now formulate the statements of global class field theory in adelic language, imitating the setup from local class field theory but using the idèle class group in place of the multiplicative group of the field. This is more than a formal similarity: it allows us to formulate a local-to-global compatibility statement that ultimately allows us to recover Artin reciprocity and the existence theorem (and a bit more) in the classical language.
Convention note.
We are going to fix an algebraic closure \(\overline{\QQ}\) of \(\QQ\text{,}\) and regard “number fields” as finite subextensions of \(\overline{\QQ}/\QQ\text{.}\) That is, we are fixing the embeddings of number fields into \(\overline{\QQ}\text{.}\) We’ll use these embeddings to decide how to embed one number field in another.
Subsection The adelic reciprocity law and existence theorem
Here are the adelic reciprocity law and existence theorem; notice that they look just like the local case except that the multiplicative group has been replaced by the idèle class group.
Proof.
Let \(S\) be any finite set of places of \(K\) containing all infinite places and all places that ramify in \(L\text{.}\) Let \(T\) be the set of places of \(L\) lying above places in \(S\text{.}\) We first compute that
\begin{equation*}
\Norm_{L/K} I_{L,S} = \prod_{v \in S} U_v \times \prod_{v \notin S} \gotho_{K_v}^*
\end{equation*}
for some open subgroups \(U_v\) of \(K_v^*\) of finite index; this amounts to a separate computation for each \(v\text{.}\) For \(v \in S\) finite, we apply Theorem 4.1.5 (or Exercise 3). For \(v \in S\) infinite, we note that \(\Norm\colon \CC^* \to \RR^*\) has image \(\RR^+\) of index \(2\text{.}\) For \(v \notin S\text{,}\) we apply Proposition 4.2.5.
Taking the union over \(S\text{,}\) we see that \(\Norm_{L/K} I_L\) is an open subgroup of \(I_K\text{.}\) By quotienting down to \(C_K\text{,}\) we see that \(\Norm_{L/K} C_L\) is open; in fact, the snake lemma on the diagram Figure 6.4.2 implies that the quotient \(I_K/(K^* \times \Norm_{L/K} I_L)\) is isomorphic to \(C_K/\Norm_{L/K} C_L\text{.}\)
We still need to check that the index \([C_K:\Norm_{L/K} C_L]\) is finite. For this, we note that the inclusion \(C_K^0 \to C_K\) induces a homeomorphism
\begin{equation*}
C_K^0/\Norm_{L/K} C_L^0 \cong C_K / \Norm_{L/K} C_L
\end{equation*}
of topological groups. Consequently, this group is both discrete (by the previous paragraph) and compact (by Proposition 6.2.8), hence finite.
Theorem 6.4.3. Adelic reciprocity law.
For each number field \(K\text{,}\) there is a map
\begin{equation*}
r_K\colon C_K \to \Gal(K^{\ab}/K) = \Gal(\overline{K}/K)^{\ab}
\end{equation*}
which induces, for each Galois extension \(L/K\) of number fields, an isomorphism
\begin{equation*}
r_{L/K}\colon C_K/\Norm_{L/K} C_L \to \Gal(L/K)^{\ab}\text{.}
\end{equation*}
Proof.
See Theorem 7.3.8 for the proof using abstract class field theory, and Remark 7.6.20 for the proof using group cohomology more directly. See also Section 6.5 for a more detailed summary of both approaches.
Theorem 6.4.4. Adelic existence theorem.
For every number field \(K\) and every open subgroup \(H\) of \(C_K\) of finite index, there exists a finite (abelian) extension \(L\) of \(K\) such that \(H = \Norm_{L/K} C_L\text{.}\)
Proof.
See Theorem 7.4.8. Again, see also Section 6.5 for a more detailed summary of the argument.
We will also obtain a global analogue of the local norm limitation theorem, which was not even suggested by the classical language. (Well, not in this treatment anyway. See Lemma 7.2.2 for an interpretation of the quotient \(C_K/\Norm_{L/K} C_L\) in ideal-theoretic terms.)
Theorem 6.4.5. Adelic norm limitation theorem.
Let \(L/K\) be a (not necessarily Galois) extension of number fields and put \(M := L \cap K^{\ab}\text{.}\) Then \(\Norm_{L/K} C_L = \Norm_{M/K} C_M\text{.}\)
Proof.
As in the local setting, this follows directly from Theorem 6.4.3 by the logic of Remark 4.3.8. See also Theorem 7.3.10.
Subsection More on the reciprocity map
The statement of Theorem 6.4.3 does not give any direct information about the map \(r_K\text{.}\) We describe this map more explicitly using local class field theory and the principle of local-global compatibility. We note in passing that this principle also lies at the heart of the extension of class field theory envisioned in the Langlands program (Remark 6.2.13).
Definition 6.4.6.
Let \(L/K\) be an abelian extension of number fields and \(v\) a place of \(K\text{.}\) Put \(G := \Gal(L/K)\) and let \(G_v\) be the decomposition group of \(v\text{,}\) that is, \(G_v := \Gal(L_w/K_v)\) for some (hence any, because \(G\) is abelian) place \(w\) above \(v\text{.}\) We define a map \(r_{K, v}\colon K_v^* \to G_v \subseteq G\) as follows.
- If \(v\) is a finite place, use the local reciprocity map (Theorem 4.1.2).
- If \(v\) is a real place, use the sign map \(\RR^* \to \{\pm 1\} \cong \Gal(\CC/\RR)\) if \(w\) is complex (otherwise there is nothing to specify).
- If \(v\) is a complex place, then \(G_v\) is trivial and so there is nothing to specify.
We obtain a well-defined product map
\begin{equation*}
\tilde{r}_K\colon I_K \to G, \qquad (\alpha_v) \mapsto \prod_v r_{K,v}(\alpha_v)
\end{equation*}
by arguing as follows. For \((\alpha_v) \in I_K\text{,}\) \(\alpha_v\) is a unit for almost all \(v\) and \(L_w/K_v\) is unramified for almost all \(v\) (we may ignore infinite places here). For the (almost all) \(v\) in both categories, \(r_{K,v}\) maps \(\alpha_v\) to the identity.
Since each of the maps \(r_{K,v}\) is continuous, so is the map \(\tilde{r}_K\text{.}\) That means the kernel of \(\tilde{r}_K\colon I_K \to \Gal(L/K)\) is an open subgroup of \(I_K\text{.}\)
Here is the subtle point, and the real source of “reciprocity” in this construction.
Proposition 6.4.7. Local-global compatibility.
For \(L/K\) an abelian extension of number fields, the map \(\tilde{r}_K\colon I_K \to \Gal(L/K)\) is trivial on \(K^*\text{;}\) the induced map \(r_K\colon C_K \to \Gal(L/K)\) factors through an isomorphism \(C_K/\Norm_{L/K} C_L \cong \Gal(L/K)\text{;}\) and the resulting map \(r_K \colon C_K \to \Gal(K^{\ab}/K)\) obeys the conclusion of Theorem 6.4.3.
Proof.
For the treatment in terms of abstract class field theory, see Proposition 7.5.6. For the treatment without abstract class field theory, see Remark 7.6.17 and Remark 7.6.20.
Remark 6.4.8.
In case \(L = K(\zeta_n)\) for some \(n\text{,}\) we can verify the first assertion of Proposition 6.4.7 by an explicit computation, similar to the direct verification of Artin reciprocity for these extensions; this will play a key role in both approaches to adelic reciprocity.
Let us spell this out explicitly for \(K = \QQ, L = K(\zeta_{p^m})\text{,}\) where we have a concrete identification \(\Gal(L/K) \cong (\ZZ/p^m\ZZ)^*\text{.}\) In this case, we are claiming that for any place \(\ell\) of \(\QQ\) and any \(a \in \QQ^*\text{,}\)
\begin{equation*}
r_{\QQ_{\ell}(\zeta_{p^m})/\QQ_{\ell}}(a) = \begin{cases}\sign(a) \amp \ell = \infty \\ \ell^{v_{\ell}(a)} \amp \ell \neq \infty, p \\ p^{v_p(a)} a^{-1} \amp \ell = p. \end{cases}
\end{equation*}
This is clear when \(\ell = \infty\text{.}\) For \(\ell\) finite and distinct from \(p\text{,}\) we are in an unramified situation, where the effect of local reciprocity is to take any uniformizer (i.e., \(\ell\)) to the local Frobenius at \(\ell\) (i.e., the automorphism \(\zeta_n \mapsto \zeta_n^\ell\)). The interesting case is when \(\ell = p\text{,}\) for which we consult Example 4.1.4.
Fom the previous paragraph, we directly obtain the first assertion of Proposition 6.4.7 for \(K = \QQ, L = K(\zeta_n)\text{.}\) The case where \(K\) is general and \(L = K(\zeta_n)\) then follows from Proposition 4.3.11.
Subsection Back to classical reciprocity
We now apply adelic reciprocity and local-global compatibility to recover the ideal-theoretic formulations of Artin reciprocity and the existence theorem.
Proposition 6.4.9.
Let \(L/K\) be an abelian extension of number fields. Given Theorem 6.4.3 and Proposition 6.4.7, let \(U\) be the kernel of \(r_K\) and identify \(C_K/U\) with a generalized ideal class group (Remark 6.2.6) of some conductor \(\gothm\text{.}\) Then the map \(C_K/U \to \Gal(L/K)\) is the Artin map; consequently, Theorem 2.2.6 holds.
Proof.
The idèle \(\alpha = (1,1, \dots, \pi, \dots)\) with a uniformizer \(\pi\) of \(\gotho_{K_\gothp}\) in the \(\gothp\) component and \(1\) elsewhere maps onto the class of \(\gothp\) in \(C_K/U\text{.}\) On the other hand, \(r_K(\alpha) = r_{K, \gothp}(\pi)\) is precisely the Frobenius of \(\gothp\) (because \(L\) is unramified over \(K\)). So indeed \(\gothp\) is being mapped to its Frobenius, and the map \(C_K/U \to \Gal(L/K)\) is indeed Artin reciprocity.
Remark 6.4.10.
We may use similar logic to recover the ideal-theoretic formulation of the existence theorem (Theorem 2.2.8): each generalized ideal class group arises as a quotient of \(C_K\) by an open subgroup (see again Remark 6.2.6, and also Definition 7.2.1) and so Theorem 6.4.4 produces the desired abelian extension.
The argument from Proposition 6.4.9 also gives some additional information about the Artin map. First, the Artin map factors through a generalized ideal class group whose conductor \(\gothm\) is divisible precisely by the ramified primes. Second, we can exactly describe the kernel of the classical Artin map: it is generated by
- all principal ideals congruent to 1 modulo \(\gothm\text{;}\)
- norms of ideals of \(L\) not divisible by any ramified primes.
