The adelic reciprocity law and Artin reciprocity

Section 6.4 The adelic reciprocity law and Artin reciprocity

We now describe the setup by which we create a reciprocity law in global class field theory, imitating the “abstract” setup from local class field theory but using the idèle class group in place of the multiplicative group of the field. We then work out why the reciprocity law and existence theorem in the adelic setup imply 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law and existence theorem; notice that they look just like the local case except that the multiplicative group has been replaced by the idèle class group.

Theorem 6.4.1. Adelic reciprocity law.

There is a canonical map \(r_K: C_K \to \Gal(K^{\ab}/K)\) which induces, for each Galois extension \(L/K\) of number fields, an isomorphism \(r_{L/K}: C_K/\Norm_{L/K} C_L \to \Gal(L/K)^{\ab}\text{.}\) Moreover, \(\Norm_{L/K} C_L\) is an open subgroup of \(C_K\text{.}\)

Proof.

We will first prove an “abstract” form of this theorem, in which we do not say much about the identity of the map \(r_K\text{;}\) see Theorem 7.3.8. We then prove a more precise version including a more specific recipe for the map; see Proposition 6.4.5 for the recipe and Proposition 7.5.7 for the comparison with the abstract version. (For the assertion that \(\Norm_{L/K} C_L\) is open in \(C_K\text{,}\) see Remark 7.1.7.)

Theorem 6.4.2. 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.

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 6.4.3. Adelic norm limitation theorem.

Let \(L/K\) be an 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.

See Theorem 7.3.10.

Subsection More on the reciprocity map

We next use local class field theory and the principle of local-global compatibility to come up with a candidate for the map \(r_K\) in the adelic reciprocity law (Theorem 6.4.1). 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 6.4.4.

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, the set of \(g \in G\) such that \(v^g = v\text{.}\) Then for any place \(w\) above \(v\text{,}\) \(G_v \cong \Gal(L_w/K_v)\text{.}\) We will define a map \(r_{K, v}: K_v^* \to G_v \subseteq G\) 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 \(\RR^* \to \{\pm 1\} \cong G_v\text{.}\)
If \(v\) is a complex place, then \(G_v\) is trivial and so there is nothing left to specify.

We obtain a well-defined product map

\begin{equation*} \tilde{r}_K: I_K \to G, \qquad (\alpha_v) \mapsto \prod_v r_{K,v}(\alpha_v): \end{equation*}

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 \(\tilde{r}_K: I_K \to \Gal(L/K)\) is an open subgroup of \(I_K\text{.}\)

Here is the subtle point, and the real source of “reciprocity” in this construction.

Proposition 6.4.5.

For \(L/K\) an abelian extension of number fields, the map \(\tilde{r}_K: I_K \to \Gal(L/K)\) is trivial on \(K^*\text{.}\) It thus factors through a map \(r_K: C_K \to \Gal(L/K)\text{.}\)

Proof.

See Proposition 7.5.7.

Remark 6.4.6.

In case \(L = K(\zeta_n)\) for some \(n\text{,}\) we can verify Proposition 6.4.5 by an explicit computation, similar to the direct verification of Artin reciprocity for these extensions. This suggests that in general, we must first prove the adelic existence theorem (Theorem 6.4.2) before establishing Proposition 6.4.5. In the interim, we will derive a makeshift form of adelic reciprocity from the framework of abstract class field theory.

Proposition 6.4.7.

Let \(L/K\) be an abelian extension of number fields. Given Theorem 6.4.1 and Proposition 6.4.5, let \(U\) be the kernel of \(r_K\text{,}\) 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 1s elsewhere maps onto the class of \(\gothp\) in \(C_K/U\text{.}\) On the other hand, \(r_K(\alpha) = r_{K, \gothp}(\pi)\) is (because \(L\) is unramified over \(K\)) precisely the Frobenius of \(\gothp\text{.}\) So indeed, \(\gothp\) is being mapped to its Frobenius, so the map \(C_K/U \to \Gal(L/K)\) is indeed Artin reciprocity.

Remark 6.4.8.

The argument from Proposition 6.4.7 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.

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