Section 7.3 An “abstract” reciprocity map
Reference.
We next manufacture a canonical isomorphism \(\Gal(L/K)^{\ab} \to C_K/\Norm_{L/K} C_L\) for any finite extension \(L/K\) of number fields, where \(C_K\) and \(C_L\) are the corresponding idèle class groups (Theorem 7.3.8). However, we won't yet know it agrees with our proposed reciprocity map, which is the product of the local reciprocity maps. We'll come back to this point after we establish the existence theorem (see Section 7.5).
Subsection Abstract unit groups and the class field axiom
We will be applying abstract class field theory with \(k = \QQ\) and \(\kbar = \overline{\QQ}\text{,}\) an algebraic closure of \(\QQ\text{.}\) We first specify the \(\Gal(\overline{\QQ}/\QQ)\)-module \(A\) which will give rise to abstract unit groups.
Definition 7.3.1.
Set \(A = \bigcup_K C_K\text{;}\) by Corollary 6.3.7, \(A_K = C_K\) for every number field \(K\text{.}\) Our earlier calculations (Theorem 7.1.2, Theorem 7.2.10) imply that the class field axiom is satisfied: for \(L/K\) a cyclic extension of number fields,
Remark 7.3.2.
In Definition 7.3.1, it will follow from the reciprocity law that the group \(H^0_T(\Gal(L/K), C_L)\) is cyclic. However, the class field axiom does not require advance knowledge of this.
Subsection Cyclotomic extensions and abstract ramification theory
The cyclotomic extensions of a number field play a role in class field theory analogous to the role played by the unramified extensions in local class field theory. This makes it essential to make an explicit study of them for use in proving the main results. However, we will not need the Kronecker-Weber theorem (Theorem 1.1.2); instead, we will recover it as part of the reciprocity law.
We first make a distinction which is of marginal significance in the totality of number theory, but is critical for our use of the machinery of abstract class field theory.
Definition 7.3.3.
The extension \(\bigcup_n \QQ(\zeta_n)\) of \(\QQ\) obtained by adjoining all roots of unity has Galois group \(\widehat{\ZZ}^* = \prod_p \ZZ_p^*\text{.}\) That group has a lot of torsion, since each \(\ZZ_p^*\) contains a torsion subgroup of order \(p-1\) (or 2, if \(p=2\)).
If we take the fixed field for the torsion subgroup of \(\ZZ^*\text{,}\) we get a slightly smaller extension, which I'll call the small cyclotomic extension of \(\QQ\) and denote \(\QQ^{\smcy}\text{.}\) Its Galois group is isomorphic to \(\prod_p \ZZ_p = \widehat{\ZZ}\text{,}\) but not canonically so.
For \(K\) a number field, define \(K^{\smcy} = K \QQ^{\smcy}\text{.}\) Then \(\Gal(K^{\smcy}/K) \cong \widehat{\ZZ}\) as well, even if \(K\) contains some extra roots of unity.
With this definition in hand, we can set up the homomorphism \(d\) needed to define abstract ramification theory for the base field \(k = \QQ\text{.}\)
Definition 7.3.4.
Choose an isomorphism of \(\Gal(\QQ^{\smcy}/\QQ)\) with \(\widehat{\ZZ}\text{;}\) our results are not going to depend on the choice (see Remark 7.3.11). That gives a continuous surjection
recall that this means we are going to regard \(\QQ^{\smcy}/\QQ\) as the “maximal unramified extension” of \(\QQ\text{.}\)
As in the general setup, for any finite extension \(L/K\) of number fields, we define the abstract ramification index \(e_{L/K}\) and the abstract inertia degree \(f_{L/K}\) by setting
Subsection An abstract henselian valuation
To complete the setup of abstract class field theory, we need an abstract henselian valuation \(v: C_{\QQ} \to \widehat{\ZZ}\) with respect to \(d\text{.}\) Recall from Definition 5.1.8 that this means:
\(v(C_{\QQ})\) is a subgroup \(Z\) of \(\widehat{\ZZ}\) containing \(\ZZ\) with \(Z/nZ \cong \ZZ/n\ZZ\) for all positive integers \(n\text{;}\)
\(v(\Norm_{K/\QQ} C_K) = f_{K/\QQ} Z\) for all finite extensions \(K/\QQ\text{.}\)
Definition 7.3.5.
To define the map \(v\text{,}\) we write
as in Remark 6.2.12. We then define \(v\) as the projection onto the third factor followed by the projection
The first condition of Definition 5.1.8 holds by construction. We will check the second condition using Artin reciprocity for cyclotomic extensions.
Lemma 7.3.6.
The map \(v\) defined in Definition 7.3.5 is a henselian valuation in the sense of Definition 5.1.8 (with respect to the map \(d\) from Definition 7.3.4).
Proof.
Since we already know from Definition 7.3.5 that \(v\) factors through \(C_\QQ\) and surjects onto \(\widehat{\ZZ}\text{,}\) it suffices to check that for every number field \(K\text{,}\) \(v(\Norm_{K/\QQ} I_K) = f_{K/\QQ} \widehat{\ZZ}\text{.}\) We may establish this by checking that the map
has image \(\Gal(K^{\cyc}/K) \subseteq \Gal(\QQ^{\cyc}/\QQ)\text{,}\) as then we get the desired condition by projecting from \(\Gal(\QQ^{\cyc}/\QQ)\) to \(\Gal(\QQ^{\smcy}/\QQ)\text{.}\) Note that for \(K = \QQ\text{,}\) this follows from Artin reciprocity for cyclotomic extensions (Definition 1.1.7).
In one direction, the fact that \(I_K\) maps into \(\Gal(K^{\cyc}/K)\) is a corollary of local reciprocity (Theorem 4.1.2) plus Artin reciprocity for cyclotomic extensions as used above.
In the other direction, the same logic shows that for each positive integer \(n\text{,}\) the image of \(I_K\) in \(\Gal(K(\zeta_n)/K)\) equals the image of the classical Artin map for \(K(\zeta_n)/K\text{;}\) it will thus suffice to check that these maps are surjective. It is convenient to deduce this from the First Inequality; see Proposition 7.3.7 below.
Here is the consequence of the First Inequality used in the proof of Lemma 7.3.6.
Proposition 7.3.7.
For \(L/K\) an abelian extension of number fields, the Artin map always surjects onto \(\Gal(L/K)\text{.}\)
Proof.
Let \(H\) be the image of the Artin map; the fixed field \(M\) of \(H\) has the property that all but finitely many primes of \(K\) split completely in \(M\text{.}\) We've already seen that this contradicts the First Inequality unless \(M = K\) (Corollary 7.1.16).
Subsection Consequences of abstract CFT
We now apply abstract class field theory to obtain an “abstract adelic reciprocity law”.
Theorem 7.3.8. Abstract adelic reciprocity law.
For every Galois extension \(L/K\) of number fields, we obtain an isomorphism
Proof.
The hypotheses of abstract class field theory are satisfied by taking \(k = \QQ\) \(\kbar = \overline{\QQ}\text{;}\) \(A = \bigcup_K C_K\) as in Definition 7.3.1 (using Theorem 7.1.2 and Theorem 7.2.10 to verify the class field axiom); \(d: \Gal(\overline{\QQ}/\QQ) \to \widehat{\ZZ}\) as in Definition 7.3.4; and \(v: C_{\QQ} \to \widehat{\ZZ}\) as in Definition 7.3.5 (using Lemma 7.3.6). We may thus apply Theorem 5.3.9 to conclude.
Definition 7.3.9.
By Proposition 5.2.7, the maps \(r'_{L/K}\) from Theorem 7.3.8 fit together to give a map \(r'_K: C_K \to \Gal(K^{\ab}/K)\text{;}\) but we do not yet know that this coincides with the product of the local reciprocity maps, so we cannot yet recover Artin reciprocity. However, we can at least deduce the norm limitation theorem (Theorem 6.4.3). See also Remark 7.3.11 below.
Theorem 7.3.10. Norm limitation theorem.
If \(L/K\) is a finite extension of number fields and \(M = L \cap K^{\ab}\text{,}\) then \(\Norm_{L/K} C_L = \Norm_{M/K} C_M\text{.}\)
Proof.
Apply Corollary 5.3.11.
Remark 7.3.11.
Although we do not have a complete description of the isomorphism \(r'_{L/K}\) coming from abstract class field theory, we do know one specific fact about this map: for “unramified” extensions \(L/K\) (i.e., \(L \subseteq K^{\smcy}\)), the “Frobenius” in \(\Gal(L/K)\) maps to a “uniformizer” in \(C_K\text{.}\) That is, the element of \(\Gal(L/K)\) coming from the element of \(\Gal(K^{\smcy}/K)\) which maps to 1 under \(d_K\) corresponds via reciprocity to the element of \(C_K\) which maps to 1 under \(v_K\text{.}\)
The broader point here is that the definitions of both \(d\) and \(v\) involve the same artificial choice of an isomorphism \(\Gal(\QQ^{\smcy}/\QQ) \cong \widehat{\ZZ}\text{,}\) which thus does not affect the reciprocity map. Compare Remark 5.1.11 and Exercise 4.