Section 5.4 A look ahead
We conclude our treatment of abstract class field theory by asking ourselves: what does the construction of an abstract reciprocity law tell us about the global Artin reciprocity law (Theorem 2.2.6)? See Section 6.5 for a continuation of this discussion with more of the details filled in.
Subsection Replacing the multiplicative group
For \(L/K\) a finite abelian extension of number fields, we need to compare \(\Gal(L/K)\) to a generalized ideal class group of \(K\text{.}\) This means that the group \(A\) must somehow be related to ideal classes. You might try taking the group of fractional ideals in \(L\text{,}\) then taking the direct limit over all finite extensions \(L\) of \(K\text{.}\) In this case, we would have to find \(H^i(\Gal(L/K), A_L)\) for \(A_L\) the group of fractional ideals in \(L\text{,}\) where \(L/K\) is cyclic and \(i=0, -1\text{.}\) Unfortunately, these groups are not so well-behaved as that!
The cohomology groups would behave better if \(A_L\) were “complete” in some sense, in the way that \(K^*\) is complete when \(K\) is a local field. But there is no good reason to distinguish one place over another in the global case. So we're going to make the target group \(A\) by “completing \(K^*\) at all places simultaneously”.
Subsection Replacing the unramified extensions and the valuation
Even without \(A\text{,}\) I can at least tell you what \(d\) is going to be over \(\QQ\text{.}\) To begin with, note that there is a surjective map \(\Gal(\overline{\QQ}/\QQ) \to \Gal(\QQ^{\cyc}/\QQ)\) that turns an automorphism into its action on roots of unity. The latter group is unfortunately isomorphic to the multiplicative group \(\widehat{\ZZ}^*\) rather than the additive group \(\widehat{\ZZ}\text{,}\) but this is a start. To make more progress, write \(\widehat{\ZZ}\) as the product \(\prod_p \ZZ_p\text{,}\) so that \(\widehat{\ZZ}^* \cong \prod_p \ZZ_p^*\text{.}\) Then recall that there exist isomorphisms
In particular, \(\ZZ_p^*\) modulo its torsion subgroup is isomorphic to \(\ZZ_p\text{,}\) but not in a canonical way. Paying this no mind, let us choose an isomorphism for each \(p\) and then obtain a surjective map \(\widehat{\ZZ}^* \to \widehat{\ZZ}\text{.}\) Composing, we get a surjective map \(\Gal(\overline{\QQ}/\QQ) \to \widehat{\ZZ}\) which in principle depends on some choices, but the ultimate statements of the theory will be independent of these choices. (Note that in this setup, every “unramified” extensions of a number field is a subfield of a cyclotomic extension, but not conversely.)
As for the valuation \(v\text{,}\) this will be more straightforward. In the situation we end up considering, the group \(A_{\QQ}\) will end up having a natural map to \(\Gal(\QQ^{\cyc}/\QQ)\text{,}\) which we can then use to map to \(\widehat{\ZZ}\text{.}\) This again involves an artificial choice, but as long as we make the same artificial choice as we did for \(d\text{,}\) we get the necessary compatibility between \(d\) and \(v\text{.}\)
Subsection Further remarks
Remark 5.4.1.
In the function field setting, we have a much more straightforward alternative to the use of cyclotomic extensions: we may take the map to the Galois group of the base finite field. The point is that in this case we have an ample supply of everywhere unramified extensions of the base field (without quotation marks).
In the number field setting, using cyclotomic extensions as a proxy for abelian, everywhere unramified extensions is a rather productive idea even outside of class field theory. For one, it is the central premise of Iwasawa theory, in which one studies the behavior of class fields in certain towers of number fields and their relationship with \(p\)-adic \(L\)-functions (and other related concepts). For another, it is the starting point of \(p\)-adic Hodge theory, in which one studies the relationship between different cohomology theories associated to algebraic varieties over local fields.
Remark 5.4.2.
One can also apply the framework of abstract class field theory to prove some forms of higher-dimensional class field theory, taking the group \(A\) to be something coming from algebraic \(K\)-theory. See the remark at the end of [37], IV.6.