Section 6.5 Adelic reciprocity: what remains to be done
We now pick up the thread from Section 5.4 to outline the proofs of the main results of class field theory, to be presented in Chapter 7. Many of these steps will be analogous to the steps in local class field theory; however, we do not directly attempt to verify Proposition 6.4.7 except for cyclotomic extensions, for which the explicit calculation suggested in Remark 6.4.6 will be an important input into the machine. Instead, we postpone this step all the way until the end.
Subsection Abstract reciprocity
Our first goal is to establish the conditions for abstract class field theory (Section 5.1), in the setup described at the end of Section 5.3 using the idèle class groups \(C_K\text{.}\) This requires verifying the class field axiom (Definition 5.1.4), plus a compatibility between the homomorphism \(d\) on the absolute Galois group and the valuation map \(v\text{.}\)
We treat the class field axiom in two steps. The first step is to show that for \(L/K\) cyclic, the Herbrand quotient of \(C_L\) as a \(\Gal(L/K)\)-module is \([L:K]\text{.}\) This implies the First Inequality (Theorem 7.1.2):
\begin{equation*}
\#H^0_T(\Gal(L/K), C_L) \geq [L:K]\text{.}
\end{equation*}
The argument will be to replace the group \(I_L\) with the subgroup \(I_{L,T}\) for some suitable set of places \(T\) of \(L\text{,}\) and reduce to studying lattices in the manner of the proof of Dirichlet’s units theorem Corollary 6.2.11).
The next step will to prove the Second Inequality (Theorem 7.2.10):
\begin{equation*}
\#H^0_T(\Gal(L/K), C_L) \leq [L:K]\text{,}
\end{equation*}
which combined with the previous point yields
\begin{equation*}
\#H^0_T(\Gal(L/K), C_L) = [L:K], \qquad \#H^1_T(\Gal(L/K), C_L) = 1\text{.}
\end{equation*}
This step is trivial in local CFT by Theorem 90 (Lemma 1.2.3), but is pretty subtle in the global case. We will first describe a proof using analytic methods (properties of \(L\)-functions); there is also an algebraic approach, more on which below.
Finally, we check the compatibility between \(d\) and \(v\) using the explicit nature of Artin reciprocity for cyclotomic extensions. Plugging into the machine then gives an “abstract” reciprocity map (Theorem 7.3.8), not yet known to be related to Artin reciprocity except for cyclotomic extensions. We also establish the norm limitation theorem (Theorem 7.3.10).
Subsection The existence theorem and local-global compatibility
Our next step is to prove the adelic existence theorem (Theorem 6.4.2). As in the local setting, having the reciprocity law (even only in abstract form, and even without any compatibility with the Artin map) and the norm limitation theorem in hand allows us to reduce to showing that every open subgroup of \(C_K\) of finite index contains a norm group (Theorem 7.4.8). This can be done after enlarging \(K\text{,}\) so we can get into the realm of Kummer theory; this is closely related to the algebraic proof of the Second Inequality mentioned above.
We then turn around and use the existence theorem to deduce that, by analogy with the relationship between the local and global Kronecker-Weber theorems over \(\QQ\text{,}\) every finite abelian extension of the completion of a number field at some place is itself the completion of a finite extension of the number field (Theorem 7.5.9). This will allow us to show, by making careful use of cyclotomic extensions, that the “abstract” global reciprocity map restricts to the usual reciprocity map from local class field theory (Proposition 7.5.7). This will finally imply Proposition 6.4.5, thus yielding the adelic reciprocity map (and also showing that it coincides with the abstract reciprocity map).
Subsection Another approach via Brauer groups
We will also briefly sketch the approach taken in [37], in which one uses Galois cohomology in place of abstract class field theory. Specifically, one first checks that \(H^2(\Gal(L/K), C_L)\) is cyclic of order \([L:K]\) in certain “unramified” (i.e., cyclotomic) cases; as in the local case, one can then deduce this result in general by induction on degree. Using Tate’s theorem (Theorem 4.1.14), one gets a reciprocity map
\begin{equation*}
\Gal(L/K)^{\ab} = H^{-2}_T(\Gal(L/K), \ZZ) \to H^0_T(\Gal(L/K), C_K/\Norm_{L/K} C_L)
\end{equation*}
which again can be reconciled with local reciprocity to get the Artin reciprocity map (Proposition 7.6.17). This approach will also yield some additional information, notably a description of the Brauer group of a number field (Theorem 7.6.10).
