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 6.4.9 except for cyclotomic extensions, for which the explicit calculation indicated in Remark 6.4.8 will be an important input into the machine.
Subsection The abstract approach to 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 7.2.9):
\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.
Subsection Norm limitation and the existence theorem
From the existence of the reciprocity map, we formally deduce the norm limitation theorem (Theorem 7.3.10) as in local class field theory. This in turn gives us a topological approach to proving the adelic existence theorem (Theorem 6.4.4), even without local-global compatibility: we just need to show 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.
Subsection Local-global compatibility
As the abstract construction of the reciprocity map does not directly yield local-global compatibility, we use a somewhat indirect approach. By analogy with the relationship between the local and global Kronecker-Weber theorems over \(\QQ\text{,}\) we can use the existence theorem to show that 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 7.5.8). 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 7.5.6). This will finally imply Proposition 6.4.7, thus yielding the adelic reciprocity map (and also showing that it coincides with the abstract reciprocity map).
Subsection The approach via Brauer groups
The reader who has chosen to skip Chapter 5 can instead follow the proofs of global class field theory as follows. We first make some computations of \(H^2(\Gal(L/K), L^*)\) to describe the Brauer group of the number field \(K\) (Theorem 7.6.12). This in turn allows us to compute that \(H^2(\Gal(L/K), C_L)\) is cyclic of order \([L:K]\) using a strategy borrowed from local class field theory: we combine an upper bound on the size of this group (derived from the First and Second Inequality) with a lower bound obtained by transferring elements from cyclotomic extensions (Proposition 7.6.14). 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 7.6.16). Note that in this approach we build local-global compatibility into the definitions, rather than having to check it separately. Also, the proofs of the norm limitation theorem and the existence theorem are agnostic as to the source of the reciprocity map, so they apply in this approach without change.
