Notes on class field theory

In the setting of abstract class field theory, $L/K$ is viewed as an “unramified” extension. Consequently, the reciprocity map $r_{L/K}^{\prime }:C_K/{\mathrm{Norm}}_{L/K}{C}_L\to \mathrm{Gal}(L/K)$ is described completely by Lemma 5.3.1: it is given by composing the valuation map $v_K:C_K\to \hat{\mathbb{Z}}$ with the inverse of the map $d_K:\mathrm{Gal}(K^{\mathrm{smcy}}/K)\cong \hat{\mathbb{Z}}\text{,}$ then projecting from $\mathrm{Gal}(K^{\mathrm{smcy}}/K)$ to $\mathrm{Gal}(L/K)\text{.}$ (Note that as per Remark 7.3.11, this does not depend on the artificial choice of the isomorphism $d_K\text{,}$ because $v_K$ is defined using the same choice.) Consequently, in this case we end up with the usual Artin map for a cyclotomic extension (Definition 1.1.7), which is compatible with local reciprocity by direct calculation (Example 4.1.4).

In the setting of abstract class field theory, $L/K$ is viewed as a “totally ramified” extension. Consequently, we may set notation as in the proof of Lemma 5.3.2, then apply Proposition 5.2.7 to obtain a commutative diagram

in which the horizontal arrows are isomorphisms (Theorem 7.3.8) and the right vertical arrow is an isomorphism, as then is the left vertical arrow. We also have a corresponding diagram on the local side:

This means that we can reduce checking the compatibility for $L/K$ to the corresponding statement for the “unramified” extension $N/M\text{,}$ to which Lemma 7.5.2 applies.

By the norm limitation theorem (Theorem 7.3.10), we may assume that $L/K$ is abelian. By Proposition 5.2.7, we may check the comparison of maps after replacing $L$ with a larger abelian extension of $K\text{.}$

We can quickly dispatch the cases where $v$ is infinite: if $v$ is complex there is nothing to prove, and if $v$ is real then we may take $L=K(\sqrt{-1})\text{.}$ So assume hereafter that $v$ is finite.

By the existence theorem (Theorem 7.4.8) plus local-to-global compatibility (Proposition 7.5.6), it suffices to produce an open subgroup $V$ of $C_K$ of finite index such that the preimage of $V$ under $K_v^{\ast }\to C_K$ is contained in $N={\mathrm{Norm}}_{M/K_v}{M}^{\ast }\text{.}$ Let $S$ be the set of infinite places and let $T=S\cup \{v\}\text{.}$ By Corollary 6.2.11, ${\mathfrak{o}}_{K,T}^{\ast }$ is a finitely generated abelian group and $G={\mathfrak{o}}_{K,T}^{\ast }\cap N$ is a subgroup of ${\mathfrak{o}}_{K,T}^{\ast }$ of finite index.

Pick a finite place $u\notin T\text{.}$ The image of ${\mathfrak{o}}_{K,T}^{\ast }$ in $K_u^{\ast }$ is a finitely generated subgroup of ${\mathfrak{o}}_{K_u}^{\ast }\text{.}$ Hence we can choose a sufficiently small neighborhood $U$ of the identity in ${\mathfrak{o}}_{K_u}^{\ast }$ so as to ensure that $U\cap {\mathfrak{o}}_{K,T}^{\ast }\subseteq G\text{.}$