(Argh. So, the spacing isn’t working as well as I would like on the post and it reads non-ideally (sorry). So I’ve also included a PDF of the post if it makes things better.  -AM)

So, we have defined this thing called the Artin map on the ideals prime to some set of primes. But we really care about the ideles. There has to be some way to relate ideals and ideles. In this post, we give a translation guide between the idealic and ideleic framework. In the good ol’ days, one apparently developed class field theory using only ideal theory, but now the language of the ideles is convenient too (and as we saw, the ideles lend themselves very nicely to computing Herbrand quotients).  But they are not as good for the Artin map, unless one already has local class field theory. We don’t—we could if we developed a lot of cohomological machinery and some delightful pieces of abstract nonsense—but that’s not what we’re going to do (at least not until I manage to muster some understanding of said machinery).

1. Some subgroups of the ideles

Fix a number field ${k}$. Let’s first look at the open subgroups of ${J_k}$. For this, we determine a basis of open subgroups in ${k_v}$ when ${v}$ is a place. When ${v}$ is real, ${k_v^+}$ will do. When ${v}$ is complex, ${k_v^*}$ (the full thing) is the smallest it gets. When ${v}$ is ${\mathfrak{p}}$-adic, we can use the subgroups ${U_i = 1 + \mathfrak{p}^i}$. Motivated by this, we define the notion of a cycle ${\mathfrak{c}}$: by this we mean a formal product of an ideal ${\mathfrak{a}}$ and real places ${v_1, \dots, v_l}$ induced by real embeddings ${\sigma_1, \dots, \sigma_l: k \rightarrow \mathbb{R}}$. Say that an idele ${(x_v)_v}$ is congruent to 1 modulo ${\mathfrak{c}}$ if ${x_{\mathfrak{p}} \equiv 1 \mod \mathfrak{p}^{ \mathrm{ord}_{\mathfrak{p}}(\mathfrak{a} )}}$ for all primes ${\mathfrak{p} \mid \mathfrak{a}}$ and ${x_{v_i} >0}$ for ${1 \leq i \leq l}$. We have subgroups ${J_{\mathfrak{c}}}$ consisting of ideles congruent to 1 modulo ${\mathfrak{c}}$. Note that ${k^* J_{\mathfrak{c}} = J_k}$ in view of the approximation theorem. We define the subgroup ${U(\mathfrak{c}) \subset J_{\mathfrak{c}}}$ consisting of ideles that are congruent to 1 modulo ${\mathfrak{c}}$ and units everywhere. Fix a finite Galois extension ${M/k}$. If ${\mathfrak{c}}$ is large enough (e.g. contains the ramified primes and to a high enough power), then ${U(\mathfrak{c})}$ consists of norms—this is because any unit is a local norm, and any idele in ${Y(\mathfrak{c})}$ is very close to 1 (or positive) at the ramified primes. These in fact form a basis of open subgroups of ${J_k}$. (more…)