(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 ${J_k}$.

2. Ideal class groups

Similarly, we can say that ${x \in k^*}$ is congruent to 1 modulo ${\mathfrak{c}}$, written ${x \equiv 1 \mod \mathfrak{c}}$. We will denote this group by ${k_{\mathfrak{c}} = J_{\mathfrak{c}} \cap k^*}$. We shall denote the principal ideals generated by elements of ${k_{\mathfrak{c}}}$ by ${P_{\mathfrak{c}}}$. First, we define the generalized ideal class groups. For a cycle ${\mathfrak{c}}$, we define ${I(\mathfrak{c})}$ as the group of ideals prime to ${\mathfrak{c}}$—or more precisely, to the associated ideal (the archimedean places mean nothing here). Then we can define the generalized ideal class group $\displaystyle I(\mathfrak{c})/P_{\mathfrak{c}}.$

Just as we expressed the regular ideal class group as a quotient ${J_k/k^*J_{S_\infty}}$, we can do the same for the generalized ideal class groups. These results are essentially exercises. I claim that $\displaystyle J_{\mathfrak{c}}/k_{\mathfrak{c}} U(\mathfrak{c}) = I(\mathfrak{c})/P_{\mathfrak{c}}.$

The proof of this is straightforward. There is an obvious map ${J_{\mathfrak{c}}}$ to the generalized ideal class group, and it is evidently surjective (by unique factorization). It also factors through ${k_{\mathfrak{c}} U(\mathfrak{c})}$. Conversely, if an idele ${(x_v)_v \in J_{\mathfrak{c}}}$ maps to the idele ${(c)}$ for ${c \in k_{\mathfrak{c}}}$, then ${(x_v)_v c^{-1}}$ is clearly a unit everywhere and congruent to 1 mod ${\mathfrak{c}}$, hence in ${U(\mathfrak{c})}$.

3. Norm class groups

The previous ideas were fairly straightforward, but now things start to get a little more subtle. Recall that, for global class field theory, we’re not so much interested in the idele class group ${J_k/k^*}$ but rather the quotient group $\displaystyle J_k/k^* NJ_L$

for ${L/k}$ a finite abelian extension. There is a way to represent this as quotient group of the ideals, which we shall explore next. This will be very useful in class field theory, because itis not at all obvious how to define the Artin map on the ideles, while it is much easier to define it on these generalized ideal class groups. So, let’s say that a cycle ${\mathfrak{c}}$ is admissible for an extension ${L/k}$ if it is divisible by the ramified primes, and also if whenever ${x \equiv 1 \mod \mathfrak{c}}$, then ${x}$ is a local norm at the primes dividing ${\mathfrak{c}}$. As a result, it follows that ${U(\mathfrak{c})}$ consists of norms only—indeed, this is true because every unit at an unramified prime is a local norm. (Cf. the computation of the local norm index, and note that an unramified extension of local fields is cyclic.) Let ${\mathbf{N}(\mathfrak{c})}$ denote the group of ideals in ${k}$ which are norms of ideals in ${L}$ prime to ${\mathfrak{c}}$ (i.e. to the primes dividing ${\mathfrak{c}}$). I claim that $\displaystyle \boxed{J_k/k^* NJ_L \simeq I(c)/P_c \mathbf{N}(\mathfrak{c})}$

when ${\mathfrak{c}}$ is admissible. In the next post, we will use facts about the (idealic, inherently) Dedekind zeta function to bound the index of the latter group. This is another reason to look at these norm class groups. How do we define this map? Well, first pick an idele ${i}$. Choose ${x \in k^*}$ such that ${xi}$ is very close to 1 at the primes dividing ${\mathfrak{c}}$, namely ${xi \equiv 1 \mod \mathfrak{c}}$. Of course, ${xi}$ need not be a unit at the other primes. Then map ${xi}$ to the associated ideal ${(xi)}$, which is prime to ${\mathfrak{c}}$. This defines a map ${J_k/k^* NJ_L \rightarrow I(c)/P_c \mathbf{N}(\mathfrak{c})}$. But need to check that this map is well-defined. Ok, so suppose we used a different ${y \in k^*}$ instead of ${x}$. Then ${xy^{-1} \equiv 1 \mod \mathfrak{c}}$, so ${(xi)= (xy^{-1}) (yi)}$ and ${(xy^{-1}) \in P_{\mathfrak{c}}}$. So it is well-defined as a map into the quotient. We do something similar if we replace an idele ${i}$ with something that differs from it by a norm (in view of the approximation theorem). We need to show, of course, that this map is both injective and surjective. First, let’s do surjectivity. Let ${\mathfrak{a}}$ be an ideal, prime to ${\mathfrak{c}}$; then we can choose an idele with component 1 at the primes dividing ${\mathfrak{c}}$, and this idele will map to the class of ${\mathfrak{a}.}$ Suppose there are two ideles ${i_1, i_2}$ that both map to the class of ${\mathfrak{a}}$. Then, choosing ${x,y \in k^*}$ appropriately, we have ${xi_1}$ and ${yi_2}$ mapping to the same ideal, which means that they must have the same valuation at the primes outside ${\mathfrak{c}}$. (At the primes in ${\mathfrak{c}}$, by definition they are both units—in fact close to 1.) So $\displaystyle x y^{-1} i_1 i_2$

is in ${U(\mathfrak{c})}$, which implies it is a norm, and ${i_1, i_2}$ map to the same place in the norm group.