In the third post of this series, we shall very briefly introduce the formalism of ideles and restate the Artin reciprocity theorem in our new language, following the ideas of Chevalley. It is a little difficult to get used to the language of adeles and ideles, but they turn out to be a very natural setting in which to study a global field (like a number field) in relation to its completions with respect to absolute values. In fact, one of the reasons for their introduction was to facilitate a proof of global class field theory from local class field theory. However, the main results of global class field theory also become much cleaner to state, as we shall see next time when we finally get to them.

Firstly, we have to define these things! Fix a global field ${K}$ (a number field or a 1-dimensional ${\mathbb{F}_p}$-function field). We want an adele to be an element of the product ${\prod_{v \in M_K} K_v}$, where ${M_K}$ is the set of valuations on ${K}$. In other words, we want to take one element of each completion of ${K}$. However, there is far too much junk in this product! We are after all studying ${K}$ itself, and the diagonal map ${K \rightarrow \prod_{v \in M_K} K_v}$ is ridiculously far from being surjective. For example, a real life element of ${K}$ will have to be integral at all but finitely many places. In fact, once we’ve noted this, we are actually ready to define the adeles.

The ring of adeles ${\mathbb{A}_K}$ is the set of ${(x_v) \in \prod_{v} K_v}$ such that ${x_v \in \mathcal{O}_v}$ for all but finitely many ${v}$ (note that there are only finitely many archimedean places, so we can ignore them and the definition still makes sense).

This is a fairly hefty ring. It had better possess some kind of interesting topology if we’re going to be able to understand it. One would like to take the subspace topology coming from the product topology, but annoyingly a product of locally compact spaces need not be locally compact (think ${{\mathbb R}^{\mathbb N}}$), so if we want local compactness (which would be nice) we need a different topology. It turns out there is a fairly natural topology. We take as a base of open sets the products ${\prod_v U_v}$ such that ${U_v}$ is open for each ${v}$ and for all but finitely many ${v}$, we have ${U_v=\mathcal{O}_v}$. So this looks kinda similar to the product topology, but is more suited to situations where we have imposed a restriction like the one we did: it is called the restricted product topology.

With this shiny new topology, ${\mathbb{A}_K}$ becomes a locally compact topological ring (noting that ${\mathcal{O}_v}$ is in fact always compact and that if you’re “locally locally compact” then you’re locally compact), and that’s absolutely fantastic! However, life just keeps getting better for us. Remember how earlier we were talking about how we’re actually studying ${K}$ and so we want the image of the diagonal map ${K \rightarrow \mathbb{A}_K}$ to not be too insignificant. Well, we’re in unbelievably good luck! Firstly the image of ${K}$ is discrete (which seems like it should be true). However, there are so many elements of ${K}$ discretely scattered in ${\mathbb{A}_K}$ that in fact the quotient ${\mathbb{A}_K/K}$ is compact! The main ingredient of the proof is again the weak approximation theorem (which is very similar in flavour and sophistication, but also ubiquity, to the chinese remainder theorem).

There is some fairly cool measure theory going on here (in fact, if you develop this theory correctly, you can bypass the classical Minkowski proofs of finiteness of the class group and Dirichlet’s unit theorem). Locally compact groups have a unique (up to scaling) Haar measure, and being compact, ${\mathbb{A}_K/K}$ has finite quotient measure. Since the effect of multiplication by ${a \in K}$ on the measure of a subset of ${\mathbb{A}_K}$ can be shown to be precisely the product of the normalised valuations, this gives an instant and satisfying proof of the classical product formula for all norms on ${K}$. We shall not explore these issues here, but some other post may, and they are definitely worth attention.

Anyway, back to the task in hand. Now we have the adeles. How about the ideles? Well, they are just the multiplicative analogue. Indeed, the idele group of ${K}$ is just the group of units ${J_K = \mathbb{A}_K^*}$. However, things aren’t quite as nice as that. We have no guarantee that inversion is continuous in the subspace topology, so we have to take a slightly refined topology: the subspace topology on ${\mathbb{A}_K^* \hookrightarrow \mathbb{A} \times \mathbb{A}}$ (where the embedding is ${x \mapsto (x,x^{-1})}$). Soon we will pass to a subgroup of the ideles which makes this issue go away, so it probably isn’t worth thinking about for too long.

So what do we know? It’s clear that the principal ideles ${K^* \hookrightarrow J_k}$ are discrete in this topology (since it’s stronger than the previous one). It can also be seen that in fact the ideles in this topology are just the elements ${(x_v) \in \prod_v K_v^*}$ such that ${x_v \in \mathcal{O}^*_v}$ for all but finitely many ${v}$, with a restricted product topology like that on the adeles, so they really are exactly the multiplicative analogue (even their topology is!).

But always we should get a nagging feeling that we should be studying real life elements of ${K}$ and maybe that ${J_K}$ is just a bit too big to be studying. What other properties do elements of ${K}$ have? Well, they satisfy a product formula

$\displaystyle \prod_v |x|_v = 1.$

We cannot guarantee this for every idele. We therefore consider the map

$\displaystyle c: (x_v) \mapsto \prod_v |x_v|_v$

which gives a continuous homomorphism ${J_k \rightarrow {\mathbb R}_{>0}^*}$, and let ${J_K^1}$ be its kernel.

This set ${J_K^1}$ turns out to be often a more natural object of study. It contains all of the principal ideles, as we remarked above. It also is a closed subgroup of ${J_K}$ and has the surprising property that its topology is equal to its subspace topology inherited from ${\mathbb{A}_K}$. However, the main reason it is a good thing to study comes from its receiving the same seal of approval as that of the ideles:

Theorem 1 The topological group ${J^1_K/K^*}$ is compact.

This is all great fun, at least for me, but lots of readers have either now probably stopped reading or started wondering how any of this is remotely related to class field theory. Well, there is a natural map ${\alpha}$ from ${J_K}$ to the ideal group ${I_K}$ of ${K}$, obtained by just trying to construct the fractional ideal which feels most likely to be generated by’ our idele. Indeed, we define

$\alpha: \displaystyle (x_v) \mapsto \prod_{v \text{ finite}} \mathfrak{p}_v^{v(x_v)}.$

This makes sense because all but finitely many of the ${x_v}$ are units in their valuation rings.

Notice how by fudging things into the archimedean valuations (which don’t get detected by this map) we get something better. In fact, restricting to ${J_K^1}$ we still get a surjection ${J_K^1 \rightarrow I_K \rightarrow 0}$. This has all kinds of cool consequences. Firstly, including the finiteness of the class group (${I_K/K^*}$ is discrete, and the above surjection implies it is compact, whence finite).

So it’s worth bearing in mind we have this nice ${J_K^1}$ thing lying around. However, for class field theory we will decide to work with ${J_K}$ instead. I am not entirely sure why – I guess it is ultimately a bigger object, and given we are able to prove stuff about it, we might as well use it (since the things proved are presumably in some sense more general’). Having not seen a proof myself it is difficult to say, but I shall try to post back here when I do find out (or you might want to tell me why we use ${J_K}$ and not ${J_K^1}$ if you know).

Now, let us take ${L/K}$ a finite abelian extension. The classical Artin map ${F_{L/K}: I^S \rightarrow G(L/K)}$ can be composed with our canonical map ${\alpha: J_K \rightarrow I_K}$ to obtain a map ${\psi_{L/K}: J_K \rightarrow G(L/K)}$ which from now on is what we mean when we say the phrase `Artin map’. The reciprocity law can be re-written in the following form:

Theorem 2 (Artin Reciprocity, idelic form) There exists a continuous map ${\psi_{L/K}: J_K \rightarrow Gal(L/K)}$ with the following properties. The kernel of ${\psi}$ must contain ${K^*}$, and any idele ${x}$ which takes the value ${1}$ at all the ramified and archimedean primes must satisfy ${\psi(x)=F_{L/K}(\alpha(x))}$.

In fact, the condition of being trivial on the principal ideles makes it natural to define the idele class group to be ${C_K = J_K/K^*}$ and regard the Artin map as a continuous homomorphism ${C_K \rightarrow Gal(L/K)}$. It is this language which turns out to seem optimal for stating the main results of global class field theory in their greatest generality, and that is what we will do next time. Indeed, we shall see that the main result of global class field theory is the now simple statement that there is an astonishing correspondence between the finite index open subgroups of ${C_K}$ and the finite abelian extensions of ${K}$.