Let’s step away from $\zeta$function stuff for a bit, and talk about something different. ^{1} In an earlier post, I mentioned these local fields like $\Q_p$ that are useful for studing things “one prime at a time” ^{2} (whatever that means). Corresponding to this local fields, one also has global objects (e.g. $\Q$) from which they arise, but in some sense, these global objects (i.e. global fields) don’t have all the information of the local objects readily available ^{3}. Because of this, it may be nice to consider different global objects that combine all the local ones in a more straightforward manner.
Definitions
Let $K$ be a global field. Technically, this means that $K$ is a number field or a function field of a curve over $\F_q$ (i.e. $K/\F_q(t)$ is finite), but the example I’ll have in mind if of $K$ as a number field. At some point I may explicitly say to let $K$ be a number field to simplify things, but know that most (all?) of what I do can be done for a general global field.
We want to construct the adele ring of $K$ which morally is just the topological ring
where $v$ ranges over all places of $K$, and $K_v$ is the completion of $K$ at $v$. However, this product is stupidbig (so maybe not the easiest thing to work), and doesn’t reflect some of the nice finiteness properties of global fields (e.g. the valuation of $x\in K$ is zero for almost all places). Because of this, we’ll replace it with a socalled restricted product.
Given this, we define the finite adele ring of $K$ (or ring of finite adeles) to be the (topological) ring
where the notation $v\nmid\infty$ means we’re ranging only over finite (i.e. nonarchimedean) places, and $\ints v\subset K_v$ is the ring of integers (i.e. elements of norm at most 1). The adele ring of $K$ is obtained from the finite adeles by throwing in the infinite places. In other words, it is
Since there are only finitely many infinite places ^{4}, if we let $\ints v=K_v$ for $v\mid\infty$, we could have just defined ^{5}
Now that we know what the adele ring is, a few remarks about why this restricted direct product is nicer than the ordinary direct product. First, topologically speaking, $\prod_vK_v$ is not locally compact ^{6} essentially because the product topology requires open sets to be entire spaces in all but finitely many factors. However,$\dots$
This means that $\A_K$ is the product of finitely many locally compact spaces, so it is itself locally compact. In this way, it is not as stupid big as the ordinary direct product even though it looks massive at a glance. It’s also worth noting that $K\into\A_K$ as a discrete subgroup via the diagonal (algebraic) embedding
since $x$ has zero valuation at all but finitely many places ^{7}. Elements of the image of this embedding are sometimes referred to as “principal adeles.” The embedding $K\into\A_K$ should be thought of as an analouge of $\Z\into\R$ (e.g it’s discrete ^{8} and we’ll later see that $\A_K/K$ is compact). The unit group $\units\A_K$ of the adele group is also important.
We similarly have a diagonal embedding $\units K\into\I_K$. Finally, we can extend the absolute value $\nabs_v$ on $K_v$ to $\A_K$ via $\abs{x}_v=\abs{x_v}_v$ for $x\in\A_K$, and then we can combine these to define a global absolute value ^{9}
on $\A_K$ which converges since $\abs{x_v}_v\le1$ for all but finitely many $v$.
Basic Properties
Let’s “prove” ^{10} some things about adeles.
One can also proove
but doing so requires saying the words “Haar measure,” and I’d rather not get into that, so I’ll skip the proof of this fact ^{11}. If you recall from before, $K$ is discrete in $\A_K$, and so certainly not dense. This results says that if we remove a single (finite) place from $\A_K$, then $K$ goes from being discrete to being dense!
We’ll next take a look at how $\A_E$ is related to $\A_K$ when $E/K$ is a(n) (finite) ^{12} extension of global fields.
Fix some (finite, separable) extensions $E/K$, and fix a place $v$ on $K$. Let
where $w\mid v$ means that $w$ is a place on $E$ which restricts to $v$ on $K$. We can use to notation to build $\A_E$ from $K$ as the following lemma illustrates.
The above isn’t the only way to get from $K$ (or $\A_K$) to $\A_E$, however.^{13}
Moving on, we claimed that the diagonal embedding $K\into\A_K$ turns $K$ into a discrete subgroup of $\A_K$. Let’s actually prove this in the case that $K$ is a number field. We’ll leave the function field case as an exercise. ^{14}
With that proven, let’s complete the second part of the $Z\into\R$ analogy by showing that $K$ is cocompact in $\A_K$. Note that, if $K$ is a number field, then $\A_K\cong(\A_{\Q})^n$ as topological groups where $n=[K:\Q]$. Hence, $(\A_K/K)\cong(\A_{\Q}/\Q)^n$ is compact iff $\A_{\Q}/\Q$ is. Thus, we can restrict our attention for the next proof (we could have use the same trick with discreteness if we wanted).
To wrap up this section, we give a proof of the Artin product formula which says that
for all $a\in\units K$. We’ll continue our trend of only proving things for number fields, so assume $K$ is one of those. Note that
so it suffices to prove this for $K=\Q$. Since $\nabs:\A_K\to\R_{\ge0}$ is multiplicative, we can further simplify to the case that $a=p$ is prime. Here, there are only two nonunit absolute values, $\abs p_p=\frac1p$ and $\abs p_\infty=p$. Thus, we win.
Class Groups
At this point we know a thing or two about adeles, but maybe we don’t know what they’re good for. One of the classic reasons for studying adeles is to give a more memorable proof of things like the finiteness of the class group of a number field. A love the geometry of numbers as much as the next guy ^{15}, but the topologist in me refuses to believe in any finiteness prove that doesn’t end with “This space is both compact and discrete and hence finite.”
It turns out that there are many “class groups” one can define using adeles. We won’t bother looking at all (most?) of them. For the remainder of this section, fix a number field $K$.
It turns out that $C_K$ is not necessarily compact, but the socalled normone idele class group of $K$ is.
We next relate these to the traditional class group $\Cl_K$ of $K$. Let $J_K$ denote its group of fraction ideals (finitely generated $\ints K$submodules of $K$), so $\Cl_K=J_K/\units K$. Recall that there is a onetoone correspondence $v\mapsto\mfp_v$ between the finite places of $K$ and the nonzero prime ideals of $\ints K$. Let $C_{K,\mrm{fin}}=\I_{K,\mrm{fin}}/\units K$. Finally, let $\tints K=\prod_{v\nmid\infty}\ints v$ and $\units{\tints K}=\prod_{v\nmid\infty}\units{\ints v}$.
Another classic use of adeles is proving Dirichlet’s theorem about the rank of the unit group of the ring of integers of a number field. It’s also worth mentioning that both this and the finiteness of the class group have generalizations to $S$integers which can be proven with adelic methods. However, instead of covering these, I think I will stop here.

Secretly, we’re not stepping that far away from it. One does Fourier analysis on R to prove nice things (i.e. existence of a functional equation) about the Riemann zeta function. Analagously, one does Fourier analysis on these adelic rings to prove nice things about more general Lfunctions. We won’t really touch on this here, but it’s lurking in the background. ↩

Better put, “one place at a time” ↩

Of course, given e.g. Q, you can complete it at various places to obtain all the Q_p’s you could want. However, if you want to study e.g. Q_2 and Q_5 at the same time, then you can’t complete Q because this will kill valuable information, but Q itself is somehow not the best place to work to understand Q_2 and Q_5 simultaneously. ↩

Exercise: prove this (hint: infinite places on number fields come from embeddings into C) ↩

If we did this, we would need to amend our definition of restricted products to not require all H_i to be compact. Instead, we’d only require all but finitely many H_i are compact. I’ll leave it up to the reader to figure out how to modify arguments for this slightly more general definition. ↩

You want this to be able to do analysisy type stuff. Locally compact topolgical groups have Haar measures which let you do Fourier Analysis (I guess in this setting it’s typically called harmonic analysis) on them (maybe just when they’re abelian). ↩

Exercise: prove this (hint: factor (x)) ↩

Exercise: prove this (hint: suffices to find a neighborhood around 1 containing no other principal adele) (hint2: It’s possible you want to hold off on proving this until you see the product formula) ↩

technically, for this to make sense we need to choose a representative of each place on K. Just choose the normal ones (e.g. for v finite, choose p =1/q where p is a uniformizer and q is the size of the residue field) 
Quotes because I won’t give all the details for most (any?) of the things here ↩

I don’t think I’ll need it for anything. If I do and it bothers you that I haven’t proved it, you can find a proof in these notes ↩

What’s the correct way to notate that the two choices are “a finite extension” and “an extension”? ↩

TODO: double check that {e_i} gives a basis for E_v/K_v for all v and not just all but finitely many v (proof should be salvagable in either case) ↩

hint: for F_q(t), the tadic absolute value (I think this is usually considered the infinite place) should play the role of the archimedean places in the number field proof. I could be wrong about this; I really haven’t spent much time with function fields. ↩

sarcasm: I’m not a fan of it ↩