While browsing through some papers, I came across some literature discussing the Arthur-Selberg trace formula. At a conceptual level I think I understand what it is doing, but when I get down to the technical details I start to get a bit lost. Part of the problem is that James Arthur's papers are all written using adelic Lie groups, instead of working over an arbitrary field or fixing something less complicated (like R or C). Clearly, he must have had a good technical reason for doing this, but for the life of me I can't see why.
Is there any real intuition behind the adeles? Outside of algebraic number theory, why would anyone ever want to use them? What do they "do" that R doesn't? (in other words, could I safely do a mental find replace of $\mathbb{A}_{\mathbb{Q}}$ with $\mathbb{R}$ and still get basically the gist of what is going here?)
Best Answer
When working on a rational problem (over $\mathbb{Q}$), you can't do much analysis - so you lack quite a few tools.
The obvious solution is then to pass to the completion, where you'll be able to do analysis ; so most people go to $\mathbb{R}$. But that isn't that natural : $\mathbb{Q}$ has several completions in fact, and choosing the absolute value to measure distances wasn't the only choice. You could have gone to the $p$-adic numbers too, and use their special properties to gain further insight on your initial problem.
So somehow you gain the idea that perhaps solving your initial problem (which is called global) might involve looking into the various problems obtained by pushing it into the various available completions (which are called local).
That means you gain huge means of study, but now there are two prices to pay : you have the question whether some kind of Hasse principle applies, which is "How equivalent is it to solve the problem locally everywhere and globally?", and you have the problem that you need to work in all of those.
That last problem is where adeles come into the picture. Working adelically means you're effectively working simultaneously in all places -- and you mostly put them on an equal footing. You're still able to go into a single local place if needs arise, but you get an object which puts the pieces together.
There is another nice thing about adeles : if your initial problem wasn't just over $\mathbb{Q}$, but other any other kind of global field (a number field or a function field), then again you'll have a notion of adeles, and most tools will work the same. In fact, getting insight on a problem for a type of global fields and pushing the idea in the adeles is a good way to know what to look for in the other type of global fields. There are problems which are thus solved for some of them, and conjectures for others, precisely for this reason.
So I think what I wrote makes clear why wanting to replace adeles by just $\mathbb{R}$, while it will give some understanding of things (in a single place!), will also pretty much destroy the whole symmetry of the matter, and hence lose much of it.