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What problem do the adeles solve?

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?)

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What is your conceptual understanding of what Arthur-Selburg is doing? I'm not an expert, but my impression is that this kind of thing is about understanding modular forms using representation theory, and to do this you have to look at groups defined over the adeles. The idea is that automorphic forms and automorphic representations are really the same thing even though they look pretty different. If you don't like adeles, you could try to translate things back into the language of modular forms, but then it's harder to use the tools of representation theory. – Max Flander Oct 16 2011 at 2:15
I literally just want to evaluate the the trace of a geometric operator numerically. In other words, my goal is to use a very direct form of the trace formula, where an integral operator defined on a group is acting on functions defined on some homogeneous space. I am evaluating this operator numerically, and if I can get the sum to converge quicker (ie via Poisson summation), then I can speed up my code. Now it seems to me that the trace formula could be a promising tool here, but I am not sure if I should invest the extra effort in trying to understand adeles. – Mikola Oct 16 2011 at 2:28
Perhaps A. Robert, Des adeles: pourquoi? Enseign. Math. 20, 133-145 (1974) will help. – Franz Lemmermeyer Oct 16 2011 at 10:45
If you want to work over $\mathbb R$ instead of adeles, why are you reading about the Arthur-Selberg trace formula as opposed to the classical Selberg trace formula? This has been studied in many contexts in its classical form. – Kimball Oct 16 2011 at 14:46
As far as the Arthur--Selberg trace formula is concerned, one reason for working with the adeles is that it's easier to talk about conjugacy classes in $G(\mathbb Q)$ than about conjugacy classes in $\Gamma$. – Faisal Oct 16 2011 at 21:56

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.

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This is (an unusually eloquent version of) the standard answer To the question. Certainly it provides some motivation for working with the adeles. But I've come across few explicit comparisons of the classical approach to a problem with the adroit version of a problem indicating where the classical approach is conceptually/aesthetically inferior.the only one's that I've seen are how class field theory can be formulated without making an arbitrary choice and how the Gamma factors in the functional equation of a Dedekind zeta function emerge naturally. – Jonah Sinick Oct 16 2011 at 19:13
(cont.) but it seems to me that it would be very desirable pedagogically to have many more explicit examples worked out. – Jonah Sinick Oct 16 2011 at 19:14