Some papers I am reading talk about an "adelic" object $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$ . This has sparked a lot of confusion since I don't know what such a quotient could mean.

A crude way of looking at the adéles is just as the product over primes:

$$ \mathbb{A}_\mathbb{Q} = \mathbb{R} \times \mathbb{Q}_{p_1} \times \dots \times \mathbb{Q}_{p_k} \dots $$

Naively one might assume this passes over to groups of fractional linear transformations. I believe the term is "strong approximation" though it doesn't make it any easier to understand.

$$ PGL(\mathbb{A}_\mathbb{Q}) = PGL(\mathbb{R}) \times PGL(\mathbb{Q}_{p_1}) \times \dots \times PGL(\mathbb{Q}_{p_k}) \dots $$

Actually even if we take just one part of that object the object is hard to understand, since $\mathbb{R}/\mathbb{Q}$ is already a nasty object:

$$PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{R})$$

I don't really understand what the rationals are doing here. The only one I kind of understood is the identificaiton of the hyperbolic plane $\mathbb{H}^3 = PGL(2, \mathbb{R}) \backslash PGL(2, \mathbb{C})$. How to understand such a complicated group action?

It seems that for any two groups $H \subset G$ we could have $PGL(2, \mathbb{H} \backslash PGL(2, \mathbb{G})$.

- Kind of similar References about $PGL(2,q^2)/PGL(2,q)$

OK. This object seems to be familiar to experts on automorphic forms - which I am definitely not:

**Partial progress** The issue of diagonal embedding $\mathbb{Q} \subset \mathbb{A}$ and the solenoid structure of $\mathbb{A}/\mathbb{Q}$ are two major points that I missed. The original question merely asked "What is $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$?"

Although these points are in books, it would be great an outline of the "adèlic solenoid" structure of $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$.

diagonallyhere, right? In this sense it is discrete because two 'real'ly close rationals have large denominators, so that there is some $\mathbb Q_p$ that sees them as being far apart. $\endgroup$ – LSpice May 27 '15 at 22:58