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](https://mathoverflow.net/questions/106709/why-are-s-arithmetic-groups-interesting) 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 $PGL(2,q^2)/PGL(2,q)$](https://math.stackexchange.com/questions/425583/references-about-pgl2-q2-pgl2-q)


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


* [Volume of PGL(2,F) \ PGL(2, A)](https://mathoverflow.net/questions/124529/volume-of-pgl2-f-pgl2-a)




---


**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 $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$.



How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
Yes, the quotient $\mathrm{PGL}\_2(\mathbb{Q})\backslash\mathrm{PGL}\_2(\mathbb{A})$ and its generalizations for other (reductive) algebraic groups is a complicated object, and this is to a large extent the reason why the theory of automorphic forms is a deep subject. The diagonal embedding of $\mathrm{PGL}\_2(\mathbb{Q})$ into $\mathrm{PGL}\_2(\mathbb{A})$ connects the quasi-factors $\mathrm{PGL}\_2(\mathbb{Q}\_v)$ in a subtle way, which otherwise would be completely independent. We would like to understand how much dependence is introduced and how much independence is lost by taking the quotient of $\mathrm{PGL}\_2(\mathbb{A})$ by $\mathrm{PGL}\_2(\mathbb{Q})$. This fits nicely in the general local-to-global philosophy of number theory. 


By the way, one cannot just throw away some quasi-factors from $\mathrm{PGL}\_2(\mathbb{A})$ and take a quotient by $\mathrm{PGL}\_2(\mathbb{Q})$, because the latter is meant to be embedded diagonally into $\mathrm{PGL}\_2(\mathbb{A})$, i.e. it appears in every quasi-factor. Hence $\mathrm{PGL}\_2(\mathbb{Q})\backslash \mathrm{PGL}\_2(\mathbb{R})$ or even $\mathrm{PGL}\_2(\mathbb{Q})\backslash \prod\_{v\neq 2}\mathrm{PGL}\_2(\mathbb{Q}\_v)$, say, have little to do with the true adelic quotient $\mathrm{PGL}\_2(\mathbb{Q})\backslash\mathrm{PGL}\_2(\mathbb{A})$.


At any rate, there are good introductions to adelic quotients. I recommend Chapter IV in Weil: Basic number theory, especially Section 2 there which explains why the adeles are separating the rationals much like the reals are separating the integers. Then one can read Sections 3.3 and 3.6 in Bump: Automorphic forms and representations, which explains in the setting of $\mathrm{PGL}\_2$ the connection of the adelic quotient to classical congruence quotients. 


