TITLE: Adelic and classical modular forms on quaternion algebras
QUESTION [9 upvotes]: Let $R$ be an Eichler order of an indefinite quaternion algebra $B/\mathbb{Q}$ (suppose B is not the collection of $2\times 2$ matrices) and $S$ the corresponding Shimura Curve. Modular forms of weight $2k$ on $S$ can be written as either
(1) Functions on the idele group $B^*_\mathbb{A}$ of $B$
(2) Holomorphic functions on the upper half plane that satisfy the required transformation property under the corresponding Fuchsian group.
It is well-known that these definitions are equivalent, but I can't find a reference that writes down both
(A) An isomorphism between (1) and (2) in this setting. (which I imagine is just given via pullback under $B^*_\mathbb{A} \xrightarrow{g \mapsto g_\infty(i)} \mathcal{H}$)
(B) Defines the Hecke operators.
Can you please give me a reference? An electronic reference, if available, would be very much appreciated.

REPLY [3 votes]: Many of these questions have to do with strong approximation; in (only somewhat) precise terms, it states that if $G$ is a simply connected semisimple group which is simple over $\mathbb Q$, and if $G(\mathbb{R})$ is not compact, then $G(\mathbb {Q})$ is a dense subgroup of $G(\mathbb{A}_f)$ ($\mathbb{A}_f$ is the ring of finite adeles over $\mathbb{Q}$). Consequently, if $K$ is a compact open subgroup of $G(\mathbb{A}_f)$, and $\Gamma =G(\mathbb{Q}\cap K$ (then $\Gamma $ is a congruence arithmetic subgroup of $G(\mathbb{Q})$), then $G(\mathbb{Q})K=G(\mathbb{A}_f)$ . In particular, functions on the quotient $G(\mathbb{A})/G(\mathbb{Q})$ which are invariant under $K$ may be identified with functions on $G(\mathbb{R})/\Gamma$. 
All this can be applied to $G=SL_1(B)$; to extend it to $GL_1(B)$ needs a little more book-keeping (since the latter is not simply connected) and is done in many places like the Corvallis volumes.
In particular, if $\pi$ is a discrete series representation of $SL_2(\mathbb{R})$ of lowest weight $k$ which occurs in $G(\mathbb{R})/\Gamma$ the lowest weight vector may be viewed as a modular form of weight $k$ on the upper half plane with some transformation property under the Fuchsian group $\Gamma$. By the preceding, this function on $G(\mathbb{R})/\Gamma$ may also be viewed as a function on $G(\mathbb{A}/G(\mathbb{Q})$, with prescribed behaviour on the archimedean component