Adelic and classical modular forms on quaternion algebras

Question

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.

Answer 1

You could try "Livin the Hida loca" and look at Theorem 2.1 of

H. Hida. On p-adic Hecke algebras for GL2 over totally real Felds. Ann. of Math. (2), 128(2):295–384, 1988.

I think this is a reference for your question.

Answer 2

People doing computations often write things more explicitly. Is the following survey explicit enough for you ?

Explicit methods for Hilbert modular forms, Lassina Dembélé and John Voight.

Answer 3

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

Answer 4

Have you tried to give a look at Section 2. of Bertolini–Darmon's paper "Heegner points on Mumford—Tate curves", Inventiones Math., 1996 vol. 126 (3) pp. 413-456? They give a very nice survey of modular forms on Shimura curves and their relation with the adelization of $B$. As main reference, they quote D. Robert's Harvard PhD thesis "Shimura curves analogous to $X_0(N)$", 1989. This seems unfortunately unavailable electronically so far.