# Adelic and classical modular forms on quaternion algebras

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.

• If you know a reference for this fact for modular curves, then just copy it verbatim? I don't know a reference though. In fact I remember having difficulty finding a reference for this statement even for modular curves (although perhaps Bump is one).
– eric
Commented Jan 4, 2016 at 18:43
• I could also say that it's hard to imagine why one would need a reference. The result is either "well-known" (in the context of needing it in a paper) or "a long but straightforward exercise" (if you're learning the area and are wondering why it's true).
– eric
Commented Jan 4, 2016 at 18:51
• @eric: Thanks, but I want to be very explicit - especially for (B).
– LMN
Commented Jan 4, 2016 at 19:11
• Then it might be your job to provide the reference :-/ I can see why you're asking though! It might just be easier to bite the bullet and try and just check everything yourself though. Hope you have a day free in the near future. To be honest if you are really after something specific then it wouldn't surprise me at all if any reference you found which covered the general problem didn't actually do the thing you needed in the details you needed. If I were you I would just do it yourself; in the time you spend waiting for a good answer to this question you could have done it...
– eric
Commented Jan 4, 2016 at 19:35
• Does section 5.3 of Miyake's book "Modular forms" not have the level of explicitness you require for (B)? Commented Jan 5, 2016 at 1:23

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.

• +1 for calling Ricky Martin inside MO, I think it's a first! Commented Sep 28, 2016 at 10:56
– user41593
Commented Sep 28, 2016 at 12:33

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.

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

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.