Explicit Jacquet-Langlands correspondence

Question

Jacquet-Langlands correspondence gives a 1-to-1 correspondence between automorphic forms on $\mathrm{GL}_{2}(\mathbb{Q})$ and automorphic forms on $\mathrm{GL}_{1}(D)$, where $D$ is a division algebra over $\mathbb{Q}$. As one can consider (holomorphic) modular forms as $\mathrm{GL}_{2}(\mathbb{Q})$ automorphic forms, I think there should exists some kind of automorphic functions on $\mathrm{GL}_{1}(D)$ corresponds to the modular form. However, I can't imagine how the automorphic function on division algebra should look like. According to this note, the corresponding function might be a function on $\mathfrak{h}^{n}$, where $\mathfrak{h}$ is a complex upper half plane and $n$ is the number of infinite places which $D$ split, with suitable transformation laws. Could anyone give more detailed explanations or references in this direction? For example, what kind of function may corresponds to the discriminant form $\Delta(z) \in S_{12}(1)$?

Answer 1

Adelically, ($L^2$-)automorphic forms on (the multiplicative group of) a division algebra $D/F$ are just elements of $L^2(G(F) \backslash G(\mathbb A_F))$, for a number field $F$. Since you're working over $F = \mathbb Q$, the number of infinite places $D$ is split is 0 or 1, according to whether $D \otimes \mathbb R$ is Hamilton's quaternions or $M_2(\mathbb R)$.

Two places to start learning about automorphic forms on quaternion algebras are Gelbart's book on GL(2) and Dembele-Voight's survey article on computing Hilbert modular forms. Sometimes these are also called "quaternionic modular forms."

Note that if you want a newform $f \in S_k(N)$ to correspond an automorphic form $\phi$ on a quaternion algebra $D$, it is necessary $D$ ramifies at each $p | N$, so level 1 forms do not transfer to (non-split) quaternion algebras.