Modular forms on central division algebra of degree $\ge 3$

Question

I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in terms of classical modular forms and quaternionic modular forms (with some keywords including basis problems). Interesting part is that quaternionic modular forms are actually functions on some "0-dimensional" space, i.e. finite set of points (for a definite quaternion algebra $D/\mathbb{Q}$, the double coset space $D^\times \backslash (D_{\mathrm{fin}}^\times) / K$ is a finite set where $D_{\mathrm{fin}} = D\otimes_{\mathbb{Q}} \mathbb{A}_{\mathbb{Q}}^\infty$). We can also define Hecke operators and dimond operators on the space of quaternionic modular forms via double cosets as usual.

I wonder if we can generalize the theory to the higher degree central division algebras (for example, over $\mathbb{Q}$), especially for the degree 3 division algebras. More precisely,

• For a given central division algebra $D$ over $\mathbb{Q}$ (maybe we need to assume that it is definite?), can we define automorphic forms on $D$ of level $K \leq D_\mathrm{fin}^\times$ as functions on the double coset space $D^\times \backslash D_{\mathrm{fin}}^\times / K$? Is the space only contains finitely many points?
• AFAIK, for a degree $D$ central division algebra over $\mathbb{Q}$ (so that $\dim_\mathbb{Q}D = d^2$) Jacquet-Langlands correspondence can be generalized to that between automorphic representations of $\mathrm{GL}_{n}(D \otimes \mathbb{A}_\mathbb{Q})$ and those of $\mathrm{GL}_{nd}(\mathbb{A}_\mathbb{Q})$ (I'm not being precise here), so for given $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$ automorphic forms/representations, e.g. $\mathrm{GL}_3$ Maass forms, or things obtained from $\mathrm{GL}_2$ automorphic forms via Gelbart-Jacquet lift ($\mathrm{Sym}^2$), there should be some automorphic forms/representations on a division algebra $D$ of degree 3 over $\mathbb{Q}$. Is there any explicit example of such correspondence for $\mathrm{GL}_3$ and degree 3 central division algebra case? Also, does there exists any "natural" map from quaternion algebra to degree 3 algebra that is compatible with Jacquet-Langlands correspondence and symmetric square lift? (My naive guess is no, but at least we can define such a map through Jacquet-Langlands and Gelbart-Jacquet...)

Thanks in advance.

Answer 1

For the first question, it is only true that if $D$ is a (totally) definite quaternion algebra over a number field $K$, then the weight 0 automorphic forms factor through a finite set (1-sided ideal class representatives of an order in the classical framework). This is because $D^\times(\mathbb R)$ is compact mod center.

If you want automorphic forms in higher rank, you need to work with groups that are (essentially) compact at infinity. You have such forms for orthogonal, symplectic and unitary groups, but not GL($n$) for $n > 2$. Such forms are called algebraic modular forms.

For the second question, by comparing symmetric power lifts with Jacquet-Langlands for GL($3$), you can get a partial map from GL(2) to $D^\times$ when $D$ has dimension 9. (Some lifts to GL(3) will have local obstructions to come from $D^\times$.)