Compactness of automorphic quotient - quaternion group and $\mathrm{GL}_{2}$

Solution 1:

Yes, even though (because a division algebra splits locally almost everywhere) the two groups are locally-almost-everywhere the same, one global quotient is non-compact, and the other is compact.

Yes, even if the division algebra is split at all archimedean places, if it is non-split at any place, then the (global) quotient is compact.

This applies to the $GL_n$ situation, and central division algebras of dimension $n^2$ over global fields. This is proven by "Fujisaki's Lemma" (as it is called in Weil's "Adeles and algebraic groups", which is just a slight non-abelian generalization of the argument that $\mathbb J^1/k^\times$ is compact (e.g., http://www.math.umn.edu/~garrett/m/v/fujisaki.pdf for the abelian case).

For other classical groups, such as orthogonal groups, again the quotient is compact if and only if the form is globally anisotropic. (I have some notes on this, based on part of Godement's Bourbaki notes on reduction theory: http://www.math.umn.edu/~garrett/m/v/cptness_aniso_quots.pdf)

By Hasse-Minkowski (and more elementary properties of $p$-adic quadratic forms), for quadratic spaces of dimension five or more, anisotropy is possible if and only if the form is anisotropic at at least one archimedean place (since $p$-adic quadratic forms in dimensions five or higher are demonstrably isotropic).

So, yes, for other classical groups, compactness at (at least one) archimedean place/s is necessary for compactness of the quotient. But over number fields with several archimedean places, many local groups can be non-compact... all it takes is a single compact factor.

And, as @AlexYoucis commented, Platonov and Rapinchuk is probably the best, most authoritative, and most encyclopedic source for this kind of thing.