TITLE: Automorphic quotient for quaternion algebras
QUESTION [5 upvotes]: Are automorphic quotient for quaternion algebras always compact (safe the totally split case)?
Is there any good reference for proof of this fact, or easy arguments to say do?

REPLY [8 votes]: Weil's "Adeles and algebraic groups" proves this for general division algebras. It is labelled "Fujisaki's Lemma", and is a natural division-algebra generalization of the analogue for number fields themselves, namely, that $\mathbb J^1/k^\times$ is compact, where $\mathbb J^1$ is the collection of ideles of idele-norm $1$. For number fields, this compactness implies finiteness of (generalized) class number, and implies the (generalized) units theorem.