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?
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?
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.
What does ``automorphic quotient'' mean?
Does my book (http://quatalg.org), Main Theorem 38.4.3 (a theorem of Hey) answer your question? A quaternion algebra $B$ over a field $F$ is either isomorphic to $\mathrm{M}_2(F)$ (is this your ``totally split case''?) or is a division algebra over $F$, and in the latter case, Hey's theorem applies.