# Automorphic quotient for quaternion algebras

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.
• Paul, I thought Fujisaki's lemma is true for any central simple algebra (not just division algebras), including in this case $B=\mathrm{M}_2(F)$. So the OP need not exclude the totally split'' case. Right? But the orbifolds obtained from quotients by discrete subgroups are only compact when $B$ is a division algebra (by the theorem of Hey). So there are two layers here, I think. – John Voight Jan 17 '18 at 0:21
• @paulgarrett, you're right, in Fujisaki's article "On the Zeta-Function of the Simple Algebra over the Field of Rational Numbers", the statement is Theorem 8.3, pg. 599, and at the beginning of section 8, he supposes that $B$ is a division algebra! I must be wrong about the matrix algebra case. – John Voight Jan 18 '18 at 3:17
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.