This is a reference request. 


Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O\_A$ be a maximal order in $A$. Then $\mathrm{SL}\_2(A)$ acts transitively on the right on the set $\mathbf P^1(A)$ of one dimensional left subspaces of $A^2$.


Let $h\_A$ be the cardinal of the *set* (1) of left ideal classes of $\mathcal O\_A$. 


It is not hard to show that when $h\_A=1$, one also has $\sharp[~\mathbf P^1(A)~/~\mathrm{SL}\_2(\mathcal O\_A)~]=1$.


More generally, computer assisted experiments for $A$ of small discriminant seem to indicate that the equality
$$\sharp[~\mathbf P^1(A)~/~\mathrm{SL}\_2(\mathcal O\_A)~]=h\_A^2$$
should hold, but after some unfruitful draft padding, I have been unable to prove it. Would someone here know where to find a proof of this undoubtedly classical fact (2), and even better, name its discoverer ?


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Notes :


(0) The inclusion $\mathrm{SL}\_2(\mathcal O\_A)\to \mathrm{GL}\_2(\mathcal O\_A)$ is an equality.


(1) In the simpler case when $A$ is a quadratic field, the ideal classes form a *group* and this point is essential in the proof of the classical equality $\sharp[~\mathbf P^1(A)~/~\mathrm{SL}\_2(\mathcal O\_A)~]=h\_A$.


(2) Some googling indicates that many authors studied automorphic cuspidal things in this quaternionic setting, so that this quotient has certainly been studied, but I could not find the statement.



Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra
The formula holds, and seems to be due to Krafft and Osenberg : *Eisensteinreihen für einige arithmetisch definierte Untergruppen von SL2(H)*. (German), Math. Z. 204 (1990), no. 3, 425–449. 


(See here : [EuDML](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0204&DMDID=dmdlog45) (free), or here : [Springer](http://link.springer.com/article/10.1007%2FBF02570884).) 


Just after Satz 2.2 : 


"Die Zahl der Spitzzenbahnen ist also das Quadrat der (einseitigen) Klassen-zahl von $\mathcal O$".


