Birational geometry of quaternions

The Hilbert class field of the quaternion algebra $B$ is an algebra $\mathscr{H}(B)$ such that every two-sided ideal of $B$ is principal in $\mathscr{H}(B)$. We study the avatars of $B$ and $\mathscr{H}(B)$, i.e. algebraic surfaces attached to the quaternion algebras. It is proved that the avatar of $\mathscr{H}(B)$ is obtained from the avatar of $B$ by a birational map. We apply this result to the function field analogy.