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 analogy between number fields and function fields.