Orders in Quaternion Algebras

Abstract

The basic algebraic theory of quaternion algebras was given in Chapter 2. That sufficed for the results obtained so far on deducing information on a Kleinian group Γ from its invariant trace field kΓ and invariant quaternion algebra AΓ. We have yet to expound on the arithmetic theory of quaternion algebras over number fields. This will be essential in extracting more information on the quaternion algebras and, more importantly, in deducing the existence of discrete Kleinian groups of finite covolume. These will be arithmetic Kleinian groups about which a great deal of the remainder of the book will be concerned. All of this is based around the structure of orders in quaternion algebras which encapsulate the arithmetic theory of quaternion algebras. These were introduced in Chapter 2, but we now give a more systematic study, particularly from a local-global viewpoint.