Lectures on Shimura Curves 9: Quaternion Orders

Our task here is to recall part of the theory of orders and ideals in quaternion algebras. Some of the theory makes sense in the context of B/K a quaternion algebra over a field K which is the quotient field of a Dedekind ring R. For our purposes K will always be a number field, or the completion of a number field at a finite prime, and R will be the ring of integers of K. (Nevertheless, we shall see that in the global case, the most important distinguishing feature of B is its non/splitting at the infinite places.)