The Arithmetic of Quaternion Algebra

14 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD
global field (ch.II and ch.III).
Let K be the fractional field of R and H/K be a quaternion algebra over K.
In sequel of this section , we fix R, K, H.
Definition 1.5. A R-lattice of a K-vector space V is a finitely generated Rmodule
contained in V . A complete R-lattice of V is a R-lattice L of V such
that K ⊗R L � V .
Definition 1.6. An element x ∈ H is an integer (over R) if R[x] is a R-lattice
of H.
Lemma 1.4.1. (Bourbaki [1]) An element x ∈ H is an integer if and only if
its reduced trace and reduced norm belong to R.
If an element is integer this lemma is valid. contrary to the commutative
case, the sum and the product of two integers are not always integer: it is the
source of several rings if we want to do some very explicit computation. It is
not a surprise ,for example, in the case of M(2, Q) the following matrices are
integers:
�
1
2
1
4
−3
� � �
1 0
, 5
5 0
1
2
, but either their sum, nor their product are integer. The set of integers does not constitute a ring,
and it leads us to consider some subrings of integers, called orders.
Definition 1.7. An ideal of H is a complete R-lattice. An order O of H is :
(1) an ideal which is a ring, or equivalently,
(2) a ring O consisting of integers and containing R, such that KO = H.
A maximal order is an order which is not contained in any other order. An Eichler order
is the intersection of two maximal orders.
There certainly exist some ideals, for example, the free R-module L = R(ai)
generated by a basis {ai} of H/K. Let I be an ideal, it is associated canonically
with two orders:
Ol = Ol(I) = {h ∈ H|hI ⊂ I},
Or = (O)d(I) = {h ∈ H|Ih ⊂ I}.
are called its left order and right order respectively. It is clear that such an
order is a ring, an R-module, and a complete lattice because of that, if a ∈
R ∩ I, Ol ⊂ a −1 I and if h is an element of H, it exists b ∈ R,such that bhI ⊂ I
,so H = KOl.
Proposition 1.4.2. (the properties of orders). The definitions (1) and (2) for
order are equivalent.It does exist orders. Every order is contained in a maximal
order.
Proof. The definition (2) shows that every order is contained in a maximal order.
It is clear that (1) implies (2). inversely, Let ai be a basis of H/K contained
in O. An element h of O can be written as h = � xiai, xi ∈ K. Since O is a
ring, so hai ∈ O and t(hai) = � xjt(aiaj) ∈ R. The Cramer rule implies that
L ⊂ O ⊂ dL where d −1 = det(t(aiaj)) �= 0. From this we obtain that O is an
ideal, hence (1) implies (2).