The Arithmetic of Quaternion Algebra

106CHAPTER 5. QUATERNION ARITHMETIC IN THE CASE WHERE THE EICHLER CONDITI
5.1 Units
If v ∈ S, then Xv = Yv is a field. Therefore for every place v,
Zv = {y ∈ Yv|||y||v ≤ 1}
is compact in Xv. It follows that ZA = XA ∩ ( � Zv) is compact in XA, and
that the group
ZA ∩ XK = {y ∈ Y |||y||v ≤ 1 ∀v ∈ V }
which is discrete in ZA by III.1.4 is a finite group. It is hence equal to the
torsion group Y 1 of Y . We have proved the
Lemma 5.1.1. The Group Y 1 of the roots of unit in Y is a finite group.
If X = K is commutative, it is a cyclic group according to the classical result
about the finiteness of the subgroup of commutative field. If X = H, it is not
commutative in general. When K is a number field it can be embedded in a
finite subgroup of the real quaternions. Its structure is well known(I.3.1).
According to III.1.4, the group X ×
K is discrete, cocompact in XA,1. Let us
proceed as in IV.1.1, and describe XA,1/X ×
K . By III.5.4 we have a finite decom-
position (at present it is not reduced to one term):
where we set
(1) XA,1 = ∪YA,1xiX ×, xi∈X A,1
K , 1 ≤ i ≤ h
YA,1 = G · C ′ with G = {x ∈ XA,1 ′ |xv = 1 if v /∈ S}
and C ′ is a compact group which is equal to � ×
v /∈S Y v . It follows from Lemma
1.1 that
Y ×=YA,1 ×
∩ X K is discrete, cocompact in G.
LEt f be the mapping which for x ∈ G associates with (||xv||v)v∈S. According
to 1.1, we have the exact sequence
1 −−−−→ Y 1 −−−−→ Y × f
−−−−→ f(G).
It follows that f(Y × ) is a discrete, cocompact subgroup of a group which is
isomorphic to Ra · Zb , a + b = CardS − 1 , supposing
f(G) = {(xv) ∈ �
||Xv|| | � xv = 1}.
v∈S
Therefore, f(Y × ) is a free group with CarsS − 1 generators.
Theorem 5.1.2. Let Y × be the unit group of Y . Then it exists an exact
sequence:
1 → Y 1 → Y × → Z CardS−1 → 1
and Y 1 which is the group of the roots of unit contained in Y is finite.
When X = K is commutative, we deduce from it that Y × is the direct
product of Y 1 by a free group with CardS − 1 generators. It is not true if
X = H as what Exercise 1.1 points out. The theorem 1.2 is an analogue of
IV.1.1
Definition 5.1. The regulator of Y is the volume of f(G)/f(Y × ) calculated for
the measure induced by Tamagawa measure. we denote it by RY .