The Arithmetic of Quaternion Algebra

80 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS
where Hv denotes the quaternion field over Kv. From this we obtain an inclusion
ϕ : H →
�
M(2, Kv) = G
v∈S,v /∈Ram(H)
which send O ∞ on a subgroup of G 1 . By abuse of notations, we identify H ′ v with
Hv in sequel. Note that two inclusion ϕ, ϕ ′ is different by an inner automorphism
of G ×
Theorem 4.1.1. (1) The group ϕ(O 1 ) is isomorphic to O 1 . It is a discrete
subgroup, of finite covolume of G 1 . It is cocompact if H is a field.
(2) The projection of ϕ(O 1 ) onto a factor G ′ = �
v SL(2, Kv) of G 1 , with
1 �= G ′ �= G 1 , is equal to O 1 . It is dense in G ′ .
Proof. The nontrivial part of the theorem is an application of the fundamental
theorem III.1.4 and III.2.3. The isomorphism with O1 is trivial since the image
of O1 in G ′ with 1 �= G ′ is � ϕv(O1 ) which is isomorphic to O1 . The idea is to
describe the group H1 A /H1 K . Set
U = G 1 �
�
· C with C =
O 1 v.
H
v∈S and v ∈ Ram(H)
1 v
v /∈S
The group U is an open subgroup in H 1 A satisfying:
H 1 A = H 1 K
From this we deduce a bijection between
and H 1 K U = O 1 .
H 1 A/H 1 KU and U/O 1 .
According to III.1.4, and III.2.3, we have
(1) H 1 K is discrete in H1 A of the finite covolume being equal to τ(H1 = 1,
cocompact if H is a field.
According to III.4.3, we have
(2) H 1 K G′′ is dense in H 1 A if G′′ = � SL(2, Kv) with i �= G ′′ .
It follows
(1) O 1 is discrete in U of finite covolume being equal to 1 for the Tamagawa
measures, cocompact if H is a field.
(2) The image of O 1 in G · C is dense.
We thus utilize the following lemma for finishing the proof of theorem 1.1.
Lemma 4.1.2. Let X be a locally compact group, Y a compact group, Z the
direct product X · Y , and T a subgroup of Z with its projection V in X. We
have the following properties:
a) If T is discrete in Z, then V is discrete in X. Moreover, T is of a finite
covolume (resp. cocompact) in Z if and only if V has the same property in X.
b) If T is dense in Z, then V is dense in X.
Proof. a) Suppose K to be discrete in Z. For every compact neighborhood D
of the unit in X, we show that V ∩ D has only a finite number of elements. In
fact, X ∩ (D · C) has a finite number of elements, being great or equal to that
of V ∩ D. Hence V is discrete in X. Let FT ⊂ Z, FV ⊂ X be the fundamental
sets of T in Z, and of V in X. It is clear that FV · C contains a fundamental