The Arithmetic of Quaternion Algebra

Chapter 4
Applications to Arithmetic
Groups
Let (K ×
i ) be a nonempty finite set of local fields. Consider the group
G 1 = �
SL(2, K ′ i).
i
we are interested in some discrete subgroups of finite covolume of G 1 . More
precisely, they are obtained in such a way: consider a quaternion algebra H/K
over a global field K such that there exist a set S of places of K satisfying
— (K × v )v∈S = (Ki) up to permutation.
— no any place v ∈ S is ramified in H. Every archimedean place not belonging
to S is ramified in H.
These groups play an important role in various domains. Their usefulness will
be well studied soon in utilizing the arithmetic of quaternions (chapter III).
4.1 Quaternion groups
Fix a global field K, a quaternion algebra H/K, a set S of places of K containing
∞ and satisfying Eichler’s condition denoted by C.E.. WE consider the group
G 1 =
�
SL(2, Kv).
v∈S,v /∈Ram(H)
This group is non-trivial because S contains at least an unramified place in H.
We denote by R = R (S) the elements of K integral relative to the places which
do not belong to S, and by Ω the se of R-orders of H. We are interested in the
quaternion groups of reduced norm 1, in the orders O ∈ Ω:
O 1 = {x ∈ O|n(x) = 1.
For each place v we fix an inclusion of K in Kv, and choose an inclusion ϕv :
H → H ′ v, where
H ′ �
v =
M(2, Kv),
Hv,
if v /∈ Ram(H)
if v ∈ Ram(H)
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