The Arithmetic of Quaternion Algebra

4.2. RIEMANN SURFACES 85
6. Hilbert’s modular group. If K is a totally real number field, and if H =
M(2, K), then the group SL(2, R) where R is the integer ring of K is called
the Hilbert modular group. It is a discrete subgroup of SL(2, R) [K:Q] , and
for Tamagawa measure we have
vol(SL(2, R)\SL(2, R) [K:Q] ) = ζK(−1)(−2π 2 ) −[K:Q] .
It can be seen by using the relation between ζK(2) and ζK(−1) obtained
by the functional equation:
ζK(2)D 3/2
K (−2π2 ) −[K:Q] = ζK(−1).
7. Let H/K be a quaternion algebra. If S is a set of places satisfying C.E.,
therefore S ′ = {v ∈ S|v /∈ Ramf (H)} satisfies C.E. If O is an order over
the ring of the elements of K which are integral at the places v ∈ S, then
O ′ = {x ∈ O|x is integral for v ∈ Ramf (H)} is an order over the ring of
the elements of K which are integral at the places v /∈ S ′ . It is easy to
check that O 1 = O ′ 1 . It follows that in the study of quaternion groups we
can suppose Ramf (H) ∩ S = ∅.
4.2 Riemann surfaces
Let H be the upper half-plane equipped with a hyperbolic metric ds 2 :
H = {z = (x, y) ∈ R 2 |y > 0}, ds 2 = y −2 (dx 2 + dy 2 ).
The group P SL(2, R) acts on H by homographies. A discrete subgroup of finite
covolume
barΓ ⊂ P SL(2, R) defines a Riemann surface ¯ Γ\H. Consider those which are
associated with the quaternion groups Γ ⊂ SL(2, R) with image ¯ Γ ⊂ P SL(2, R).
The results of III.5, IV.1 allow us conveniently obtain – the genus,
– the number of the elliptic points of a given order,
– the number of the minimal geodesic curves of a given length.
We shall deduce them with simple examples and the explicit expression of the
isospetral (for laplacian) but not isometric riemannian surfaces(§3)
Definition 4.4. A complex homography is a mapping of C ∪ ∞ in C ∪ ∞ of the
form
z ↦→ (az + b)(cz + d) −1 �
a
= t, where g =
c
�
b
∈ GL(2, C).
d
We set t = ¯g(z) and ¯ X = {¯x|x ∈ X} for every set X ⊂ GL(2, C).
We are interested henceforth only in real homographies induced by SL(2, R).
We have
Y = y|cz + d| −2 .
These homographies preserve the upper (lower) half-plane H and the real axis.
Differentiating the relation of t, we have
dt = (cz + d) −2 dz.