The Arithmetic of Quaternion Algebra

4.2. RIEMANN SURFACES 89
double points in C ∩ ∞:
(1) distinct, real if (a + d) 2 > 4,
(2) distinct, complex conjugation, if (a + d) 2 < 4,
(3) mingling if (a + d) 2 = 4.
We obtain the above statement easily in virtue of the equalities:
z = (az + b)(cz + d) −1 is equivalent to cz 2 + (d − a)z − b = 0.
The discriminant of the quadratic equation is (d + a) 2 − 4.
Definition 4.6. In the case (1), the homography is said to be hyperbolic. Its
norm or its multiplicator is equal to N = λ 2 , where λ is the proper value of g
which is great than 1 strictly.
In case (2), it said to be elliptic. Ita angle or its multiplicatoris equal to N = λ 2 ,
where λ = e iθ is the proper value of g such that 0 ≤ θ ≤ π.
In case (3), it is said to be parabolic.
These definitions depend only on the conjugate class of g in GL(2, R), and
hence can be extended to the conjugate classes.
Proposition 4.2.8. Let ¯g be a homography of norm N. We have
log N = d(z0, ¯g(z0) = inf d(z, ¯g(z))
z∈H
for every element z0 belonging to the geodesic joining the double points of ¯g.
Proof. Since GL(2, R) acts by isometry, it follows that ¯g(z) = Nz, and then use
2.2.
compactificaton of H. We shall compactify H by embeding it in the space
H ∪ R ∪ ∞ which is equipped a topology as follows: the system of basic neighborhoods
at the infinity is the open neighborhoods Vy, y > 0 defined as below :
two pictures here!!!
for ∞ : Vy = {z ∈ H|Imz > 0}, for A ∈ R : Vy = {z ∈ H|d(B − z) < y}.
Fundamental domains. We recall a certain number of classic results about the
construction of fundamental domains.
References: Poincare [1], Siegal [3].
Let Γ be a discrete subgroup of SL(2, R) of finite covolume, and ¯ Γ be the group
of the homographies associate with Γ.
1. For every element z0∈H which is not the double point of any elliptic matrix
of Γ ( the existence of such point is easy to prove), the set
F = {z ∈ H|d(z, z0 ≤ d(¯g(z), z0) ∀¯g ∈ ¯ Γ}
is a hyperbolic polygon and a fundamental set of Γ in H2.
2. The edges of F are even in number and congruent pairwise modulo ¯ Γ. We
can also rearrange them in pairs (Ci, ¯gi(Ci)), 1 ≤ i ≤ n.