The Arithmetic of Quaternion Algebra

4.2. RIEMANN SURFACES 87
Corollary 4.2.2. Let N be a positive real number. For every point z 0 ∈ H of
its real part zero, we have
log N = d(z0, Nz0) = inf d(z, Nz).
z∈H
Proof. d(z, Nz) = arccosh(1 + (N−1)2 x2
2N (1 + y2 )) is minimal for x = 0 and equal
to log N.
Proposition 4.2.3. The area of a triangle whose vertices are at the infinity is
equal to π.
Proof. (A picture here!!!)
� �
y −2 dxdy =
� π
0
� ∞
− sin θdθ y
sin θ
−2 dy = π.
The common area of these triangles can be taken as the definition of the value
π.
Proposition 4.2.4. The area of a hyperbolic triangle of the angles at the vertices
being θ1, θ2, θ3 is equal to π − θ1 − θ2 − θ3,
Proof. The formula is true if every vertex is at infinity. We use the Green formula
if no any vertex is at the infinity: if Ci, i = 1, 2, 3 are the edges of the
triangle , then � � y−2dxdy = � �
i Ci dx/y
Here are two pictures!!!
�
C dx/y = � θ 2
r sin u/(−r sin u)du = α.
θ1
The area is thus I = α1 + α2 + α3. The total rotation of the turning normal
along is the triangle 2π, and that around a vertex of angle θ is π − θ. It follows
2π = �
i (π − θi) + �
i αi, from this we have I = π − θ1 − θ2 − θ3. It turns back
to one of these two cases when one of the angles is zero (its vertex corresponds
to the infinity). By the triangulation we can compute the area of a polygon.
Corollary 4.2.5. The area of a hyperbolic polygon with the angles at vertices
θ1, ..., θn equals (n − 2)π − (θ1 + ... + θ2).
EXAMPLE. A fundamental domain of P Sl(2, Z). The group P SL(2, Z) is
generated by the homographies t = z + 1 and t = −1/z. We show that the
hatching domain in the picture is a fundamental set
F = {z ∈ C|Imz > o, |z| ≥ 1, −1/2 ≤ Rez ≤ 1/2}.
It ia a triangle with one of its vertices being at the infinity. Its area is π−2π/3 =
π/3. It equals the area of the triangle without hatching, which is also a fundamental
set of SL(2, Z) in H.
Here is a picture!!!