The Arithmetic of Quaternion Algebra

94 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS
Proof. The assertion for Γ comes from Example 2.1. As for G, it suffices to verify
that the possible values of the order of the cyclic groups which are contained in
G are 1, 2, 4, 6, 8, 12. It can be obtained at once from the structure of G\Γ and
the order of the cyclic groups in Γ.
The formulae for eq(G) are not so simple as for eq(Γ) but can be obtained
by elementary method.
The following table gives the list of all surfaces barΓ\H of genus 0,1,2.
here is Table 1 !!!
Using the results of Ogg about the hyperelliptic Riemann surfaces of genus
g ≥ 2, we can determine the surfaces ¯ Γ\H of genus g ≥ 2 which are hyperelliptic.
In all these cases, the hyperelliptic involution is induced by an element of
G.
We denote by πi the element of O of reduced norm pi (1 ≤ i ≤ 2m) and gd
the element of G defined by
gd = d −1/2 π ε1
1 ...πε2m 2m
for d = πε1 1 ...πε2m 2m , εi = 0 or 1
The table below gives the list of the hyperelliptic surfaces with their genus and
the element of G which induces the hyperelleptic involution:
Here is a talbe!!!
C The construction of a fundamental domain for Γ and G in The case of D = 15.(Michon
[1]). The quaternion algebra is generated by i, j satisfying
The order O generated over Z by
i 2 = 3, j 2 = 5, ij = −ji.
1, i, (i + j)/2, (i + k)/2
is maximal. It has the matrix representation
O = { 1
�
√
x
2 5¯y
√ �
5y
, |x, y ∈ Q(
¯x
√ 3) are integer, and x ≡ y mod (2)}.
The group Γ = O 1 is consists of the above matrices such that
(1) n(x) − 5n(y) = 4.
The group G normalizing Γ is consists of the matrices satisfying
(2) n(x) − 5n(y) = 4, 12, 20, or 60
divided by the square root of their determinants. The fixed points in C of an
element of G are distinct and given by
z = b√ 3 ∓ √ a 2 − 4
√ 5 ¯ (y)
if x = a + b √ 3, a, b ∈ Z.
The elliptic fixed points corresponds to a = −1, or 1. It can be restricted
to a = 0 or 1, since the change of sign of the matrix does not change the