The Arithmetic of Quaternion Algebra

92 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS
Corollary 4.2.12. If K is a totally real field, then ζK(−1) is rational.
It is a special case of a theorem of Siegel which asserts that the numbers
ζK(1 − n) for n ≥ 1 are the rational number.
The number e∞ of points for an arithmetic group is not zero if and only if
the group is commensurable to P SL(2, Z). For the congruent group Γ(N) and
Γ0(N), the formula for the number of points can be found in the book of Shimura
[6],p.25.
Exercise
. Let Γ be the group of the proper automorphisms of the quadratic form
x 2 + y 2 − D(z 2 + t 2 )
where D is an integer great or equal to 1. Prove
1) Γ is the the unit group of reduced norm 1 and of order O = Z[1, i, j, ij] of
the quaternion algebra H/Q generated by the elements i, j satisfying
i 2 = −1, j 2 = D, ij = −ji.
2) The volume V of a fundamental domain of Γ in H2 for the hyperbolic metric
given by Humbert formula:
V = D �
p|D,p�=2
Hint: Write V = pi 3 �
p|D Vp, where
(1 + ( −1
p )p−1 ).
V2 = 2 m−1 (1 + 1
2 ) if 2m ||D,
Vp = p m (1 + ( −1
p )p−1 ) if p m ||D.
Then compare Vp with the volume of O 1 p for Tamagawa measure.
4.3 Examples and Applications
A. congruent groups. Let H/Q be a quaternion algebra contained in M(2, R), of
reduced discriminant D, and Γ be a congruence group of level N = N0, N1, N2,
for the definition see example 4) in §1. The genus of ¯ Γ\H ∗ is given by
2 − 2g = vola( ¯ Γ\H2) + e2/2 + 2e3/3 + e∞.
The volume of ¯ Γ\H2 which is calculated for Euler-Poicare measure is equl to
vola( ¯ Γ\H2) = − 1
6
�
(p − 1) · N0N 2 1 N 3 2 · �
(1 + p −1 ) · �
p|D
p|N0
p|N1N2
(1 − p −2 ) · (1/2)
by putting (1/2) = 1 if N1N2 ≤ 2 and (1/2) = 1/2 otherwise.
Let the quadratic cyclotomic extension of Q be Q(x) and Q(y) with x, y being