The Arithmetic of Quaternion Algebra

112CHAPTER 5. QUATERNION ARITHMETIC IN THE CASE WHERE THE EICHLER CONDITI
The notations used here are that in I.3.7. Suppose D �= 2, 3 and denote by
hi the class number of ideals I to the left of O such that the unit group of I −1 I
is of order 2i. Applying 2.4 and the formula for mass M(B), with B = Z[ √ −1]
and Z[ √ −3], we have
Proposition 5.3.2. The class number h, h2, h3 are equal to
h = 1 �
12
h2 = 1 �
2 �
h3 = 1
2
p|D
p|D
p|D
� 1 (p − 1) + 4 p|D
(1 − ( −4
p ))
(1 − ( −3
p ))
�
−4 1
−3
(1 − ( p )) + 3 p|D (1 − p ))
We can give another proof of the formula for h in a pure algebraic way .
It will use the relation between the quaternion algebras and the elliptic curves
(Igusa [1]). We shall give a table for h and for the type number of maximal
orders t in the end of §3.
B.Arithmetic graphes.
We shall give a geometric interpretation of the class numbers h, h2, h3, and the
Brandt matrices i terms of graphs. Let p be a prime number which does not
divide D. The tree X = P GL(2, Zp\P GL(2, Qp) admits a description by the
orders and the ideals of H: we fix first a maximal order O,
– The vertices of X correspond bijectively to the maximal orders O ′ such that
Oq = O ′ q, ∀q �= p;
– The edges of X of the starting O ′ are correspond bijectively to the integral
ideals which are to the left of O ′ and with reduced norm pZ.
Precisely, to x ∈ X, with the representative a ∈ GL(2, Qp) = H × p , we associate
the order O ′ such that O ′ p = a −1 Opa, and O ′ q = Oq if q �= p. SEE II.2.5 and
2.6.
Let Z (p) be the set of the rational number of the form a/p n , a ∈ Z, n ∈ N.
The maximal orders O ′ , i.e. the vertices of X, generate the same Z (p) -order
O (p) = �
q�=p (Oq ∩ H). The unit group O p× defines an isometric group Γ =
O (p)× /Z (p)× of the tree X of which the quotient graph is finite. The group
ΓO ′of the isometries of Γ fixing a vertex O′ is equal to O ′× /Z × . It is from the
above results :
—A cyclic group of order 1, 2, 3;
—A4, if H = {−1, −1};
—D3, if H = {−1, −3}.
By definition Card(ΓO ′) is the order of vertex of X/Γ defined by O′ .
Proposition 5.3.3. The number of the vertices of the quotient graph X/Γ is
equal to the number of class h of H.
If H = {−1, −1} resp. H = {−1, −3}, the quotient graph has a single vertex
of order 12, resp, of order 6. In the other cases, the number of the vertices of
order i of the quotient graph is equal to hi.
In fact, this is the consequence of a formal computation in the ideles : Since
{∞, p} satisfies the Eichler condition, and Z (p) is principal, the order O (p) is
principal(Ch.III), thus it has the decomposition H ×
A = � O × q H × ∞H × p H × , the
product is taken on all the prime numbers q �= p. By using the decomposition
Q ×
A = Q× Z × � ×
p Z q expressing that Z is principal, we see that the class number
of maximal orders (over Z) of H is the cardinal of one of the sets which is
.