The Arithmetic of Quaternion Algebra

110CHAPTER 5. QUATERNION ARITHMETIC IN THE CASE WHERE THE EICHLER CONDITI
if B = R[g] ⊂ H in the notations of III.5. It follows that if wi = [O (i) : R × ] and
w(B) = [B × : R × ], we then have
It follows
Definition 5.2. We call
mi = m (i)
O
M =
wi/w(B) = mi(B).
�
mi/wi = h(B) �
mp.
w(B)
p/∈S
h�
1/wi, M(B) =
i=1
the mass of O and the mass of B in O.
h�
mi/wi
We consider then the Eichler-Brandt matrices P (A) defined in III. exercise
5.8 for the integral ideal of R. these are the matrices in M(h, N). The entries at
i- place of the diagonal αi,i is equal to the number of the principal ideals in O (i)
with reduced norm A. When C.E. is not satisfied the traces of the matrices can
be calculate in use of III.5.11 and V.2.3. The result is given below. Suppose K
to be a number field.
Proposition 5.2.4. (The trace of Eichler-Brandt matrix). The trace of matrix
P (A) is null if A in not a principal ideal. If A is not the square of a principal
ideal, it is equal to
Otherwise, it is equal to
1
2
�
M(B).
(x,B)
M + 1
2
i=1
�
M(B)
(x,B)
where (x, B) runs through all the pairs formed by an element x ∈ Ks, and a
commutative order B satisfying:
– x is the root of an irreducible polynomial X 2 − tX + a, where (a) is a system
of representatives of the generators of A modulo R ×2 , and t ∈ R; – R[x] ⊂ B ⊂
K(x).
Proof. If A is not principal, this is clear. Otherwise, we utilize
2wiαi,i = �
Card{x ∈ O (i) |n(x) = 1}.
a
Introduce now the pairs (x, B). By the definitions of III.5.11, we see that
2wiαi,i = �
�
0 if A is not a square
+
.
2 if A is a square
(x,B)
We then use the precedent definition for mass.