10 CHIH-YUN CHUANG, TING-FANG LEE, FU-TSUN WEI, and JING YU
Then Ri is also an Eichler A-order of type
(N+, N−),
as
Ri = D biRbi
−1
.
Given an element b in D, its reduced trace and the reduced norm are denoted
by Tr(b) and Nr(b), respectively. We call b is a pure quaternion if Tr(b) = 0. Define
σ to be the permutation of {1, ..., n} such that for 1 i n,
¯−1
I
i
is equivalent to
Iσ(i). Here
¯
I
i
:=

b : b Ii}
and
¯
b := Tr(b) b is the conjugate involution of D. It is clear that
σ2
= 1.
3. Brandt matrices
Let R be an Eichler A-order of type (N+, N−). Let I1, ..., In be representatives
of locally-principal right ideal classes of R. For 1 i n, let Ri be the left order
of Ii. We denote the reduced ideal norm of Ii by Nr(Ii), i.e. Nr(Ii) is the fractional
ideal of A generated by Nr(b) for all elements b in Ii. For 1 i, j n, set
Nij := Nr(Ii)
Nr(Ij)−1.
Then for every m Divf,≥0(k), the m-th Brandt matrix B(m) is defined to be
(
Bij(m)
)
1≤i,j≤n
Matn(Z), where
Bij(m) :=
#{b IiIj
−1
: Nr(b)Nij
−1
= Mm}
#(Rj
×)
.
Recall that Mm is the ideal of A corresponding to m. It is clear that Bij(m) depends
only on the ideal classes of Ii, Ij, and the divisor m. For each divisor a Divf (k),
We set the permutation matrix
L(a) :=
(
Lij(a)
)
1≤i,j≤n
Matn(Z)
where
Lij(a) =
1, if MaIi is equivalent to Ij,
0, otherwise.
Then it is observed that
Proposition 2.1. (1) For every m and m in Divf,≥0(k) which are relatively
prime,
B(mm ) = B(m)B(m ).
(2) For m in Divf,≥0(k) and a in Divf (k),
B(m)L(a) = L(a)B(m).
(3) When v
N+N−,
B(vr+2)
=
B(vr+1)B(v)

qvL(v)B(vr).
(4) B(vr) = B(v)r if v | N−.
(5) L(a) = L(a ) if Ma and Ma are in the same ideal classes of A.
(6) The summation

j
Bij(m) is independent of the choice of i. Moreover, let
b(m) :=
j
Bij(m),
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