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),