2. BRANDT MATRICES AND DEFINITE SHIMURA CURVES 9
and let δ = δψ Div(k) be the canonical divisor associated to ψ. For each place v
of k, let ψv be the additive character on kv such that
ψv(av) := ψ(ιv(av)), ∀av kv.
Then ordv(δ) is the maximal integer r such that πv
−rOv
is contained in the kernel
of ψv. It is known that deg δ = 2gk 2, where gk is the genus of k. To keep records
on δ, we introduce
v(δ) :=
0 if ordv(δ) is even,
1 if ordv(δ) is odd,
and
Ω :=
v=∞
v
v
(δ)
Divf,≥0(k).
2. Definite quaternion algebra over function fields
Let D be a definite (with respect to ∞) quaternion algebra, i.e. D is a central
simple algebra over k with dimk D = 4 and D ⊗k k∞ is a division algebra. Let
N−
= ND

Divf,≥0(k) be the product of finite places v of k where D is ramified,
i.e. Dv := D ⊗k kv is division. For each place v of k, we choose an element Πv in
Dv
×
such that Πv
2
= πv.
Given a positive divisor
N+
Divf,≥0(k) which is prime to
N−.
We call a
ring R an Eichler A-order of type
(N+, N−)
if R is an A-order of D such that
Rv := R ⊗A Ov is a maximal Ov-order for each v
N+;
and when v |
N+,
there
exists an isomorphism i : Dv

= Mat2(kv) such that
i(Rv) =
a b
c d
Mat2(Ov) c πv
ordv(N+)Ov
.
We note that R is unique up to local conjugacy. Since D is definite, the cardinality
of the multiplicative group of R is finite.
A locally-principal (fractional) right ideal I of R is an A-lattice in D such that
I · R = I and for each finite place v of k, there exists αv in Dv
×
such that
Iv(:= I ⊗A Ov) = αvRv.
Two locally-principal right ideals I1 and I2 are called equivalent if there exists an
element b in such that I1 = b · I2. Let
DA∞ := D ⊗k
A∞
and R := R ⊗A OA∞ .
Then the set of locally-principal right ideal classes of R can be identified with the
finite double coset space D×\DA∞
×
/R×. More precisely, let b1, ..., bn DA∞
×
be
representatives of the double cosets. Then
{Ii := D biR | 1 i n}
is a set of representatives of locally-principal right ideal classes of R.
For 1 i n, let Ri be the left order of Ii, i.e.
Ri := {b D : bIi Ii}.
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