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 R× 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 D× 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}.