CHAPTER 2

Brandt matrices and definite Shimura curves

Given a global function field k together with a fixed place ∞, we consider ”def-

inite” quaternion algebras over k and introduce the Brandt matrices. The entries

are non-negative integers which count the number of ideals of given norm in an

ideal class of an Eichler A-order. Then we define the definite Shimura curves asso-

ciated to Eichler A-orders and introduce Hecke correspondences on these definite

Shimura curves. We describe the connection between these correspondences and

Brandt matrices.

1. Basic setting

Let k be a global function field with finite constant field Fq, i.e. k is a finitely

generated field extension of transcendence degree one over Fq and Fq is algebraically

closed in k. For each place v of k, the completion of k at v is denoted by kv, and Ov is

the valuation ring in kv. We choose a uniformizer πv in Ov and set Fv := Ov/πvOv,

the residue field of kv. The degree deg v of v is [Fv : Fq], and the cardinality of Fv

is denoted by qv. For each av ∈ kv, the absolute value |av|v of av is normalized to

be qv

−ordv

(av).

Let A be the adele ring of k, which is the restricted direct product

v

kv with respect to Ov. The maximal compact subring

v

Ov of A is denoted

by OA. The idele group

A×

of k is the restricted direct product

v

kv

×

with respect

to Ov

×,

and for a = (av)v ∈

A×

we set

|a|A :=

v

|av|v.

Embedding k into A diagonally, the product formula says that

|α|A = 1, ∀α ∈

k×.

Let Div(k) be the divisor group of k. We adopt the multiplicative notation so

that every element m in Div(k) is written as

m =

v

vordv(m).

Given m ∈ Div(k), we define

m :=

v

qv

ordv(m)

=

qdeg m,

where

deg m :=

v

deg v · ordv(m)

is the degree of m. There is a canonical group epimorphism

div :

A×

→ Div(k)

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