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

of k is the restricted direct product
v
kv
×
with respect
to Ov
×,
and for a = (av)v

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 :

Div(k)
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