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such that
∞
+
)
∞
−
)
.
div(
f
(
x,y
))
=
D
1
+
ι
(
D
2
)
−
(
n
2
+
deg(
D
2
)
−
n
1
)(
−
(
n
1
+
deg(
D
1
)
−
n
2
)(
Since
f
(
x,y
) has poles only at infinity it follows that
f
(
x,y
)
=
a
(
x
)
+
yb
(
x
) where
a
(
x
)
,b
(
x
)
∈ k
[
x
]. Now, 0
≤
n
i
≤
n
i
+
deg(
D
i
)
≤
g
and so
−
g
≤
v
∞
+
(
f
(
x,y
))
=−
(
n
2
+
deg(
D
2
)
−
n
1
)
≤
g
and
−
g
≤
v
∞
−
(
f
(
x,y
))
=−
(
n
1
+
deg(
D
1
)
−
n
2
)
≤
g
.But
v
∞
+
(
y
)
=
=−
+
=
=
=
+
−
v
∞
−
(
y
)
(
g
1) and so
b
(
x
)
0 and
f
(
x,y
)
a
(
x
). But div(
a
(
x
))
D
ι
(
D
)
∞
+
)
+
∞
−
)) and so
D
1
=
D
2
,
n
1
+
−
n
2
=
n
2
+
−
deg(
a
(
x
))((
(
deg(
D
1
)
deg(
D
2
)
n
1
and
n
1
=
n
2
.
Exercise 10.4.20
Let
C
be a hyperelliptic curve over
k
of genus
g
=
d
−
1 with split model.
∞
+
)
∞
−
) is not a principal divisor and that this divisor is represented as
Show that (
−
(
(1
,
0
,
g/
2
+
1).
10.5 Jacobians, Abelian varieties and isogenies
As mentioned in Section
7.8
, we can consider Pic
0
k
(
C
) as an algebraic group, by considering
the
Jacobian variety
J
C
of the curve. The fact that the divisor class group is an algebraic
group is not immediate from our description of the group operation as an algorithm (rather
than a formula).
Indeed,
J
C
is an Abelian variety (namely, a projective algebraic group). The dimension
of the variety
J
C
is equal to the genus of
C
. Unfortunately, we do not have space to introduce
the theory of Abelian varieties and Jacobians in this topic. We remark that the Mumford
representation directly gives an affine part of the Jacobian variety of a hyperelliptic curve
(see Propositions 1.2 and 1.3 of Mumford [
399
] for the details).
An explicit description of the Jacobian variety of a curve of genus 2 has been given by
Flynn; we refer to Chapter 2 of Cassels and Flynn [
115
] for details, references and further
discussion.
There are several important concepts in the theory of Abelian varieties that are not able
to be expressed in terms of divisor class groups.
3
Hence, our treatment of hyperelliptic
curves will not be as extensive as the case of elliptic curves. In particular, we do not give a
rigorous discussion of isogenies (i.e., morphisms of varieties that are group homomorphisms
with finite kernel) for Abelian varieties of dimension
g>
1. However, we do mention one
important result. The Poincare reducibility theorem (see Theorem 1 of Section 19 (page 173)
of Mumford [
398
]) states that if
A
is an Abelian variety over
and
B
is an Abelian subvariety
of
A
(i.e.,
B
is a subset of
A
that is an Abelian variety over
k
k
) then there is an Abelian
subvariety
B
⊆
B
is finite and
B
B
=
A
over
k
such that
B
∩
+
A
. It follows that
A
is
k
×
B
. If an Abelian variety
A
over
k
isogenous over
to
B
has no Abelian subvarieties
k
over
then we call it
simple
. An Abelian variety is
absolutely simple
if is has no Abelian
subvarieties over
k
.
3
There are two reasons for this: first, the divisor class group is merely an abstract group and so does not have the geometric
structure necessary for some of these concepts; second, not every Abelian variety is a Jacobian variety.
