Cryptography Reference
In-Depth Information
Definition 10.3.6
Let
D
be a non-zero semi-reduced divisor. The polynomials (
u
(
x
)
,v
(
x
))
of Lemma
10.3.5
are the
Mumford represent
a
tion
of
D
.If
D
=
0 then take
u
(
x
)
=
1 and
v
(
x
)
[
x
] is called a
Mumford representation
if
u
(
x
) is monic, deg(
v
(
x
))
<
deg(
u
(
x
)) and if equation (
10.5
) holds.
=
0. A pair of polynomials
u
(
x
)
,v
(
x
)
∈ k
We have shown that every semi-reduced divisor
D
has a Mumford representation and
that the polynomials satisfying the conditions in Definition
10.3.6
are unique. We now
show that one can easily recover an affine divisor
D
from the pair (
u
(
x
)
,v
(
x
)): write
u
(
x
)
=
l
i
=
1
e
i
(
x
i
,v
(
x
i
)).
Exercise 10.3.7
Show that the processes of associating a Mumford representation to a
divisor and associating a divisor to a Mumford representation are inverse to each other.
More precisely, let
D
be a semi-reduced divisor on a hyperelliptic curve. Show that if one
represents
D
in Mumford representation, and then obtains a corresponding divisor
D
as
explained above, then
D
=
=
l
i
=
1
(
x
x
i
)
e
i
and let
D
−
D
.
Exercise 10.3.8
Let
u
(
x
)
,v
(
x
)
[
x
] be such that equation (
10.5
) holds. Let
D
be the
corresponding semi-reduced divisor. Show that
∈ k
D
=
min
{
v
P
(
u
(
x
))
,v
P
(
y
−
v
(
x
))
}
(
P
)
.
P
∈
(
C
∩A
2
)(
k
)
−
This is called the
greatest common divisor
of div(
u
(
x
)) and div(
y
v
(
x
)) and is denoted
div(
u
(
x
)
,y
−
v
(
x
)).
Exercise 10.3.9
Let (
u
1
(
x
)
,v
1
(
x
)) and (
u
2
(
x
)
,v
2
(
x
)) be the Mumford representations of
two semi-reduced divisors
D
1
and
D
2
. Show that if gcd(
u
1
(
x
)
,u
2
(
x
))
=
1 then Supp(
D
1
)
∩
Supp(
D
2
)
= ∅
.
Lemma 10.3.10
Let C be a hyperelliptic curve over
k
and let
D
be a semi-reduced divisor
∈
on C with Mumford representation
(
u
(
x
)
,v
(
x
))
. Let σ
Gal(
k
/
k
)
.
1. σ
(
D
)
is semi-reduced.
2. The Mumford representation of σ
(
D
)
is
(
σ
(
u
(
x
))
,σ
(
v
(
x
)))
.
3. D is defined over
k
if and only if u
(
x
)
,v
(
x
)
∈ k
[
x
]
.
Exercise 10.3.11
Prove Lemma
10.3.10
.
Exercise
10.3.8
shows that the Mumford representation of a semi-reduced divisor
D
is
natural from the point of view of principal divisors. This explains why condition (
10.5
)isthe
natural definition for the Mumford representation. There are two other ways to understand
condition (
10.5
). First, the divisor
D
corresponds to an ideal in the ideal class group of the
affine coordinate ring
k
[
x,y
], and condition (
10.5
) shows this ideal is equal to the
k
[
x,y
]-
ideal (
u
(
x
)
,y
v
(
x
)). Second, from a purely algorithmic point of view, condition (
10.5
)is
needed to make the Cantor reduction algorithm work (see Section
10.3.3
).
A divisor class contains infinitely many divisors whose affine part is semi-reduced. Later
we will define a reduced divisor to be one whose degree is sufficiently small. One can then
−
