Cryptography Reference
In-Depth Information
classifies ramified hyperelliptic models. Exercise
10.1.14
shows that if
C
(
then one
may transform
C
into a ramified or split model. Hence, when working over finite fields, it
is not usually necessary to deal with curves having an inert model.
k
)
= ∅
∞
+
)
=∞
−
.
Exercise 10.1.12
With notation as in Definition
10.1.11
show that
ι
(
Exercise 10.1.13
Let
C
:
y
2
+
H
(
x
)
y
=
F
(
x
) be a hyperelliptic curve over
k
satisfying
all the conditions above. Let
d
=
max
{
deg(
H
(
x
))
,
deg(
F
(
x
))
/
2
}
. Show that this is a
ramified model if and only if (deg(
H
(
x
))
<d
and deg(
F
(
x
))
=
2
d
−
1) or (char(
k
)
=
2,
(
H
d
/
2)
2
).
deg(
F
(
x
))
=
2
d
,deg(
H
(
x
))
=
d
and
F
2
d
=−
Exercise 10.1.14
Let
C
:
y
2
+
H
(
x
)
y
=
F
(
x
) be a hyperelliptic curve over
k
and let
P
∈
C
(
k
). Define the rational map
x
P
)
d
)
.
ρ
P
(
x,y
)
=
(1
/
(
x
−
x
P
)
,y/
(
x
−
C
where
C
is also a hyperelliptic curve. Show that
ρ
P
is just the translation
Then
ρ
P
:
C
→
(0
,y
P
)followedbythemap
ρ
and so is an isomorphism from
C
to
C
.
Show that if
P
map
P
→
=
ι
(
P
) then
C
is birational over
k
(using
ρ
P
) to a hyperelliptic curve
with ramified model. Show that if
P
=
ι
(
P
) then
C
is birational over
k
to a hyperelliptic
curve with split model.
We now indicate a different projective model for hyperelliptic curves.
Exercise 10.1.15
Let the notation and conditions be as above. Assume
C
:
y
2
=
F
(
x
) is irreducible and non-singular as an affine curve. Let
Y,X
d
,X
d
−
1
,...,X
1
,X
0
be
coordinates for
+
H
(
x
)
y
P
d
+
1
(one interprets
X
i
=
x
i
). The
projective hyperelliptic equation
is
d
+
1
the projective algebraic set in
P
given by
Y
2
F
2
d
X
d
+
+
(
H
d
X
d
+
H
d
−
1
X
d
−
1
+···+
H
0
X
0
)
Y
=
F
2
d
−
1
X
d
X
d
−
1
+···
F
0
X
0
,
+
F
1
X
1
X
0
+
X
i
=
X
i
−
1
X
i
+
1
,
for 1
≤
i
≤
d
−
1
,
X
d
X
i
=
≤
≤
−
X
(
d
+
i
)
/
2
X
(
d
+
i
)
/
2
,
for 0
i
d
2
.
(10.3)
1. Give a birational map (assuming for the moment that the above model is a variety)
between the affine algebraic set
C
and the model of equation (
10.3
).
2. Show that the hyperelliptic involution
ι
extends to equation (
10.3
)as
ι
(
Y
:
X
d
:
···
:
X
0
)
=
(
−
Y
−
H
d
X
d
−
H
d
−
1
X
d
−
1
−···−
H
0
X
0
:
X
d
:
···
:
X
0
)
.
3. Show that the points at infinity on equation (
10.3
) satisfy
X
0
=
X
1
=
X
2
=···=
0 and
Y
2
F
2
d
X
d
=
X
d
−
1
=
+
H
d
X
d
Y
−
0. Show that if
F
2
d
=
H
d
=
0 then there is a
single point at infinity.
4. Show that if the conditions of Lemma
10.1.6
or Lemma
10.1.8
hold then equation (
10.3
)
is non-singular at infinity.
5. Show that equation (
10.3
)isavariety.
