Cryptography Reference
In-Depth Information
7.2 Weierstrass equations
Definition 7.2.1
Let
a
1
,a
2
,a
3
,a
4
,a
6
∈ k
.A
Weierstrass equation
is a projective algebraic
set
E
over
k
given by
y
2
z
a
3
yz
2
x
3
a
2
x
2
z
a
4
xz
2
a
6
z
3
.
+
a
1
xyz
+
=
+
+
+
(7.1)
The
affine Weierstrass equation
is
y
2
x
3
a
2
x
2
+
a
1
xy
+
a
3
y
=
+
+
a
4
x
+
a
6
.
(7.2)
Exercise 7.2.2
Let
E
be a Weierstrass equation as in Definition
7.2.1
.Let
ι
(
x
:
y
:
z
)
=
(
x
:
−
y
−
a
1
x
−
a
3
z
:
z
). Show that if
P
∈
E
(
k
) then
ι
(
P
)
∈
E
(
k
) and that
ι
is an isomorphism
over
k
from
E
to itself.
y
2
Lemma 7.2.3
Let H
(
x
)
,F
(
x
)
∈ k
[
x
]
,
deg(
F
)
=
3
,
deg(
H
)
≤
1
. Then E
(
x,y
)
=
+
H
(
x
)
y
−
F
(
x
)
is irreducible over
k
.
Theorem
5.3.8
therefore implies that a Weierstrass equation describes a projective vari-
ety. By Exercise
5.6.5
the variety has dimension 1. Not every Weierstrass equation gives
a curve, since some of them are singular. We now give conditions for when a Weierstrass
equation is non-singular.
=
Exercise 7.2.4
Show that a Weierstrass equation has a unique point with
z
0. Show that
this point is not a singular point.
Definition 7.2.5
Let
E
be a Weierstrass equation over
k
. The point (0 : 1 : 0)
∈
E
(
k
)is
denoted by
O
E
and is called the
point at infinity
.
Exercise 7.2.6
Show that if char(
k
)
=
2
,
3 then every Weierstrass equation over
k
is
isomorphic over
k
to a Weierstrass equation
y
2
z
x
3
a
4
xz
2
a
6
z
3
=
+
+
(7.3)
for some
a
4
,a
6
∈ k
. This is called the
short Weierstrass form
. Show that this equation is
non-singular if and only if the
discriminant
4
a
4
+
27
a
6
=
0in
k
.
Exercise 7.2.7
Show that if char(
k
)
=
2 then every Weierstrass equation over
k
is isomor-
phic over
k
to a Weierstrass equation
y
2
z
x
3
a
2
x
2
z
a
6
z
3
or
y
2
z
yz
2
x
3
a
4
xz
2
a
6
z
3
.
+
xyz
=
+
+
+
=
+
+
(7.4)
The former is non-singular if
a
6
=
0 and the latter is non-singular for all
a
4
,a
6
∈ k
.
Formulae to determine whether a general Weierstrass equation is singular are given in
Section III.1 of [
505
].
Definition 7.2.8
An
elliptic curve
is a curve given by a non-singular Weierstrass equation.
The following easy result is useful for explicit calculations.
