Cryptography Reference
In-Depth Information
Lemma 7.2.9
Let E be an elliptic curve over
(
E
)
restricts
to a function on the affine Weierstrass equation of E that is equivalent to a function of the
form
k
. Then every function f
∈ k
a
(
x
)
b
(
x
)
y
c
(
x
)
+
(7.5)
where a
(
x
)
,b
(
x
)
,c
(
x
)
∈ k
[
x
]
. Conversely, every such function on the affine curve corre-
sponds to a unique
2
function on the projective curve.
7.3 Uniformisers on curves
Let
C
be a curve over
(
C
). It is necessary to formalise the notion
of multiplicity of a zero or pole of a function at a point. The basic definition will be
that
f
k
with function field
k
∈
m
m
+
1
P,
k
∈
m
P,
k
. However, there are
a number of technicalities to be dealt with before we can be sure this definition makes
sense. We introduce uniformisers in this section as a step towards the rigorous treatment of
multiplicity of functions.
First we recall the definition of non-singular from Definition
7.1.10
:let
C
be a non-
singular curve over
∈
O
P,
k
(
C
) has multiplicity
m
at
P
if
f
and
f
k
∈
k
), then the quotient m
P,
k
(
C
)
/
m
P,
k
(
C
)
2
and
P
C
(
(which is a
k
-vector space by Lemma
7.1.6
) has dimension one as a
k
-vector space.
Lemma 7.3.1
Let C be a non-singular curve over a field
k
and let P
∈
C
(
k
)
. Then the
ideal
m
P,
k
(
C
)
is principal as an
O
P,
k
(
C
)
-ideal.
Proof
This result needs Nakayama's Lemma. We refer to Section I.6 of Hartshorne [
252
]
for the details; also see Proposition II.1.1 of Silverman [
505
] for references.
Definition 7.3.2
Let
C
be a non-singular curve over
k
and
P
∈
C
(
k
). A
uniformiser
(or
uniformising parameter
)at
P
is an element
t
P
∈
O
P,
k
(
C
) such that m
P,
k
(
C
)
=
(
t
P
)asan
O
P,
k
(
C
)-ideal.
(
C
)
2
; in other words, the uni-
One can choose
t
P
to be any element of m
P,
k
(
C
)
−
m
P,
k
−
m
P,
k
(
C
)
2
,
formiser is not unique. If
P
is defined over
k
then one can take
t
P
∈
m
P,
k
(
C
)
i.e. take the uniformiser to be defined over
k
; this is typically what one does in
practice.
For our presentation it is necessary to know uniformisers on
1
P
and on a Weierstrass
equation. The next two examples determine such uniformisers.
= P
1
. For a point (
a
:1)
∈
U
1
⊆ P
1
Example 7.3.3
Let
C
one can work instead with the
ϕ
−
1
1
point
a
on the affine curve
A
1
=
(
U
1
). One has m
a
=
(
x
−
a
) and so
t
a
=
(
x
−
a
)is
2
By unique we mean that there is only one function on the projective curve corresponding to a given function on the affine curve.
The actual polynomials
a
(
x
)
,b
(
x
)and
c
(
x
) are, of course, not unique.
