Cryptography Reference
In-Depth Information
8
Rational maps on curves and divisors
The purpose of this chapter is to develop some tools in the theory of algebraic curves that
are needed for the applications (especially, hyperelliptic curve cryptography). The technical
machinery in this chapter is somewhat deeper than the previous one and readers can skip
this chapter if they wish.
The reader should note that the word “curve” in this chapter always refers to a non-
singular curve.
8.1 Rational maps of curves and the degree
Lemma 8.1.1
Let C be a curve over
k
and f
∈ k
(
C
)
. One can associate with f a rational
1
over
map φ
:
C
→ P
k
by φ
=
(
f
:1)
. (Indeed, this is a morphism by Lemma
7.3.6
.)
Denote by
∞
the constant map
∞
(
P
)
=
(1 : 0)
. Then there is a one-to-one correspondence
1
.
between
k
(
C
)
∪{∞}
and the set of morphisms φ
:
C
→ P
Exercise 8.1.2
Prove Lemma
8.1.1
.
Note that since
k
(
C
)
∪{∞}
is not a field, it does not make sense to interpret the set of
1
rational maps
φ
:
C
→ P
as a field.
Lemma 8.1.3
Let C
1
and C
2
be curves over
k
(in particular, non-singular and projec-
tive) and let φ
:
C
1
→
C
2
be a non-constant rational map over
k
. Then φ is a dominant
morphism.
Proof
See Proposition II.6.8 of Hartshorne [
252
] or Proposition II.2.1 of Silverman
[
505
].
The notion of degree of a mapping is fundamental in algebra and topology; a degree
d
map is “
d
-to-one on most points”.
1
(
Example 8.1.4
Let
k
be a field of characteristic not equal to 2. The morphism
φ
:
A
k
)
→
1
(
x
2
A
k
) given by
φ
(
x
)
=
is clearly two-to-one away from the point
x
=
0. We say that
φ
has degree 2.
