Cryptography Reference
In-Depth Information
7
Curves and divisor class groups
The purpose of this chapter is to develop some basic theory of divisors and functions on
curves. We use this theory to prove that the set of points on an elliptic curve over a field is
a group. There exist more elementary proofs of this fact, but I feel the approach via divisor
class groups gives a deeper understanding of the subject.
We start by introducing the theory of singular points on varieties. Then we define
uniformisers and the valuation of a function at a point on a curve. When working over a
field
that is not algebraically closed it turns out to be necessary to consider not just points
on
C
defined over
k
(alternatively, one can generalise the
notion of point to places of degree greater than one; see [
529
] for details). We then discuss
divisors, principal divisors and the divisor class group. The hardest result is that the divisor
of a function has degree zero; the proof for general curves is given in Chapter
8
. Finally,
we discuss the “chord and tangent” group law on elliptic curves.
k
but also those defined over
k
7.1 Non-singular varieties
The word “local” is used throughout analysis and topology to describe any property that
holds in a neighbourhood of a point. We now develop some tools to study “local” properties
of points of varieties. The algebraic concept of “localisation” is the main technique used.
Definition 7.1.1
Let
X
be a variety over
k
.The
local ring
over
k
of
X
at a point
P
∈
X
(
k
)
is
O
P,
k
(
X
)
={
f
∈ k
(
X
):
f
is regular at
P
}
.
Define
m
P,
k
(
X
)
={
f
∈
O
P,
k
(
X
):
f
(
P
)
=
0
}⊆
O
P,
k
(
X
)
.
When the variety
X
and field
k
are clear from the context we simply write
O
P
and m
P
.
Lemma 7.1.2
Let the notation be as above. Then:
1.
O
P,
k
(
X
)
is a ring;
2.
m
P,
k
(
X
)
is an
O
P,
k
(
X
)
-ideal;
