Cryptography Reference
In-Depth Information
8.7 Riemann-Roch theorem and Hurwitz genus formula
In this section we state, without proof, two very important results in algebraic geometry.
Neither will play a crucial role in this topic.
Lemma 8.7.1
Let C be a curve over
k
of genus g and let ω
∈
k
(
C
)
. Then
1.
deg(div(
ω
))
=
2
g
−
2
.
2.
k
(div(
ω
))
=
g.
Proof
See Corollary I.5.16 of Stichtenoth [
529
] or Corollary 11.16 of Washington [
560
].
For non-singular plane curves see Sections 8.5 and 8.6 of Fulton [
199
].
k
Theorem 8.7.2
(Riemann-Roch) Let C be a non-singular projective curve over
of genus
g, ω
∈
k
(
C
)
a differential and D a divisor. Then
k
(
D
)
=
deg(
D
)
+
1
−
g
+
k
(div(
ω
)
−
D
)
.
Proof
There are several proofs. Section 8.6 of Fulton [
199
] gives the Brill-Noether proof
for non-singular plane curves. Theorem I.5.15 of Stichtenoth [
529
] and Theorem 2.5 of
Moreno [
395
] give proofs using repartitions.
Some standard applications of the Riemann-Roch theorem are to prove that every genus
1 curve with a rational point is birational to an elliptic curve in Weierstrass form, and to
prove that every hyperelliptic curve of genus
g
is birational to an affine curve of the form
y
2
+
=
≤
+
≤
+
H
(
x
)
y
F
(
x
) with deg(
H
(
x
))
g
1 and deg(
F
(
x
))
2
g
2.
Theorem 8.7.3
(Hurwitz genus formula) Let φ
:
C
1
→
C
2
be a rational map of curves over
k
. Let g
i
be the genus of C
i
. Suppose that
k
is a field of characteristic zero or characteristic
coprime to all e
φ
(
P
)
. Then
2
g
1
−
2
=
deg(
φ
)(2
g
2
−
2)
+
(
e
φ
(
P
)
−
1)
.
P
∈
C
1
(
k
)
Proof
See Theorem III.4.12 and Corollary III.5.6 of Stichtenoth [
529
], Theorem II.5.9 of
Silverman [
505
] or Exercise 8.36 of Fulton [
199
].
A variant of the above formula is known in the case where some of the
e
φ
(
P
) are divisible
by char(
k
).
