Cryptography Reference
In-Depth Information
Proof
Part 1 is Satz 1 of Stichtenoth [
528
]. Part 2 is Proposition 1 of Stichtenoth and
Xing [
530
].
We refer to Yui [
572
] for a survey of the Cartier-Manin matrix and related criteria for
the
p
-rank.
Exercise 10.8.6
Let
A
be an Abelian variety of dimension 2 over
F
p
that has
p
-rank zero.
Show that
A
is supersingular.
In fact, the result of Exercise
10.8.6
holds when
F
p
is replaced by any finite field; see
page 9 of Li and Oort [
348
].
Exercise 10.8.7
Let
C
:
y
2
+
=
F
2
n
where deg(
F
(
x
))
=
5 be a genus 2 hyper-
elliptic curve. Show that
C
has 2-rank zero (and hence is supersingular).
y
F
(
x
) over
Example
10.8.8
shows that, once the genus is at least 3,
p
-rank zero does not imply
supersingularity.
Example 10.8.8
Define
C
:
y
2
x
7
over
T
6
2
T
3
2
3
and so by
+
y
=
F
2
. Then
P
(
T
)
=
−
+
Theorem
10.8.5
the 2-rank of
C
is zero but
C
is not supersingular.
Example 10.8.9
(Hasse/Hasse-Davenport/Duursma [
171
]) Let
p>
2 be prime and
C
:
y
2
x
p
=
−
x
+
1 over
F
p
. One can verify that
C
is non-singular and the genus of
C
is
p
((
−
p
)
pT
) where
p
(
T
)isthe
p
-th cyclotomic polynomial. It follows that the roots of
P
(
T
) are roots of unity and so
C
is supersingular.
−
F
p
2
,
L
(
T
)
=
(
p
1)
/
2. It is shown in [
171
] that, over
