Cryptography Reference
In-Depth Information
PROPOSITION 8.31
An exactsequence
0
→
M
1
→
M
2
→
M
3
→
0
of
G
-m o d u les induces a long exact sequence
0
→ H
0
(
G, M
1
)
→ H
0
(
G, M
2
)
→ H
0
(
G, M
3
)
→
H
1
(
G, M
1
)
H
1
(
G, M
2
)
H
1
(
G, M
3
)
→
→
of cohom ology groups.
For a proof, see any topic on group cohomology, for example [132], [21],
or [6]. The hardest part of the proposition is the existence of the map from
H
0
(
G, M
3
)to
H
1
(
G, M
1
).
Suppose now that we have an elliptic curve defined over
Q
.L t
n
be
a positive integer. Multiplication by
n
gives a
n
endomo
rp
hism of
E
.By
Theorem 2.22, it is surjective from
E
(
Q
)
→ E
(
Q
), since
Q
is algebraically
closed. Therefore, we have an exact sequence
n
→ E
(
Q
)
→
0
.
0
→ E
[
n
]
→ E
(
Q
)
(8.17)
Let
G
=Gal(
Q
/
Q
)
be the Galois group of
Q
/
Q
. The reader who doesn't know what this group
looks like should not worry. No one does. Much of modern number theory
can be interpreted as trying to understand the structure of this group. The
onepropertyweneedatthemomentisthat
H
0
(
G, E
(
Q
)) =
E
(
Q
)
G
=
E
(
Q
)
.
Applying Proposition 8.31 to the exact sequence (8.17) yields the long exact
sequence
n
→
0
→
E
(
Q
)
→ H
1
(
G, E
[
n
])
→ H
1
(
G, E
(
Q
))
E
(
Q
)[
n
]
→
E
(
Q
)
n
→ H
1
(
G, E
(
Q
))
.
This induces the short exact sequence
H
1
(
G, E
[
n
])
H
1
(
G, E
(
Q
))[
n
]
0
→
E
(
Q
)
/nE
(
Q
)
→
→
→
0
,
(8.18)
wherewehavewritten
A
[
n
]forthe
n
-torsion in an abelian group
A
.This
sequence is similar to the sequence
0
→ E
(
Q
)
/
2
E
(
Q
)
→ S
2
→
→
0
2
that we met in Section 8.7. In the remainder of this section, we'll show how the
two sequences relate when
n
= 2 and also consider the situation for arbitrary
n
.
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