Cryptography Reference
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elliptic curve, as in Section 2.5.4. A lengthy calculation, using the formulas of
Theorem 2.17, shows that this elliptic curve is the original curve
E
.If
C
a,b,c
does not have any rational points, then the triple (
a, b, c
) is eliminated.
The problem is how to decide which curves
C
a,b,c
have rational points. In
the examples of Section 8.2, we used considerations of sign and congruences
mod powers of 2 and 5. These can be interpreted as showing that the curves
C
a,b,c
that are being eliminated have no real points, no 2-adic points, or no
5-adic points (for a summary of the relevant properties of
p
-adic numbers, see
Appendix A). For example, when we used inequalities to eliminate the triple
(
a, b, c
)=(
−
1
,
1
, −
1) for the curve
y
2
=
x
(
x−
2)(
x
+ 2), we were showing that
the curve
u
2
v
2
=2
,
u
2
+
w
2
=
C
−
1
,
1
,−
1
:
−
−
−
−
2
has no real points. When we eliminated (
a, b, c
)=(1
,
2
,
2), we used congru-
ences mod powers of 2. This meant that
C
1
,
2
,
2
:
u
2
2
v
2
=2
,
2
2
w
2
=
−
−
−
2
has no 2-adic points.
The
2-Selmer group
S
2
is defined to be the set of (
a, b, c
) such that
C
a,b,c
has a real point and has
p
-adic points for all
p
. For notational convenience,
the real numbers are sometimes called the
∞
-adics
Q
∞
. Instead of saying
that something holds for the reals and for all the
p
-adics
Q
p
,wesaythatit
holds for
Q
p
for all
p ≤∞
. Therefore,
S
2
=
{
(
a, b, c
)
| C
a,b,c
(
Q
p
) is nonempty for all
p ≤∞}.
Therefore,
S
2
is the set of (
a, b, c
) that cannot be eliminated by sign or congru-
ence considerations. It is a group under multiplication mod squares. Namely,
we regard
S
2
⊂
(
Q
×
/
Q
×
2
)
⊕
(
Q
×
/
Q
×
2
)
⊕
(
Q
×
/
Q
×
2
)
.
The prime divisors of
a, b, c
divide (
e
1
− e
2
)(
e
1
− e
3
)(
e
2
− e
3
), which implies
that
S
2
is a finite group.
The descent map
φ
gives a map
φ
:
E
(
Q
)
/
2
E
(
Q
)
→ S
2
.
The 2-torsion in the
Shafarevich-Tate group
is the cokernel of this map:
2
=
S
2
/
Im
φ.
The symbol is the Cyrillic letter “sha,” which is the first letter of “Shafare-
vich” (in Cyrillic). We'll define the full group in Section 8.9. The group
2
represents those triples (
a, b, c
) such that
C
a,b,c
has a
p
-adic point for all
p ≤∞
, but has no rational point. If
= 1, then it is much more dicult
to find the points on the elliptic curve
E
.If(
a, b, c
) represents a nontrivial
2
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