Cryptography Reference
In-Depth Information
This is nonsingular at 2 (since the partial derivative with respect to
y
is
2
y
+1
0 (mod 2)). Therefore,
E
has good reduction at 2. We conclude
that
E
has good reduction at all primes except
p
= 11, where it has bad
reduction. The equation
y
3
+
y
3
=
x
3
−
≡
1
≡
x
3
is the minimal Weierstrass equation
for
E
.
Let's analyze the situation at 11 more closely. The polynomial in
x
2
factors
as
x
2
−
4
x
2
+16=(
x
2
+1)
2
(
x
2
+5)
.
Therefore,
E
has multiplicative reduction at 11. The method of Section 2.10
shows that the slopes of the tangent lines at the singular point (
x
2
,y
2
)=
(
−
1
,
0) are
±
2, which lie in
F
11
. Therefore,
E
has split multiplicative reduc-
tion at 11.
We now give the full definition of the
L
-series of
E
.Foraprime
p
of bad
reduction, define
⎨
0if
E
has additive reduction at
p
1if
E
has split multiplicative reduction at
p
a
p
=
⎩
−
1if
E
has nonsplit multiplicative reduction at
p
.
The numbers
a
p
for primes of good reduction are those given above:
a
p
=
p
+1
−
#
E
(
F
p
). Then the
L
-function of
E
is the Euler product
L
E
(
s
)=
bad
p
1
− a
p
p
−s
−
1
1
− a
p
p
−s
+
p
1
−
2
s
−
1
.
good
p
The estimate
|a
p
| <
2
√
p
easily implies that the product converges for
(
s
)
>
3
/
2 (see Exercise 14.3).
Each good factor can be expanded in the form
a
p
p
−s
+
p
1
−
2
s
)
−
1
=1+
a
p
p
−s
+
a
p
2
p
−
2
s
+
(1
−
···
,
where the
a
p
on the left equals the
a
p
on the right (so this is not bad notation)
and
a
p
2
=
a
p
−
p.
(14.1)
The product over all
p
yields an expression
L
E
(
s
)=
∞
a
n
n
−s
.
n
=1
If
n
=
j
p
e
j
,then
j
a
n
=
j
a
p
e
j
j
.
(14.2)
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