Cryptography Reference
In-Depth Information
Example 14.2
Let
E
be the elliptic curve
y
2
+
y
=
x
3
x
2
considered in the previous example.
If we compute the number
N
p
of points on
E
mod
p
for various primes, we
obtain, with
a
p
=
p
+1
−
−
N
p
,
a
2
=
−
2
,
3
=
−
1
,
5
=1
,
7
=
−
2
,
13
=4
,...
(except for
p
=2
,
3
,
5, the numbers
a
p
can be calculated using any of the
equations in the previous example). The value
a
11
=1
is specified by the formulas for bad primes. We then calculate the coecients
for composite indices. For example,
4
=
a
2
−
2=2
a
6
=
a
2
a
3
=2
,
(see (14.2) and (14.1)). Therefore,
f
E
(
τ
)=
q −
2
q
2
− q
3
+2
q
4
+
q
5
+2
q
6
−
2
q
7
+
··· .
It can be shown that
f
(
τ
)=
q
∞
(1
− q
j
)
2
(1
− q
11
j
)
2
j
=1
is a cusp form of weight 2 and level
N
= 11. In fact, it is the only such form,
up to scalar multiples. The product for
f
can be expanded into an infinite
series
f
(
τ
)=
q −
2
q
2
− q
3
+2
q
4
+
q
5
+2
q
6
−
2
q
7
+
··· .
It can be shown that
f
=
f
E
(see [61]).
The
L
-series for
E
satisfies the functional equation
(
√
11
/
2
π
)
s
Γ(
s
)
L
E
(
s
)=+(
√
11
/
2
π
)
2
−s
Γ(2
− s
)
L
E
(2
− s
)
.
In the early 1960s, Birch and Swinnerton-Dyer performed computer exper-
iments to try to understand the relation between the number of points on
an elliptic curve mod
p
as
p
ranges through the primes and the number of
rational points on the curve. Ignoring the fact that the product for
L
E
(
s
)
doesn't converge at
s
= 1, let's substitute
s
= 1 into the product (we'll ignore
the finitely many bad primes):
p
−
1
1
− a
p
p
−
1
+
p
−
1
−
1
=
p
=
p
−
a
p
+1
p
p
N
p
.
p
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