Cryptography Reference
In-Depth Information
Substituting this into the definition of
Z
(
T
), we obtain
log
Z
(
T
)=
∞
N
n
n
T
n
n
=1
=
∞
n
T
n
m
1
m
n
=1
|
n
S
P
deg(
P
)=
m
=
∞
∞
1
mj
mT
mj
(where
mj
=
n
)
j
=1
m
=1
S
P
deg(
P
)=
m
=
∞
1
j
T
j
deg(
P
)
j
=1
S
P
log(1
− T
deg(
P
)
)
.
=
−
S
P
Let
T
=
q
−s
and exponentiate to obtain the result.
14.2 Elliptic Curves over Q
Let
E
be an elliptic curve defined over
Q
. By changing variables if necessary,
we may assume that
E
is defined by
y
2
=
x
3
+
Ax
+
B
with
A, B ∈
Z
.For
aprime
p
, we can reduce the equation
y
2
=
x
3
+
Ax
+
B
mod
p
.If
E
mod
p
is an elliptic curve, then we say that
E
has
good reduction
mod
p
.This
happens for all but finitely many primes. For each such
p
,wehave
#
E
(
F
p
)=
p
+1
−
a
p
,
as in Section 14.1. The
L
-function
of
E
is defined to be approximately the
Euler product
1
− a
p
p
−s
+
p
1
−
2
s
−
1
.
good
p
This definition is good enough for many purposes. However, for completeness,
we say a few words below about what happens at the primes of bad reduction.
The factor 1
a
p
p
−s
+
p
1
−
2
s
perhaps seems to be rather artificially constructed.
However, it is just the numerator of the zeta function for
E
mod
p
,asin
Section 14.1. It might seem more natural to use the whole mod
p
zeta function,
but the factors arising from the denominator yield the Riemann zeta function
(with a few factors removed) evaluated at
s
and at
s
+ 1. Since the presence
of the zeta function would complicate matters, the denominators are omitted
in the definition of
L
E
(
s
).
−
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