Cryptography Reference
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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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