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§2. Zeta Functions of Curves over Q 311

P p (p −s )

∗
ζ = ζ (s) =
C C p (1 − p −s )(1 − p 1−s )
p/∈S p/∈S
and the crude Hasse–Weil L-function of C is

∗ −s −1
L (s) = P p (p ) .
C
p/∈S
We know what the reduction mod p of an elliptic curve is in terms of the normal
form. For a general curve C, one needs scheme theoretical techniques. For C over
Q, it has good reduction outside a finite set S.
The Hasse–Weil zeta and L-functions are the analytic objects which store the
diophantine data of C related to congruences modulo p. In this definition, we have
referred to the crude zeta and L-functions because the zeta function and L-function
will have factors associated with p ∈ S. In this form, we can make some elementary
assertions about convergence which also apply to the more precise versions of these
functions.
(2.2) Remark. For each finite set of primes S, we can consider a modified Riemann
zeta function ζ S (s) defined by
1
−s
ζ S (s) = = (1 − p ) · ζ(s)
(1 − p −s )
p/∈S p∈S
1

= s ,
n
1≤n,(S,n)=1
where ζ(s) is the ordinary Riemann zeta function correspondingto empty S. In terms
of the modified zeta function ζ S , we have the relation
∗
∗
L (s) · ζ (s) = ζ S (s) · ζ S (s − 1).
C C
Since the Riemann zeta function, and therefore also the modified versions, are fairly
well understood relative to convergence and functional equation, we consider the
∗
∗
elementary study of the Hasse–Weil zeta function ζ (s) and L-function L (s) as
C C
effectively equivalent. The L-function L (s) is an Euler product
∗
C
1

,
(1 − α 1 (p)p −s ) ··· (1 − α 2g (p)p −s )
p
where we take all α j (p) = 0 for p ∈ S, j = 1,..., 2g. For an elliptic curve E over
Q as in the previous definition,
1

∗
L (s) =
E −s −s
p (1 − α(p)p )(1 − β(p)p )
since g = 1. Now the considerations of 13(1.7) apply because the Riemann hypoth-
esis asserts that