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310 16. L-Function of an Elliptic Curve and Its Analytic Continuation

an Euler factor and form the product of these Euler factors into a Dirichlet series. The
study of Euler products was started in Chapter 11 where they arose in connection
with modular forms.
A remarkable feature of the theory is that certain of these L-functions associated
to modular forms by the Mellin transform, and which have Euler product decompo-
sitions, also arise as the L-functions of elliptic curves, which are defined as Euler
products from the local arithmetic data of the elliptic curve defined over the rational
numbers. This is the theory of Eichler–Shimura.

§2. Zeta Functions of Curves over Q

In Chapter 13, §2, we discussed the zeta function ζ C of a smooth curve over F q .It
took two forms, (2.2) and (2.5) respectively

N m −ms
ζ C (s) = exp q ,
m
i≤m
where N m = #C(F q ) the number of points of C over F q ,and
m
m
P(q −s )
ζ C (s) = ,
(1 − q −s )(1 − q 1−s )
where P(T ) = (1 − α 1 T ) ··· (1 − α 2g T ) and the reciprocal roots α 1 ,...,α 2g are
√
paired off in such a way that α 2 j−1 = α 2 j so that |α i |= q and α 2 j−1 α 2 j = q.The
numbers of points N m and the reciprocal roots, which are all algebraic integers, are
related by

2q
m m
N m = q + 1 − α .
i
i=1
The number g is the genus of C.
a
Since q = p , the terms (1 − q −s ), (1 − q · q −s ) and P(q −s ) are the kinds of
expressions used to make Euler products provided we are given these expressions
for all primes. This comes up in the study of smooth curves C over a number field
of genus g. Roughly, as in Chapter 5, we wish to choose equations of C over Z in
such a way that the reduction C p modulo p of C is a smooth curve of genus g for all
primes p not in a finite set S. The curve C is said to have good reduction at p /∈ S.

More precisely, we extend C to a smooth scheme over Z[1/n] where n = p/∈S p.
Then the curves C p are the special fibres. As with elliptic curves, good reduction can
be described in terms of a projective model over Q.

(2.1) Definition. Let C be a curve over Q extending to a smooth scheme over
Z[1/n] such that the reduction C p (mod p) is smooth over F p for p /∈ S, where
S is the set of prime divisors of n. The crude Hasse–Weil zeta function of C is