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§2. Generalities on Zeta Functions of Curves over a Finite Field 257

C(k n ) for some n. For the minimal such n, we write k 1 (P) = k n and n = deg(P),
the local degree of P.If σ generates the cyclic Galois group Gal(k n /k 1 ) of order n,
σ
then p = P + P +···+ P σ n−1 is an example of a prime divisor over k 1 on C which
is rational over k 1 . In fact, all prime divisors on C over k 1 are of this form and the
n
norm of the divisor is defined Np = q . Using multiplicative notation, in constrast
to Chapter 12, §3, we denote a positive rational divisor over k 1 by
n(1) n(r)
a = p ... p ,
1 r
where n(1), ... , n(r) are natural numbers and p 1 ,... , p r are prime divisors rational
over k 1 . Then norm of this a is given by
Na = (Np 1 ) n(1) ··· (Np r ) n(r) .

Through the above examples, we circumvent the general theory of rational divisors.
(2.1) Definition. Let C be an algebraic curve defined over k 1 = F q . The zeta func-
tion ζ C (s) of C/k 1 is defined as a sum or as a product involving divisors rational
over k 1
−s −s −1
ζ C (s) = (Na) = (1 − (Np) ) .
positive a prime p

At this point these two formal expressions for the zeta function are equivalent
from the unique decomposition of divisors as products of prime divisors. Now we
begin a discussion which will show that this zeta function is the same zeta function
as in Definition (1.5) in the case of an elliptic curve. We will also outline its basic
properties for a general complete nonsingular curve.
Let A m denote the number of positive divisors a rational over k 1 with norm q m
on C and let P m denote the number of prime divisors y rational over k 1 with norm
q m on C. The equality of the two expressions in the definition of the zeta function
leads to two expressions for the related function
∞ ∞

m
m −P m
Z C (u) = A m u = (1 − u )
m=0 m=0
and Z C (q −2 ) = ζ C (s).If N m is the number of points on C(k m ) where k m = F q as
m
above, then

N m = dP d
d|m
from the above description of prime divisors rational over k m . Now calculate
∞
∞
∞
d 1 dP d u d 1 dd 1 m
log Z C (u) = = dP d u = N m u .
du u 1 − u d u u
d=1 d,d =1 m=1

Hence we have with the above notations the following result.