Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 428

t )= t n /n ,wehave Therefore, using the expansion − log(1 −

Z E ( T )=exp ∞ T n

N n
n

n =1

=exp ∞

− β n ) T n
n

( q n +1 − α n

n =1

=exp( − log(1 − qT ) − log(1 − T )+log(1 − αT )+log(1 − βT ))

(1 − αT )(1 − βT )
(1 − T )(1 − qT )

=

qT 2 − aT +1

(1 − T )(1 − qT ) .

=

Note that the numerator of Z E ( T ) is the characteristic polynomial of the
Frobenius endomorphism, as in Chapter 4, with the coecients in reverse
order.
A function Z C ( T ) can be defined in a similar way for any curve C over a
finite field, and, more generally, for any variety over a finite field. It is always
a rational function (proved by E. Artin and F. K. Schmidt for curves and by
Dwork for varieties).
The zeta function of E is defined to be

ζ E ( s )= Z E ( q −s ) ,

where s is a complex variable. As we'll see below, ζ E ( s ) can be regarded as
an analogue of the classical Riemann zeta function

ζ ( s )= ∞

1
n s .

n =1

One of the important properties of the Riemann zeta function is that it sat-
isfies a functional equation relating the values at s and 1 − s :

π −s/ 2 Γ( s/ 2) ζ ( s )= π − (1 −s ) / 2 Γ((1 − s ) / 2) ζ (1 − s ) .

A famous conjecture for ζ ( s ) is the Riemann Hypothesis, which predicts that
if ζ ( s )=0with0 ≤ ( s ) ≤ 1then ( s )=1 / 2 (there are also the “trivial”
zeros at the negative even integers). The elliptic curve zeta function ζ E ( s ) also
satisfies a functional equation, and the analogue of the Riemann Hypothesis
holds.