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§1. The Riemann Hypothesis for Elliptic Curves over a Finite Field 255

2
3
(1.4) Example. The elliptic curve E defined by y + y = x + x has five points
over F 2 , namely
{∞ = 0,(0, 0), (0, 1), (1, 0), (1, 1)}.

Hence E(F 2 ) is isomorphic to Z/5and N 1 = 5. Note that N 1 −1−q = 5−1−2 =
√
2 ≤ 2 2.
(1.5) Definition. Let E be an elliptic curve defined over a finite field F q . The char-
acteristic polynomial of Frobenius is
f E (T ) = det(1 − π E T ) = 1 − (Tr(π))T + qT 2 in Z[T ],

and the zeta function ζ E (s) is the rational function in q s

f E (q −s ) 1 − (Tr(π))q −s + q 1−2s
ζ E (s) = = .
(1 − q −s )(1 − q 1−s ) (1 − q −s )(1 − q 1−s )
(1.6) Remark. The zeta function ζ E (s) has poles at s = 0and s = 1. The inequality
in (1.2), called the Riemann hypothesis, is equivalent to the assertion that the roots of
f E (T ) are complex conjugates of each other. These roots have absolute value equal
√
to 1/ q. In turn this condition is equivalent to the assertion that the zeta function
ζ E (s) has zeros only on the line Re(s) = 1/2.

For f E (q −s ) = 1 − (Tr π)q −s + q 1−2s , we replace s by 1 − s, and we obtain the
relation

f E (q −(1−s) ) = 1 − (Tr π)q s−1 + q 2s−1 = q 2s−1 (1 − (Tr π)q −s + q 1−2s )
= q 2s−1 f E (q −s ).

This functional equation for f E and the invariance of the denominator of the zeta
function under s changedto1 − s yield the following immediately.

(1.7) Proposition. The zeta function ζ E (s) of E over F q satisfies the functional
equation
2s−1
ζ E (1 − s) = q ζ E (s).
There is a far reaching generalization of this zeta function to zeta functions for
any projective variety over a finite field due to A. Weil. In fact, it is the focus of
a vast conjectural program announced by Weil [1949], carried out up to the Rie-
mann hypothesis primarily by Grothendieck and Artin [1963–64], and completed
with Deligne’s [1973] proof of the Riemann hypothesis for smooth projective vari-
eties. Note, for example, the generalization of (1.7) is a consequence of Poincar´ e du-
ality in -adic cohomology theory. For a survey of the whole story, see Katz [1976].