Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 429

THEOREM 14.2
Let E be an elliptic curve defined over a finitefie d.

1. ζ E ( s )= ζ E (1

− s )

2. If ζ E ( s )=0 ,then

( s )=1 / 2 .

PROOF The proof of the first statement follows easily from Proposi-

tion 14.1:

− aq −s +1
(1 − q −s )(1 − q 1 −s )

q 1 − 2 s

ζ E ( s )=

aq s− 1 + q − 1+2 s 1 −

=

( q s − 1)( q s− 1 − 1)

= ζ E (1 − s ) .

Since the numerator of Z E ( T )is(1 − αT )(1 − βT ), we have

q s = α or β. ζ E ( s )=0 ⇐⇒

By the quadratic formula,

α, β = a ± a 2

− 4 q

.

2

Hasse's theorem (Theorem 4.2) says that

2 √ q,

| a |≤

hence a 2

− 4 q ≤ 0. Therefore, α and β are complex conjugates of each other,

and

|α| = |β| = √ q.

If q s = α or β ,then

= √ q. q ( s ) = q s

| |

Therefore, ( s )=1 / 2.

There are infinitely many solutions to q s = α . However, if s 0 is one such
solution, all others are of the form s 0 +2 πin/ log q with n ∈ Z . A similar

situation holds for β .
If C is a curve, or a variety, over a finite field, then an analogue of Theo-
rem 14.2 holds. For curves, the functional equation was proved by E. Artin
and F. K. Schmidt, and the Riemann Hypothesis was proved by Weil in the
1940s. In 1949, Weil announced what became known as the Weil conjectures,
which predicted that analogues of Proposition 14.1 and Theorem 14.2 hold for