ABSTRACT
Let E be an elliptic curve defined over Fq, and let #E(Fq) = q+1−a. Then
ZE(T ) = qT 2 − aT + 1
(1− T )(1− qT ) .
PROOF Factor X2 − aX + q = (X − α)(X − β). Theorem 4.12 says that
Nn = qn + 1− αn − βn.
t) =∑ t we have
ZE(T ) = exp
( ∞∑ n=1
Nn n
Tn
)
= exp
( ∞∑ n=1
(qn + 1− αn − βn)T n
n
)
= 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 ) .
