Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 427

Chapter 14

Zeta Functions

14.1 Elliptic Curves over Finite Fields

Let E be an elliptic curve over a finite field F q .Let

N n =# E ( F q n )

be the number of points on E over the field F q n .The Z -function of E is
defined to be

Z E ( T )=exp ∞ T n .

N n
n

n =1

Here exp( t )= t n /n ! is the usual exponential function. The Z -function
encodes certain arithmetic information about E as the coe cients of a gen-
erating function. The presence of the exponential function is justified by the
simple form for Z E ( T ) in the following result.

PROPOSITION 14.1

Let E be an elliptic curve defined over F q ,and let # E ( F q )= q +1 −a .Then

qT 2 − aT +1

(1 − T )(1 − qT ) .

Z E ( T )=

Factor X 2 PROOF − aX + q =( X − α )( X − β ). Theorem 4.12 says that

N n = q n +1 − α n − β n .