Elliptic Curves over Finite Fields

Abstract

In this chapter we study elliptic curves defined over a finite field. The most important arithmetic quantity associated with such a curve is its number of rational points. We start by proving a theorem of Hasse which says that if K is a field with q elements, and E/K is an elliptic curve, then E(K) contains approximately q points, with an error of no more than $2\sqrt {q}$. Following Weil, we then reinterpret and extend this result in terms of a certain generating function, the zeta-function of the curve. In the final two sections we study in some detail the endomorphism ring of an elliptic curve defined over a finite field, and in particular give the relationship between End(E) and the existence of non-trivial p-torsion points. The notation for chapter V is: