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13

Elliptic Curves over Finite Fields

In this chapter we carry further the algebraic theory of elliptic curves over fields of
characteristic p > 0. We already pointed out that the p-division points in character-
istic p form a group isomorphic to Z/pZ or zero while the -division points form a
2
group isomorphic to (Z/ Z) for p = . Moreover, the endomorphism algebra has
rank 1 or 2 in characteristic 0 but possibly also rank 4 in characteristic p > 0.
A key issue for elliptic curves in characteristic p is whether or not a curve can
be defined over a finite field. A basic result of Deuring is the E has complex multi-
plication if and only if E can be defined over a finite field. Thus the elliptic curves
E with rk(End(E)) equal to 2 or 4 are the curves defined over a finite field, and
among the curves E in characteristic p the case rk(End(E)) is equal to 4 occurs if
¯
and only if the group p E(k) is zero. Curves E with these equivalent properties are
called supersingular, and we go further to give ten characteriszations for a curve to be
supersingular. Supersingular curves are all defined over F p or F 2 in characteristic
p
p, and they form a single isogeny class of finitely many curves up to isomorphism.
We derive a formula for the number of these curves for given p.
The number of rational points #E(F q ) on an elliptic curve E over a finite field
F q is estimated by the Riemann hypothesis
√
|#E(F q ) − q − 1|≤ 2 q.

This inequality is equivalent to the assertion that the zeros of the zeta function of an
elliptic curve E over F q are all on the line Re(s) = 1/2. This is the first topic taken
up in this chapter.
The reader should refer to Silverman [1986], Chapter V in this chapter.

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

In part 1 the group E(Q) was studied for elliptic curves over the rational numbers
Q. A similar problem is the determination of E(F q ) for E an elliptic curve defined