The torsion of an elliptic curve over a finite field

Question

There is a result for $p$ prime, $E$ an elliptic curve over $\mathbb F_p$, then $E(\overline{\mathbb{F}_p})[m]\cong (\mathbb{Z}/m\mathbb{Z})^2$ for $m \nmid p$. The book on cryptography I am using says the result is classical and doesn't give a reference. Anyone know how one goes about proving this?

Answer 1

This follows from the following facts, none of which is very hard (but still some work). You can look in Silverman for details, as others have suggested. The outline is as follows:

1. An isogeny $E \to E'$ is separable if and only if it pulls back a nonzero differential on $E'$ to a nonzero differential on $E$;

2. The multiplication-by-$m$ isogeny $E \to E$ induces multiplication by $m$ on differentials; together with (1), this implies that multiplication by $m$ is separable whenever $p\nmid m$;

3. The kernel of a separable isogeny of degree $d$ consists of $d$ points (over the algebraic closure).

4. Multiplication by $m$ has degree $m^2$.

This gives the order of the $m$-torsion for $p \nmid m$, and an easy argument from there gives the group structure.

As for the case $m=p$: the $p$-torsion is either $0$ ("supersingular" case) or $\mathbf Z/p\mathbf Z$ ("ordinary" case).

Answer 2

This is Corollary 6.4 (part (b)) in Chapter III of Silverman's "The Arithmetic of Elliptic Curves". You can find a proof there.