5.1 Ordinary and Supersingular Curves

Theorem 15.1. Let E/Fq be an elliptic curve over a finite field, and let πE be the Frobenius endomorphism of E. Then E is supersingular if and only if trπE ≡ 0 mod p. Proof. Let q = pn and let π be the p-power Frobenius map π(x, y) = (xp, yp) (note that π is an isogeny, but not necessarily an endomorphism, since E need not be defined over Fp). We have π̂π = [p], where [p] denotes the multiplication-by-p endomorphism on E. We first suppose that E is supersingular. The kernel of π̂ must then be trivial, since the kernel of [p] is trivial, and π̂ is therefore inseparable, since it has degree p > 1. The map π̂n = π̂n = π̂E is also inseparable, as is πE , so trπE = πE + π̂E is a sum of inseparable endomorphisms. Thus the endomorphism [trπE ] is inseparable, which means that p divides trπE , since [m] is separable ⇔ p m, by Theorem 6.9. So trπE ≡ 0 mod p. Conversely, if trπE ≡ 0 mod p, then [trπE ] is inseparable, and π̂E = trπE−πE is a sum of inseparable isogenies and therefore inseparable. This means that π̂n and therefore π̂ is inseparable. Therefore π̂ must have trivial kernel, since its degree is prime, and the same is true of π. So the kernel of [p] = π̂π is trivial and E is supersingular.