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 multiplicat...

1. Introduction Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as U-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian JS-groups by presenting defining relations for a family of p-groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2....