Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:

(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [correction: $\equiv 0 \pmod{p}$]


Also, are there generalisations of this to abelian varieties?