# supersingular

An elliptic curve $E$ over a field of characteristic $p$ defined by the cubic equation $f(w,x,y)=0$ is called supersingular if the coefficient of $(wxy)^{p-1}$ in $f(w,x,y)^{p-1}$ is zero.

A supersingular elliptic curve is said to have Hasse invariant $0$; an ordinary (i.e. non-supersingular) elliptic curve is said to have Hasse invariant $1$.

This is equivalent to many other conditions. $E$ is supersingular iff the invariant differential is exact. Also, $E$ is supersingular iff $F^{*}:H^{1}(E,\mathcal{O}_{E})\to H^{1}(E,\mathcal{O}_{E})$ is nonzero where $F^{*}$ is induced from the Frobenius morphism $F:E\to E$.

Title supersingular Supersingular 2013-03-22 12:18:30 2013-03-22 12:18:30 nerdy2 (62) nerdy2 (62) 5 nerdy2 (62) Definition msc 14H52