An elliptic curveMathworldPlanetmath 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to many other conditions. E is supersingular iff the invariant differential is exact. Also, E is supersingular iff F*:H1(E,𝒪E)H1(E,𝒪E) is nonzero where F* is induced from the Frobenius morphism F:EE.

Title supersingular
Canonical name Supersingular
Date of creation 2013-03-22 12:18:30
Last modified on 2013-03-22 12:18:30
Owner nerdy2 (62)
Last modified by nerdy2 (62)
Numerical id 5
Author nerdy2 (62)
Entry type Definition
Classification msc 14H52