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 $F^{*}:H^1(E,{𝒪}_E)\to H^1(E,{𝒪}_E)$ is nonzero where $F^{*}$ is induced from the Frobenius morphism $F:E\to E$.

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