# 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 |
---|---|

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 |