Supersingular 2002-02-10 15:17:03 2002-02-10 15:21:46 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here An elliptic curve $E$ over a field of characteristic $p$ defined by the cubic equation $f(w,x,y) = 0$ is called {\em 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 $F : E\to E$.