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supersingular (Definition)

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 $F : E\to E$




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Cross-references: Frobenius morphism, induced, invariant differential, iff, equivalent, invariant, coefficient, cubic equation, characteristic, field, elliptic curve
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This is version 2 of supersingular, born on 2002-02-10, modified 2002-02-10.
Object id is 1891, canonical name is Supersingular.
Accessed 4027 times total.

Classification:
AMS MSC14H52 (Algebraic geometry :: Curves :: Elliptic curves)

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