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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$.



"supersingular" is owned by nerdy2.
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Cross-references: Frobenius morphism, induced, invariant differential, iff, equivalent, invariant, coefficient, cubic equation, characteristic, field, elliptic curve
There are 2 references to this entry.

This is version 2 of supersingular, born on 2002-02-10, modified 2002-02-10.
Object id is 1891, canonical name is Supersingular.
Accessed 3073 times total.

Classification:
AMS MSC: 14H52 (Algebraic geometry :: Curves :: Elliptic curves)
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