Supersingular elliptic curve

In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects. Hasse (1936) discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and Deuring (1941) developed their basic theory.

The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular. The primes p such that every supersingular elliptic curve in characteristic p can be defined over the prime subfield ${\displaystyle F_{p}}$ rather than ${\displaystyle F_{p^{m}}}$ are called supersingular primes.

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