Mathematical concept
In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all "n" the slopes of the Newton polygon of the "n"th crystalline cohomology are all "n"/2 . For special classes of varieties such as elliptic curves it is common to use various ad hoc definitions of "supersingular", which are (usually) equivalent to the one given above.
The term "singular elliptic curve" (or "singular "j"-invariant") was at one times used to refer to complex elliptic curves whose ring of endomorphisms has rank 2, the maximum possible. Helmut Hasse discovered that, in finite characteristic, elliptic curves can have larger rings of endomorphisms of rank 4, and these were called "supersingular elliptic curves". Supersingular elliptic curves can also be characterized by the slopes of their crystalline cohomology, and the term "supersingular" was later extended to other varieties whose cohomology has similar properties. The terms "supersingular" or "singular" do not mean that the variety has singularities.
Examples include:
References.
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