Supersingular variety

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 nth crystalline cohomology are all n/2 (de Jong 2014). 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 singularites.

Examples include:

References

This article is issued from Wikipedia - version of the 8/21/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.