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 singularities.
Examples include:
Supersingular elliptic curve. Elliptic curves in non-zero characteristic with an unusually large ring of endomorphisms of rank 4.
Supersingular Abelian variety Sometimes defined to be an abelian variety isogenous to a product of supersingular elliptic curves, and sometimes defined to be an abelian variety of some rank g whose endomorphism ring has rank (2g)2.
A surface is called Shioda supersingular if the rank of its Néron–Severi group is equal to its second Betti number.
A surface is called Artin supersingular if its formal Brauer group has infinite height.
Referencesedit
de Jong (2014), Shioda’s conjecture
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December 18, 2023
supersingular, variety, mathematics, supersingular, variety, usually, smooth, projective, variety, nonzero, characteristic, such, that, slopes, newton, polygon, crystalline, cohomology, jong, 2014, special, classes, varieties, such, elliptic, curves, common, v. 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 singularities Examples include Supersingular elliptic curve Elliptic curves in non zero characteristic with an unusually large ring of endomorphisms of rank 4 Supersingular Abelian variety Sometimes defined to be an abelian variety isogenous to a product of supersingular elliptic curves and sometimes defined to be an abelian variety of some rank g whose endomorphism ring has rank 2g 2 Supersingular K3 surface Certain K3 surfaces in non zero characteristic Supersingular Enriques surface Certain Enriques surfaces in characteristic 2 A surface is called Shioda supersingular if the rank of its Neron Severi group is equal to its second Betti number A surface is called Artin supersingular if its formal Brauer group has infinite height References editde Jong 2014 Shioda s conjecture nbsp This article includes a list of related items that share the same name or similar names If an internal link incorrectly led you here you may wish to change the link to point directly to the intended article Retrieved from https en wikipedia org w index php title Supersingular variety amp oldid 925022615, wikipedia, wiki, book, books, library,
