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 Néron–Severi group is equal to its second Betti number.
A surface is called Artin supersingular if its formal Brauer group has infinite height.
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In mathematics, a supersingularvariety is (usually) a smooth projective variety in nonzero characteristic such that for all n the slopes of the Newton...
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large...
Supersingular isogeny Diffie–Hellman key exchange (SIDH or SIKE) is an insecure proposal for a post-quantum cryptographic algorithm to establish a secret...
In particular, this gives the standard notion of a supersingular abelian variety. For a variety X over a finite field Fq, it is equivalent to say that...
surface. List of algebraic surfaces Enriques–Kodaira classification Supersingularvariety Artin, Michael (1960), On Enriques surfaces, PhD thesis, Harvard...
Kummer variety of any non-supersingular abelian surface over Fp with p odd has these properties.) It is not known whether varieties with these properties...
the curve is ordinary) or one connected component (if the curve is supersingular). If we consider a family of elliptic curves, the p-torsion forms a...
quartic Modular curve Modular equation Modular function Modular group Supersingular primes Fermat curve Bézout's theorem Brill–Noether theory Genus (mathematics)...
cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p. The endomorphism...
reasonable properties. The classic reason (due to Serre) is that if X is a supersingular elliptic curve, then its endomorphism ring is a maximal order in a quaternion...
August 2004. ([1]) E.R. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems, in B. Pfitzmann (ed.) EUROCRYPT 2001,...
related to the representations of elliptic curves with ordinary (non-supersingular) reduction. More precisely, they are 2-dimensional representations that...
field of characteristic p > 0, there is a special class of K3 surfaces, supersingular K3 surfaces, with Picard number 22. The Picard lattice of a K3 surface...
what Monstrous Moonshine is in one sentence, it is the voice of God." Supersingular prime, the prime numbers that divide the order of the monster Bimonster...
Weddle surfaces, birational to Kummer surfaces Smooth quartic surfaces Supersingular K3 surfaces Reye congruences, the locus of lines that lie on at least...
cyclotomic field. Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant X {\displaystyle X}...
those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this...
Oslo 1970, Wolters-Noordhoff 1972 with Ke-Zheng Li: Moduli of supersingular abelian varieties, Springer 1998 as editor with Steenbrink and van der Geer:...
outer automorphism group are both trivial. Since 37 and 67 are not supersingular primes, the Lyons group cannot be a subquotient of the monster group...
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