Algebraic geometry

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. 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. (Wikipedia).

Video thumbnail

MegaFavNumbers: 1.5 Billion

#MegaFavNumbers What’s your Mega Favourite Number?

From playlist MegaFavNumbers

Video thumbnail

Valentijn Karemaker, Mass formulae for supersingular abelian varieties

VaNTAGe seminar, Jan 18, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Oort: https://link.springer.com/chapter/10.1007/978-3-0348-8303-0_13 Honda: https://doi.org/10.2969/jmsj/02010083 Tate: https://link.springer.com/article/10.1007/BF01404549 Tate: https

From playlist Curves and abelian varieties over finite fields

Video thumbnail

Factoring the GCF from a binomial, 4x^2 + 24x

👉Learn how to factor quadratics. A quadratic is an algebraic expression having two as the highest power of its variable(s). To factor an algebraic expression means to break it up into expressions that can be multiplied together to get the original expression. To factor a quadratic, all we

From playlist Factor Quadratic Expressions | GCF

Video thumbnail

There are a lot more numbers than I thought there were - MegaFavNumbers

A short video detailing my favorite number larger than 1 million! There are so many numbers out there it was hard to choose from, but I’m glad I could participate in the #MegaFavNumbers series

From playlist MegaFavNumbers

Video thumbnail

Summary for classifying polynomials

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials

Video thumbnail

Is it a monomial, binomial, trinomial, or polynomial

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials

Video thumbnail

Classifying a polynomial expression by subtraction

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials | Simplify First

Video thumbnail

Ben Howard: Supersingular points on som orthogonal and unitary Shimura varieties

To an orthogonal group of signature (n,2), or to a unitary group of any signature, one can attach a Shimura variety. The general problem is to describe the integral models of these Shimura varieties, and their reductions modulo various primes. I will give a conjectural description of the s

From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"

Video thumbnail

Learn how to classify a polynomial based on the degree

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials

Video thumbnail

Factoring a binomial by distributive property

👉Learn how to factor quadratics. A quadratic is an algebraic expression having two as the highest power of its variable(s). To factor an algebraic expression means to break it up into expressions that can be multiplied together to get the original expression. To factor a quadratic, all we

From playlist Factor Quadratic Expressions | GCF

Video thumbnail

Valentijn Karemaker - Mass formula for supersingular abelian threefolds - WAGON

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field k of characteristic p. We combine these results with computations of the automorphism groups to study Oort

From playlist WAGON

Video thumbnail

Tomoyoshi Ibukiyama: Survey on quaternion hermitian lattices and its application to supersingular...

CIRM HYBRID EVENT The theme of the survey is how arithmetic theory of quatenion hermitian lattices can be applied to the theory of supersingular abelian varieties. Here the following geometric objects will be explained by arithmetics. Principal polarizations of superspecial abelian varieti

From playlist Number Theory

Video thumbnail

Chole Martindale, Torsion point attacks on the SIDH key exchange protocol

VaNTAGe Seminar, November 8, 2022 License: CC-BY-NC-SA Links to papers mentioned in the video: Jao-De Feo-Plut (2011): https://eprint.iacr.org/2011/506.pdf Galbraith-Petit-Shani-Ti (2016): https://eprint.iacr.org/2016/859 Petit (2017): https://eprint.iacr.org/2017/571 dQKLMPPS (2020): h

From playlist New developments in isogeny-based cryptography

Video thumbnail

Classifying a polynomial

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials

Video thumbnail

Christelle Vincent, Exploring angle rank using the LMFDB

VaNTAGe Seminar, February 15, 2022 License: CC-NC-BY-SA Links to some of the papers mentioned in the talk: Dupuy, Kedlaya, Roe, Vincent: https://arxiv.org/abs/2003.05380 Dupuy, Kedlaya, Zureick-Brown: https://arxiv.org/abs/2112.02455 Zarhin 1979: https://link.springer.com/article/10.100

From playlist Curves and abelian varieties over finite fields

Related pages

Helmut Hasse | F-crystal | Elliptic curve | Supersingular K3 surface | Slope | Characteristic (algebra) | Projective variety | Endomorphism | Crystalline cohomology | Supersingular elliptic curve