|
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 ''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 ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-crystal
In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter ''F'' stands for Frobenius, indicating that ''F''-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny". F-crystals and F-isocrystals over perfect fields Suppose that ''k'' is a perfect field, with ring of Witt vectors ''W'' and let ''K'' be the quotient field of ''W'', with Frobenius automorphism σ. Over the field ''k'', an ''F''-crystal is a free module ''M'' of finite rank over the ring ''W'' of Witt vectors of ''k'', together with a σ-linear injective endomorphism of ''M''. An ''F''-isocrystal is defined in the same way, except that ''M'' is a module for the quotient field ''K'' of ''W'' rather than ''W''. Dieudonné–Manin classification theorem The Dieudonné–Manin classification theorem was proved by and . It describes the structure of ''F''-isocrystals over an algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystalline Cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H''''n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . Crystalline cohomology is partly inspired by the ''p''-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety ''X'' in characteristic ''p'' is the de Rham cohomology of a smooth lift of ''X'' to characteristic 0, while de Rham cohomology of ''X'' is the crystalline cohomology reduced mod ''p'' (after taking into account higher ''Tor''s). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Life Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse, also written Haße (12 April 1868 – 1 June 1940, son of Friedrich Ernst Hasse and his wife Anna Von Reinhard) and his wife Margarethe Louise Adolphine Quentin (born 5 July 1872 in Milwaukee, daughter of retail toy merchant Adolph Quentin (b. May 1832, probably Berlin, Kingdom of Prussia) and Margarethe Wehr (b. about 1840, Prussia), then raised in Kassel). After serving in the Imperial German Navy in World War I, he studied at the University of Göttingen, and then at the University of Marburg under Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Elliptic Curve
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ''ordinary'' and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Abelian 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 ''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 ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular K3 Surface
In algebraic geometry, a supersingular K3 surface is a K3 surface over a field ''k'' of characteristic ''p'' > 0 such that the slopes of Frobenius on the crystalline cohomology ''H''2(''X'',''W''(''k'')) are all equal to 1. These have also been called Artin supersingular K3 surfaces. Supersingular K3 surfaces can be considered the most special and interesting of all K3 surfaces. Definitions and main results More generally, a smooth projective variety ''X'' over a field of characteristic ''p'' > 0 is called supersingular if all slopes of Frobenius on the crystalline cohomology ''H''a(''X'',''W''(''k'')) are equal to ''a''/2, for all ''a''. In particular, this gives the standard notion of a supersingular abelian variety. For a variety ''X'' over a finite field ''F''''q'', it is equivalent to say that the eigenvalues of Frobenius on the l-adic cohomology ''H''a(''X'',''Q''''l'') are equal to ''q''''a''/2 times roots of unity. It follows that any variety in positive characteristic who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Enriques Surface
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity ''q'' = 0 and the canonical line bundle ''K'' is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0. Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by as an answer to a question discussed by about whether a surface with ''q'' = ''p''''g'' = 0 is necessarily rational, though some of the Reye congruences introduced earlier by are also examples of Enriques surfaces. Enriques surfaces can also be defined over other fields. Over fields of characteristic other than 2, showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Projective Variety")
