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159 relations: Abelian group, Abelian variety, Absolute convergence, Absolute value, Alfred Menezes, Algebra, Algebraic closure, Algebraic curve, Algebraic geometry, Algebraic number field, Algorithm, Analytic continuation, Andrew Wiles, Annals of Mathematics, Arithmetic dynamics, Atkin–Lehner theory, Étale cohomology, Barry Mazur, Birch and Swinnerton-Dyer conjecture, Carl Ludwig Siegel, Carl Pomerance, Characteristic (algebra), Clay Mathematics Institute, Cohen structure theorem, Compact space, Complex multiplication, Complex number, Complex projective plane, Conductor of an abelian variety, Congruent number, Coprime integers, Counting points on elliptic curves, Cryptography, Cubic plane curve, Cusp (singularity), Degree of a field extension, Dirichlet L-function, Discrete logarithm, Discriminant, Doubling-oriented Doche–Icart–Kohel curve, Dual abelian variety, Dual EC DRBG, EdDSA, Edwards curve, Ellipse, Elliptic algebra, Elliptic Curve Digital Signature Algorithm, Elliptic curve primality, Elliptic function, Elliptic integral, ..., Elliptic surface, Elliptic-curve cryptography, Elliptic-curve Diffie–Hellman, Equivalence relation, Euclidean algorithm, Euler product, Factorization, Fermat's Last Theorem, Field (mathematics), Field extension, Finite field, Finite morphism, Finitely generated abelian group, Finitely generated group, Functional equation, Fundamental pair of periods, Galois module, Generalized Riemann hypothesis, Generating function, Genus (mathematics), Geometry, Gerhard Frey, Glossary of arithmetic and diophantine geometry, Graduate Texts in Mathematics, Group (mathematics), Group isomorphism, Hasse's theorem on elliptic curves, Hasse–Weil zeta function, Hesse configuration, Hessian form of an elliptic curve, Homogeneous coordinates, Homomorphism, Inflection point, Integer factorization, Inventiones Mathematicae, J-invariant, J-line, Jacobian curve, Jean-Pierre Serre, John Tate, Ken Ribet, Lenstra elliptic-curve factorization, List of computer algebra systems, Loïc Merel, Local zeta-function, Logarithm, Louis J. Mordell, Mathematics, Meromorphic function, Millennium Prize Problems, Modular arithmetic, Modular curve, Modular form, Modular lambda function, Montgomery curve, Mordell–Weil theorem, Nagell–Lutz theorem, Nicolas Bourbaki, Noam Elkies, Number theory, Order (group theory), Plane curve, Point at infinity, Power series, Prime number, Projective plane, Projective variety, Proof by infinite descent, Quadric, Quotient group, Rank of an abelian group, Rational function, Rational mapping, Rational point, Real number, Richard Crandall, Riemann hypothesis, Riemann surface, Riemann zeta function, Riemann–Hurwitz formula, Ring of integers, Sato–Tate conjecture, Schoof's algorithm, Scott Vanstone, Serge Lang, Singular point of a curve, Singular point of an algebraic variety, Singularity (mathematics), Splitting field, Springer Science+Business Media, Square-free integer, Square-free polynomial, Stella octangula number, Subgroup, Supersingular isogeny key exchange, Symmetry, Torsion conjecture, Torsion subgroup, Torus, Tripling-oriented Doche–Icart–Kohel curve, Tunnell's theorem, Twisted Edwards curve, Twisted Hessian curves, Twists of curves, Uniformization theorem, Up to, Upper half-plane, Weierstrass's elliptic functions, Weil conjectures. Expand index (109 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.
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Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.
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Absolute value
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
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Alfred Menezes
Alfred Menezes is co-author of several books on cryptography, including the Handbook of Applied Cryptography, and is a professor of mathematics at the University of Waterloo in Canada.
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Algebra
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
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Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
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Algebraic curve
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
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Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.
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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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Arithmetic dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory.
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Atkin–Lehner theory
In mathematics, Atkin–Lehner theory is part of the theory of modular forms, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level.
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Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.
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Barry Mazur
Barry Charles Mazur (born December 19, 1937) is an American mathematician and a Gerhard Gade University Professor at Harvard University.
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Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.
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Carl Ludwig Siegel
Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a German mathematician specialising in number theory and celestial mechanics.
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Carl Pomerance
Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Peterborough, New Hampshire, United States.
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Cohen structure theorem
In mathematics, the Cohen structure theorem, introduced by, describes the structure of complete Noetherian local rings.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules).
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space.
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Conductor of an abelian variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is.
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Congruent number
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Counting points on elliptic curves
An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.
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Cryptography
Cryptography or cryptology (from κρυπτός|translit.
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Cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation.
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Cusp (singularity)
In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward.
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Degree of a field extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.
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Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.
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Discrete logarithm
In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.
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Discriminant
In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.
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Doubling-oriented Doche–Icart–Kohel curve
In mathematics, the doubling-oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written.
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Dual abelian variety
In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.
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Dual EC DRBG
Dual_EC_DRBG (Dual Elliptic Curve Deterministic Random Bit Generator) is an algorithm that was presented as a cryptographically secure pseudorandom number generator (CSPRNG) using methods in elliptic curve cryptography.
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EdDSA
In public-key cryptography, Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based on Twisted Edwards curves.
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Edwards curve
x^2+y^2.
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Ellipse
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
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Elliptic algebra
In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2.
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Elliptic Curve Digital Signature Algorithm
In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.
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Elliptic curve primality
In mathematics elliptic curve primality testing techniques are among the quickest and most widely used methods in primality proving.
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Elliptic function
In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions.
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Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.
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Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1.
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Elliptic-curve cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.
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Elliptic-curve Diffie–Hellman
Elliptic-curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Euclidean algorithm
. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.
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Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
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Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field extension
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Finite morphism
In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi.
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Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.
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Finitely generated group
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.
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Functional equation
In mathematics, a functional equation is any equation in which the unknown represents a function.
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Fundamental pair of periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane.
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Galois module
In mathematics, a Galois module is a ''G''-module, with G being the Galois group of some extension of fields.
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Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics.
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Generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.
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Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Gerhard Frey
Gerhard Frey (born 1944) is a German mathematician, known for his work in number theory.
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Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Hasse's theorem on elliptic curves
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.
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Hasse–Weil zeta function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.
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Hesse configuration
In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by, is a configuration of 9 points and 12 lines with three points per line and four lines through each point.
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Hessian form of an elliptic curve
In geometry, the Hessian curve is a plane curve similar to folium of Descartes.
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Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.
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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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Inflection point
In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
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Integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.
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Inventiones Mathematicae
Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.
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J-invariant
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for defined on the upper half-plane of complex numbers.
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J-line
In the study of the arithmetic of elliptic curves, the j-line over any ring R is the coarse moduli scheme attached to the moduli problem Γ(1): with the j-invariant normalized a la Tate: j.
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Jacobian curve
In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation).
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Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
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John Tate
John Torrence Tate Jr. (born March 13, 1925) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.
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Ken Ribet
Kenneth Alan "Ken" Ribet (born June 28, 1948) is an American mathematician, currently a professor of mathematics at the University of California, Berkeley.
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Lenstra elliptic-curve factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.
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List of computer algebra systems
The following tables provide a comparison of computer algebra systems (CAS).
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Loïc Merel
Loïc Merel (born 13 August 1965) is a French mathematician.
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Local zeta-function
In number theory, the local zeta function Z(V,s) (sometimes called the congruent zeta function) is defined as where N_m is the number of points of V defined over the degree m extension \mathbf_ of \mathbf_q, and V is a non-singular n-dimensional projective algebraic variety over the field \mathbf_q with q elements.
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Logarithm
In mathematics, the logarithm is the inverse function to exponentiation.
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Louis J. Mordell
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.
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Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.
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Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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Modular curve
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).
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Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
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Modular lambda function
In mathematics, the elliptic modular lambda function λ(&tau) is a highly symmetric holomorphic function on the complex upper half-plane.
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Montgomery curve
In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987.
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Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.
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Nagell–Lutz theorem
In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.
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Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.
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Noam Elkies
Noam David Elkies (born August 25, 1966) is an American mathematician and professor of mathematics at Harvard University.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane.
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Point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn 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.
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Proof by infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle.
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Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.
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Rational function
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
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Rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties.
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Rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Richard Crandall
Richard E. Crandall (December 29, 1947 – December 20, 2012) was an American physicist and computer scientist who made contributions to computational number theory.
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Riemann hypothesis
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.
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Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
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Riemann–Hurwitz formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other.
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Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
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Sato–Tate conjecture
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If Np denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np.
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Schoof's algorithm
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.
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Scott Vanstone
Scott A. Vanstone was a mathematician and cryptographer in the University of Waterloo Faculty of Mathematics.
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Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.
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Singular point of a curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter.
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Singular point of an algebraic variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.
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Singularity (mathematics)
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.
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Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Square-free integer
In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.
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Square-free polynomial
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, a unique factorization domain) that does not have as a factor any square of a non-unit factor.
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Stella octangula number
In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Supersingular isogeny key exchange
Supersingular isogeny Diffie–Hellman key exchange (SIDH) is a post-quantum cryptographic algorithm used to establish a secret key between two parties over an otherwise insecure communications channel.
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Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.
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Torsion conjecture
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field.
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Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A).
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Tripling-oriented Doche–Icart–Kohel curve
The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve.
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Tunnell's theorem
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.
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Twisted Edwards curve
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008.
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Twisted Hessian curves
In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic.
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Twists of curves
In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny.
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Uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
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Up to
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
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Upper half-plane
In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.
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Weierstrass's elliptic functions
In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass.
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Weil conjectures
In mathematics, the Weil conjectures were some highly influential proposals by on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.
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References
[1] https://en.wikipedia.org/wiki/Elliptic_curve
