Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that...
Abelian variety - Wikipedia
History of manifolds and varieties
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of m...
History of manifolds and varieties - Wikipedia
Abelian integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Abel, is an integral in the complex plane of the formwhere is an arbitrary rational function of the two variables ...
Elliptic curve
In mathematics, an elliptic curve (EC) is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety – that is, it has a ...
Elliptic curve - Wikipedia
Theta function
In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic fo...
Theta function - Wikipedia
Theta divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the asso...
Kummer variety
In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse.The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface.<...
Weil pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More...
Coble hypersurface
In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curveof genus 2 or 3 by Arthur Coble. There are two similar but different types of Coble...
Coble hypersurface - Wikipedia
Complex multiplication
In mathematics, complex multiplication 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 havin...
Complex multiplication - Wikipedia
Semistable abelian variety
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.For an abelian varie...
Torelli theorem
In mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is det...
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N ...
Prym variety
In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it wa...
Mock theta function
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta ...
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.
Suppose that the e...
Bogomolov conjecture
In mathematics, the Bogomolov conjecture, named for Fedor Bogomolov, is the following statement:Let C be an algebraic curve of genus g at least two defined over a number field K, let denote the algeb...
Albanese variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
The Albanese variety is the abelian variety generated by a variety V...
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity:for complex numbers x and y, with |x| < 1 and y ≠ 0.It was introduced by Jacobi (1829) in his work Fundamenta Nova Theori...
Riemann form
In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:(The hermitian form written here is linear in the first variable.)Riemann forms are important...
Potential good reduction
In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally, prime ideal, of an algebraic variety.
Good reduction refers to the reduced variety having...
Theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class, In terms of holomorphic line bundles L on a connected compact Riema...
Hesse pencil
In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation...
Jacobian variety
In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group...
Raynaud's isogeny theorem
In mathematics, Raynaud's isogeny theorem, proved by Raynaud (1985), relates the Faltings heights of two isogeneous elliptic curves.
Raynaud's isogeny theorem - Wikipedia
Picard group
In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product...