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Appell–Humbert theorem

In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety.It was proved for 2-dimensional tori by Appell and Humbert, and in general by Lefsch

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 grou

Albanese variety

In mathematics, the Albanese variety , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

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.

Fourier–Mukai transform

In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a ker

Mordell–Weil theorem

In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group

Grothendieck–Ogg–Shafarevich formula

In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data i

Torelli theorem

In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (co

Seesaw theorem

In algebraic geometry, the seesaw theorem, or seesaw principle, says roughly that a limit of trivial line bundles over complete varieties is a trivial line bundle. It was introduced by André Weil in a

Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space over characteristic 0 constructed

Shimura subgroup

In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X0(N) of level N, given by the kernel of the natural map to the Jacobian of X1(N). It is named after Goro S

Height function

A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica

Néron–Ogg–Shafarevich criterion

In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field

Bogomolov conjecture

In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometr

Complex multiplication

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions wi

Complex multiplication of abelian varieties

In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex m

Dual abelian variety

In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.

Mordell–Weil group

In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field , it is an arithmetic invariant of the Abelian variety. It is simply th

Torsion conjecture

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian vari

Tate's isogeny theorem

In mathematics, Tate's isogeny theorem, proved by Tate, states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations

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. It is connected to the

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 vari

Eisenstein ideal

In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihi

Theorem of the cube

In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence,

Serre–Tate theorem

In algebraic geometry, the Serre–Tate theorem says that an abelian scheme and its p-divisible group have the same infinitesimal deformation theory. This was first proved by Jean-Pierre Serre when the

Arithmetic of abelian varieties

In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what a

Zarhin trick

In mathematics, the Zarhin trick is a method for eliminating the polarization of abelian varieties A by observing that the abelian variety A4 × Â4 is principally polarized. The method was introduced b

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

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

Kuga fiber variety

In algebraic geometry, a Kuga fiber variety, introduced by Kuga, is a fiber space whose fibers are abelian varieties and whose base space is an arithmetic quotient of a Hermitian symmetric space.

Fay's trisecant identity

In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by Fay . Fay's identity holds for theta functions of Jacobians of curves, but not

Riemann form

In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:
* A lattice Λ in a complex vector space Cg.
* An alternating bilinear form α from Λ to the

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

Tate module

In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative

Timeline of abelian varieties

This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.

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.

Schottky problem

In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.

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

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 g

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

Abelian integral

In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form where is an arbitrary rational function of the two varia

Equations defining abelian varieties

In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian varie

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