Abelian variety

In mathematics, Abelian varieties in the context of algebraic geometry and number theory are investigated. Abelian varieties simultaneously have two mathematical structures: the structure of an algebraic variety ( that is, the elements of an Abelian variety are determined by polynomials ) and the structure of a group (that is, the elements of an Abelian variety can thus link together that as known from the addition of whole numbers arithmetic laws apply ). In addition, an Abelian variety still has certain topological conditions satisfy (completeness, coherence ). The term of the Abelian variety originated by appropriate generalization of the properties of elliptic curves.

Definition

An Abelian variety is a complete, coherent Gruppenvarietät.

Explaining the definition

In this definition, the term " variety " shows the property of abelian varieties, to consist of the solutions of polynomial systems of equations. These solutions are often referred to as dots. In the case of an abelian variety, which is an elliptical curve reason, this system of equations can be composed of only an equation approximately. The associated abelian variety then consists of all projective points and with the point, which is often symbolized by.

The part of " group " in the definition of abelian varieties refers to the fact that you can have two points of an Abelian variety always so mapped to a third point that computational rules apply as for the addition of integers: this operation is associative, there is a neutral element and to each element of an inverse element. In the definition of abelian varieties is not required that this group operation must be abelian ( commutative ). However, it can be shown that the group operation on an Abelian variety always - as the name suggests - is abelian.

The terms "complete" and " contiguous " refer to topological properties of the algebraic variety, which are an Abelian variety is based. The following sections specify the three components " Gruppenvarietät ", "complete" and " contiguous " to the definition of abelian varieties.

For the term ' Gruppenvarietät "

Be an arbitrary, not necessarily algebraically closed field. A Gruppenvarietät about is an algebraic variety over along with two regular figures and well over a defined element, and so on define a group structure with neutral element over an algebraic degree of consideration, algebraic variety. The regular mapping defines the group operation of Gruppenvarietät and the inversion. A Gruppenvarietät is therefore a Viertupel with the mentioned properties.

For the term "complete"

An algebraic variety is called complete if for all algebraic varieties the projection imaging is complete (with respect to the Zariski topology). This means that maps every closed subset of a closed subset of. For example, projective algebraic varieties are always complete; but a complete algebraic variety may not be necessary projective.

For the term " contiguous "

A topological space is called connected if it can not be represented as a union of two disjoint, non- empty, open subsets.

Properties

From the definition of abelian varieties is important, rather surprising properties can be derived:

• The group operation of an Abelian variety is always commutative ( Abelian ).
• The one Abelian variety underlying algebraic variety is projective, non-singular and irreducible.

Examples

The following mathematical structures are Abelian varieties:

• Elliptic Curves
• Jacobian varieties
• Albanese varieties