In mathematics, Abelian varieties are examined in the context of algebraic geometry , complex analysis and number theory . Abelian varieties have two mathematical structures at the same time : the structure of an algebraic variety (i.e. the elements of an Abelian variety are determined by polynomials) and the structure of a group (i.e. the elements of an Abelian variety can be linked together in such a way that those of the addition of whole numbers, the usual laws of calculation apply). In addition, an Abelian variety must have certain topologicalFulfill conditions (completeness, context). Abelian varieties are special algebraic groups .
The concept of the Abelian variety emerged from an appropriate generalization of the properties of elliptic curves .
An Abelian variety is a complete, contiguous group variety .
Explanation of the definition
In this definition, the term “variety” indicates the property of Abelian varieties to consist of the solutions of polynomial systems of equations. These solutions are often referred to as points. In the case of an Abelian variety based on an elliptic curve, this system of equations can consist of only one equation, for example . The corresponding Abelian variety then consists of all projective points with as well as the point which is often symbolized by.
The component "group" in the definition of Abelian varieties refers to the fact that two points of an Abelian variety can always be mapped to a third point in such a way that arithmetic laws apply as for the addition of whole numbers: This connection is associative , there is a neutral element and an inverse element for each element . In the definition of Abelian varieties is not required that this group operation abelian is (commutative). However, it can be shown that the group operation on an Abelian variety is always - as the name suggests - Abelian.
The terms “complete” and “connected” refer to topological properties of the algebraic variety on which an Abelian variety is based. The following sections specify the three components "group variety", "complete" and "connected" of the definition of Abelian varieties.
On the term "group variety"
Let be an arbitrary, not necessarily algebraically closed field . A group variety over is an algebraic variety over together with two regular mappings and as well as an element defined over , so that and define a group structure with a neutral element on the algebraic variety considered over the algebraic closure of . The regular mapping defines the group operation of the group variety and the inversion. A group variety is therefore a quadruple with the properties mentioned.
About the term "complete"
An algebraic variety is called complete if the projection mapping is completed for all algebraic varieties (with regard to the Zariski topology ). That means: maps each closed subset of to a closed subset of . For example, projective algebraic varieties are always complete; a complete algebraic variety need not be projective.
About the term "connected"
From the definition of Abelian varieties, important, quite surprising properties can be derived:
- The group operation of an Abelian variety is always commutative (Abelian).
- The algebraic variety on which an Abelian variety is based is projective , non-singular, and irreducible .
The following mathematical structures are Abelian varieties:
- James S. Milne : Abelian Varieties . Course Notes, Version 2.00, 2008, Chapter I, Section 1, Page 8, middle.