A projective algebraic variety that is also an algebraic group
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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. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.
An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be holomorphically embedded into a complex projective space.
Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.
Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.
complex analysis and algebraic number theory, an abelianvariety is a projective algebraic variety that is also an algebraic group, i.e., has a group...
mathematics, the arithmetic of abelianvarieties is the study of the number theory of an abelianvariety, or a family of abelianvarieties. It goes back to the...
mathematics, a dual abelianvariety can be defined from an abelianvariety A, defined over a field k. A 1-dimensional abelianvariety is an elliptic curve...
Abelianvarieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural...
In algebraic geometry, a semistable abelianvariety is an abelianvariety defined over a global or local field, which is characterized by how it reduces...
This is a timeline of the theory of abelianvarieties in algebraic geometry, including elliptic curves. c. 1000 Al-Karaji writes on congruent numbers...
conjecture holds for sufficiently general abelianvarieties, for products of elliptic curves, and for simple abelianvarieties of prime dimension. However, Mumford...
Jacobian variety is an example of an abelianvariety, a complete variety with a compatible abelian group structure on it (the name "abelian" is however...
contained in the concept of abelianvariety, or more precisely in the way an algebraic curve can be mapped into abelianvarieties. Abelian integrals were later...
In mathematics, an abelianvariety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring...
Another class is formed by the abelianvarieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem...
component of the identity in the Picard group of C, hence an abelianvariety. The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete...
Pic0(S) non-reduced, and hence not an abelianvariety. The quotient Pic(V)/Pic0(V) is a finitely-generated abelian group denoted NS(V), the Néron–Severi...
On the other hand, an abelian scheme may not be projective. Examples of abelianvarieties are elliptic curves, Jacobian varieties and K3 surfaces. Let...
In mathematics, in Diophantine geometry, the conductor of an abelianvariety defined over a local or global field F is a measure of how "bad" the bad...
an abelianvariety A to another one B is a surjective morphism with finite kernel. Some theorems on abelianvarieties require the idea of abelian variety...
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements...
the concept of abelianvariety is the higher-dimensional generalization of the elliptic curve. The equations defining abelianvarieties are a topic of...
ring of endomorphisms of rank 4. Supersingular Abelianvariety Sometimes defined to be an abelianvariety isogenous to a product of supersingular elliptic...
If G and H are abelian (i.e., commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h...
generally there is a similar Weil pairing between points of order n of an abelianvariety and its dual. It was introduced by André Weil (1940) for Jacobians...
algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelianvarieties, then any morphism f : A → B of...
ways. Abelianvarieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly...
conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelianvariety). On 6 April 1929, four Norwegian stamps were issued...
solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose...
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