complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group...

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...

mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field k. A 1-dimensional abelian variety is an elliptic curve...

Abelian varieties 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 abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces...

This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves. 3rd century AD Diophantus of Alexandria studies...

Jacobian variety is an example of an abelian variety, a complete variety with a compatible abelian group structure on it (the name "abelian" is however...

conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension. However, Mumford...

abelian Abelianisation Abelian variety, a complex torus that can be embedded into projective space Abelian surface, a two-dimensional abelian variety...

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...

contained in the concept of abelian variety, or more precisely in the way an algebraic curve can be mapped into abelian varieties. Abelian integrals were later...

component of the identity in the Picard group of C, hence an abelian variety. The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete...

Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem...

an abelian variety A to another one B is a surjective morphism with finite kernel. Some theorems on abelian varieties require the idea of 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...

Pic0(S) non-reduced, and hence not an abelian variety. The quotient Pic(V)/Pic0(V) is a finitely-generated abelian group denoted NS(V), the Néron–Severi...

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 abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of...

In mathematics, the Mordell–Weil theorem states that for an abelian variety A {\displaystyle A} over a number field K {\displaystyle K} , the group A...

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...

On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces. Let...

algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism f : A → B of...

variety over a field is called an abelian variety. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian...

generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians...

conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety). On 6 April 1929, four Norwegian stamps were issued...

In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but...

variety of a curve. The Albanese variety is the abelian variety A {\displaystyle A} generated by a variety V {\displaystyle V} taking a given point of V...

ring of endomorphisms of rank 4. Supersingular Abelian variety Sometimes defined to be an abelian variety isogenous to a product of supersingular elliptic...

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...

because abelian groups are modules over the ring of integers. (In fact, this is the origin of the terminology, which was introduced for abelian groups...