In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a dual abelian variety can be defined from an abelian variety

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and 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...

 A, defined over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K.

Definition

To an abelian variety A over a field k, one associates a dual abelian variety A^v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a k-variety T is defined to be a line bundle L on
A×T such that

1. for all , the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety A^v and a family of degree 0 line bundles P, the Poincaré bundle, parametrized by A^v such that a family L on T is associated a unique morphism f: T → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A×T → A×A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v given by tensor product of line bundles, which makes it into an abelian variety.

In the language of representable functor

Representable functor

In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...

 one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles on T and to each k-morphism f: T → T the mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (A^v, P).

This association is a duality in the sense that there is a natural isomorphism between the double dual A^vv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms f^v: B^v → A^v in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual

Dual group

In mathematics, the dual group may be:* The Pontryagin dual of a locally compact abelian group* The Langlands dual of a reductive algebraic group* The Deligne-Lusztig dual of a reductive group over a finite field....

 to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group scheme

Group scheme

In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...

s of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing

Weil pairing

In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing on the torsion subgroup of E...

 for elliptic curves.

History

The theory was first put into a good form when K was the field of complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s. In that case there is a general form of duality between the Albanese variety

Albanese variety

In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...

 of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil

André Weil

André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

 had given a general definition of Albanese variety. For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic⁰(A), the connected component

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 of what in contemporary terminology is the Picard scheme.

For the case of the Jacobian variety

Jacobian variety

In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

 J of a compact Riemann surface

Compact Riemann surface

In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...

 C, the choice of a principal polarization of J gives rise to an identification of J with its own Picard variety. This in a sense is just a consequence of Abel's theorem

Abel–Jacobi map

In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus....

. For general abelian varieties, still over the complex numbers, A is in the same isogeny

Isogeny

In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....

 class as its dual. An explicit isogeny can be constructed by use of an invertible sheaf

Invertible sheaf

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...

 L on A (i.e. in this case a holomorphic line bundle), when the subgroup

K(L)

of translations on L that take L into an isomorphic copy is itself finite. In that case, the quotient

A/K(L)

is isomorphic to the dual abelian variety Â.

This construction of  extends to any field K of characteristic zero. In terms of this definition, the Poincaré bundle, a universal line bundle can be defined on

A × Â.

The construction when K has characteristic p uses scheme theory. The definition of K(L) has to be in terms of a group scheme

Group scheme

In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...

 that is a scheme-theoretic stabilizer

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

, and the quotient taken is now a quotient by a subgroup scheme.

Dual isogeny (elliptic curve case)

Given an isogeny


of elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

s of degree , the dual isogeny is an isogeny


of the same degree such that


Here denotes the multiplication-by- isogeny which has degree

Construction of the dual isogeny

Often only the existence of a dual isogeny is needed, but it can be explicitly given as the composition


where is the group of divisors of degree 0. To do this, we need maps given by where is the neutral point of and given by

To see that , note that the original isogeny can be written as a composite


and that since is finite of degree , is multiplication by on

Alternatively, we can use the smaller Picard group  , a quotient of The map descends to an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

, The dual isogeny is


Note that the relation also implies the conjugate relation Indeed, let Then But is surjective, so we must have