Dual abelian variety
Encyclopedia
In mathematics
, a dual abelian variety can be defined from an abelian variety
A, defined over a field
K.
A×T such that
Then there is a variety Av and a family of degree 0 line bundles P, the Poincaré bundle, parametrized by Av such that a family L on T is associated a unique morphism f: T → Av so that L is isomorphic to the pullback of P along the morphism 1A×f: A×T → A×Av. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety.
In the language of representable functor
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 (Av, P).
This association is a duality in the sense that there is a natural isomorphism between the double dual Avv 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 fv: Bv → Av in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual
to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group scheme
s of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing
for elliptic curves.
s. In that case there is a general form of duality between the Albanese variety
of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil
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
of what in contemporary terminology is the Picard scheme.
For the case of the Jacobian variety
J of a compact Riemann surface
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
. For general abelian varieties, still over the complex numbers, A is in the same isogeny
class as its dual. An explicit isogeny can be constructed by use of an invertible sheaf
L on A (i.e. in this case a holomorphic line bundle), when the subgroup
of translations on L that take L into an isomorphic copy is itself finite. In that case, the quotient
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
The construction when K has characteristic p uses scheme theory. The definition of K(L) has to be in terms of a group scheme
that is a scheme-theoretic stabilizer
, and the quotient taken is now a quotient by a subgroup scheme.
of elliptic curve
s of degree
, the dual isogeny is an isogeny
of the same degree such that
Here
denotes the multiplication-by-
isogeny 
which has degree 
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
,
The dual isogeny is
Note that the relation
also implies the conjugate relation 
Indeed, let 
Then 
But 
is surjective, so we must have 
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 Av (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 onA×T such that
- for all
, the restriction of L to A×{t} is a degree 0 line bundle, - the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).
Then there is a variety Av and a family of degree 0 line bundles P, the Poincaré bundle, parametrized by Av such that a family L on T is associated a unique morphism f: T → Av so that L is isomorphic to the pullback of P along the morphism 1A×f: A×T → A×Av. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av 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 (Av, P).
This association is a duality in the sense that there is a natural isomorphism between the double dual Avv 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 fv: Bv → Av 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 numberComplex 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 isogenyof 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 isogenyof 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 compositionwhere
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 compositeand 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 isomorphismIsomorphism
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 isNote that the relation
also implies the conjugate relation Indeed, let Then But is surjective, so we must have