“Dual abelian variety”的意思、由来-开放百科全书

In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field 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

Then there is a variety A^v and a family of degree 0 line bundles P,{{clarify|Bundles on what?|date=July 2014}} 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 functors 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 to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing for elliptic curves.

History

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

This construction of Â extends to any field K of characteristic zero.^[1] 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.^[2]

Dual isogeny (elliptic curve case)

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

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

Poincaré line bundle

The product of an abelian variety and its dual has a canonical line bundle, called the Poincaré line bundle.^[3] The corresponding height for varieties defined over number fields is sometimes called the Poincaré height.

Notes