hold for some other classes of fields k. Products The product of an abelian variety A of dimension m, and an abelian variety B of dimension n, over the same field, is an abelian variety of dimension m + n. An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties. Polarisation and dual abelian variety Dual abelian variety 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 parametrised by a k-variety T is defined to be a line bundle L on A×T such that for all t in T, 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, parametrised 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. 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 schemes of dual abelian varieties are Cartier duals of each other. This generalises the Weil pairing for elliptic curves. Polarisations A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect