to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have finite automorphism groups. A principal polarisation is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is > 1. Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem. A polarisation induces a Rosati involution on the endomorphism ring of A. Polarisations over the complex numbers Over the complex numbers, a polarised abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H1 and H2 are called equivalent if there are positive integers n and m such that nH1=mH2. A choice of an equivalence class of Riemann forms on A is called a polarisation of A. A morphism of polarised abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A. Abelian scheme One can also define abelian varieties scheme-theoretically and relative to a base. This allows for a uniform treatment of phenomena such as reduction mod p of abelian varieties (see Arithmetic of abelian varieties), and parameter-families of abelian varieties. An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by S. For an abelian scheme A / S, the group of n-torsion points forms a finite flat group scheme. The union of the pn-torsion points, for all n, forms a p-divisible group. Deformations of abelian schemes are, according to the Serre–Tate theorem, governed by the deformation properties of the associated p-divisible groups. Example Let be such that has no repeated complex roots. Then the discriminant is nonzero. Let , so is an open subscheme of . Then is an abelian scheme over . It can be extended to