Explicit Arithmetic on Abelian Varieties

We describe linear algebra algorithms for doing arithmetic on an abelian variety which is dual to a given abelian variety. The ideas are inspired by Khuri-Makdisi's algorithms for Jacobians of curves. Let chi(0) be the Euler characteristic of the line bundle associated with an ample divisor H on an abelian variety A. The Hilbert scheme of effective divisors D such that O (D) has Hilbert polynomial (1 + t)(g) chi(0) is a projective bundle (with fibres P chi 0-1) over the dual abelian variety (A) over cap via the Abel-Jacobi map. This Hilbert scheme can be embedded in a Grassmannian, so that points on it (and hence, via the above-mentioned Abel-Jacobi map, points on (A) over cap can be represented by matrices. Arithmetic on (A) over cap can be worked out by using linear algebra algorithms on the representing matrices.