Explicit Arithmetic on Abelian Varieties

Abstract

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 ${\mathscr {O}}(D)$ has Hilbert polynomial $(1+t)^g\chi _{0}$ is a projective bundle (with fibres ${\mathbb {P}}^{\chi _{0}-1}$) over the dual abelian variety $\widehat{A}$ 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 $\widehat{A}$) can be represented by matrices. Arithmetic on $\widehat{A}$ can be worked out by using linear algebra algorithms on the representing matrices.