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.