Sequences of rational torsions on abelian varieties

Abstract

We address the question of how fast the available rational torsion on abelian varieties over ℚ increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results in §2) are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval [a g ,a g +b g ]] wherea g is quadratic ing andb g is linear ing. As well as providing an improvement from linear to quadratic, these results provide a wide selection of torsion orders for potential use by those involved in computer integration. We conclude by considering possible techniques for divisors of non-hyperelliptic curves, and for general abelian varieties.