Bounding Mordell-Weil ranks using Visualisation
by
Nils Bruin
Simon Fraser University
Coauthors: Victor Flynn (University of Liverpool)

The rational points of an Abelian variety form a finitely generated abelian group. While there is no known algorithm to compute the free rank of this group effectively, on can compute a finite upper bound on the rank via Selmer-groups. The obstruction for such bounds to be sharp is contained in the Tate-Shafarevich of the Abelian variety.

An interesting fact is that Tate-Shafarevich groups of isogenous varieties are often different. This means that one can often compute different bounds on the Mordell-Weil rank by computing Selmer-groups of isogenous Abelian varieties.

If isogenies are not available, then one can form a product variety with an appropriate cofactor, such that the resulting variety does have a suitable isogeny, which kills part of the Tate-Shafarevich group. In Cremona-Mazur's terminology, such a set-up is said to visualise that part of the Tate-Shafarevich group.

In this talk, we will show how one can construct such visualising abelian varieties for arbitrary hyperelliptic Jacobians. As an example, we will show an Abelian surface with non-trivial 2-torsion in its Tate-Shafarevich group and an elliptic curve with non-trivial 4-torsion.

Date received: April 30, 2004