SFU Number Theory and Algebraic Geometry Seminar: Emiel Haakma

The rational points of an abelian variety form a finitely generated group and computing the rank of this group is a hard and central problem in arithmetic geometry. One method, which follows the original proof of Mordell and Weil of the finiteness of this rank, is explicit finite descent. It approximates it using Selmer groups, which bounds the rank using local information. The Tate-Shafarevich group measures the failure of this bound to be sharp. It is one of the most mysterious objects in arithmetic geometry.

Tate-Shafarevich groups have been shown to grow arbitrarily large in certain families by comparing different but related Selmer groups. Results on this have been primarily for Jacobians of hyperelliptic and superelliptic curves, which have additional automorphisms.

We discuss generalizations of these methods to curves of genus 3, which has the important distinction that not all curves are hyperelliptic. This will give us computational access to various Selmer groups of abelian threefolds with minimal endomorphism ring and that are not hyperelliptic Jacobians, and potentially allow us to show that the 2-torsion of Tate-Shafarevich groups for them is unbounded.