A method of 2-descent on a genus 3 hyperelliptic curve

Abstract: 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.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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