BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Emiel Haakma (SFU) DTSTART:20231019T230000Z DTEND:20231020T000000Z DTSTAMP:20260422T054411Z UID:SFUQNTAG/98 DESCRIPTION:Title: A method of 2-descent on a genus 3 hyperelliptic curve\nby Emiel Haa kma (SFU) as part of SFU NT-AG seminar\n\n\nAbstract\nThe rational points of an abelian variety form a finitely generated group and computing the r ank of this group is a hard and central problem in arithmetic geometry. On e method\, which follows the original proof of Mordell and Weil of the fin iteness of this rank\, is explicit finite descent. It approximates it usin g Selmer groups\, which bounds the rank using local information. The Tate- Shafarevich group measures the failure of this bound to be sharp. It is on e of the most mysterious objects in arithmetic geometry.\n\nTate-Shafarevi ch 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.\n\nWe discuss generalizations of these meth ods to curves of genus 3\, which has the important distinction that not al l curves are hyperelliptic. This will give us computational access to vari ous Selmer groups of abelian threefolds with minimal endomorphism ring and that are not hyperelliptic Jacobians\, and potentially allow us to show t hat the 2-torsion of Tate-Shafarevich groups for them is unbounded.\n LOCATION:https://researchseminars.org/talk/SFUQNTAG/98/ END:VEVENT END:VCALENDAR