91Ö±²¥

Title: "Torsion Points on Elliptic and Genus Two Curves"

Abstract:
An elliptic curve E is a cubic curve such as y^2 = x^3 + 17. The ancients discovered a way to "add" two points (x,y) and (x',y') on E to get a third, by drawing a line between them. Adding a point P to itself n times typically produces n distinct points. But occasionally it turns out that {nP : n in N} is a finite set; we call such P "torsion points". I'll show the beautiful classification of torsion points on elliptic curves over the rational numbers, due to Mazur and Ogg (1973). I'll explain a similarly geometric way to add (tuples of) points on higher degree curves as well. A difficult open question in number theory is to find the largest possible collection of torsion points on a curve of fixed degree. I'll end with some recent examples in higher degree, including the 96 different torsion points on the curve 
y^2 = x(x −120^2)(x - 143^2)(x - 266^2)(x - 218^2)(x - 241^2).