The set of rational points on an elliptic curve has the structure of a finitely generated abelian group, but many of the most basic questions about this group remain answered. For instance, Poincaré in 1901 implicitly asked whether there is a uniform upper bound on the number of generators required as one varies the elliptic curve, and this question is still open. Inspired by the Cohen-Lenstra heuristics for class groups, we will give a heuristic that suggests that such a bound exists. This is joint work with Jennifer Park, John Voight, and Melanie Matchett Wood.
(The four lectures will be mostly independent of each other.)
Also, more details on these topics are contained in the following survey articles written by Bjorn Poonen: