Torsion Points on Families of Abelian Varieties
Points of finite order are sparse on a subvariety of an abelian variety, except for algebraic subgroups and their components. This is the Manin-Mumford Conjecture, proved by Raynaud in the 1980s, when sparse is interpreted as not Zariski dense. Later, Pink formulated a relative variant of the conjecture in a family of abelian varieties. Masser and Zannier ob- tained the first results for a curve in a family of abelian varieties which were later extended to surfaces together with Corvaja and Tsimerman. I will give a overview of the history of this question and speak about joint work with Ziyang Gao in higher dimensions. Our work relies on the Pila-Zannier counting strategy. We rely on definability results of Peterzil-Starchenko in o-minimal geometry which built upon earlier work of Wilkie, van den Dries, and Miller. Additionally, we use new height bounds from joint work with Dimitrov and Gao.