Abstract:
It is a fundamental question in math to solve equations; for example to find rational solutions of a given system of polynomials. In modern language such questions become studying certain kind of points on algebraic varieties; in the example mentioned above we are studying rational points.
This question is already very deep for projective algebraic curves defined over Q. Depending on the genus of the curve, our focus is different. Genus 0 curve either has no rational points or is birational to the projective line. For genus 1 curves, we have the BSD conjecture for the infinite part and torsion part is made explicit by Mazur. For genus 2 curves, the problem has several grades: finiteness, bounds, uniform bounds, distribution of rational points, effectiveness. The finiteness part, known as the Mordell conjecture, was proved by Faltings. A second proof using Diophantine estimates was given by Vojta, and it led to several explicit bounds on the number of the rational points, by Bombieri, de Diego, David, Philippon, Rémond, etc.
In my talk, I will present the following results regarding the question mentioned at the beginning:
(1) a rather uniform bound on the number of rational points on curves of genus at least 2, known as Mazur’s Conjecture B; more generally rational points in subvarieties of abelian varieties (Uniform Mordell-Lang Conjecture).
(2) small point on abelian varieties over function fields of characteristic 0. (Geometric Bogomolov Conjecture).
(3) special points on mixed Shimura varieties (mixed Andre-Oort Conjecture).
(4) torsion points in families of abelian varieties (relative Manin-Mumford conjecture), and its consequence on the uniformity of the size of torsion packets.
(5) two common tools used in these work: functional transcendence and unlikely intersections.
Some of these works are in collaboration, with (in chronological order) Philipp Habegger, Serge Cantat, Junyi Xie, Vesselin Dimitrov, Tangli Ge, and Lars Kühne.