Bounding the number of rational points on curves

Abstract: Mazur conjectured, after Faltings’s proof of the Mordell conjecture, that the number of rational points on a curve of genus g at least 2 defined over a number field of degree d is bounded in terms of g, d and the Mordell-Weil rank. In particular the height of the curve is not involved. In this talk I will explain how to prove this conjecture and some generalizations. I will focus on how functional transcendence and unlikely intersections are applied in the proof. If time permits, I will talk about how the dependence on d can be furthermore removed if we moreover assume the relative Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.

Mathematics

Audience: researchers in the topic

ICCM 2020