Ziyang Gao - Number of Points on Curves: a Conjecture of Mazur
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2019-07-06

With Philipp Habegger we recently proved a height inequality, using which one can bound the number of rational points on 1-parameter families of curves in terms of the genus, the degree of the number field and the Mordell-Weil rank (but no dependence on the Faltings height). This gives an affirmative answer to a conjecture of Mazur for pencils of curves. In this talk I will give a blueprint to generalize this method to an arbitrary family of curves. In particular I will focus on:
(1) how establishing a criterion for the Betti map to be immersive leads to the desired bound;
(2) how to apply mixed Ax-Schanuel to establish such a criterion.
This is work in progress, partly joint with Vesselin Dimitrov and Philipp Habegger.