Appears in collection : Journée Gretchen & Barry Mazur

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.