July 5, 2019
Europe/Paris timezone

Application of Functional Transcendence to Counting Rational Points on Curves

Jul 5, 2019, 2:00 PM
Centre de conférences Marilyn et James Simons (IHES)

Centre de conférences Marilyn et James Simons


Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette


Prof. Ziyang Gao (IMJ-PRG)


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.

Presentation materials