Modul: MAT770 Oberseminar: Algebraische Geometrie

## Application of mixed Ax-Schanuel to bounding the number of rational points on curves

Dr. Ziyang Gao talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 17.12.18 **Time:** 13.15 - 14.45 **Room:** Y27H25

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). In this talk I will give a blueprint to generalize this method to arbitrary 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.