Speaker: Gal Binyamini.
Date: 10:30 28/08/2020
Title: Point counting for foliations over number fields and Diophantine applications
Abstract:
I will talk about two types of results concerning (compact pieces of) leafs of foliations defined over a number field:
counting the number of intersections between a leaf and an algebraic variety of complementary dimension;
and counting the number of algebraic points of a given degree and height in the leaf.
For most foliations that arise in Diophantine geometry, particularly those associated to the flat structure on a
principal G-bundle for some algebraic group G, our results depend polynomially on the degrees and log-heights involved.
We thus sharpen the bounds of the Pila-Wilkie counting theorem in these cases. I will outline how this gives
effective bounds, as well as polynomial time algorithms, for various problems in Diophantine geometry.
Specifically, time permitting I will discuss implications for the Andre-Oort conjecture, unlikely intersections in abelian
schemes, and Pell's equation over polynomial rings.