Gal Binyamini: Point counting for foliations in Diophantine geometry (NTWS 046)
Channel: Number Theory Web Seminar
Date: 2023-08-06

Abstract: I will discuss "point counting" in two broad senses: counting the intersections between a trascendental variety and an algebraic one; and counting the number of algebraic points, as a function of degree and height, on a transcendental variety. After reviewing the fundamental results in this area - from the theory of o-minimal structures and the Pila-Wilkie theorem, I will restrict attention to the case that the transcendental variety is given in terms of a leaf of an algebraic foliation, and everything is defined over a number field. It turns out that in this case far stronger estimates can be obtained.
Applying the above to foliations associated to principal G-bundles on various moduli spaces, many classical application of the Pila-Wilkie theorem can be sharpened and effectivized. In particular I will discuss issues around effectivity and polynomial-time solvability for the Andre-Oort conjecture, unlikely intersections in abelian schemes, and some related directions.

Link to slides: https://drive.google.com/file/d/1pGk-HS4u_HQhuYN97trC9NTU8-nA30AE

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Tuesday, October 27, 2020 (3am PDT, 6am EDT, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)