Point counting for foliations in Diophantine geometry

Speaker: 

Gal Binyamini

Affiliation: 

Weizmann Institute of Science

Date: 

Tue, 27/10/2020 - 9:00pm

Venue: 

RC-4082, The Red Centre, UNSW

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.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

To attend the talks, registration is necessary. To register please visit our website

www.ntwebseminar.org/home

Registered users will receive an email before each talk with a link to the Zoom meeting.

Organisers:
Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)

School Seminar Series: