Rational points on analytic varieties

TL;DR

This paper provides an overview of Pila's work on rational points on analytic varieties, including key theorems, extensions, and applications to major conjectures like Manin-Mumford and André-Oort.

Contribution

It compiles and explains the Bombieri-Pila theorem, its extensions, and their applications to significant conjectures in number theory.

Findings

Overview of Bombieri-Pila theorem and its extensions

Applications to Manin-Mumford and André-Oort conjectures

Insights into rational points on analytic varieties

Abstract

These are expanded notes of the mini-courses on Pila's work that Yuri Bilu gave in Basel in April 2011, Yaroslavl in August 2011 and Chennai in February 2012. The topics covered include the Bombieri-Pila theorem, its extensions and applications to some prominent conjectures (Manin-Mumford, Andr\'e-Oort).