An effective Pila-Wilkie theorem for sets definable using Pfaffian functions, with some diophantine applications

TL;DR

This paper develops an effective version of the Pila-Wilkie theorem for sets definable with Pfaffian functions, enabling explicit bounds on algebraic points and applications to diophantine problems like Manin-Mumford and André-Oort conjectures.

Contribution

It introduces effective estimates for algebraic points on Pfaffian-definable sets and extends these results to important diophantine conjectures with uniform bounds.

Findings

Effective bounds for algebraic points of bounded height and degree.

Polynomial dependence of estimates on the set’s complexity.

Applications to Manin-Mumford and André-Oort conjectures.

Abstract

We prove an effective version of the Pila-Wilkie Theorem for sets definable using Pfaffian functions, providing effective estimates for the number of algebraic points of bounded height and degree lying on such sets. We also prove effective versions of extensions of this result due to Pila and Habegger-Pila . In order to prove these counting results, we obtain an effective version of Yomdin-Gromov parameterization for sets defined using restricted Pfaffian functions. Furthermore, for sets defined in the restricted setting, as well as for unrestricted sub-Pfaffian sets, our effective estimates depend polynomially on the degree (one measure of complexity) of the given set. The level of uniformity present in all the estimates allows us to obtain several diophantine applications. These include an effective and uniform version of the Manin-Mumford conjecture for products of elliptic curves with complex multiplication, and an effective, uniform version of a result due to Habegger which characterizes the set of special points lying on an algebraic variety contained in a fibre power of an elliptic surface. We also show that if Andr\'e-Oort for $Y(2)^g$ can be made effective, then Andr\'e-Oort for a family of elliptic curves over $Y(2)^g$ can be made effective.