Counting algebraic points on definable sets
Margaret Thomas (Purdue University)
MSRI: Simons Auditorium
The counting theorem of Pila and Wilkie (2006) has led to a lively interaction between o-minimality and diophantine geometry in recent years. Part of this interaction has focussed further on the problem of bounding the density of rational and algebraic points lying on transcendental sets, seeking to sharpen the Pila-Wilkie bound in certain cases. We shall survey some results in this area, focussing on a conjecture of Wilkie which concerns the real exponential field. Time permitting, we shall also look at some more recent results (by, amongst others, Besson, Boxall, Jones, Masser, and myself) which concern a variety of classical functions, including the Riemann zeta function, Weierstrass zeta functions and Euler's gamma function
Pila and Wilkie's influential counting theorem provides a bound on the density of rational points of bounded height lying on the 'transcendental parts' of sets definable in o-minimal expansions of the real field. This result has brought about a lively interaction in recent years between o-minimality and diophantine geometry, including several important applications to arithmetical conjectures which will be explored further in Ya'acov Peterzil's tutorial. As a prelude to this, we will provide an introduction to the Pila-Wilkie Theorem, indicating the main ingredients involved in the proof. In particular, we will focus on the key step known as the Pila-Wilkie Reparameterization Theorem. This is a model theoretic statement about the geometry of sets definable in o-minimal expansions of real closed fields - namely that they can be covered by the images of finitely many sufficiently differentiable functions with bounded derivatives. Following the Pila-Wilkie Theorem, subsequent work carried out by a number of authors, including Pila, Besson, Boxall, Butler, Jones, Masser and myself, has focussed on establishing that a sharper bound holds in certain situations. One important goal is a conjecture of Wilkie concerning sets definable in the real exponential field. We shall explore some of the cases of this conjecture already established and the methods involved, indicating how a suitable modification of the Pila-Wilkie notion of parameterization could play an important role in the pursuit of this conjecture.