Margaret Thomas (Purdue University)

Effective Pila–Wilkie bounds for Pfaffian sets with some diophantine applications

Note location change: Roger Stevens LT23 (8.23)

Following critical insights of Pila and Zannier, there are by now many applications of model theory to diophantine geometry arising from the celebrated counting theorem of Pila and Wilkie and its variants. The original Pila–Wilkie Theorem bounds the number of rational points of bounded numerator and denominator lying on (the transcendental parts of) sets definable in o-minimal expansions of the real field. However, the proof of this theorem (and that of its variants) does not provide an effective bound, which limits the precision of its applications. I will discuss some joint work with Gal Binyamini, Gareth O. Jones and Harry Schmidt in which we obtained effective forms of the Pila–Wilkie Theorem and its variants for sets definable in various structures described by Pfaffian functions (including an effective Yomdin–Gromov parameterization result for sets defined using restricted Pfaffian functions), and then used these effective estimates to derive several effective diophantine applications, including an effective, uniform Manin–Mumford statement for products of elliptic curves with complex multiplication.