O-Minimality and Diophantine Geometry London Mathematical Society Lecture Note Series

This collection of articles, originating from a short course held at the University of Manchester, explores the ideas behind Pila's proof of the Andre?Oort conjecture for products of modular curves. The basic strategy has three main ingredients: the Pila?Wilkie theorem, bounds on Galois orbits, and functional transcendence results. All of these topics are covered in this volume, making it ideal for researchers wishing to keep up to date with the latest developments in the field. Original papers are combined with background articles in both the number theoretic and model theoretic aspects of the subject. These include Martin Orr's survey of abelian varieties, Christopher Daw's introduction to Shimura varieties, and Jacob Tsimerman's proof via o-minimality of Ax's theorem on the functional case of Schanuel's conjecture.

Preface A. J. Wilkie and G. O. Jones; 1. The Manin–Mumford conjecture, an elliptic curve, its torsion points and their Galois orbits P. Habegger; 2. Rational points on definable sets A. J. Wilkie; 3. Functional transcendence via o-minimality Jonathan Pila; 4. Introduction to abelian varieties and the Ax–Lindemann–Weierstrass theorem Martin Orr; 5. The André–Oort conjecture via o-minimality Christopher Daw; 6. Lectures on elimination theory for semialgebraic and subanalytic sets A. J. Wilkie; 7. Relative Manin–Mumford for abelian varieties D. Masser; 8. Improving the bound in the Pila–Wilkie theorem for curves G. O. Jones; 9. Ax–Schanuel and o-minimality Jacob Tsimerman.

A. J. Wilkie is the Fielden Professor of Pure Mathematics at the University of Manchester. He has twice been joint winner of the Association for Symbolic Logic's Karp Prize. He is a Fellow of the Royal Society, of the American Mathematical Society and a member of Academia Europaea. Wilkie recently served as President of the Association for Symbolic Logic from 2010 to 2013.
G. O. Jones is a Researcher in the School of Mathematics at the University of Manchester.