O-Minimality and Diophantine Geometry

G. O. Jones, A. J. Wilkie

• Publisher: Cambridge University Press
• Year: 2015
• Print ISBN: 9781107462496
• eBook ISBN: 9781316311080
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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.

Table of contents

• Cover
• Series information
• Title page
• Copyright information
• Table of contents
• Preface
• 1 The Manin-Mumford Conjecture, an elliptic Curve, its Torsion Points & their Galois Orbits
• 2 Rational points on definable sets
• 3 Functional transcendence via o-minimality
• 4 Introduction to abelian varieties and the Ax–Lindemann–Weierstrass theorem
• 5 The André-Oort conjecture via o-minimality
• 6 Lectures on elimination theory for semialgebraic andsubanalytic sets
• 7 Relative Manin-Mumford for abelian varieties
• 8 Improving the bound in the Pila-Wilkie theoremfor curves
• 9 Ax-Schanuel and o-minimality