O-Minimality and Diophantine Geometry

doi:10.1017/CBO9781316106839

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Van Hille, Siegfried 2021. On a family of mild functions. International Journal of Number Theory, Vol. 17, Issue. 06, p. 1379.

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Chiu, Kenneth Chung Tak 2025. Ax–Schanuel for variations of mixed Hodge structures. Mathematische Annalen, Vol. 391, Issue. 2, p. 1681.

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Edited by G. O. Jones, University of Manchester, A. J. Wilkie, University of Manchester

Publisher:: Cambridge University Press
Online publication date:: August 2015
Print publication year:: 2015
Online ISBN:: 9781316106839
DOI:: https://doi.org/10.1017/CBO9781316106839

Subjects:: Mathematics (general), Mathematics, Logic, Categories and Sets, Number Theory
Series:: London Mathematical Society Lecture Note Series (421)

57.99 (GBP)

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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.

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O-Minimality and Diophantine Geometry

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

Frontmatter
pp i-vi

Contents
pp vii-x

Preface
pp xi-xii

1 - The Manin-Mumford Conjecture, an elliptic Curve, its Torsion Points & their Galois Orbits
pp 1-40

2 - Rational points on definable sets
pp 41-65

3 - Functional transcendence via o-minimality
pp 66-99

4 - Introduction to abelian varieties and the Ax–Lindemann–Weierstrass theorem
pp 100-128

5 - The André-Oort conjecture via o-minimality
pp 129-158

6 - Lectures on elimination theory for semialgebraic and subanalytic sets
pp 159-192

7 - Relative Manin-Mumford for abelian varieties
pp 193-203

8 - Improving the bound in the Pila-Wilkie theorem for curves
pp 204-215

9 - Ax-Schanuel and o-minimality
pp 216-221

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O-Minimality and Diophantine Geometry

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