Point-Counting and the Zilber-Pink Conjecture

Jonathan Pila (University of Oxford)

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English

Cambridge University Press
09 June 2022

Mathematical logic; Number theory; Algebraic geometry

Series: Cambridge Tracts in Mathematics

Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the Andre-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research in this area.

By: Jonathan Pila (University of Oxford)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Edition: New edition
Dimensions: Height: 235mm, Width: 157mm, Spine: 22mm
Weight: 543g
ISBN: 9781009170321
ISBN 10: 1009170325
Series: Cambridge Tracts in Mathematics
Pages: 268
Publication Date: 09 June 2022
Audience: General/trade , ELT Advanced
Format: Hardback
Publisher's Status: Active

1. Introduction; Part I. Point-Counting and Diophantine Applications: 2. Point-counting; 3. Multiplicative Manin-Mumford; 4. Powers of the Modular Curve as Shimura Varieties; 5. Modular Andre-Oort; 6. Point-Counting and the Andre-Oort Conjecture; Part II. O-Minimality and Point-Counting: 7. Model theory and definable sets; 8. O-minimal structures; 9. Parameterization and point-counting; 10. Better bounds; 11. Point-counting and Galois orbit bounds; 12. Complex analysis in O-minimal structures; Part III. Ax-Schanuel Properties: 13. Schanuel's conjecture and Ax-Schanuel; 14. A formal setting; 15. Modular Ax-Schanuel; 16. Ax-Schanuel for Shimura varieties; 17. Quasi-periods of elliptic curves; Part IV. The Zilber-Pink Conjecture: 18. Sources; 19. Formulations; 20. Some results; 21. Curves in a power of the modular curve; 22. Conditional modular Zilber-Pink; 23. O-minimal uniformity; 24. Uniform Zilber-Pink; References; List of notation; Index.

Jonathan Pila is Reader in Mathematical Logic and Professor of Mathematics at the University of Oxford, and a Fellow of the Royal Society. He has held posts at Columbia University, McGill University, and the University of Bristol, as well as visiting positions at the Institute for Advanced Study, Princeton. His work has been recognized by a number of honours and he has been awarded a Clay Research Award, a London Mathematical Society Senior Whitehead Prize, and shared the Karp Prize of the Association for Symbolic Logic. This book is based on the Weyl Lectures delivered at the Institute for Advanced Study in Princeton in 2018.