Point-Counting and the Zilber–Pink Conjecture

ISBN-13: 9781009170321 / Angielski / Twarda / 2022 / 268 str.

Jonathan Pila (University of Oxford)

Opis

Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André–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.

Spis treści

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 André–Oort; 6. Point-Counting and the André–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.