Point-Counting and the Zilber–Pink Conjecture

Explores the recent spectacular applications of point-counting in o-minimal structures to functional transcendence and diophantine geometry.

Jonathan Pila (Author)

9781009170321, Cambridge University Press

Hardback, published 9 June 2022

268 pages
23.5 x 15.7 x 2.2 cm, 0.543 kg

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.

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.

Subject Areas: Algebraic geometry [PBMW], Number theory [PBH], Mathematical logic [PBCD]