Written by an expert in the field, this book is intended for postgraduate students and researchers in number theory and model theory who want to become familiar with point-counting techniques and their application to the Andre-Oort and Zilber-Pink conjectures, together with their model-theoretic context and connections with transcendence theory.
Written by an expert in the field, this book is intended for postgraduate students and researchers in number theory and model theory who want to become familiar with point-counting techniques and their application to the Andre-Oort and Zilber-Pink conjectures, together with their model-theoretic context and connections with transcendence theory.
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.
Inhaltsangabe
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.
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.
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