Contents

pp v-vi

Preface

pp vii-ix

Conventions

pp x-x

Acknowledgments

pp xi-xii

1 - Introduction

pp 1-6

Part I - Point-Counting and Diophantine Applications

pp 7-8

2 - Point-Counting

pp 9-17

3 - Multiplicative Manin–Mumford

pp 18-25

4 - Powers of the Modular Curve as Shimura Varieties

pp 26-37

5 - Modular André–Oort

pp 38-43

6 - Point-Counting and the André–Oort Conjecture

pp 44-58

Part II - O-Minimality and Point-Counting

pp 59-60

7 - Model Theory and Definable Sets

pp 61-68

8 - O-Minimal Structures

pp 69-81

9 - Parameterization and Point-Counting

pp 82-93

10 - Better Bounds

pp 94-103

11 - Point-Counting and Galois Orbit Bounds

pp 104-106

12 - Complex Analysis in an O-Minimal Structure

pp 107-110

Part III - Ax–Schanuel Properties

pp 111-112

13 - Schanuel’s Conjecture and Ax–Schanuel

pp 113-118

14 - A Formal Setting

pp 119-125

15 - Modular Ax–Schanuel

pp 126-129

16 - Ax–Schanuel for Shimura Varieties

pp 130-139

17 - Quasi-Periods of Elliptic Curves

pp 140-144

Part IV - The Zilber–Pink Conjecture

pp 145-146

18 - Sources

pp 147-155

19 - Formulations

pp 156-163

20 - Some Results

pp 164-179

21 - Curves in a Power of the Modular Curve

pp 180-191