Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181) [NOOK Book]

Preface ix

Notation and Conventions xi

Introduction: An Overview of Some Problems of Unlikely Intersections 1

1 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture 15

1.1 Torsion points on subvarieties of G_mⁿ 16

1.2 Higher multiplicative rank 22

1.3 Remarks on Theorem 1.3 and its developments 29

1.3.1 Fields other than Q 29

1.3.2 Weakened assumptions 29

1.3.3 Unlikely intersections of positive dimension and height bounds 31

1.3.4 Unlikely intersections of positive dimension and Zilber's conjecture 33

1.3.5 Unlikely intersections and reducibility of lacunary polynomials (Schinzel's conjecture) 35

1.3.6 Zhang's notion of dependence 36

1.3.7 Abelian varieties (and other algebraic groups) 36

1.3.8 Uniformity of bounds 37

Notes to Chapter 1 39

Sparseness of multiplicatively dependent points 39

Other unlikely intersections 39

A generalization of Theorem 1.3 40

An application of the methods to zeros of linear recurrences 40

Comments on the Methods 41

2 An Arithmetical Analogue 43

2.1 Some unlikely intersections in number fields 43

2.2 Some applications of Theorem 2.1 48

2.3 An analogue of Theorem 2.1 for function fields 50

2.4 Some applications of Theorem 2.2 52

2.5 A proof of Theorem 2.2 54

Notes to Chapter 2 58

Simplifying the proof of Theorem 1.3 58

Rational points on curves over F_p 58

Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory 60

3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser 62

3.1 A method for the Manin-Mumford conjecture 62

3.2 Masser's questions on elliptic pencils 66

3.3 A finiteness proof 70

3.4 Related problems, conjectures, and developments 77

3.4.1 Pink's and related conjectures 77

3.4.2 Extending Theorem 3.3 from Q to C 80

3.4.3 Effectivity 83

3.4.4 Extending Theorem 3.3 to arbitrary pairs of points on families of elliptic curves 84

3.4.5 Simple abelian surfaces and Pell's equations over function fields 85

3.4.6 Further extensions and analogues 87

3.4.7 Dynamical analogues 89

Notes to Chapter 3 92

Torsion values for a single point: other arguments 92

A variation on the Manin-Mumford conjecture 93

Comments on the Methods 94

4 About the André-Oort Conjecture 96

4.1 Generalities about the André-Oort Conjecture 96

4.2 Modular curves and complex multiplication 99

4.3 The theorem of Andre 105

4.3.1 An effective variation 111

4.4 Pila's proof of Andre's theorem 112

4.5 Shimura varieties 118

Notes to Chapter 4 123

Remarks on Edixhoven's approach to André's theorem 123

Some unlikely intersections beyond André-Oort 124

Definability and o-minimal structures 125

Appendix A Distribution of Rational Points on Subanalytic Surfaces Umberto Zannier 128

Appendix B Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions David Masser 136

Appendix C Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof David Masser 138

Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence Approach David Masser 140

Appendix E A Transcendence Measure for a Quotient of Periods David Masser 143

Appendix F Counting Rational Points on Analytic Curves: A Transcendence Approach David Masser 145

Appendix G Mixed Problems: Another Approach David Masser 147

Bibliography 149

Index 159