Table of Contents
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 Gmn 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 Fp 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