Cover

Title

Copyright

Contents

Preface

Notation and Conventions

Introduction: An Overview of Some Problems of Unlikely Intersections

1 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture

1.1 Torsion points on subvarieties of G

1.2 Higher multiplicative rank

1.3 Remarks on Theorem 1.3 and its developments

1.3.1 Fields other than Q

1.3.2 Weakened assumptions

1.3.3 Unlikely intersections of positive dimension and height bounds

1.3.4 Unlikely intersections of positive dimension and Zilber’s conjecture

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

1.3.6 Zhang’s notion of dependence

1.3.7 Abelian varieties (and other algebraic groups)

1.3.8 Uniformity of bounds

Notes to Chapter 1

Sparseness of multiplicatively dependent points

Other unlikely intersections

A generalization of Theorem 1.3

An application of the methods to zeros of linear recurrences

Comments on the Methods

2 An Arithmetical Analogue

2.1 Some unlikely intersections in number fields

2.2 Some applications of Theorem 2.1

2.3 An analogue of Theorem 2.1 for function fields

2.4 Some applications of Theorem 2.2

2.5 A proof of Theorem 2.2

Notes to Chapter 2

Simplifying the proof of Theorem 1.3

Rational points on curves over F

Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory

3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser

3.1 A method for the Manin-Mumford conjecture

3.2 Masser’s questions on elliptic pencils

3.3 A finiteness proof

3.4 Related problems, conjectures, and developments

3.4.1 Pink’s and related conjectures

3.4.2 Extending Theorem 3.3 from Q to C

3.4.3 Effectivity

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

3.4.5 Simple abelian surfaces and Pell’s equations over function fields

3.4.6 Further extensions and analogues

3.4.7 Dynamical analogues

Notes to Chapter 3

Torsion values for a single point: other arguments

A variation on the Manin-Mumford conjecture

Comments on the Methods

4 About the André-Oort Conjecture

4.1 Generalities about the André-Oort Conjecture

4.2 Modular curves and complex multiplication

4.3 The theorem of André

4.3.1 An effective variation

4.4 Pila’s proof of André’s theorem

4.5 Shimura varieties

Notes to Chapter 4

Remarks on Edixhoven’s approach to André’s theorem

Some unlikely intersections beyond André-Oort

Definability and o-minimal structures

Appendix A Distribution of Rational Points on Subanalytic Surfaces

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

Appendix C Silverman’s Bounded Height Theorem for Elliptic Curves: A Direct Proof

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

Appendix E A Transcendence Measure for a Quotient of Periods

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

Appendix G Mixed Problems: Another Approach

Bibliography

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

P

R

S

T

U

V

W

Y

Z