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Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

Book Description

This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010.

The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Notation and Conventions
  7. Half title
  8. Introduction: An Overview of Some Problems of Unlikely Intersections
  9. 1 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture
    1. 1.1 Torsion points on subvarieties of
    2. 1.2 Higher multiplicative rank
    3. 1.3 Remarks on Theorem 1.3 and its developments
      1. 1.3.1 Fields other than
      2. 1.3.2 Weakened assumptions
      3. 1.3.3 Unlikely intersections of positive dimension and height bounds
      4. 1.3.4 Unlikely intersections of positive dimension and Zilber’s conjecture
      5. 1.3.5 Unlikely intersections and reducibility of lacunary polynomials (Schinzel’s conjecture)
      6. 1.3.6 Zhang’s notion of dependence
      7. 1.3.7 Abelian varieties (and other algebraic groups)
      8. 1.3.8 Uniformity of bounds
    4. Notes to Chapter 1
      1. Sparseness of multiplicatively dependent points
      2. Other unlikely intersections
      3. A generalization of Theorem 1.3
      4. An application of the methods to zeros of linear recurrences
      5. Comments on the Methods
  10. 2 An Arithmetical Analogue
    1. 2.1 Some unlikely intersections in number fields
    2. 2.2 Some applications of Theorem 2.1
    3. 2.3 An analogue of Theorem 2.1 for function fields
    4. 2.4 Some applications of Theorem 2.2
    5. 2.5 A proof of Theorem 2.2
    6. Notes to Chapter 2
      1. Simplifying the proof of Theorem 1.3
      2. Rational points on curves over Fb
      3. Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory
  11. 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser
    1. 3.1 A method for the Manin-Mumford conjecture
    2. 3.2 Masser’s questions on elliptic pencils
    3. 3.3 A finiteness proof
    4. 3.4 Related problems, conjectures, and developments
      1. 3.4.1 Pink’s and related conjectures
      2. 3.4.2 Extending Theorem 3.3 from Q to C
      3. 3.4.3 Effectivity
      4. 3.4.4 Extending Theorem 3.3 to arbitrary pairs of points on families of elliptic curves
      5. 3.4.5 Simple abelian surfaces and Pell’s equations over function fields
      6. 3.4.6 Further extensions and analogues
      7. 3.4.7 Dynamical analogues
    5. Notes to Chapter 3
      1. Torsion values for a single point: other arguments
      2. A variation on the Manin-Mumford conjecture
      3. Comments on the Methods
  12. 4 About the André-Oort Conjecture
    1. 4.1 Generalities about the André-Oort Conjecture
    2. 4.2 Modular curves and complex multiplication
    3. 4.3 The theorem of André
      1. 4.3.1 An effective variation
    4. 4.4 Pila’s proof of André’s theorem
    5. 4.5 Shimura varieties
    6. Notes to Chapter 4
      1. Remarks on Edixhoven’s approach to André’s theorem
      2. Some unlikely intersections beyond André-Oort
      3. Definability and o-minimal structures
  13. Appendix A Distribution of Rational Points on Subanalytic Surfaces
  14. Appendix B Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions
  15. Appendix C Silverman’s Bounded Height Theorem for Elliptic Curves: A Direct Proof
  16. Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence Approach
  17. Appendix E A Transcendence Measure for a Quotient of Periods
  18. Appendix F Counting Rational Points on Analytic Curves: A Transcendence Approach
  19. Appendix G Mixed Problems: Another Approach
  20. Bibliography
  21. Index