Some problems of unlikely intersections in arithmetic and geometry (Annals of mathematics studies number 181)
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...
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| Hlavní autor: | |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Princeton
Princeton University Press
2012
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| Vydání: | 1 |
| Edice: | Annals of Mathematics Studies |
| Témata: | |
| ISBN: | 9780691153704, 069115371X, 0691153701, 9780691153711, 1400842719, 9781400842711 |
| On-line přístup: | Získat plný text |
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- Some problems of unlikely intersections in arithmetic and geometry (Annals of mathematics studies number 181) -- Contents -- Preface -- Notation and Conventions -- Introduction: An Overview of Some Problems of Unlikely Intersections -- Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture -- Chapter 2: An Arithmetical Analogue -- Chapter 3: Unlikely Intersections in Elliptic Surfaces and Problems of Masser -- Chapter 4: About the Andre-Oort Conjecture -- 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.
- Front matter Table of Contents Preface Notation and Conventions Introduction: Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture Chapter 2: An Arithmetical Analogue Chapter 3: Unlikely Intersections in Elliptic Surfaces and Problems of Masser Chapter 4: About the André-Oort Conjecture Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Bibliography Index
- 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
- Appendix E: A Transcendence Measure for a Quotient of Periods
- 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 F: Counting Rational Points on Analytic Curves: A Transcendence Approach
- Chapter 4: About the André-Oort Conjecture
- Index
- Notation and Conventions
- Introduction: An Overview of Some Problems of Unlikely Intersections
- -
- Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser
- /
- Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture
- Chapter 2: An Arithmetical Analogue
- Appendix G: Mixed Problems: Another Approach
- Contents
- Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach
- Umberto Zannier --
- Frontmatter --
- Preface
- Appendix A: Distribution of Rational Points on Subanalytic Surfaces
- Bibliography
- David Masser --