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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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Princeton
Princeton University Press
2012
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| Vydání: | 1 |
| Edice: | Annals of Mathematics Studies |
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| ISBN: | 9780691153704, 069115371X, 0691153701, 9780691153711, 1400842719, 9781400842711 |
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| Abstract | 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 ofunlikelydimension. 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). |
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| AbstractList | 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 ofunlikelydimension. 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). 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). No detailed description available for "Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-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 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). |
| Author | Zannier, Umberto |
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| Copyright | 2012 Princeton University Press |
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| Keywords | Polynomial Lecture Subset Complex torus Limit point Diophantine equation Subgroup Endomorphism ring Endomorphism Rational point Complex multiplication Shimura variety Zariski topology Finitely generated group Algebraic curve Contradiction Finite field Cardinality Algebraic group Estimation Determinant Analytic continuation Conjecture Existential quantification Division by zero Coset Mean value theorem Codimension Big O notation Special case Summation Prime number Natural number Sign (mathematics) Semialgebraic set Exponential function Upper and lower bounds Root of unity Subspace theorem Linear map Multiplicative group Moduli space Monomial Theorem Abelian variety Prime factor Hypersurface Automorphism Coefficient Equation Quantity Analytic function Elliptic curve Modular curve Dimension Complex number Integer Variable (mathematics) Union (set theory) Schwarz lemma Algebraic variety Coprime integers Finite set Weierstrass function Algebraic equation Parameter Divisor Siegel's lemma Algebraic number Irreducibility (mathematics) |
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| Notes | Includes bibliographical references (p. [149]-158) and index |
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| Snippet | 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... No detailed description available for "Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)". |
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| SubjectTerms | Abelian variety Algebraic curve Algebraic equation Algebraic geometry Algebraic group Algebraic number Algebraic varieties Algebraic variety Analytic continuation Analytic function Arithmetic Automorphism Big O notation Cardinality Codimension Coefficient Complex multiplication Complex number Complex torus Conjecture Contradiction Coprime integers Coset Determinant Dimension Diophantine equation Division by zero Divisor Elliptic curve Endomorphism Endomorphism ring Equation Estimation Existential quantification Exponential function Finite field Finite set Finitely generated group Geometry Geometry, Algebraic Group Theory Hypersurface Integer Intersection theory Irreducibility (mathematics) Lecture Limit point Linear map MATHEMATICS MATHEMATICS / Arithmetic MATHEMATICS / Geometry / General MATHEMATICS / Group Theory MATHEMATICS / Number Theory Mean value theorem Modular curve Moduli space Monomial Multiplicative group Natural number Number Theory Parameter PBG PBH Polynomial Prime factor Prime number Quantity Rational point Root of unity Schwarz lemma Semialgebraic set Shimura variety Siegel's lemma Sign (mathematics) Special case Subgroup Subset Subspace theorem Summation Theorem Union (set theory) Upper and lower bounds Variable (mathematics) Weierstrass function Zariski topology |
| SubjectTermsDisplay | Arithmetic Geometry Group Theory Mathematics Number Theory PBG PBH |
| TableOfContents | 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 -- |
| Title | Some problems of unlikely intersections in arithmetic and geometry (Annals of mathematics studies number 181) |
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