Some problems of unlikely intersections in arithmetic and geometry / Umberto Zannier ; with appendixes by David Masser.

Format:

Book

Author/Creator:

Zannier, U. (Umberto), 1957-

Contributor:

Masser, David William, 1948-

Series:

Annals of mathematics studies ; no. 181.

Annals of mathematics studies ; no. 181

Language:

English

Subjects (All):

Intersection theory (Mathematics).

Algebraic varieties.

Geometry, Algebraic.

Physical Description:

1 online resource (175 p.)

Edition:

Course Book

Place of Publication:

Princeton : Princeton University Press, 2012.

Language Note:

English

Summary:

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).

Contents:

Frontmatter

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 André-Oort Conjecture

Appendix A: Distribution of Rational Points on Subanalytic Surfaces / Zannier, Umberto

Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions / Masser, David

Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof / Masser, David

Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach / Masser, David

Appendix E: A Transcendence Measure for a Quotient of Periods / Masser, David

Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach / Masser, David

Appendix G: Mixed Problems: Another Approach / Masser, David

Bibliography

Index

Notes:

Description based upon print version of record.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

Includes bibliographical references and index.

ISBN:

9786613457059

9781283457057

1283457059

9781400842711

1400842719

OCLC:

775873005