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

by Umberto Zannier

Overview

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).

Product Details

ISBN-13:	9781400842711
Publisher:	Princeton University Press
Publication date:	03/25/2012
Series:	Annals of Mathematics Studies , #181
Sold by:	Barnes & Noble
Format:	eBook
Pages:	176
File size:	4 MB

About the Author

Umberto Zannier is professor of mathematics at the Scuola Normale Superiore di Pisa in Pisa, Italy. He is the author of Lecture Notes on Diophantine Analysis and the editor of Diophantine Geometry.

Table of Contents

• FrontMatter, pg. i
• Contents, pg. v
• Preface, pg. ix
• Notation and Conventions, pg. xi
• Introduction: An Overview of Some Problems of Unlikely Intersections, pg. 1
• Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture, pg. 15
• Chapter 2: An Arithmetical Analogue, pg. 43
• Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser, pg. 62
• Chapter 4: About the André-Oort Conjecture, pg. 96
• Appendix A: Distribution of Rational Points on Subanalytic Surfaces, pg. 128
• Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions, pg. 136
• Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof, pg. 138
• Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach, pg. 140
• Appendix E: A Transcendence Measure for a Quotient of Periods, pg. 143
• Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach, pg. 145
• Appendix G: Mixed Problems: Another Approach, pg. 147
• Bibliography, pg. 149
• Index, pg. 159

Editorial Reviews

"Zannier's book is well written and a pleasure to read. . . . [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject. . . . [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry."—-Joseph H. Silverman, Bulletin of the AMS

"This book is indeed a great source of knowledge and inspiration for everybody interested in the unlikely intersection problems. The author must be commended for doing this job, and doing it so well."—-Yuri Bilu, Mathematical Reviews Clippings

From the Publisher

This book is indeed a great source of knowledge and inspiration for everybody interested in the unlikely intersection problems. The author must be commended for doing this job, and doing it so well.

Mathematical Reviews Clippings - Yuri Bilu

Zannier's book is well written and a pleasure to read. . . . [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject. . . . [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry.

Bulletin of the AMS - Joseph H. Silverman

Zannier's book is well written and a pleasure to read. . . . [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject. . . . [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry.
— Joseph H. Silverman

Bulletin of the AMS