Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181) [NOOK Book]

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 ...

See more details below
Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

Available on NOOK devices and apps  
  • NOOK Devices
  • Samsung Galaxy Tab 4 NOOK
  • NOOK HD/HD+ Tablet
  • NOOK
  • NOOK Color
  • NOOK Tablet
  • Tablet/Phone
  • NOOK for Windows 8 Tablet
  • NOOK for iOS
  • NOOK for Android
  • NOOK Kids for iPad
  • PC/Mac
  • NOOK for Windows 8
  • NOOK for PC
  • NOOK for Mac
  • NOOK for Web

Want a NOOK? Explore Now

NOOK Book (eBook - Course Book)
$44.99
BN.com price
(Save 42%)$78.50 List Price

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).

Read More Show Less

Editorial Reviews

Bulletin of the AMS
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 - 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.
Mathematical Reviews Clippings - Yuri Bilu
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.
From the Publisher

"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

Read More Show Less

Product Details

  • ISBN-13: 9781400842711
  • Publisher: Princeton University Press
  • Publication date: 3/25/2012
  • Series: Annals of Mathematics Studies
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition description: Course Book
  • Pages: 176
  • File size: 5 MB

Meet 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."
Read More Show Less

Table of Contents

Preface ix

Notation and Conventions xi

Introduction: An Overview of Some Problems of Unlikely Intersections 1

1 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture 15

1.1 Torsion points on subvarieties of Gmn 16

1.2 Higher multiplicative rank 22

1.3 Remarks on Theorem 1.3 and its developments 29

1.3.1 Fields other than Q 29

1.3.2 Weakened assumptions 29

1.3.3 Unlikely intersections of positive dimension and height bounds 31

1.3.4 Unlikely intersections of positive dimension and Zilber's conjecture 33

1.3.5 Unlikely intersections and reducibility of lacunary polynomials (Schinzel's conjecture) 35

1.3.6 Zhang's notion of dependence 36

1.3.7 Abelian varieties (and other algebraic groups) 36

1.3.8 Uniformity of bounds 37

Notes to Chapter 1 39

Sparseness of multiplicatively dependent points 39

Other unlikely intersections 39

A generalization of Theorem 1.3 40

An application of the methods to zeros of linear recurrences 40

Comments on the Methods 41

2 An Arithmetical Analogue 43

2.1 Some unlikely intersections in number fields 43

2.2 Some applications of Theorem 2.1 48

2.3 An analogue of Theorem 2.1 for function fields 50

2.4 Some applications of Theorem 2.2 52

2.5 A proof of Theorem 2.2 54

Notes to Chapter 2 58

Simplifying the proof of Theorem 1.3 58

Rational points on curves over Fp 58

Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory 60

3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser 62

3.1 A method for the Manin-Mumford conjecture 62

3.2 Masser's questions on elliptic pencils 66

3.3 A finiteness proof 70

3.4 Related problems, conjectures, and developments 77

3.4.1 Pink's and related conjectures 77

3.4.2 Extending Theorem 3.3 from Q to C 80

3.4.3 Effectivity 83

3.4.4 Extending Theorem 3.3 to arbitrary pairs of points on families of elliptic curves 84

3.4.5 Simple abelian surfaces and Pell's equations over function fields 85

3.4.6 Further extensions and analogues 87

3.4.7 Dynamical analogues 89

Notes to Chapter 3 92

Torsion values for a single point: other arguments 92

A variation on the Manin-Mumford conjecture 93

Comments on the Methods 94

4 About the André-Oort Conjecture 96

4.1 Generalities about the André-Oort Conjecture 96

4.2 Modular curves and complex multiplication 99

4.3 The theorem of Andre 105

4.3.1 An effective variation 111

4.4 Pila's proof of Andre's theorem 112

4.5 Shimura varieties 118

Notes to Chapter 4 123

Remarks on Edixhoven's approach to André's theorem 123

Some unlikely intersections beyond André-Oort 124

Definability and o-minimal structures 125

Appendix A Distribution of Rational Points on Subanalytic Surfaces Umberto Zannier 128

Appendix B Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions David Masser 136

Appendix C Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof David Masser 138

Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence Approach David Masser 140

Appendix E A Transcendence Measure for a Quotient of Periods David Masser 143

Appendix F Counting Rational Points on Analytic Curves: A Transcendence Approach David Masser 145

Appendix G Mixed Problems: Another Approach David Masser 147

Bibliography 149

Index 159

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)