Some Problems of Unlikely Intersections in Arithmetic and Geometry

174 Pages·2012·3.72 MB·English

by  Umberto Zannier, David Masser

Annals of Mathematics Studies
Number 181
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Some Problems of Unlikely Intersections in
Arithmetic and Geometry
Umberto Zannier
with Appendixes by David Masser
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2012
LibroPrincetonPrepages January 11, 2012 7x10
Copyright c 2012 by Princeton University Press
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Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire
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Library of Congress Cataloging-in-Publication Data
Zannier, U. (Umberto), 1957-
Some problems of unlikely intersections in arithmetic and geometry / Umberto Zannier ; with
appendixes by David Masser.
p. cm. – (Annals of mathematics studies ; no. 181)
Includes bibliographical references and index.
ISBN978-0-691-15370-4(hardcover: acid-freepaper)–ISBN978-0-691-15371-1(pbk. : acid-free
paper) 1. Intersection theory. 2. Algebraic varieties. 3. Algebraic geometry. I. Masser, David
William, 1948- II. Title.
QA564.Z36 2012
516.35–dc23
￿
2011037619
British Library Cataloging-in-Publication Data is available
This book has been composed in LATEX.
Thepublisherwouldliketoacknowledgetheauthorofthisvolumeforprovidingthecamera-ready
copy from which this book was printed.
Printed on acid-free paper
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Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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 n . . . . . . . . . . . . . . . . . . . . . . . . . . 16
m
1.2 Higher multiplicative rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Remarks on Theorem 1.3 and its developments . . . . . . . . . . . . . . . . . . . . 29
1.3.1 Fields other than . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 poly- . . . . . . . .
nomials (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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
p
Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory . . . . . . . . 60
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vi CONTENTS
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 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 Andr´e-Oort Conjecture 96
4.1 Generalities about the Andr´e-Oort Conjecture. . . . . . . . . . . . . . . . . . . . . 96
4.2 Modular curves and complex multiplication . . . . . . . . . . . . . . . . . . . . . . 99
4.3 The theorem of Andr´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.1 An effective variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 Pila’s proof of Andr´e’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Notes to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Remarks on Edixhoven’s approach to Andr´e’s theorem . . . . . . . . . . . . . . . . 123
Some unlikely intersections beyond Andr´e-Oort . . . . . . . . . . . . . . . . . . . . 124
Definability and o-minimal structures . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix A Distribution of Rational Points on Subanalytic
Surfaces 128
by Umberto Zannier
Appendix B Uniformity in Unlikely Intersections: An Example for Lines in
Three Dimensions 136
by David Masser
Appendix C Silverman’s Bounded Height Theorem for Elliptic Curves:
A Direct Proof 138
by David Masser
Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence
Approach 140
by David Masser
Appendix E A Transcendence Measure for a Quotient of Periods 143
by David Masser
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CONTENTS vii
Appendix F Counting Rational Points on Analytic Curves:
A Transcendence Approach 145
by David Masser
Appendix G Mixed Problems: Another Approach 147
by David Masser
Bibliography 149
Index 159
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Preface
ThepresentmonographarosefromtheHermannWeylLectures,whichIhadthehonorandpleasure
to deliver during May 2010 at the Institute for Advanced Study in Princeton.
The series in question consisted of four lectures, entitled, respectively:
1. “An Overview of Some Problems of Unlikely Intersections”
2. “Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture”
3. “Unlikely Intersections in Elliptic Surfaces and Problems of Masser”
4. “About the Andr´e-Oort Conjecture”
The denomination “unlikely intersections” roughly speaking refers to varieties which we do
not expect to intersect, due to natural dimensional reasons: for instance, if X,Y are varieties of
dimensionsr,sinaspaceofdimensionn>r+s, weusuallyexpectX∩Y tobeempty. Moreover,
if this emptiness does not occur for whole families of varieties which arisefor independent reasons,
we should expect some (simple) structural reason behind this; discovering and proving such a
motivationfortheexistenceofthesaidunexpectedintersectionsisthebasicpatternoftheproblems
in this realm. We stress that the families considered here are not part of continuous or algebraic
ones, but are genuinely discrete ones, a fact which introduces certain arithmetical aspects into the
picture, seemingly not of the most common diophantine type.
Inthelectures,Ifocusedonsomeknownproblemsthatcanbeviewedinthisperspective,which
sometimes unifies them. I tried to offer an overview of some of the problems and especially of a
method that arose recently, without any attempt to be complete, but limited mainly to the issues
with which I am more familiar. Also, according to the general spirit of these lectures, I tried to
give some survey and sketch of the proofs, rather than fine details on technical points.
In these notes I have basically followed similar principles, and I have presented essentially the
same general topics and theorems as in the lectures, with a few exceptions. Naturally, I have
added a substantial amount of detail. Similarly to the lectures, these notes are addressed to a
“general” reader: they may be considered rather elementary, and known facts are often recalled
for convenience. Also, the subject shows rapid evolution, so probably several results shall soon be
superseded; nevertheless, hopefully this should not affect too much the said spirit of the notes.
The introduction, which roughly corresponds to the first of the lectures, gives an overview of
thetopicsandofthecontentofthesubsequentchapters,whichshouldcorrespondtotheremaining
threelectures;however,Chapter2presentsmaterialwhichinasenseismorespecificandcouldnot
be mentioned in the lectures. The chapters are concluded with notes, where other, more specific
questions are included. Moreover, the volume contains seven (short) appendixes (six of which are
by David Masser), which in particular illustrate some essentials of the proofs of certain auxiliary
tools needed for the main results.
Acknowledgments. It is a real pleasure to thank the Institute for Advanced Study for the
invitation to deliver the Hermann Weyl Lectures, and for the generous hospitality. In particular, I
x Preface
deeplyandheartilythankEnricoBombieriandPeterSarnak,alsofortheirveryimportantgeneral
help and advice and for rewarding encouragement. I am also much indebted to David Masser
for several illuminating discussions, revisions, and the precious appendixes. I further thank Yves
Andr´e, Matt Baker, Daniel Bertrand, Yuri Bilu, Paula Cohen-Tretkoff, Pietro Corvaja, Philipp
Habegger, Ben Hutz, Lars Ku¨hne, Aaron Levin, Vincenzo Mantova, Jonathan Pila, Francesco
Veneziano and Shou-Wu Zhang for very helpful clarifications and references. I further thank the
staffofthePrincetonUniversityPress,especiallyVickieKearnandBenHolmes,forverykindand
helpful assistance.