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Some Problems of Unlikely Intersections in Arithmetic and Geometry
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Umberto Zannier
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In collaboration with:
David Masser
Language:
English
Published/Copyright:
2012
About this book
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 / Editor information
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.
Reviews
"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
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"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
Topics
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Frontmatter
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Contents
v -
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Preface
ix -
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Notation and Conventions
xi -
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Introduction: An Overview of Some Problems of Unlikely Intersections
1 -
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Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture
15 -
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Chapter 2: An Arithmetical Analogue
43 -
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Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser
62 -
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Chapter 4: About the André-Oort Conjecture
96 -
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Appendix A: Distribution of Rational Points on Subanalytic Surfaces
128 -
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Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions
136 -
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Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof
138 -
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Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach
140 -
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Appendix E: A Transcendence Measure for a Quotient of Periods
143 -
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Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach
145 -
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Appendix G: Mixed Problems: Another Approach
147 -
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Bibliography
149 -
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Index
159
Publishing information
Pages and Images/Illustrations in book
eBook published on:
March 25, 2012
eBook ISBN:
9781400842711
Edition:
Course Book
Pages and Images/Illustrations in book
eBook ISBN:
9781400842711
Keywords for this book
Theorem; Conjecture; Elliptic curve; Polynomial; Integer; Rational point; Contradiction; Upper and lower bounds; Variable (mathematics); Analytic function; Algebraic variety; Special case; Complex multiplication; Schwarz lemma; Abelian variety; Estimation; Algebraic number; Coefficient; Subset; Determinant; Subgroup; Coset; Parameter; Equation; Algebraic curve; Complex number; Existential quantification; Irreducibility (mathematics); Codimension; Hypersurface; Zariski topology; Modular curve; Siegel's lemma; Algebraic equation; Algebraic group; Sign (mathematics); Cardinality; Root of unity; Linear map; Weierstrass function; Subspace theorem; Union (set theory); Finitely generated group; Endomorphism; Finite field; Semialgebraic set; Divisor; Division by zero; Shimura variety; Finite set; Diophantine equation; Dimension; Summation; Monomial; Endomorphism ring; Big O notation; Exponential function; Complex torus; Moduli space; Prime number; Analytic continuation; Lecture; Coprime integers; Multiplicative group; Natural number; Automorphism; Quantity; Prime factor; Mean value theorem; Limit point
Audience(s) for this book
For an expert adult audience, including professional development and academic research
