Zannier Some Problems of Unlikely Intersections in Arithmetic and Geometry

Course Book
ISBN: 978-1-4008-4271-1
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 176 Seiten

Reihe: Annals of Mathematics Studies

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).

Autoren/Hrsg.

Zannier, Umberto

Weitere Mitwirkende

Masser, David

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

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

Über Autor(innen)

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.