Some problems of unlikely intersections in arithmetic and geometry (Annals of mathematics studies number 181)

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...

Full description

Saved in:

Bibliographic Details
Main Author:	Zannier, Umberto
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2012
Edition:	1
Series:	Annals of Mathematics Studies
Subjects:	Abelian variety Algebraic curve Algebraic equation Algebraic geometry Algebraic group Algebraic number Algebraic varieties Algebraic variety Analytic continuation Analytic function Arithmetic Automorphism Big O notation Cardinality Codimension Coefficient Complex multiplication Complex number Complex torus Conjecture Contradiction Coprime integers Coset Determinant Dimension Diophantine equation Division by zero Divisor Elliptic curve Endomorphism Endomorphism ring Equation Estimation Existential quantification Exponential function Finite field Finite set Finitely generated group Geometry Geometry, Algebraic Group Theory Hypersurface Integer Intersection theory Irreducibility (mathematics) Lecture Limit point Linear map MATHEMATICS MATHEMATICS / Arithmetic MATHEMATICS / Geometry / General MATHEMATICS / Group Theory MATHEMATICS / Number Theory Mean value theorem Modular curve Moduli space Monomial Multiplicative group Natural number Number Theory Parameter PBG PBH Polynomial Prime factor Prime number Quantity Rational point Root of unity Schwarz lemma Semialgebraic set Shimura variety Siegel's lemma Sign (mathematics) Special case Subgroup Subset Subspace theorem Summation Theorem Union (set theory) Upper and lower bounds Variable (mathematics) Weierstrass function Zariski topology Polynomial Lecture Subset Complex torus Limit point Diophantine equation Subgroup Endomorphism ring Endomorphism Rational point Complex multiplication Shimura variety Zariski topology Finitely generated group Algebraic curve Contradiction Finite field Cardinality Algebraic group Estimation Determinant Analytic continuation Conjecture Existential quantification Division by zero Coset Mean value theorem Codimension Big O notation Special case Summation Prime number Natural number Sign (mathematics) Semialgebraic set Exponential function Upper and lower bounds Root of unity Subspace theorem Linear map Multiplicative group Moduli space Monomial Theorem Abelian variety Prime factor Hypersurface Automorphism Coefficient Equation Quantity Analytic function Elliptic curve Modular curve Dimension Complex number Integer Variable (mathematics) Union (set theory) Schwarz lemma Algebraic variety Coprime integers Finite set Weierstrass function Algebraic equation Parameter Divisor Siegel's lemma Algebraic number Irreducibility (mathematics)
ISBN:	9780691153704, 069115371X, 0691153701, 9780691153711, 1400842719, 9781400842711
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Description
Summary:	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 ofunlikelydimension. 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).
Bibliography:	Includes bibliographical references (p. [149]-158) and index
ISBN:	9780691153704 069115371X 0691153701 9780691153711 1400842719 9781400842711
DOI:	10.1515/9781400842711