eBook

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

Umberto Zannier

2012

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 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).

• Publisher: Princeton University Press
• Subject: 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
• ISBN: 9780691153704, 069115371X, 0691153701, 9780691153711, 1400842719, 9781400842711