Some Problems of Unlikely Intersections in Arithmetic and Geometry (eBook)

Umberto Zannier (Autor)

eBook Download: EPUB

2012
176 Seiten
Princeton University Press (Verlag)
978-1-4008-4271-1 (ISBN)

Infos

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 Andre-Oort conjecture (outlining work by Pila).

Autor

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.

Produkt-Details

• Erscheint lt. Verlag: 25.3.2012
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Mitarbeit: Anhang von: David Masser
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber ► Natur / Technik ► Naturwissenschaft
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: abelian analogue • abelian variety • absolute value • algebraic closure • Algebraic Curve • algebraic equation • algebraic group • algebraic number • algebraic number field • algebraic variety • analytic continuation • analytic function • AndrЏort conjecture • arithmetic • arithmetical analogue • automorphism • Big O notation • boundedness • Calculation • Cardinality • Codimension • coefficient • complex multiplication • complex number • complex plane • Complex torus • conjecture • Connected component (graph theory) • Contradiction • Coprime integers • Coset • counterexample • Determinant • Dimension • Diophantine equation • Division by zero • Divisor • Elliptic Curve • elliptic surfaces • Endomorphism • endomorphism ring • Equation • Equidistribution theorem • estimation • existential quantification • exponential function • exponential polynomial • Extrapolation • finite field • Finitely generated group • Finite set • Galois group • holomorphic function • hypergeometric function • hypersurface • Integer • Irreducibility (mathematics) • irreducible polynomial • isogeny • Lecture • Limit point • Linear map • ManinЍumford conjecture • ManinЍumford issue • ManinЍumfordВaynaud • mean value theorem • minkowski's theorem • model Theory • modular curve • modular curves • moduli space • Monomial • multiplicative algebraic groups • Multiplicative group • Natural number • Number Fields • Parameter • Pell's equation • Pila • Plane curve • polynomial • prime factor • Prime number • projective space • projective variety • Quantity • rational function • Rational point • resultant • Riemann hypothesis • Root of unity • roots of unity • Schwarz lemma • Semialgebraic set • shimura varieties • Shimura variety • Siegel's lemma • Sign (mathematics) • singular invariants • S. Lang • Special case • SUBGROUP • Subset • Subspace theorem • Summation • Theorem • theory • Tori • torsion points • transcendence degree • uniformization theorem • Union (set theory) • unlikely intersections • Upper and lower bounds • Variable (mathematics) • Vojta's conjecture • Weierstrass function • Without loss of generality • Zariski topology
• ISBN-10: 1-4008-4271-9 / 1400842719
• ISBN-13: 978-1-4008-4271-1 / 9781400842711