Leseprobe
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Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)
Umberto Zannier
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Sachbuch / Naturwissenschaft
Beschreibung
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).
Kundenbewertungen
Endomorphism ring, Complex number, Monomial, Semialgebraic set, Parameter, Siegel's lemma, Root of unity, Integer, Diophantine equation, Union (set theory), Analytic function, Abelian variety, Coefficient, Rational point, Linear map, Schwarz lemma, Contradiction, Moduli space, Automorphism, Natural number, Hypersurface, Existential quantification, Subgroup, Lecture, Irreducibility (mathematics), Summation, Multiplicative group, Dimension, Equation, Analytic continuation, Coset, Coprime integers, Endomorphism, Prime number, Weierstrass function, Upper and lower bounds, Modular curve, Prime factor, Algebraic variety, Zariski topology, Finite field, Finite set, Polynomial, Complex multiplication, Determinant, Divisor, Complex torus, Finitely generated group, Quantity, Mean value theorem, Algebraic group, Elliptic curve, Subset, Subspace theorem, Conjecture, Variable (mathematics), Algebraic number, Sign (mathematics), Theorem, Algebraic equation, Algebraic curve, Cardinality, Division by zero, Exponential function, Big O notation, Special case, Limit point, Codimension, Shimura variety, Estimation