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Curves, Elliptic. --- Rational points (Geometry). --- Diophantine analysis.

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The classical descent on curves of genus one can be interpreted as providing conditions on the set of rational points of an algebraic variety X defined over a number field, viewed as a subset of its adelic points. This is the natural set-up of the Hasse principle and various approximation properties of rational points. The most famous among such conditions is the Manin obstruction exploiting the Brauer-Grothendieck group of X. It emerged recently that a non-abelian generalization of descent sometimes provides stronger conditions on rational points. An all-encompassing 'obstruction' is related to the X-torsors (families of principal homogenous spaces with base X) under algebraic groups. This book, first published in 2001, is a detailed exposition of the general theory of torsors with key examples, the relation of descent to the Manin obstruction, and applications of descent: to conic bundles, to bielliptic surfaces, and to homogenous spaces of algebraic groups.

Torsion theory (Algebra) --- Rational points (Geometry)

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Algebraic geometry --- Curves, Elliptic --- Diophantine analysis --- Courbes elliptiques --- Analyse diophantienne --- Rational points (Geometry) --- Curves, Elliptic. --- Diophantine analysis. --- Rational points (Geometry). --- 511 --- 511 Number theory --- Number theory --- Indeterminate analysis --- Forms, Quadratic --- Points, Rational (Geometry) --- Arithmetical algebraic geometry --- Elliptic curves --- Curves, Algebraic

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Ordered algebraic structures --- 512.75 --- Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Rational points (Geometry) --- Torsion theory (Algebra) --- Rational points (Geometry). --- Torsion theory (Algebra). --- 512.75 Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Commutative rings --- Ideals (Algebra) --- Modules (Algebra) --- Points, Rational (Geometry) --- Arithmetical algebraic geometry

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Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer-Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.

Homotopy groups --- Torsors. --- Étale homotopy. --- Rational points. --- Torsion theory (Algebra) --- Homotopy theory. --- Rational points (Geometry) --- Points, Rational (Geometry) --- Arithmetical algebraic geometry --- Deformations, Continuous --- Topology --- Commutative rings --- Ideals (Algebra) --- Modules (Algebra) --- Homotopy theory --- Homogeneous spaces --- Geometry, Algebraic --- Spaces, Homogeneous --- Lie groups