TY  - GEN
AU  - Skorobogatov, Alexei
TI  - Torsors and rational points
SN  - 9780521802376
U1  - 512.4 
PY  - 2001///
CY  - Cambridge
PB  - Cambridge University Press
KW  - Torsion theory--Algebra
KW  - Rational points--Geometry
KW  - Real and complex analysis
KW  - Number theory
KW  - X-torsors
N1  - Includes index and references
N2  - 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.

https://www.cambridge.org/core/books/torsors-and-rational-points/76C9B8890C39601665082CFA8258E20E#fndtn-information
ER  -