{"product_id":"torsors-and-rational-points","title":"Torsors and Rational Points","description":"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.","brand":"Gardners","offers":[{"title":"Default Title","offer_id":57481611805045,"sku":"9780521802376","price":121.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0612\/7193\/3106\/files\/9780521802376.jpg?v=1786089891","url":"https:\/\/backstory.london\/products\/torsors-and-rational-points","provider":"Backstory","version":"1.0","type":"link"}