Theory Of Torsors And Its Applications On The Arithmetic Of Rational Points

In algebraic geometry,the theory of torsors is applied to studying the arithmetic of rational points of varieties over number fields,for example it can be used to describe descent method,to construct descent obstruction,to prove the existence of a solubility algorithm for some systems equations,and so on.The concept of torsors is similar to the concept of principal bundles in algebraic topology,and is also a generalization of Galois extension in abstract algebra.In some classical literature,torsors also called the principal homogeneous spaces.The main document for this article is the book "Torsors and Rational Points" by Skorobogatov,and we mainly introduce the theory of torsors,briefly introduce the theory of Manin obstruction,and finally provide some of its applications in finding an algorithm for rational points.Specifically,in the second chapter,we introduce the basic theory and research tools of torsors,examples of torsors,and relate the obstruction defined by torsors to the obstruction to existence of rational points over fields.The abelian torsors in the third chapter are an important and useful class of torsors,which we describe by explicit equations in order to be able to apply them to arithmetic.In the fourth chapte,we focus on methods used to study rational points,such as the local-to-global principle,weak approximation,Manin obstruction,descent theory,etc.In the fifth chapter,we combine the methods and theories of the first four chapters to study the algorithms for rational points of conic bundle surfaces and some intersection of two quadrics in Pk5.