The Hasse Principle For Certain Hyperelliptic Curves And Forms

This paper is about Hasse principle and Brauer-Manin obstruction for varieties over number fields.We study the Hasse principle for some hyperelliptic curves and forms,and construct counterexamples to Hasse principle explained by Brauer-Manin obstruction.By manipulating the parameters,we construct an infinite number of counterexamples to Hasse principle.Firstly,we recall the concept of Brauer groups of fields and group cohomology theory,which give the characterization of Brauer groups of global fields and local fields as a consequence in Class Field Theory.Then we recall properties of quaternion algebras,which are a class of certain elements in the Brauer groups of fields.Next,we recall how to generalize the concept of Brauer group to any scheme and obtain properties of Brauer groups of some special schemes with the help of étale cohomology theory.In particular,properties of Brauer groups of varieties over fields will be shown,and we recall how to give another characterization of Brauer groups of varieties over fields by unramified Brauer groups.Applying the relevant conclusions about Brauer groups of fields and schemes to number fields and varieties over number fields,we recall the definition of Hasse principle and Brauer-Manin obstruction for varieties over number fields and properties of Brauer-Manin obstruction,which shows an useful way of constructing counterexamples to Hasse principle.Finally,we put the above theory into practice and recall N.N.Dong Quan’s results about Hasse principle for hyperelliptic curves of genus g≥2 and forms of degree of n≡2(mod 4)in PQ2.He proves that,for each positive integer g≥2,there is an infinite arithmetic family of hyperelliptic curves of genus g violating the Hasse principle explained by the BrauerManin obstruction,and by using these families of curves and Q-morphisms,it can be shown that for any positive integer k≥1,there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4k+2 such that they are counterexamples to Hasse principle explained by Brauer-Manin obstruction.Using these methods and conclusions,we consturct some hyperelliptic curves and forms,which are counterexamples to Hasse principle explained by Braue-Manin obstruction.