The Growth Of Tate-Shafarevich Groups In Cyclic Extensions

The Tate-Shafarevich groups of abelian varieties over global fields,which contain rich arithmetic information,are extremely important in number theory study.However,because of the complexity of these groups,people know still little about them.The well-known Tate-Shafarevich conjecture claims that the TateShafarevich groups are finite.This is one of the most important conjecture in number theory.Only very special cases of this conjecture are confirmed and the general case is still far form being solved.This thesis mainly studies the growth of Tate-Shafarevich groups in cyclic extensions.More concretely,we solve a problem raised by K.Cesnavicius concerning the unboundedness of Tate-Shafarevich groups of a fixed abelian variety in cyclic extensions,and prove the unboundedness of Tate-Shafarevich groups of elliptic curves in fixed cyclic extensions.In Chapter 1,we introduce the background of this thesis and state our main results.In Chapter 2,we recall some basic knowledge of number theory and arithmetic geometry needed in this thesis,including the definition and basic properties of abelian varieties,the reduction,the Selmer groups and the Tate-Shafarevich groups of abelian varieties over global fields,and the twists of commutative algebraic groups.In Chapter 3,we revise the machinery developed by Mazur-Rubin about cyclic twists of abelian varieties,which provides the key tool to prove our main results.In Chapter 4,based on the idea of Cesnavicius,we prove a quantitative relation between the growth of Tate-Shafarevich groups and the number of ramified primes in cyclic extensions,which is the basis of proving the unboundedness of Tate-Shafarevich groups.In Chapter 5,we firstly recall Class Field Theory,and then study the Tramified,∑-split Z/pkZ-extensions introduced by Mazur-Rubin,which will help us to overcome a technical difficulty in Chapter 6.In Chapter 6,based on the results of Chapter 3 to Chapter 5,we prove the unboundedness of Tate-Shafarevich groups of fixed abelian variety in cyclic extensions.In Chapter 7,we generalize the 2-descent method in odd prime degree cyclic extensions by Mazur-Rubin to odd degree cyclic extensions,and use this method to prove the unboundedness of Tate-Shafarevich groups of elliptic curves in any fixed cyclic extensions of number fields.Based on the method developed in Chapter 6,we prove the unboundedness of Tate-Shafarevich groups of quadratic family of principle polorized abelian varieties in fixed Z/2kZ-extensions.