In this talk we prove two unboundedness results about the Tate-Shafarevich groups of abelian varieties in a fixed nontrivial cyclic extension $L/K$ of global fields, firstly in the case that $K$ is a number field and the abelian varieties are elliptic curves, secondly in the case that $K$ is a global field, $[L : K]$ is a $2$-power and the abelian varieties are principally polarized. This is a joint work with Jianfeng Xie.