Unboundedness of Tate–Shafarevich groups in fixed cyclic extensions

Abstract

In this paper 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.