Unboundedness of Tate-Shafarevich groups in fixed cyclic extensions (Q6564155)

This paper presents two unboundedness results for the Tate-Shfarevich groups of abelian varieties in a fixed non-trivial cyclic extension of global fields.\N\NTheorem 1: Let \(K\) be a number field, and let \(L/K\) be a non-trivial cyclic extension of degree \(n\). Then for any integer \(m\), there exists an elliptic curve \(E/K\) such that \(\operatorname{rk}_N(\Sha(E/L))>m\).\N\NTheorem 2: Let \(K\) be a global field with char \(K\neq 2\), and let \(L/K\) be a fixed unramified cyclic extension of \(2\)-power degree. Let \(A\) be a principally polarized abelian variety over \(K\). Then for any integer \(m\), there exists a quadratic extension \(F/K\) such that \(\dim_{F_2}\Sha(A^F/L)[2]-\dim_{F_2}\Sha(A^F/K)[2]>m\).\N\NThe proofs rest on their earlier paper ``The growth of Tate-Shafarevich groups in cyclic extensions'' [Compos. Math. 158, No. 10, 2014--2032 (2022; Zbl 1516.11064)].