Locally trivial torsors that are not Weil-Châtelet divisible (Q2854222)

The paper addresses the question of Cassels whether for a given abelian variety \(A\) defined over a number field \(k\), the elements of the Shafarevich-Tate group \(\text{Ш}(A)\) of \(A\) are infinitely divisible in the Weil-Châtelet group \(H^1(k,A(\overline{k}))\).NEWLINENEWLINEFor elliptic curves over number fields, one knows that the elements in the Shafarevich-Tate group are \(p\)-divisible for all primes \(p> 7\) (see [\textit{M. Çiperiani, J. Stix}, ``Weil-Châtelet divisible elements in Tate-Shafarevich groups. II: On a question of Cassels'', Preprint, \url{arXiv:1308.4528}]). Nevertheless, the author shows that the answer to Cassels' question is in general no. The main result of the paper shows that there exists an elliptic curve \(E/\mathbb Q\) such that \(\text{Ш}(E)\) is not contained in \(H^1(\mathbb Q,E(\overline{\mathbb Q}))\), thus showing that there are elements in the Tate-Shafarevich group of \(E\) which are not \(4\)-divisible.NEWLINENEWLINEFor general abelian varieties over \(\mathbb Q\), one knows that the elements of \(\text{Ш}(A)\) are infinitely \(p\)-divisible if \(p\) is sufficiently large (depending on \(A\)); this is a result due to \textit{M. I. Bashmakov} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 661--664 (1964; Zbl 0148.41501)]. On the opposite direction, the author shows that for any given prime number \(p\), there are infinitely many non-isomorphic abelian varieties \(A\) defined over \(\mathbb Q\) such that \(\text{Ш}(A)\) is not contained in \(pH^1(\mathbb Q,A(\overline{\mathbb Q}))\).