Locally trivial torsors that are not Weil–Châtelet divisible

Brendan Creutz

Abstract

For every prime \(p\) we give infinitely many examples of torsors under abelian varieties over \(\mathbb{Q}\) that are locally trivial but not divisible by \(p\) in the Weil–Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over \(\mathbb{Q}\) which is not divisible by 4 in the Weil–Châtelet group. This gives a negative answer to a question of Cassels

Keywords: Shafarevich–Tate group, Weil–Châtelet group.

AMS Subject Classification: Primary 11G05,14K14.