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.