## 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.

: Primary 11G05,14K14.

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 Monday, October 29, 2012