Locally trivial torsors that are not Weil-Ch\^atelet divisible

TL;DR

The paper constructs explicit examples of locally trivial torsors under abelian varieties over Q that are not divisible by a prime p or by 4, providing counterexamples to a question by Cassels.

Contribution

It provides the first known infinite families of such torsors that are locally trivial but not divisible in the Weil-Châtelet group, answering a longstanding open question.

Findings

Infinite examples of torsors not divisible by p for any prime p

An example of a torsor not divisible by 4 under an elliptic curve

Negative answer to Cassels' question about divisibility of torsors

Abstract

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