A **principal homogeneous space** (or **torsor**) $X$ of an abelian variety $A$ is a variety equipped with an $A$-action that is both free and transitive.

If $X$ has a rational point $O$ then we can view $X$ as an abelian variety with $O$ as its identity element by defining a group operation on $X$ in terms of the $A$-action on $O$.

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**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2016-03-29 23:11:30

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**History:**(expand/hide all)

- 2022-03-24 17:55:59 by Bjorn Poonen
- 2022-03-24 17:18:40 by Bjorn Poonen
- 2016-03-29 23:11:30 by Andrew Sutherland (Reviewed)

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