The **Tate-Shafarevich group** of an abelian variety $A$ over a number field $K$ is
$$
Ш(A) = \ker\Bigl(H^1(G_K,A) \to \prod_v H^1(G_{K_v},A_{K_v})\Bigr),
$$
where $G_K$ is the absolute Galois group of $K$,
and $v$ ranges over all places of $K$, including the archimedean places.

It classifies locally solvable principal homogeneous spaces of $A$.

It is a torsion abelian group that is conjectured to be finite.

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**Knowl status:**

- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-24 17:49:49

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

- 2022-03-24 17:49:49 by Bjorn Poonen (Reviewed)
- 2020-10-13 18:20:20 by Andrew Sutherland (Reviewed)
- 2020-10-13 16:19:46 by Andrew Sutherland
- 2020-10-13 16:19:23 by Andrew Sutherland
- 2020-10-13 16:18:48 by Andrew Sutherland
- 2020-10-10 06:36:53 by Andrew Sutherland (Reviewed)
- 2020-10-09 18:12:03 by Andrew Sutherland (Reviewed)
- 2020-10-09 18:07:31 by Andrew Sutherland
- 2020-10-09 15:54:05 by Andrew Sutherland
- 2020-10-09 15:53:42 by Andrew Sutherland

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