For an abelian variety $A$ over a number field $K$ with absolute Galois group $G_K$, the **Tate-Shafarevich** group Ш$(A)$ consists of elements of the Weil-Châtelet group $H^1(G_K,A))$ that are trivial in every completion $K_v$ of $K$; here $v$ denotes a place of $K$. In other words
$$
Ш(A) = \ker\Bigl(H^1(G_K,A) \to \prod_v H^1(G_{K_v},A_{K_v})\Bigr).
$$
It is a torsion abelian group that is conjectured to be finite.

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- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-13 18:20:20

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- 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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