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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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