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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with algebraic closure $\\overline{K}$ and absolute [[Galois group|0XTY86BZKY]] $G_K = \\Gal(\\overline{K}/K)$, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $K$.\nFor each [[place|BT5V3TSNY6]] $v$ of $K$, let $K_v$ be the completion of $K$ at $v$.\nFix an embedding $\\overline{K} \\to \\overline{K_v}$; it identifies $G_{K_v}$ with a decomposition subgroup of $G_K$.\nLet $H^1(K, E) = H^1(G_K, E(\\overline{K}))$ be the first [[Galois cohomology|9MPCZ3EAJM]] group, and similarly $H^1(K_v, E) = H^1(G_{K_v}, E(\\overline{K_v}))$.\nLet the map below have as components the restrictions, with the product over all places $v$ of $K$.\nThe *Tate–Shafarevich group* of $E$ over $K$ is\n$$\\text{Ш}(E/K) = \\ker\\left( H^1(K, E) \\to \\prod_{v} H^1(K_v, E) \\right).$$\n\n## Remarks\n\nThe group $H^1(K, E)$ classifies principal homogeneous spaces of $E$: curves $C$ over $K$ with a simply transitive action of $E$ defined over $K$.\nA class is trivial exactly when $C$ has a point over $K$, so $\\text{Ш}(E/K)$ consists of the classes of such curves $C$ that have a point over every $K_v$.\n\nExample: Selmer's curve $C \\colon 3x^3 + 4y^3 + 5z^3 = 0$ in $\\mathbb{P}^2_{\\Q}$ has points over $\\R$ and over every $\\Q_p$, but no point over $\\Q$.\nIt is a principal homogeneous space for its Jacobian $E \\colon x^3 + y^3 + 60z^3 = 0$, so it gives a nonzero element of $\\text{Ш}(E/\\Q)$.\n\nNon-example: let $C$ be a principal homogeneous space of $E$ over $\\Q$ with $C(\\R) = \\emptyset$.\nIts class in $H^1(\\Q, E)$ has nonzero restriction at the real place, so it is not in $\\text{Ш}(E/\\Q)$.\n\nThe group $\\text{Ш}(E/K)$ is a torsion abelian group, and each of its $n$-torsion subgroups is finite, by the finiteness of the [[Selmer group|36YNSTESMQ]].\nIt is conjectured to be finite; when it is finite, the Cassels–Tate pairing shows that its order is a perfect square.\nThe letter Ш is the Cyrillic letter sha.\n",
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