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  "body": "\n## Summary\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over a [[number field|MWBDFS2KJ5]] $K$.\nIts Tate–Shafarevich group $\\text{Ш}(E/K)$ measures the failure of the [[Hasse principle|EMAR4X3W8D]] for twisted forms of $E$.\nIts elements are classes of genus one curves that become isomorphic to $E$ over an algebraic closure.\nThese curves have points over every completion of $K$, and the trivial class is that of curves with a point over $K$.\nIt is conjectured to be finite, and its order enters the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\n\n## Explanations\n\n### L1 human\n\nTo find rational points on a curve, one first checks whether the curve has points over the real numbers and over every $p$-adic field.\nFor some curves of genus one, all these local tests succeed, and yet there is no rational point.\nThe Tate–Shafarevich group collects exactly these deceptive curves, attached to a given elliptic curve.\nA classical example is Selmer's curve $3x^3 + 4y^3 + 5z^3 = 0$: it has points over $\\R$ and every $\\Q_p$, but no rational point other than $(0,0,0)$.\nWhen the group is trivial, local information is enough to decide such questions for that elliptic curve.\n",
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