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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over the field $\\Q$ of [[rational numbers|0NCJ5WXN6Z]].\nThen the group $E(\\Q)$ of rational points of $E$, with the [[chord-and-tangent law|P7V1BQJPSS]], is a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].\n\n## Remarks\n\nBy the structure theorem for finitely generated abelian groups, $E(\\Q) \\cong \\Z^r \\oplus T$, where $T$ is the finite torsion subgroup and $r \\ge 0$ is the rank of $E$.\nThe proof has two steps.\nThe weak Mordell theorem says that $E(\\Q)/2E(\\Q)$ is finite; it is proved with a descent map into a finite group built from square classes of rationals.\nThen a height function $h$ on $E(\\Q)$, with finitely many points of bounded height and $h(2P) \\approx 4h(P)$, lets one show by descent that finitely many points together with $2E(\\Q)$ generate everything.\nExample: for $y^2 = x^3 - x$ one has $E(\\Q) = \\set{O, (0,0), (1,0), (-1,0)}$, of rank $0$.\nWeil extended the theorem to abelian varieties over number fields; the general result is the Mordell–Weil theorem.\n\n## Explanations\n\n### L1 human\n\nAn elliptic curve over the rationals can have infinitely many rational points.\nMordell's theorem says that they are never a shapeless infinite mess: finitely many of them generate all others by the chord-and-tangent construction.\n\n### L2 human\n\nThe theorem gives the two basic invariants of an elliptic curve over $\\Q$.\nThe torsion subgroup is described by [[Mazur's torsion theorem|RQS6YHAR8N]], and the rank is the subject of the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\nThe descent step uses [[height functions|F1FF505PWH]], and the obstruction to making descent effective is measured by the [[Selmer group|36YNSTESMQ]] and the [[Tate–Shafarevich group|BK9WBMF56G]].\nNo algorithm is known that provably computes the rank in all cases.\n",
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