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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $A$ be an [[abelian variety|R7W03TXRS9]] over $K$.\nThen the group $A(K)$ of $K$-[[rational points|SWJYG87QQH]] of $A$ is a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].\n\n## Remarks\n\nThe proof has two parts.\nThe weak Mordell–Weil theorem says that $A(K)/mA(K)$ is finite for an integer $m \\ge 2$.\nIt is proved with Kummer theory and the finiteness of the class group and of the unit group modulo $m$-th powers.\nThen a [[height function|F1FF505PWH]] on $A(K)$ has finitely many points of bounded height and satisfies $h(mP) \\approx m^2 h(P)$.\nA descent with it shows that representatives of $A(K)/mA(K)$ and the points of small height generate $A(K)$.\nMordell proved the case of an elliptic curve over $\\Q$, which is [[Mordell's theorem|BZHGKN6Q3K]]; Weil proved the general case.\nSo $A(K) \\cong \\Z^r \\oplus T$ with $T$ finite; the integer $r$ is the rank of $A$ over $K$.\nThe hypothesis on $K$ matters.\nOver $\\overline{\\Q}$ the group $E(\\overline{\\Q})$ of an [[elliptic curve|427N6S6QQW]] contains points of every finite order, so its torsion subgroup is infinite and it is not finitely generated.\nOver $\\R$ the group $E(\\R)$ is uncountable.\n\n## Explanations\n\n### L1 human\n\nAn abelian variety, such as an elliptic curve, has a group law on its points.\nThe theorem says that over a number field all rational points are built from finitely many of them with this group law.\nSo an infinite set of solutions to a system of equations has a finite description.\n\n### L2 human\n\nThe theorem gives the rank and the torsion subgroup as the basic invariants of $A(K)$.\nThe rank is hard to compute: the descent gives only an upper bound through the Selmer group, and the gap is measured by the Tate–Shafarevich group.\nThe [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]] predicts the rank from an $L$-function.\nLet $C$ be a curve of genus at least $2$ embedded in its Jacobian.\nThe theorem places $C(K)$ inside a finitely generated group, and [[Faltings's theorem|H0DB07WM20]] shows that $C(K)$ is finite.\n",
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