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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]], and let $C$ be a smooth projective geometrically irreducible curve over $K$ of genus $g \\ge 2$.\nThen the set $C(K)$ of points of $C$ defined over $K$ is finite.\n\n## Remarks\n\nThe statement was conjectured by Mordell in 1922 and proved by Faltings in 1983; it was long called the Mordell conjecture.\nFaltings's proof goes through the Tate conjecture for abelian varieties over number fields.\nIt also proves the Shafarevich conjecture: there are finitely many abelian varieties of given dimension with good reduction outside a given finite set of primes, up to isomorphism.\nA construction of Parshin passes from curves to abelian varieties.\nVojta later gave a different proof by Diophantine approximation, simplified by Bombieri.\nThe genus hypothesis is necessary.\nA curve of genus $0$ with one rational point is isomorphic to $\\mathbb{P}^1$ and has infinitely many rational points.\nAn elliptic curve, of genus $1$, has infinitely many rational points when the rank in [[Mordell's theorem|BZHGKN6Q3K]] is positive.\nExample: for $n \\ge 4$, the Fermat curve $x^n + y^n = z^n$ has genus $(n-1)(n-2)/2 \\ge 3$, so it has only finitely many rational points.\nThe proofs are not effective: they bound the number of points but give no algorithm to find them all.\n\n## Explanations\n\n### L1 human\n\nA polynomial equation in two variables describes a curve, and the shape of the curve over the complex numbers has a number of holes, its genus.\nFaltings's theorem says that once the genus is at least two, the equation has only finitely many rational solutions.\nSo the shape of a curve controls how many rational points it can have.\n\n### L2 human\n\nA curve of genus $0$ has no rational point or infinitely many, and for genus $1$ with a point the points form a finitely generated group.\nFaltings's theorem completes this qualitative picture of rational points on curves.\nFor integral points on affine curves, the analogous finiteness is [[Siegel's theorem on integral points|4KTBRP27P6]], which already applies from genus $1$.\n",
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