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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_K$.\nLet $C$ be a smooth projective geometrically irreducible curve over $K$ of genus $g \\ge 1$.\nLet $f$ be a nonconstant rational function on $C$ defined over $K$.\nThen the set $\\set{P \\in C(K) : P \\text{ is not a pole of } f \\text{ and } f(P) \\in \\mathcal{O}_K}$ is finite.\n\n## Remarks\n\nEquivalently, an affine curve of genus at least $1$ over $K$, embedded in affine space, has only finitely many points with coordinates in $\\mathcal{O}_K$.\nFor genus $0$ the same conclusion holds when $f$ has at least three distinct poles over $\\overline{K}$.\nThe hypothesis $g \\ge 1$ is necessary: the Pell equation $x^2 - 2y^2 = 1$ defines a curve of genus $0$ with infinitely many integral points, here $f = x$ has only two poles.\nExample: for each integer $k \\ne 0$, the Mordell equation $y^2 = x^3 + k$ defines an elliptic curve, so it has only finitely many solutions with $x, y \\in \\Z$.\nThe proof combines the weak Mordell–Weil theorem for the Jacobian with a Diophantine approximation theorem of Thue–Siegel type, now usually Roth's theorem; it is not effective in general.\nSiegel proved the theorem in 1929.\n\n## Explanations\n\n### L1 human\n\nA curve of genus at least one can have infinitely many rational points, but only finitely many of them have integer coordinates.\nFor instance, $y^2 = x^3 + k$ has only finitely many integer solutions for each fixed nonzero $k$.\n\n### L2 human\n\nThe theorem applies to [[elliptic curves|427N6S6QQW]], where rational points may be infinite in number by [[Mordell's theorem|BZHGKN6Q3K]] but integral points are finite.\nBaker's method in transcendence theory later gave effective bounds in many cases, such as Mordell's equation and [[Thue equations|3Q6CFPQZDF]].\nFor genus at least $2$, [[Faltings's theorem|H0DB07WM20]] gives the stronger statement that even rational points are finite.\n",
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  "name": "Siegel's theorem on integral points",
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