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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $X$ be a projective variety of general type over $K$.\nThis means that the [[Kodaira dimension|ZW00Q0SAGR]] of $X$ equals its dimension.\n\n1. The set $X(K)$ of $K$-[[rational points|SWJYG87QQH]] is not dense in $X$ for the [[Zariski topology|8SQK7RTB63]].\n2. There is a nonempty Zariski open subset $U \\subseteq X$ such that $U(L)$ is finite for every finite extension $L$ of $K$.\n\n## Remarks\n\nPart 1 is often called the Bombieri–Lang conjecture.\nBombieri posed it for surfaces in a 1980 lecture, and Lang posed it in general.\nFor curves, general type means genus at least $2$, and both parts follow from [[Faltings's theorem|H0DB07WM20]].\nThe conjecture is open for surfaces.\nIt follows from [[Vojta's conjecture|QQ717VSFNN]].\nCaporaso, Harris and Mazur showed in 1997 that it implies a uniform bound for curves.\nThe number of $K$-rational points on a curve of genus $g \\ge 2$ over $K$ would be bounded in terms of $g$ and $K$ only.\n\n## Explanations\n\n### L1 human\n\nFaltings proved that a curve of genus at least two has only finitely many rational points.\nLang's conjectures predict the same kind of scarcity in higher dimension, for varieties \"of general type\", the most curved ones.\nRational points may be infinite, but they should all lie on a smaller subvariety, outside of which there are only finitely many.\n",
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    "history": {
      "first_stated": "1980",
      "posed_by": "Enrico Bombieri and Serge Lang"
    },
    "kind": "conjecture",
    "msc": [
      "14G05",
      "11G35",
      "14G40"
    ],
    "name": "Lang's conjectures",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bombieri%E2%80%93Lang_conjecture"
      }
    ],
    "tag": "TQ1QDAK2BR",
    "uses": [
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/TQ1QDAK2BR/",
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  "markdown": "https://tamnd.github.io/math2/node/TQ1QDAK2BR.md",
  "msc": [
    "14G05",
    "11G35",
    "14G40"
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  "name": "Lang's conjectures",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/TQ1/TQ1QDAK2BR.md",
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  "uri": "m2://TQ1QDAK2BR",
  "url": "https://tamnd.github.io/math2/wiki/langs-conjectures/",
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    "tag": "TQ1QDAK2BR",
    "kind": "conjecture",
    "level": "W0",
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        "note": "the row of the inventory that the node accounts for",
        "have": 0,
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