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  "body": "\n## Statement\n\nLet $N \\ge 1$, $d \\ge 2$ and $D \\ge 1$ be integers.\nThere is a constant $B = B(N, d, D)$ with the following property.\nLet $K$ be a [[number field|MWBDFS2KJ5]] with $[K : \\Q] \\le D$.\nLet $\\phi \\colon \\mathbb{P}^N \\to \\mathbb{P}^N$ be a morphism of degree $d$ defined over $K$.\nThen the number of preperiodic points of $\\phi$ in $\\mathbb{P}^N(K)$ is at most $B$.\n\n## Remarks\n\nHere a point $P$ of [[projective space|ZT9A6JK8QF]] is preperiodic for $\\phi$ when its forward orbit $\\set{\\phi^n(P) : n \\ge 0}$ is finite, as in [[arithmetic dynamics|BE4X0QG8XW]].\nFor each single $\\phi$, the set of preperiodic points in $\\mathbb{P}^N(K)$ is finite by Northcott's theorem on heights.\nThe conjecture asks for a bound that depends only on $N$, $d$ and $D$.\n\nThe conjecture is open, even for the maps $\\phi(z) = z^2 + c$ with $c \\in \\Q$ acting on $\\mathbb{P}^1(\\Q)$.\n\nThe case $N = 1$, $d = 4$ implies the uniform boundedness of torsion of elliptic curves, proved by Merel in 1996.\nIndeed, let $E$ be an elliptic curve over $K$.\nThe $x$-coordinates of its torsion points are preperiodic points of the degree $4$ Lattès map that describes doubling on $E$.\nAt most two points of $E$ share an $x$-coordinate.\n\n## Explanations\n\n### L1 human\n\nFix a kind of polynomial or rational map, like $z \\mapsto z^2 + c$, and a kind of number system, like the fractions.\nFor each single map, only finitely many fractions eventually repeat under iteration.\nThe conjecture says more: there is one bound on how many such fractions there can be, the same for all maps of that degree.\nIt is a dynamical version of famous finiteness results about elliptic curves.\n",
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    "history": {
      "first_stated": "1994",
      "posed_by": "Patrick Morton and Joseph Silverman"
    },
    "kind": "conjecture",
    "msc": [
      "37P35",
      "37P15",
      "11G05"
    ],
    "name": "Uniform boundedness conjecture for preperiodic points",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Arithmetic_dynamics"
      }
    ],
    "tag": "3T9TT5SSGD",
    "uses": [
      "MWBDFS2KJ5",
      "ZT9A6JK8QF",
      "BE4X0QG8XW"
    ]
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/3T9TT5SSGD/",
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  "markdown": "https://tamnd.github.io/math2/node/3T9TT5SSGD.md",
  "msc": [
    "37P35",
    "37P15",
    "11G05"
  ],
  "name": "Uniform boundedness conjecture for preperiodic points",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3T9/3T9TT5SSGD.md",
  "status": null,
  "tag": "3T9TT5SSGD",
  "uri": "m2://3T9TT5SSGD",
  "url": "https://tamnd.github.io/math2/wiki/uniform-boundedness-conjecture-for-preperiodic-points/",
  "writing": {
    "tag": "3T9TT5SSGD",
    "kind": "conjecture",
    "level": "W0",
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