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  "body": "\n## Definition\n\nLet $\\overline{\\Q}$ be an algebraic closure of $\\Q$ and let $n \\ge 1$.\nLet $P = [x_0 : \\cdots : x_n] \\in \\mathbb{P}^n(\\overline{\\Q})$ be a point of projective space, and let $K$ be an [[algebraic number field|MWBDFS2KJ5]] that contains $x_0, \\ldots, x_n$.\nFor each [[place|BT5V3TSNY6]] $v$ of $K$, let $\\abs{\\cdot}_v$ be the absolute value of $v$ that extends the usual or a $p$-adic absolute value of $\\Q$.\nLet $n_v = [K_v : \\Q_v]$ be the local degree.\nThe *absolute logarithmic height* of $P$ is\n$$h(P) = \\frac{1}{[K : \\Q]} \\sum_{v} n_v \\log \\max\\left(\\abs{x_0}_v, \\ldots, \\abs{x_n}_v\\right).$$\nIt does not depend on the choice of $K$ or of homogeneous coordinates.\n\nMore generally, let $V$ be a projective variety over $\\overline{\\Q}$ and let $\\phi \\colon V \\to \\mathbb{P}^n$ be a morphism.\nThe *height function* relative to $\\phi$ is $h_\\phi = h \\circ \\phi \\colon V(\\overline{\\Q}) \\to \\R$.\nThe height of an [[algebraic number|0MNF0BYFQC]] $\\alpha$ is $h(\\alpha) = h([1 : \\alpha])$.\n\n## Remarks\n\nIndependence of the coordinates follows from the product formula $\\prod_v \\abs{x}_v^{n_v} = 1$ for $x \\in K^\\times$.\n\nExample: for $P = [a_0 : \\cdots : a_n] \\in \\mathbb{P}^n(\\Q)$ with $a_i \\in \\Z$ and $\\gcd(a_0, \\ldots, a_n) = 1$, take $K = \\Q$.\nEach $p$-adic term is $\\log 1 = 0$, since some $a_i$ is not divisible by $p$, so $h(P) = \\log \\max_i \\abs{a_i}$.\nFor the rational number $2/3$, that is $[3 : 2]$, this gives $h = \\log 3$.\n\nNon-example: the rule $P \\mapsto \\log \\max_i \\abs{a_i}$ for arbitrary integer coordinates, without the coprimality condition, is not a function on $\\mathbb{P}^n(\\Q)$.\nThe coordinates $[1 : 1]$ and $[2 : 2]$ give the same point but the values $\\log 1$ and $\\log 2$.\n\nNorthcott's theorem: for all $B$ and $d$, the set of $P \\in \\mathbb{P}^n(\\overline{\\Q})$ with $h(P) \\le B$ and $[\\Q(P) : \\Q] \\le d$ is finite.\nThe multiplicative height is $H = e^h$.\nOn an elliptic curve, the Néron–Tate height $\\hat{h}(P) = \\lim_{m \\to \\infty} 4^{-m} h_x(2^m P)$, with $h_x$ the height relative to the $x$-coordinate, is a positive semi-definite quadratic form on $E(\\overline{\\Q})$; some authors include a factor $\\frac{1}{2}$.\n",
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