Height function

Remarks

Independence of the coordinates follows from the product formula $\prod_v \abs{x}_v^{n_v} = 1$ for $x \in K^\times$.

Example: 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 $K = \Q$. Each $p$-adic term is $\log 1 = 0$, since some $a_i$ is not divisible by $p$, so $h(P) = \log \max_i \abs{a_i}$. For the rational number $2/3$, that is $[3 : 2]$, this gives $h = \log 3$.

Non-example: the rule $P \mapsto \log \max_i \abs{a_i}$ for arbitrary integer coordinates, without the coprimality condition, is not a function on $\mathbb{P}^n(\Q)$. The coordinates $[1 : 1]$ and $[2 : 2]$ give the same point but the values $\log 1$ and $\log 2$.

Northcott'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. The multiplicative height is $H = e^h$. On 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 $\frac{1}{2}$.