Diophantine Geometry: An Introduction
《Diophantine Geometry: An Introduction》 · 第 195 页 · Part B，Section B.4「Canonical Height Functions」，Theorem B.4.1
Let $V/k$ be a smooth variety defined over a number field, let $D\in\operatorname{Div}(V)$, and let $\varphi:V\to V$ be a morphism. Suppose that $\varphi^*D\sim\alpha D$ for some number $\alpha>1$. Then there is a unique function, called the canonical height on $V$ relative to $\varphi$ and $D$, $\widehat h_{V,\varphi,D}:V(\overline{k})\to\mathbb R$, with the following two properties: (i) $\widehat h_{V,\varphi,D}(P)=h_{V,D}(P)+O(1)$; (ii) $\widehat h_{V,\varphi,D}(\varphi(P))=\alpha\widehat h_{V,\varphi,D}(P)$.