Weil height (reviewed)

The (logarithmic) Weil height of a nonzero rational number $a/b\in\mathbb{Q}$ in lowest terms is the quantity $$ h(a/b) = \log\max\bigl\{|a|,|b|\bigr\}. $$ The height of $0$ is taken to be $0.$

The (absolute logarithmic) Weil height of an element $\alpha$ in a number field $K$ is the quantity $$ h(\alpha) = \frac{1}{[K:\mathbb Q]} \sum_{v\in M_K} [K_v:\mathbb Q_v]\log\max\bigl\{\|\alpha\|_v,1\bigr\}, $$ where $M_K$ is an appropriately normalized set of inequivalent absolute values on $K$. More generally, the height of a point $P=[\alpha_0,\alpha_1,\ldots,\alpha_n]$ in projective space $\mathbb P^n(K)$ is given by $$ h(P) = \frac{1}{[K:\mathbb Q]} \sum_{v\in M_K} [K_v:\mathbb Q_v]\log\max_{0\le i\le n}\bigl\{\|\alpha_i\|_v\bigr\}. $$

If $\mathcal{L}$ is a very amply line bundle on a projective variety $V$ inducing an embedding $\iota \colon V \hookrightarrow \mathbb P^n$, then the Weil height associated on $X$ associated to $\mathcal{L}$ is given by $$ h_{\mathcal{L}}(P) = h(\iota(P)).$$ This definition can be extended to all line bundles by using the following linearity: $$ h_{\mathcal{L}_1 \otimes \mathcal{L}_2}(P) = h_{\mathcal{L}_1}(P) + h_{\mathcal{L}_2}(P).$$