Proposition 8.1 . [02Y1]

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Proposition 8.1.

Let $H\colon N\to\mathbb{Z}^{r}$ be an injective map such that $H(N)$ is a saturated sublattice of $\mathbb{Z}^{r}$ , $p\in\mathbb{P}^{r}_{0}(K)$ . Consider the map $\varphi_{H,p}:\mathbb{T}\to\mathbb{P}^{r}$ , and set ${\overline{L}}=\varphi_{H,p}^{*}{\overline{{\mathcal{O}}(1)}}$ and $s=\varphi_{H,p}^{*}s_{\infty}$ . Let $\psi_{{\overline{L}},s}\colon N_{\mathbb{R}}\to\mathbb{R}$ be the associated concave function, $m_{i}=e_{i}^{\vee}\circ H\in M$ , $i=1,\dots,r$ , and $p=(1:p_{1}:\dots:p_{r})$ with $p_{i}\in K^{\times}$ . Then, for $u\in N_{\mathbb{R}}$ ,

\psi_{{\overline{L}},s}(u)=\begin{cases}-\frac{1}{2}\log(1+\sum_{i=1}^{r}|p_{i}|^{2}\operatorname{e}^{-2\langle m_{i},u\rangle}),&\text{in the Archimedean case},\\ \min_{1\leq i\leq r}\{0,\langle m_{i},u\rangle+{\operatorname{val}}_{K}(p_{i})\}&\text{in the non-Archimedean case}.\end{cases}