Let $\mathfrak{a}$ be a vertical coherent fractional ideal on the algebraic ${K^{\circ}}$-model ${\mathscr{X}}$ of the proper scheme $X$ over $K$. We define the function

$$\log|\mathfrak{a}|(x)\coloneqq\max\{\log|f(x)|\ \big|\ f\in\mathfrak{a}_{\pi(x)}\}$$

where $\pi:{X^{\rm an}}\to{\mathscr{X}}_{s}$ is the reduction map.