Let $f$ be an invertible meromorphic function on $X$. Let us view it as an invertible meromorphic section of the trivial metrized line bundle $\overline{\mathscr{O}_{X}}$. Let $\overline{L}$ be any line bundle on $X$ with an admissible adelic metric. Then,

$$({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{\mathscr{O}_{X}})|X)=0.$$

Moreover, according to Theorem 1.3 of [19] (see Section 1.3),

$$({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{\mathscr{O}_{X}})|X)=({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))+\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|_{v}^{-1}c_{1}(\overline{L})_{v}.$$

In other words, this furnishes an integral formula for the height (relative to $\overline{L}$) of any divisor which is rationally equivalent to $0$ :

$$({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))=\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|_{v}c_{1}(\overline{L})_{v}.$$

We want to apply this formula to a specific metrized line bundle on $X$. The Jacobian $J$ of $X$ is an Abelian variety of dimension $g$. We also choose a divisor $D$ of degree $1$ on $X$ and correspondingly fix an embedding $\iota$ of $X$ into $J$. (For this, we may need to enlarge the ground field $F$.) Finally, we let $\Theta$ be the theta divisor of $J$, defined as the image of $X^{g-1}$ by the map $(x_{1},\dots,x_{g-1})\mapsto\sum_{j=1}^{g-1}\iota(x_{j})$.

As described above, the line bundle $\mathscr{O}_{J}(\Theta)$ admits a canonical metrization ; this induces a metrization on its inverse image $L=\iota^{*}\mathscr{O}_{J}(\Theta)$ on $X$. The metrized line bundle $\overline{\mathscr{O}_{J}(\Theta)}$ gives rise to the (theta) Néron–Tate height on $J$. Consequently, decomposing $\operatorname{div}(f)=\sum n_{P}P$, we obtain

$$({\widehat{c}}_{1}(\overline{L})|\operatorname{div}(f))=\sum n_{P}[F(P):F]\widehat{h}_{\Theta}(\iota(P))=\sum n_{P}[F(P):F]\widehat{h}_{\Theta}([P-D]),$$

where $D$ is the fixed divisor of degree $1$ on $X$.

Since $L$ has degree $g$, the measure $c_{1}(\overline{L})_{v}$ on $X_{v}$ has total mass $g$ ; let us define a measure of total mass $1$ on $X_{v}$ by

$$\mu_{v}=\frac{1}{g}c_{1}(\overline{L})_{v}.$$

When $v$ is archimedean, the measure $\mu_{v}$ is the Arakelov measure on the Riemann surface $X_{v}({\mathbf{C}})$. Let us recall its definition. Consider an orthonormal basis $(\omega_{1},\dots,\omega_{g})$ of $H^{0}(X,\Omega^{1}_{X})$, i.e., a basis satisfying the relations

$$\int_{X_{v}({\mathbf{C}})}\omega_{j}\wedge\overline{\omega_{k}}=\delta_{j,k}=\begin{cases}1&\text{if $j=k$ ;}\\ 0&\text{otherwise.}\end{cases}$$

Then,

$$\mu_{v}=\frac{1}{g}\sum_{j=1}^{g}\omega_{j}\wedge\overline{\omega_{j}}.$$

Let us now assume that $v$ is ultrametric. By a theorem of Heinz [40], the metric on the line bundle $\overline{L}$ coincides with the canonical metric defined by Zhang [57] using the reduction graph of the minimal regular model of $X$. This allows in particular to compute the measure $\mu_{v}$ : the reader will find in [20, 57, 5]) a quite explicit formula for $\mu_{v}$, involving the physical interpretation of the graph as an electric network.

The formulas of this section combine to the following : if $\operatorname{div}(f)=\sum n_{P}P$ is a divisor of an invertible meromorphic function on $X$,

$$\sum n_{P}\widehat{h}_{\Theta}([P-D])=\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f(x)}\right|_{v}\,\mathrm{d}\mu_{v}(x).$$

As pointed out by R. De Jong [22], the case of superelliptic curves is particularly interesting. Indeed, such curves are presented as a ramified $\mu_{N}$-covering $x\colon X\rightarrow{\mathbf{P}}^{1}$ of the projective line, which is totally ramified over the point at infinity, given by an equation $y^{N}=a(x)$, where $a$ is a polynomial of degree $m>N$, prime to $N$. One has $g=\frac{1}{2}(N-1)(m-1)$.

Let us take for the divisor $D$ the single point $O$ over the point at infinity. For each point $P$ in $X(F)$, $x-x(P)$ is a rational function on $X$ which has a single pole of order $N$ at infinity, and which vanishes along the fiber $x^{-1}(x(P))$ of $x$. The group of automorphisms of $X$ acts transitively on this fiber, and respects the metrics, so that all of these points have the same Néron-Tate height. This implies the following formula

$$\widehat{h}_{\Theta}(P-O)=\frac{1}{N}\sum_{v\in M(F)}\int_{X_{v}}\log\left|{x-x(P)}\right|_{v}\mu_{v}$$

of [22]. The elliptic Mahler measure, defined by [27, 26] as a Shnirelman integral is therefore a natural integral when viewed on Berkovich spaces.