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.