Let $X$ be a projective smooth curve over $F$, let $\overline{L}$ be a line bundle of degree $0$ on $X$, with an admissible metric. Let $\overline{L}_{0}$ be the same line bundle with the canonical metric : if $X$ has genus $\geq 1$, this is the metric induced by an embedding of $X$ into its Jacobian, if $X$ is of genus $0$, then $\overline{L}_{0}$ is the trivial metrized line bundle. The metrized line bundle $\overline{L}\otimes\overline{L}_{0}^{-1}$ is the trivial line bundle, together with an admissible metric which is given by a function $f_{v}$ at the place $v$ of $F$.

A formula of Faltings–Hriljac expresses $({\widehat{c}}_{1}(\overline{L}_{0})^{2}|X)$ as twice minus the Néron–Tate height of the point of $J$ corresponding to $L$. More generally,

$$({\widehat{c}}_{1}(\overline{L})^{2}|X)=-2\widehat{h}_{\mathrm{NT}}([L])+\sum_{v\in M(F)}\mathscr{D}(f_{v}),$$

where for each $v\in M(F)$,

$$\mathscr{D}(f_{v})=\int_{X_{v}}f_{v}\mathop{\mathrm{d}\mathrm{d}^{c}}(f_{v})$$

is the Dirichlet energy of $f_{v}$. This is a non positive quadratic form which vanishes if and only if $f_{v}$ is constant. For more details, I refer to [15] at archimedean places and [51] at ultrametric places. (When $X$ has genus $0$, $L\simeq\mathscr{O}_{X}$ and the term $\widehat{h}_{\mathrm{NT}}([L])$ has to be interpreted as $0$.)

As a consequence, $({\widehat{c}}_{1}(\overline{L})^{2}|X)\leq 0$. Let us analyse the case of equality. Since they are nonpositive, all terms in the formula above have to vanish. Consequently, $[L]$ is a torsion point in the Jacobian, and all functions $f_{v}$ are constant. We will say that some power of $\overline{L}$ is constant