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

Assume that $(x_{n})$ is a generic sequence of points such that $h_{\overline{L}}(x_{n})$ tends to $0$. By Theorem 3.3.1, $h_{\overline{M}}(x_{n})$ converges to

$$\frac{1}{\ell}({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{M})|X).$$

(3.4.2)

Except when both lower bounds are zero, this is strictly bigger than the lower bound of the proposition, which is equal to

$$\frac{1}{\ell+m}({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{M})|X).$$

In other words, the greedy obvious method to find points of small height for $\overline{L}+\overline{M}$ that first minimizes the height $h_{\overline{L}}$, only works up to the factor $(\ell+m)/\ell>1$.

Let us give some explicit formulae for the lower-bound above, in some particular cases. We consider $X=\mathbf{P}^{1}$ over ${\mathbf{Q}}$ and the metrized line bundle $\overline{\mathscr{O}(1)}_{\mathrm{W}}$. Let $\varphi$ and $\psi$ be polynomials with integral coefficients, of degrees $\ell$ and $m$ respectively ; let us pose $\overline{L}=\varphi^{*}\overline{\mathscr{O}(1)}_{\mathrm{W}}$, $\overline{M}=\psi^{*}\overline{\mathscr{O}(1)}_{\mathrm{W}}$. The line bundle $\overline{L}^{m}\otimes\overline{L}^{-\ell}$ is trivial and its metric is given by a family of functions $(f_{v})$. Since $\varphi$ and $\psi$ have integral coefficients, $f_{v}=0$ at all finite places. Moreover, since $g_{\overline{L}}(x)=\log\max(\left|{\varphi(x)}\right|,1)$ and $g_{\overline{M}}(x)=\log\max(\left|{\psi(x)}\right|,1)$ are the Green functions for the divisors $\ell[\infty]$ and $m[\infty]$ respectively, one has

$$f_{\infty}(x)=\log\frac{\max(\left|{\varphi(x)}\right|^{m},1)}{\max(\left|{\psi(x)}\right|^{\ell},1)}.$$

Then,

$$\mathop{\mathrm{d}\mathrm{d}^{c}}f_{\infty}=\frac{m}{2\pi}\mathrm{d}\operatorname{Arg}\varphi(x)\wedge\delta_{\left|{\varphi(x)}\right|=1}-\frac{\ell}{2\pi}\mathrm{d}\operatorname{Arg}\psi(x)\wedge\delta_{\left|{\psi(x)}\right|=1}$$

From this, we deduce that

	$\displaystyle\mathscr{D}(f_{\infty})$	$\displaystyle=\frac{\ell m}{2\pi}\left(\int_{\left|{\psi(x)}\right|=1}\log\max(\left|{\varphi(x)}\right|,1)\mathrm{d}\operatorname{Arg}\psi(x)\right.$
		$\displaystyle\qquad{}\left.+\int_{\left|{\varphi(x)}\right|=1}\log\max(\left|{\psi(x)}\right|,1)\mathrm{d}\operatorname{Arg}\varphi(x)\right),$

the two others terms vanishing. In fact, Stokes’s formula implies that the two terms within the parentheses in the previous formula are equal and we have

$$\mathscr{D}(f_{\infty})=\frac{\ell m}{\pi}\int_{\left|{\varphi(x)}\right|=1}\log\max(\left|{\psi(x),1}\right|)\mathrm{d}\operatorname{Arg}\varphi(x).$$

The simplest case to study is for $\varphi(x)=x^{\ell}$. Then,

$$\mathscr{D}(f_{\infty})=\frac{\ell m}{\pi}\int_{0}^{2\pi}\log\max(\left|{\psi(e^{i\theta})}\right|,1)\,\mathrm{d}\theta$$

is $2\ell m$ times the logarithm of the variant $\mathrm{M}^{+}(\psi)$ of the Mahler measure of $\psi$ :

$$\mathrm{M}^{+}(\psi)=\exp\left(\frac{1}{2\pi}\int_{0}^{2\pi}\log\max(\left|{\psi(e^{i\theta})}\right|,1)\,\mathrm{d}\theta\right).$$

In fact, Jensen’s formula implies that

$$\mathrm{M}^{+}(\psi)=\exp\left(\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\log\left|{\psi(e^{i\theta_{1}})-e^{i\theta_{2}}}\right|\,\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}\right)$$

is the Mahler measure $\mathrm{M}(\psi(x)-y)$ of the 2-variables polynomial $\psi(x)-y$.

Consequently, except for finitely many exceptions, any algebraic point $x\in{\mathbf{P}}^{1}(\overline{{\mathbf{Q}}})$ satisfies

$$\ell h(x)+h(\psi(x))\geq\frac{1}{\ell+m}\log\mathrm{M}(\psi(x)-y).$$

For $\ell=1$ and $\psi(x)=1-x$, we obtain that up to finitely many exceptions,

$$h(x)+h(1-x)\geq\frac{1}{2}\log\mathrm{M}(1-x-y)\approx 0.161538,$$

In that particular case, Zagier [56] has proved a much more precise result : except for 5 explicit points in ${\mathbf{P}}^{1}$,

$$h(x)+h(1-x)\geq\frac{1}{2}\log(\frac{1+\sqrt{5}}{2})\approx 0.240606.$$

Observe also that if $(x_{j})$ is a sequence of points such that $h(x_{j})\rightarrow 0$, Theorem 3.3.1 implies that $h(1-x)\rightarrow\log\mathrm{M}(1-x-y)\approx 0.323076$.

Let us assume that $\overline{L}$ and $\overline{M}$ are the metrized line bundles $\overline{\mathscr{O}(1)}_{\varphi}$ and $\overline{\mathscr{O}(1)}_{\psi}$ attached to rational functions $\varphi$ and $\psi$ of degres $d$ and $e$ respectively, with $d\geq 2$ and $e\geq 2$. Let us write $h_{\varphi}$ and $h_{\psi}$ for the height relative to these metrized line bundles ; we call them the canonical heights. The isometry $\varphi^{*}\overline{\mathscr{O}(1)}_{\varphi}\simeq\overline{\mathscr{O}(1)}_{\varphi}^{d}$ and the functorial properties of the height imply that for any $x\in{\mathbf{P}}^{1}(\overline{F})$, $h_{\varphi}(\varphi(x))=dh_{\varphi}(x)$ and $h_{\psi}(\psi(x))=eh_{\psi}(x)$. In particular, preperiodic points for $\varphi$ (i.e., points with finite forward orbit) satisfy $h_{\varphi}(x)=0$. Moreover,

$$d^{2}({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\varphi})^{2}|{\mathbf{P}}^{1})=({\widehat{c}}_{1}(\varphi^{*}\overline{\mathscr{O}(1)}_{\varphi})^{2}|{\mathbf{P}}^{1})=({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\varphi})^{2}|\varphi_{*}{\mathbf{P}}^{1})=d({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\varphi})^{2}|{\mathbf{P}}^{1}),$$

hence $({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\varphi})^{2}|{\mathbf{P}}^{1})=0$ since $d\neq 0,1$. Similarly, preperiodic points of $\psi$ satisfy $h_{\psi}(x)=0$, and $({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\psi})^{2}|{\mathbf{P}}^{1})=0$.

In the arithmetic case, or over function fields over a finit field, Northcott’s finiteness theorem implies easily that points $x$ such that $h_{\varphi}(x)=0$ are preperiodic for $\varphi$, and similarly for $\psi$. This is not true in general : for example, if $\varphi$ is constant, all constant points have height $0$ but only countably many of them are preperiodic ; more generally isotrivial rational functions, i.e. rational functions which are constant after conjugacy by an automorphism of ${\mathbf{P}}^{1}$ will furnish counterexamples. The best known result is restricted to (non-isotrivial) polynomials : by Benedetto [10], a point of height zero is then preperiodic ; the proof relies on a detailed analysis of the Julia set.

Let us show how Prop. 3.4.1 implies results of Baker and DeMarco [4], and of Petsche, Szpiro and Tucker [45].

1) The restrictive hypotheses on $\psi$ have only been used to establish the implication 1)$\Rightarrow$2).

2) Of course, many other results can be established by the same reasoning, in particular the number field case of Theorem 1.1 of [4]. Let us also recall that the support of the equilibrium measure $\mu_{\varphi}$ is the Julia set $J(\varphi)$. If $J(\varphi)\neq J(\psi)$ at some place, then none of the assertions of Prop. 3.4.3 can possibly hold.

3) The main result of [54] is that a variant of the implication (4)$\Rightarrow$(5) also holds in a more general setting : two semi-positive metrics on a line bundle which define the same measure at a place $v$ differ by multiplication by a constant. The given proof works for curves.

4) We also recall that an implication similar to (1)$\Rightarrow$(5) holds for general metrized line bundles on arithmetic varieties, as proven by [1] : if $\overline{L}$ and $\overline{M}$ are line bundles with adelic metrics such that $h_{\overline{L}}=h_{\overline{M}}$, then $\overline{L}\otimes\overline{M}^{-1}$ is torsion in the Arakelov Picard group $\overline{\operatorname{Pic}}(X)$ : the heights determine the metrics.