Let $F$ be either a number field (arithmetic case), or a finite extension of the field of rational functions over a constant field (geometric case). Let $X$ be a projective variety over $F$, Let $M(F)$ be the set of normalized absolute values on $F$. Any $v\in M(F)$ gives rise to a complete valued field $F_{v}$, and to an analytic space $X_{v}$ over $F_{v}$ : if $v$ is archimedean, $X_{v}=X(\overline{F_{v}})$, while $X_{v}$ is the Berkovich analytic space attached to $X_{F_{v}}$ if $v$ is ultrametric.

If $L$ is a line bundle on $X$, an adelic metric on $L$ is a family $(\left\|{\cdot}\right\|_{v})_{v\in M(F)}$ of continuous metrics on the induced line bundles over the analytic spaces $X_{v}$. We require the following supplementary compatibility assumption : there exists a model $(\mathfrak{X},\mathscr{L},e)$ over the ring of integers of $F$ inducing the given metrics at almost all places $v$. An adelic metric is said to be semi-positive, resp. admissible if it is so at all places of $F$.

Line bundles on $X$ endowed with an adelic metric form a group $\overline{\operatorname{Pic}}(X)$ ; admissible line bundles form a subgroup $\overline{\operatorname{Pic}}_{\text{ad}}(X)$. If $f\colon Y\rightarrow X$ is any morphism, there is a natural morphism of groups $f^{*}\colon\overline{\operatorname{Pic}}(X)\rightarrow\overline{\operatorname{Pic}}(Y)$ ; it maps $\overline{\operatorname{Pic}}_{\text{ad}}(X)$ into $\overline{\operatorname{Pic}}_{\text{ad}}(Y)$.

Consider line bundles $\overline{L}_{0},\dots,\overline{L}_{n}$ with admissible adelic metrics. Let $Z$ be a subvariety of $X$ of dimension $k$ and $s_{0},\dots,s_{k}$ invertible meromorphic sections of $L_{0},\dots,L_{k}$ whose divisors hace no common intersection point on $Z$. For any $v\in M(F)$, we have recalled in Sections 1.2, 1.2 and 1.3 the definitions of the local height pairing

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}$$

where the index $v$ indicates the corresponding place of $F$. The global height is the sum, over all $v\in M(F)$, of these local heights :

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}.$$

It inherits from the local heights their multilinear symmetric character.

Let us replace $s_{k}$ by another invertible meromorphic section $fs_{k}$. Then,

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(fs_{k})|Z)$	$\displaystyle=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(fs_{k})|Z)_{v}$
		$\displaystyle=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}$
		$\displaystyle\qquad+\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|^{-1}c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}_{v}}.$

In particular, if $Z$ is a point $z\in X(F)$, then $\delta_{\mathrm{Z}_{v}}=\delta_{z}$ is the Dirac mass at $z$ and

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})|Z)=\sum_{v\in M(F)}\log\left\|{s_{0}}\right\|^{-1}_{v}(z).$$

Let us observe that it is independent on the choice of the chosen meromorphic section $s_{0}$, provided it is regular at $z$. Any other section has the form $fs_{0}$, for some invertible meromorphic function $f$ on $X$. Then,

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(fs_{0})|Z)$	$\displaystyle=\sum_{v\in M(F)}\log\left\|{fs_{0}}\right\|^{-1}_{v}(z)$
		$\displaystyle=\sum_{v\in M(F)}\log\left\|{s_{0}}\right\|^{-1}_{v}(z)+\sum_{v\in M(F)}\log\left|{f}\right|_{v}^{-1}(z)$
		$\displaystyle=(\mathop{\widehat{\operatorname{div}}}(fs_{0})|Z)$

since, by the product formula, the second term vanishes.

By induction on the dimension of $Z$, and using the commutativity of the local height pairings, it follows that the global height only depends on the metrized line bundles, and not on the actual chosen sections $s_{0},\dots,s_{k}$. We denote it by

$$({\widehat{c}}_{1}(\overline{L}_{0})\dots{\widehat{c}}_{1}(\overline{L}_{k})|Z).$$

Again, it is multilinear symmetric in the metrized line bundles $\overline{L}_{0},\dots,\overline{L}_{k}$. By the same argument, it only depends on their isomorphism classes in $\overline{\operatorname{Pic}}_{\text{ad}}(X)$.

It satisfies a projection formula : for any morphism $f\colon Y\rightarrow X$ and any $k$-dimensional subvariety $Z$ of $Y$,

$$({\widehat{c}}_{1}(f^{*}\overline{L}_{0})\dots{\widehat{c}}_{1}(f^{*}\overline{L}_{k})|Z)=({\widehat{c}}_{1}(\overline{L}_{0})\dots{\widehat{c}}_{1}(\overline{L}_{k})|f_{*}(Z),$$

where the cycle $f_{*}(Z)$ is defined as $\deg(Z/f(Z))f(Z)$ if $Z$ and $f(Z)$ have the same dimension, so that $f\colon Z\rightarrow f(Z)$ is generically finite, of some degree $\deg(Z/f(Z))$. If $Z$ and $f(Z)$ don’t have the same dimension, one sets $f_{*}(Z)=0$.

The height of an algebraic point is an important tool in Diophantine geometry. If $\overline{L}$ is a line bundle with an adelic metric on $X$, then for any point $P\in X(F)$, viewed as a closed subscheme of $X$, one has

$$h_{\overline{L}}(P)=({\widehat{c}}_{1}(\overline{L})|P)=\sum_{v}\log\left\|{s}\right\|^{-1}_{v}(P),$$

where $s$ is any meromorphic section on $L$ which has neither a zero nor a pole at $P$. More generally, let $P\in X(\overline{F})$ be an algebraic point and let $[P]$ be the corresponding closed point of $X$. Then,

$$h_{\overline{L}}(P)=\frac{1}{[F(P):F]}({\widehat{c}}_{1}(\overline{L})|[P])$$

is the height of $P$ with respect to the metrized line bundle $\overline{L}$. In fact, restricted to points, these definitions apply to any, not necessary admissible,

Observe also the following functorial property of the height : If $f\colon Y\rightarrow X$ is a morphism and $P\in Y(\overline{F})$, then $h_{f^{*}\overline{L}}(P)=h_{\overline{L}}(f(P))$. Finally, recall that if $F$ is a global field, then the height with respect to a metrized ample line bundle $\overline{L}$ satisfies Northcott’s finiteness property : for any integers $d$ and $B$, there are only finitely many points $P\in X(\overline{F})$ such that $[F(P):F]\leq d$ and $h_{\overline{L}}(P)\leq B$.

The essential minimum of the height $h_{\overline{L}}$ is defined as

$$e(\overline{L})=\sup_{\begin{subarray}{c}\emptyset\neq U\subset X\end{subarray}}\inf_{P\in U(\overline{F})}h_{\overline{L}}(P),$$

where the supremum runs over non-empty open subsets of $X$. If $L$ is big, then $e(\overline{L})$ is a real number. Another way to state its definition is the following : for any real number $B$, then the set

$$\{P\in X(\overline{F})\,;\,h_{\overline{L}}(P)\leq B\}$$

is Zariski dense if $B>e(\overline{L})$, and is not Zariski dense if $B<e(\overline{L})$.

Assume that $\overline{L}$ is an ample line bundle on $X$, equipped with a semi-positive adelic metric. The (geometric/arithmetic) Hilbert-Samuel theorem implies the following inequality

$$e(\overline{L})\geq\frac{({\widehat{c}}_{1}(\overline{L})^{n+1}|X)}{(n+1)(c_{1}(L)^{n}|X)}.$$

(See Zhang [59], as well as [38, 28] for more details in the geometric case). When $X$ is a curve and $F$ is a number field, Autissier [3] proved that the inequality holds for any ample line bundle with an admissible adelic metric (see [18]) ; this extends to the geometric case.

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.

Let $X$ be a projective smooth curve of genus $g\geq 2$ and let $\overline{L}$ be an ample line bundle on $X$ with a canonical metric inducing the Néron–Tate height. When $F$ is a number field, Bogomolov conjectured in [14] that $e(\overline{L})>0$ ; this conjecture has been shown by Ullmo [53]. Its generalization to a subvariety $X$ of an Abelian variety $A$, $L$ being an ample line bundle on $A$ with a canonical metric, asserts that $e(X,\overline{L})>0$ when $X$ is not the translate of an abelian subvariety by a torsion point ; it has been shown by Zhang [60].

Since $h_{\overline{L}}(P)=0$ for any algebraic point $P\in A(\overline{F})$ which is a torsion point, these theorems imply in turn a theorem of Raynaud [46, 47] (formerly, a conjecture of Manin and Mumford) that the torsion points lying in a subvariety $X$ of an abelian variety are not Zariski dense in $X$, unless $X$ is itself the translate of an abelian subvariety by a torsion point.

The analogues of Bogomolov’s and Zhang’s conjecture in the geometric case is still open in general ; see [37, 21] and the references therein for partial results.

The proofs by Ullmo and Zhang of Bogomolov’s conjecture make a fundamental use of an equidistribution principle which had been discovered together with Szpiro [50]. Let us first introduce a terminology : say a sequence (or a net) of algebraic points in a variety $X$ over a number field is generic if any strict subvariety of $X$ contains at most finitely terms of the sequence.

Let $\overline{L}$ be a line bundle on $X$ with a semi-positive adelic metric. The idea of the equidistribution principle is to consider a generic sequence $(x_{j})$ such that $h_{\overline{L}}(x_{j})\rightarrow e(\overline{L})$, i.e., realizing the equality in Zhang’s inequality, and to use this inequality further, as a variational principle. Let $v$ be a place of $F$ ; for any $n$, let $\delta(x_{j})_{v}$ be the probability measure on $X_{v}$ which gives any conjugate of $x_{j}$ the same mass, $1/[F(x_{j}):F]$. The equidistribution theorem states that for a generic sequence $(x_{j})$, the sequence of measures $(\delta(x_{j})_{v})$ on $X_{v}$ converges vaguely towards the measure $c_{1}(\overline{L})^{n}_{v}/(c_{1}(L)^{n}|X)$.

In these papers, the equidistribution property was only investigated at an archimedean place, but the introduction of the measures on Berkovich spaces was motivated by potential equidistribution theorems on those. In [18], I was able to prove general results on curves only. Indeed, unless $X$ is a curve, I needed an ampleness assumption on the metrized line bundle $\overline{L}$ in order to apply Zhang’s inequality to slight variations of it. This requirement has been removed by a paper of Yuan [55] who could understand arithmetic volumes beyond the ample case. Yuan’s proof is an arithmetic analogue of an inequality of Siu [49] which Faber [28] and Gubler [38] used to prove the geometric case of the equidistribution theorem.

In [19], we considered more general variations of the metrized line bundles. The discussion in that article was restricted to the arithmetic case but the arguments extend to the geometric case.

The particular case stated above is equivalent to loc.cit., Lemma 6.1, as one can see by by multiplying the metric on $\overline{L}$ by an adequate constant at some place of $F$. Taking for $M$ the trivial line bundle $\mathscr{O}_{X}$, with an admissible metric, one recovers the equidistribution theorems of Yuan, Faber and Gubler.

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.