is an injective morphism of ${\mathcal{O}}_{{\widehat{\mathcal{V}}}}$-modules,
and its cokernel ${\rm coker\,}\iota$ is a coherent
${\mathcal{O}}_{\widehat{\mathcal{V}}}$-module supported by a finite set of
closed points in $|{\widehat{\mathcal{V}}}|\simeq{\rm
Spec\,}{{\mathcal{O}}_{K}}$.

In other words, ${\rm coker\,}\iota$ is the
${\mathcal{O}}_{{\widehat{\mathcal{V}}}}$-module associated to a _finite_
${\mathcal{O}}({\widehat{\mathcal{V}}})$-module $C$. In particular, for every
embedding $\sigma:K\hookrightarrow\mathbb{C}$, the vectors bundles
${\widehat{E}}^{\vee\vee}_{\sigma}$ and ${\widehat{E}}_{\sigma}$ on the formal
curve ${\widehat{\mathcal{V}}}_{\sigma}$ may be identified.

Associated to the vector bundle ${\widehat{E}}^{\vee\vee}$ over
${\widehat{\mathcal{V}}}$, we may define the Hermitian vector bundle over
${\widetilde{\mathcal{V}}}$:

(6.1.7)
${\widetilde{\overline{E}}}^{\vee\vee}:=({\widehat{E}}^{\vee\vee},(E_{\sigma},\varphi_{\sigma},\|.\|_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

The short exact sequence of topological ${{\mathcal{O}}_{K}}$-modules:

(6.1.8)
$0\longrightarrow\Gamma({\widehat{\mathcal{V}}},{\widehat{E}})\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\Gamma({\widehat{\mathcal{V}}},{\widehat{E}}^{\vee\vee})\longrightarrow
C\longrightarrow 0$

shows that $\Gamma({\widehat{\mathcal{V}}},{\widehat{E}})$, like
$\Gamma({\widehat{\mathcal{V}}},{\widehat{E}}^{\vee\vee})$, is an object of
the category $\mathbf{CTC}_{{{\mathcal{O}}_{K}}}$, with the notation of
[Bos20, Chapter 4]. Since the direct image of
$\widetilde{\overline{E}}^{\vee\vee}$, namely:

$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}^{\vee\vee}:=\Big{(}\Gamma({\widehat{\mathcal{V}}},\widehat{E}),(\Gamma_{L^{2}}(V_{\sigma},\mu_{\sigma};E_{\sigma},\|.\|_{\sigma}),\widehat{\eta}_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\Big{)},$

is a pro-Hermitian vector bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}},$ this
shows that the direct image of $\widetilde{\overline{E}}$, defined as:

$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}:=\Big{(}\Gamma({\widehat{\mathcal{V}}},\widehat{E}^{\vee\vee}),(\Gamma_{L^{2}}(V_{\sigma},\mu_{\sigma};E_{\sigma},\|.\|_{\sigma}),\widehat{\eta}_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\Big{)},$

is also a pro-Hermitian vector bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}}$.

Actually the short exact sequence (6.1.8) show that the properties of the pro-
Hermitian vector bundles
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}^{\vee\vee}$
and
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}$
are closely related. For instance, we have:

######  Proposition 6.1.1.

For every Hermitian torsion free coherent sheaf $\widetilde{\overline{E}}$
over ${\widetilde{\mathcal{V}}}$, the pro-Hermitian vector bundles
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}^{\vee\vee}$
is $\theta$-finite if and only if
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}$ is
$\theta$-finite. When this holds, using the notation (6.1.6), their
$\theta$-invariants satisfy the following estimates:

(6.1.9)
${h^{0}_{\theta}}({\widetilde{\mathcal{V}}},\mu;\widetilde{\overline{E}}^{\vee\vee})-\log|C|\leq{h^{0}_{\theta}}({\widetilde{\mathcal{V}}},\mu;\widetilde{\overline{E}})\leq{h^{0}_{\theta}}({\widetilde{\mathcal{V}}},\mu;\widetilde{\overline{E}}^{\vee\vee}).$

Proposition 6.1.1 is a direct consequence of the following result of
independent interest concerning pro-Hermitian vector bundles over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$.

######  Proposition 6.1.2.

Let
${\widehat{\overline{F}}}:=({\widehat{F}},(F^{\mathrm{Hilb}}_{\sigma},\|.\|_{\sigma},i_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$
be a pro-Hermitian vector bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}}$. If
${\widehat{F}}^{\prime}$ is an open ${{\mathcal{O}}_{K}}$-submodule of finite
index in ${\widehat{F}},$ then:

(6.1.10)
${\widehat{\overline{F}}}^{\prime}:=({\widehat{F}}^{\prime},(F^{\mathrm{Hilb}}_{\sigma},\|.\|_{\sigma},i_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

is a pro-Hermitian vector bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}}$.
Moreover ${\widehat{\overline{F}}}^{\prime}$ is $\theta$-finite if and only if
${\widehat{\overline{F}}}$ is, and when this holds, the following estimates
are satisfied:

${h^{0}_{\theta}}({\widehat{\overline{F}}}^{\prime})\leq{h^{0}_{\theta}}({\widehat{\overline{F}}})\leq{h^{0}_{\theta}}({\widehat{\overline{F}}}^{\prime})+\log|{\widehat{F}}/{\widehat{F}}^{\prime}|.$

Observe that the right-hand side of (6.1.10) indeed defines a pro-Hermitian
vector bundle since, for every embedding $\sigma:K\hookrightarrow\mathbb{C}$,
${\widehat{F}}^{\prime}_{\sigma}:={\widehat{F}}^{\prime}\widehat{\otimes}_{{{\mathcal{O}}_{K}},\sigma}\mathbb{C}$
may be identified to
${\widehat{F}}_{\sigma}:={\widehat{F}}\widehat{\otimes}_{{{\mathcal{O}}_{K}},\sigma}\mathbb{C}$
by the completed base change of the inclusion map
${\widehat{F}}^{\prime}\hookrightarrow{\widehat{F}}.$

Proposition 6.1.2 is a simple consequence of its special case when
${\widehat{\overline{F}}}$ is a Hermitian vector bundle, established in
[Bos20, Corollary 2.3.4], and of the characterizations of strongly summable
and $\theta$-finite pro-Hermitian vector bundles in [Bos20, Section 7.7].

#####  6.1.3. The algebra ${\mathcal{O}}({\widetilde{\mathcal{V}}})$ and the
field ${\mathcal{M}}({\widetilde{\mathcal{V}}})$

We may apply the construction of the space of sections
$\Gamma({\widetilde{\mathcal{V}}},\widetilde{E})$ to the vector bundle
$\mathcal{O}_{{\widetilde{\mathcal{V}}}}$ over ${\widetilde{\mathcal{V}}}$
defined by:

$\mathcal{O}_{{\widetilde{\mathcal{V}}}}:=(\mathcal{O}_{{\widehat{\mathcal{V}}}},(\mathcal{O}^{\mathrm{an}}_{V_{\sigma}},\varphi_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}),$

where $\varphi_{\sigma}$ is the tautological isomorphism:

$\mathcal{O}_{{\widehat{\mathcal{V}}}}\widehat{\otimes}_{{{\mathcal{O}}_{K}},\sigma}\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\iota_{\sigma}^{*}\mathcal{O}_{\widehat{V}_{\sigma,P_{\sigma}}}.$

This defines an ${{\mathcal{O}}_{K}}$-algebra:

$\mathcal{O}({\widetilde{\mathcal{V}}}):=\Gamma({\widetilde{\mathcal{V}}},\mathcal{O}_{{\widetilde{\mathcal{V}}}})\subseteq{\mathcal{O}}({\widehat{\mathcal{V}}})\times\prod_{\sigma:K\hookrightarrow\mathbb{C}}{\mathcal{O}}^{{\rm
an}}(V_{\sigma}^{+}),$

the algebra of _regular functions on ${\widetilde{\mathcal{V}}}$,_ whose
elements may be described as those formal functions on
${\widehat{\mathcal{V}}}$ that extend to functions analytic up to the boundary
on the Riemann surfaces $V_{\sigma}$. It is readily seen that the canonical
morphism:

${{\mathcal{O}}_{K}}\longrightarrow\mathcal{O}({\widetilde{\mathcal{V}}}),\quad
a\longmapsto\left(\pi^{\ast}a,(\sigma(a))_{\sigma:K\hookrightarrow\mathbb{C}}\right)$

is injective, and that $\mathcal{O}({\widetilde{\mathcal{V}}})$ is an
integrally closed domain.

In the same spirit, we may consider the field
${\mathcal{M}}({\widehat{\mathcal{V}}})$ of formal meromorphic functions on
${\widehat{\mathcal{V}}}$, as defined in [HM68],555The field
${\mathcal{M}}({\widehat{\mathcal{V}}})$ coincides with the fraction field of
the domain
${\mathcal{O}}({\widehat{\mathcal{V}}}):=\Gamma(|{\widehat{\mathcal{V}}}|,{\mathcal{O}}_{{\widehat{\mathcal{V}}}})$.
For instance, when ${{\mathcal{O}}_{K}}=\mathbb{Z}$, the formal scheme
${\widehat{\mathcal{V}}}$ is isomorphic to ${\rm Spf\,}\mathbb{Z}[[T]]$ and
${\mathcal{M}}({\widehat{\mathcal{V}}})$ to $\mathrm{Frac}\,\mathbb{Z}[[T]]$.
and the fields ${\mathcal{M}}(V_{\sigma}^{+})$ of meromorphic functions up to
the boundary on the Riemann surfaces $V_{\sigma}$, and define the field of
_meromorphic functions on ${\widetilde{\mathcal{V}}}$_ as follows:

${\mathcal{M}}({\widetilde{\mathcal{V}}}):=\Big{\\{}(\hat{f},(f_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})\in{\mathcal{M}}({\widehat{\mathcal{V}}})\times\prod_{\sigma:K\hookrightarrow\mathbb{C}}{\mathcal{M}}(V_{\sigma}^{+})\,\big{|}\,\forall\sigma:K\hookrightarrow\mathbb{C},\,\widehat{\eta}_{\sigma}(f_{\sigma})=\hat{f}{\otimes}_{\sigma}1\Big{\\}}.$

Here $\widehat{\eta}_{\sigma}$ denotes the Laurent expansion at $P_{\sigma}$:

$\widehat{\eta}_{\sigma}:{\mathcal{M}}(V_{\sigma}^{+})\longrightarrow\mathrm{Frac}\,{\mathcal{O}}^{\rm
an}_{V_{\sigma}\\!,P_{\sigma}}\hookrightarrow\mathrm{Frac}\,{\mathcal{O}}_{\widehat{V}_{\sigma}\\!,P_{\sigma}},$

and $\hat{f}\otimes_{\sigma}\\!1$ is an element of the $\mathbb{C}$-algebra
${\mathcal{M}}({\widehat{\mathcal{V}}})\otimes_{{{\mathcal{O}}_{K}}\\!,\sigma}\mathbb{C}$,
which is canonically embedded in
$\mathrm{Frac}\,(\mathcal{O}_{{\widehat{\mathcal{V}}}}\widehat{\otimes}_{{{\mathcal{O}}_{K}}\\!,\sigma}\mathbb{C})$,
which in turn is isomorphic to
$\mathrm{Frac}\,{\mathcal{O}}_{\widehat{V}_{\sigma}\\!,P_{\sigma}}$ by
$\iota_{\sigma}^{\ast}$. If $z$ denotes a local analytic coordinate on some
neighborhood of $P_{\sigma}$ in $\mathring{V}_{\sigma}$ that vanishes at
$P_{\sigma},$ the field
$\mathrm{Frac}\,{\mathcal{O}}^{\mathrm{an}}_{V_{\sigma},P_{\sigma}}$ (resp.
$\mathrm{Frac}\,{\mathcal{O}}_{{\widehat{V}}_{\sigma},P_{\sigma}}$) may be
identified with the field of Laurent series $\mathbb{C}\\{z\\}[z^{-1}]$ (resp.
$\mathbb{C}[[z]][z^{-1}]$).

In brief, an element of the field ${\mathcal{M}}({\widetilde{\mathcal{V}}})$
is a formal meromorphic function on ${\widehat{\mathcal{V}}}$ that extends to
meromorphic functions up to the boundary on the Riemann surfaces
$V_{\sigma}^{+}$.

The ${{\mathcal{O}}_{K}}$-algebra ${\mathcal{O}}({\widetilde{\mathcal{V}}})$
naturally embeds into ${\mathcal{M}}({\widetilde{\mathcal{V}}})$, which is
therefore an extension of the fraction field
$\mathrm{Frac}\,{\mathcal{O}}({\widetilde{\mathcal{V}}})$, hence of the number
field $K$.

####  6.2. Arakelov divisors and intersection numbers on f.-a. arithmetic
surfaces

In this section, we still denote by
${\widetilde{\mathcal{V}}}:=({\widehat{\mathcal{V}}},(V_{\sigma},O_{\sigma},\iota_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$
a smooth f.-a. arithmetic surface as defined in 6.1.1.

#####  6.2.1. The group ${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$

We define an _Arakelov divisor with compact support_ on
${\widetilde{\mathcal{V}}}$ as a pair $(D,g)$ where:

  * •

$D$ is a $1$-dimensional algebraic cycle on $|{\widehat{\mathcal{V}}}|$;

  * •

$g$ is a Green function (with ${\mathcal{C}}^{{\mathrm{b}\Delta}}$ regularity)
for the divisor $D_{\mathbb{C}}$ on the Riemann surface $V^{+}$ that is
invariant under complex conjugation, and whose support satisfies:

(6.2.1) ${\mathrm{supp}\,}g\subseteq
V_{\mathbb{C}}:=\coprod_{\sigma:K\hookrightarrow\mathbb{C}}V_{\sigma}.$

The first condition on $D$ simply means that $D$ is of the form: $D=nP$ for
some $n\in\mathbb{Z}.$

For every field embedding $\sigma:K\hookrightarrow\mathbb{C}$, we shall use
the notation:

$g_{\sigma}:=g_{\mid V_{\sigma}^{+}}.$

The support condition (6.2.1) on $g$ is equivalent to the vanishing of the
each of these functions $g_{\sigma}$ on
$V^{+}_{\sigma}\setminus\mathring{V}_{\sigma}$. In particular the restriction
$g_{\sigma\mid V_{\mathbb{C}}}$ of $g_{\sigma}$ to the the compact submanifold
$V_{\sigma}$ satisfies the “Dirichlet boundary condition”:

$g_{\mid\partial V_{\sigma}}=0.$

The Arakelov divisors with compact supports on ${\widetilde{\mathcal{V}}}$
define an additive group which will be denoted by
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}}).$ There is a canonical
morphism of commutative groups:

(6.2.2)
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})\longrightarrow\overline{{\rm
Pic}}_{{\mathcal{C}}^{{\mathrm{b}\Delta}}}({\widetilde{\mathcal{V}}}),$

that maps the Arakelov divisor $(D,g)$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ to the isomorphism class
of the Hermitian line bundle over ${\widetilde{\mathcal{V}}}$:

$\widetilde{\overline{\mathcal{O}}}(D,g):=({\mathcal{O}}_{{\widehat{\mathcal{V}}}}(D),({\mathcal{O}}^{{\rm
an}}_{V^{+}_{\sigma}}(D_{\sigma}),\|.\|_{g_{\sigma}},\varphi_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}),$

where the isomorphism $\varphi_{\sigma}$ is the tautological isomorphism
induced by $\iota_{\sigma}$.

Let us emphasize that the properties of the morphism (6.2.2) differ strikingly
from those of the analogue morphism:

${\overline{Z}}^{1}_{c}(X)\longrightarrow\overline{{\rm
Pic}}_{{\mathcal{C}}^{{\mathrm{b}\Delta}}}(X),$

relating Arakelov divisors and Hermitian line bundles on a regular projective
arithmetic surface $X$. For instance, when ${{\mathcal{O}}_{K}}=\mathbb{Z}$,
the morphism (6.2.2) is easily seen to be injective. Moreover its image is
expected to be a “very small” subgroup of $\overline{{\rm
Pic}}_{{\mathcal{C}}^{{\mathrm{b}\Delta}}}({\widetilde{\mathcal{V}}})$ when
${\widetilde{\mathcal{V}}}$ belongs to the class of pseudoconcave f.-a.
arithmetic surfaces, which are investigated in the next chapters.

#####  6.2.2. The Arakelov divisor
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$ and the line bundle
${\overline{N}}_{P}{\widetilde{\mathcal{V}}}$

The divisor

$P_{\mathbb{C}}:=\sum_{\sigma:K\hookrightarrow\mathbb{C}}P_{\sigma}$

in $V^{+}_{\mathbb{C}}$ admits a distinguished Green function
${g}_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}$ that satisfies the support
condition (6.2.1), namely the Green function defined by the equilibrium
potentials for the points $P_{\sigma}$ in the connected compact Riemann
surfaces with boundary $V_{\sigma}$:

$g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}\mid
V_{\sigma}^{+}}:=g_{V_{\sigma},P_{\sigma}},$

introduced in 4.3.1 above.

The Green function $g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}$ is invariant
under complex conjugation, and the pair
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$ defines an element of
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, canonically attached to
${\widetilde{\mathcal{V}}}$.

To this Arakelov divisor is attached the Hermitian line bundle
$\widetilde{\overline{\mathcal{O}}}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$
on ${\widetilde{\mathcal{V}}}$. Its restriction to
$P=|{\widehat{\mathcal{V}}}|$ may be identified with the Hermitian line
bundle:

(6.2.3)
$\overline{N}_{P}{\widetilde{\mathcal{V}}}:=\left(N_{P}{\widehat{\mathcal{V}}},(\|.\|^{\mathrm{cap}}_{V_{\sigma},P_{\sigma}})_{\sigma:K\hookrightarrow\mathbb{C}}\right)$

defined by the normal bundle $N_{P}{\widehat{\mathcal{V}}}$ of $P$ in
${\widehat{\mathcal{V}}}$ equipped with the capacitary metrics
$\|.\|^{\mathrm{cap}}_{V_{\sigma},P_{\sigma}}$ on the complex lines:

$(N_{P}{\widehat{\mathcal{V}}})_{\sigma}\stackrel{{\scriptstyle
T_{\iota_{\sigma}}}}{{\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}}N_{P_{\sigma}}V_{\sigma}=T_{P_{\sigma}}V_{\sigma},$

introduced in 3.2.1 and 4.2.1 above.

The group ${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ admits a simple
description in terms of the Arakelov divisor
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$.

Let us define
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$ as
the subspace of ${\mathcal{C}}^{{\mathrm{b}\Delta}}(V^{+}_{\mathbb{C}})$
consisting in the ${\mathcal{C}}^{{\mathrm{b}\Delta}}$-functions
$\varphi:V^{+}_{\mathbb{C}}\rightarrow\mathbb{R}$ that are invariant under
complex conjugation acting on $V^{+}_{\mathbb{C}}$ and satisfy the following
“Dirichlet boundary condition:”

${\mathrm{supp}\,}\varphi\subseteq V_{\mathbb{C}}.$

The following proposition is then a straightforward consequence of the
definitions:

######  Proposition 6.2.1.

The morphism of commutative groups:

$\mathbb{Z}\oplus{\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}}\longrightarrow{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}}),\quad(n,\varphi)\longmapsto
n(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})+(0,\varphi)$

is an isomorphism.

#####  6.2.3. Arithmetic intersection theory on ${\widetilde{\mathcal{V}}}$

Let ${\overline{L}}$ be a Hermitian line bundle on
${\widetilde{\mathcal{V}}}$. Its restriction to $|{\widehat{\mathcal{V}}}|=P$
is a Hermitian line bundle on a one-dimensional scheme proper over ${\rm
Spec\,}\mathbb{Z}$, and admits a well defined arithmetic degree:

$\widehat{\rm
deg}\,({\overline{L}}|P)=0pt_{{\overline{L}}}(|{\widehat{\mathcal{V}}}|):=\widehat{\rm
deg}\,P^{\ast}{\overline{L}}.$

More generally, a divisor $D$ in ${\widetilde{\mathcal{V}}}$ supported by
$|{\widehat{\mathcal{V}}}|$ may be written $D=nP,$ with $P\in\mathbb{Z},$ and
we have:

$\widehat{\rm deg}\,({\overline{L}}|D):=n\,\widehat{\rm
deg}\,P^{\ast}{\overline{L}}.$

For every Hermitian line bundle $\overline{L}$ on ${\widetilde{\mathcal{V}}}$,
defined by a metric of regularity $\mathcal{C}^{{\mathrm{b}\Delta}}$, and any
Arakelov divisor in ${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, we
may define their arithmetic intersection number by the same formula as in
Subsections 3.3.2 and 4.5.2:

(6.2.4) $\overline{L}\cdot(D,g):=\widehat{\rm
deg}\,({\overline{L}}|D)+\int_{V^{+}_{\mathbb{C}}}g\,c_{1}(\overline{L}_{\mathbb{C}})\quad\in\mathbb{R}.$

The product $g\,c_{1}(\overline{L}_{\mathbb{C}})$ is a product of a function
in $C^{{\mathrm{b}\Delta}}(V^{+}_{\mathbb{C}})$ and of a measure in
${\mathbf{M}}^{{\mathrm{cp}}}(V^{+}_{\mathbb{C}})$, and is supported by the
compact submanifold $V_{\mathbb{C}}$. The integral in the right-hand side of
(6.2.4) is therefore well-defined, as discussed in 4.2.2, and could also be
written:

$\int_{V_{\mathbb{C}}}g\,c_{1}(\overline{L}_{\mathbb{C}}).$

We may specialize the definition (6.2.4) of arithmetic intersection numbers to
Hermitian line bundles whose isomorphism class belong to the image of the
morphism (6.2.2). Thus we define, for any two Arakelov divisors $(D,g)$ and
$(D^{\prime},g^{\prime})$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$:

(6.2.5)
$(D^{\prime},g^{\prime})\cdot(D,g):=\widetilde{\overline{\mathcal{O}}}(D^{\prime},g^{\prime})\cdot(D,g)=\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(D^{\prime},g^{\prime})|D)+\int_{V_{\mathbb{C}}}g\,\omega(g^{\prime}).$

Observe that, for every
$(D,g)\in{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ and any
$\varphi\in{\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}},$ we have:

$\displaystyle(0,\varphi)\cdot(D,g)$ $\displaystyle=\widehat{\rm
deg}\,(\overline{\mathcal{O}}(0,\varphi)\mid
D)+\int_{V_{\mathbb{C}}}g\,\frac{i}{\pi}\partial\overline{\partial}\varphi$
(6.2.6)
$\displaystyle=\int_{V_{\mathbb{C}}}\varphi\,\delta_{D_{\mathbb{C}}}+\pi^{-1}\int_{V_{\mathbb{C}}}g\,i\partial\overline{\partial}\varphi.$

In particular, for every $\varphi_{1}$ and $\varphi_{2}$ in
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$,
we have:

(6.2.7)
$(0,\varphi_{1})\cdot(0,\varphi_{2})=-\pi^{-1}\langle\varphi_{1},\varphi_{2}\rangle_{\rm
Dir}.$

The fact that the measure $\omega(g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$
is supported by $\partial V_{\mathbb{C}}$ implies that arithmetic intersection
numbers involving the Arakelov divisor
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$ admit simple expressions:

######  Proposition 6.2.2.

For every $n\in\mathbb{Z}$ and every Arakelov divisor of the form $(nP,g)$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, we have:

(6.2.8)
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot(nP,g)=n\,\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}.$

In particular:

(6.2.9)
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}},$

and, for every
$\varphi\in{\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}},$ we have:

(6.2.10) $(0,\varphi)\cdot(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=0.$

###### Proof.

According to the definition (6.2.5) of the arithmetic intersection number, we
have:

$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot(nP,g)=n\,\widehat{\rm
deg}\,P^{\ast}\widetilde{\overline{\mathcal{O}}}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})+\int_{V_{\mathbb{C}}}g\,\omega(g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}).$

As observed in 6.2.2, the restriction of
$\widetilde{\overline{\mathcal{O}}}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$
is canonically isomorphic to $\overline{N}_{P}{\widetilde{\mathcal{V}}}$, and
therefore:

$\widehat{\rm
deg}\,P^{\ast}\widetilde{\overline{\mathcal{O}}}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=\widehat{\rm
deg}\,P^{\ast}\overline{N}_{P}{\widetilde{\mathcal{V}}}.$

Moreover the product $g\,\omega(g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$
is zero. This establishes (6.2.8). The relations (6.2.9) and (6.2.10) are
special cases of (6.2.8). ∎

######  Corollary 6.2.3.

For every $n_{1}$ and $n_{2}$ in $\mathbb{Z}$ and $\varphi_{1}$ and
$\varphi_{2}$ in
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$,
we have:

(6.2.11)
$(n_{1}P,n_{1}\,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}+\varphi_{1})\cdot(n_{2}P,n_{2}\,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}+\varphi_{2})=n_{1}n_{2}\,\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}-\pi^{-1}\langle\varphi_{1},\varphi_{2}\rangle_{\rm
Dir}.$

In particular, the arithmetic intersection pairing on
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ defined by (6.2.5) is
symmetric. Moreover, for every $\varphi$ in
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$,
we have:

(6.2.12)
$(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}+\varphi)\cdot(P,\,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}+\varphi)=\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}-\pi^{-1}\langle\varphi,\varphi\rangle_{\rm
Dir}.$

This shows that the self-intersection of an Arakelov divisor in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ of the form $(P,h)$
satisfies:

(6.2.13)
$(P,h)\cdot(P,h)\geq(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}),$

and that equality holds in (6.2.13) if and only if
$h=g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}$.

####  6.3. Pseudoconcavity and pseudoconvexity

The next two propositions establish an analogue, concerning f.-a. arithmetic
surfaces, of the results about $\mathbf{CNB}$ divisors on analytic surfaces
fibered over a projective curve in Proposition 2.2.1 and Corollary 2.2.2.

######  Proposition 6.3.1.

Let $h$ be a Green function $h$ for $P_{\mathbb{C}}$ in $V^{+}_{\mathbb{C}}$
such that $(P,h)$ belongs to
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$.

The following two conditions are equivalent:

  1. (i)

for every $(D,g)\in{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$:

$D\geq 0\quad\mbox{and}\quad g\geq 0\,\,\Longrightarrow(P,h)\cdot(D,g)\geq 0;$

  2. (ii)

$\omega(h)_{\mid\mathring{V}_{\mathbb{C}}}\geq 0\mbox{ and }\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)\geq 0.$

###### Proof.

Let us assume that $(i)$ holds. Then, for every non-negative function
$\psi\in{\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}},$ we have:

$0\leq(P,h)\cdot(0,\psi)=\int_{V_{\mathbb{C}}}\psi\,\omega(h).$

This is readily seen to imply that, over $\mathring{V}_{\mathbb{C}}$, the
measure $\omega(h)$ is non-negative.

Moreover, for any non-negative Green function $g$ for $P_{\mathbb{C}}$ in
$V_{\mathbb{C}}^{+}$ such that $(P,g)$ belongs to
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, we have:

(6.3.1) $0\leq(P,h)\cdot(P,g)=\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)+\int_{V_{\mathbb{C}}}g\,\omega(h).$

Clearly there exists such a non-negative Green function $g,$ and the
construction in 4.2.2 of $\mathcal{C}^{{\mathrm{b}\Delta}}$ functions
approximating a Green function with $\mathcal{C}^{{\mathrm{b}\Delta}}$
regularity establishes the existence of a sequence $(g_{n})$ in
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$
such that:

$0\leq g_{n}\leq g,$

and:

$\lim_{n\rightarrow+\infty}\int_{V_{\mathbb{C}}}(g-g_{n})\,\omega(h)=0.$

The inequality (6.3.1), applied to $g-g_{n}$ instead of $g$, implies:

$0\leq\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)+\int_{V_{\mathbb{C}}}(g-g_{n})\,\omega(h),$

and by letting $n$ go to infinity, we obtain:

$\widehat{\rm deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)\geq 0.$

Conversely, assume that (ii) holds, and let $(D,g)$ be an element of
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ such that $D\geq 0$ and
$g\geq 0.$ Then $D=nP$ for some $n\in\mathbb{N}$, and:

$(P,h)\cdot(D,g)=(P,h)\cdot(nP,g)=n\,\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)+\int_{V_{\mathbb{C}}}g\,\omega(h)$

is clearly non-negative. ∎

Let us denote :

$\mathcal{SH}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}}:=\left\\{\varphi\in{\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}}\mid
i\partial\overline{\partial}\varphi_{\mid\mathring{V}_{\mathbb{C}}}\geq
0\right\\}.$

The functions in $\mathcal{SH}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$ are
continuous on $V_{\mathbb{C}}$, subharmonic on $\mathring{V}_{\mathbb{C}}$,
and vanish on $\partial V_{\mathbb{C}}.$ Consequently, they are non-positive
on $V_{\mathbb{C}}$.

######  Proposition 6.3.2.

The following two conditions are equivalent:

  1. (i)

There exists a Green function $h$ for $P_{\mathbb{C}}$ in $V^{+}_{\mathbb{C}}$
such that $(P,h)$ belongs to
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ and such that, for every
$(D,g)\in{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$:

(6.3.2) $D\geq 0\quad\mbox{and}\quad g\geq
0\,\,\Longrightarrow(P,h)\cdot(D,g)\geq 0.$

  2. (ii)

$\widehat{\rm deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}\geq
0.$

When these conditions are satisfied, the set of Green functions $h$ as in (i)
is $g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}+{\mathcal{C}}$, where:

${\mathcal{C}}:=\Big{\\{}\varphi\in\mathcal{SH}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}}\mid\sum_{\sigma:K\hookrightarrow\mathbb{C}}\varphi(P_{\sigma})+\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}\geq 0\Big{\\}}.$

In particular, (i) holds with $h=g_{{\widetilde{\mathcal{V}}}_{C}}$, and every
Green function $h$ for $P_{\mathbb{C}}$ as in (i) satisfies: $h\leq
g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}.$

Since $g_{{\widetilde{\mathcal{V}}}_{C}}$ is non-negative, the conditions (i)
and (ii) are also equivalent to the strengthened variant of (i) where $h$ is
required to be non-negative.

###### Proof.

The Green functions $h$ for $P_{\mathbb{C}}$ such that $(P,h)$ belongs to
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ are precisely of the form:

$h=g_{{\widetilde{\mathcal{V}}}_{C}}+\varphi$

with $\varphi$ in
${\mathcal{C}}^{{\mathrm{b}\Delta}}(V_{\mathbb{C}})_{\mathbb{R},{\rm Dir}}$.
When this holds, we have:

$\omega(h)_{\mid\mathring{V}_{\mathbb{C}}}\geq
0\Longleftrightarrow\varphi\in\mathcal{SH}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}},$

and:

$\widehat{\rm
deg}\,(\widetilde{\overline{\mathcal{O}}}(P,h)|P)=\int_{V_{\mathbb{C}}}\varphi\,\delta_{P_{\mathbb{C}}}+\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}=\sum_{\sigma:K\hookrightarrow\mathbb{C}}\varphi(P_{\sigma})+\widehat{\rm
deg}\,P^{\ast}{\overline{N}}_{P}{\widetilde{\mathcal{V}}}.$

The proposition therefore follows from Proposition 6.3.1 and from the non-
positivity of the functions in $\mathcal{SH}(V_{\mathbb{C}})_{\mathbb{R},{\rm
Dir}}.$ ∎

######  Definition 6.3.3.

We shall say that the smooth f.-a. arithmetic surface
${\widetilde{\mathcal{V}}}$ is _pseudoconcave_ (resp. _pseudoconvex_) when:

(6.3.3) $\widehat{\rm
deg}\,{\overline{N}}_{P}{\widetilde{\mathcal{V}}}>0\quad\mbox{( resp.
$\widehat{\rm deg}\,{\overline{N}}_{P}{\widetilde{\mathcal{V}}}<0$)}.$

Proposition 6.3.2 and the inequality (6.2.13) show that
${\widetilde{\mathcal{V}}}$ is pseudoconcave if and only if there exists a
non-negative Green function $h$ for $P_{\mathbb{C}}$ in $V^{+}_{\mathbb{C}}$
such that $(P,h)$ belongs to
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$ and satisfies the
condition of “numerical effectivity” (6.3.2) and the “bigness” condition:

$(P,h)\cdot(P,h)>0.$

Moreover, when ${\widetilde{\mathcal{V}}}$ is pseudoconcave, these conditions
are satisfied by $h=g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}}.$

####  6.4. The arithmetic surfaces
${\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)$

In this section, we discuss some simple examples of smooth f.-a. arithmetic
surfaces ${\widetilde{\mathcal{V}}}$ and of the associated algebra
${\mathcal{O}}({\widetilde{\mathcal{V}}})$. Our aim is to demonstrate, on some
simple but significant examples, the crucial role played the numerical
conditions (6.3.3) on the “size” of the ring
${\mathcal{O}}({\widetilde{\mathcal{V}}})$.

For simplicity, we restrict ourselves to f.-a. arithmetic surfaces over
$\mathbb{Z}$, and leave it to the interested reader to extend the results of
this sections when $\mathbb{Z}$ is replaced by the ring ${\mathcal{O}_{K}}$ of
integers of an arbitrary number field $K$.

Indeed, when ${{\mathcal{O}}_{K}}=\mathbb{Z},$ the definitions in 6.1.1 and
6.1.3 above become especially simple. Namely, a smooth f.-a. arithmetic
surface ${\widetilde{\mathcal{V}}}=(\widehat{{\mathcal{V}}},(V,O,i))$ over
$\mathbb{Z}$ is the data of a formal scheme $\widehat{{\mathcal{V}}}$ over
${\rm Spec\,}\mathbb{Z}$, isomorphic to $\mathrm{Spf\,}\mathbb{Z}[[X]]$, of a
connected compact Riemann surface with non-empty boundary $V$ equipped with a
real structure, of a real point $O$ in ${\mathring{V}}$, and of an isomorphism
of complex formal curves, compatible with the real structures:

$i:\widehat{{\mathcal{V}}}_{\mathbb{C}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{V}_{O}.$

The section $P:{\rm Spec\,}\mathbb{Z}\rightarrow\mathrm{Spf\,}\mathbb{Z}[[X]]$
is the inverse of the tautological isomorphism
$\big{|}\mathrm{Spf\,}\mathbb{Z}[[X]]\big{|}\simeq{\rm Spec\,}\mathbb{Z}$.

Moreover the elements of the ring ${\mathcal{O}}({\widetilde{\mathcal{V}}})$
(resp. of the field ${\mathcal{M}}({\widetilde{\mathcal{V}}})$) may be
described as pairs $\alpha:=(\hat{\alpha},\alpha^{{\rm an}})$ where
$\hat{\alpha}$ and $\alpha^{{\rm an}}$ belong respectively to
${\mathcal{O}}(\widehat{{\mathcal{V}}})$ and ${\mathcal{O}}^{{\rm an}}(V^{+})$
(resp. to
${\mathcal{M}}(\widehat{{\mathcal{V}}})\simeq\mathrm{Frac}\,\mathbb{Z}[[X]]$
and ${\mathcal{M}}(V^{+})$) and satisfy the following gluing condition:

$i^{\ast}\alpha^{{\rm an}}_{\mid\widehat{V}_{O}}=\hat{\alpha}_{C}.$

#####  6.4.1. The f.-a. arithmetic surfaces
${\widetilde{\mathcal{V}}}(D,\psi)$

######

Let $D$ be a compact domain with ${\mathcal{C}}^{\infty}$ boundary666that is,
a compact ${\mathcal{C}}^{\infty}$ submanifold with boundary, of dimension 2.
in $\mathbb{C}$, that is invariant under complex conjugation and contains $0$
in its interior, and let $\psi$ be a formal series in $\mathbb{R}[[X]]$ such
that:

(6.4.1) $\psi(0)=0\quad\mbox{and}\quad\psi^{\prime}(0)\neq 0.$

To $(D,\psi)$, we may attach the smooth f.-a. arithmetic surface
${\widetilde{\mathcal{V}}}=(\widehat{{\mathcal{V}}},(V,P,i))$ over
$\mathbb{Z}$ defined as follows:

$\widehat{{\mathcal{V}}}:=\mathrm{Spf\,}\mathbb{Z}[[X]],\quad V:=D,\quad O=0,$

and:777In other words, for any
$f\in\mathbb{C}[[z]]\simeq{\mathcal{O}}_{\hat{D}_{0}},$$i^{\ast}f:=f\circ\psi.$

$i:=\psi:\mathrm{Spf\,}\mathbb{C}[[X]]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Spf\,}\mathbb{C}[[z]]\simeq\widehat{D}_{0}.$

We shall denote it by ${\widetilde{\mathcal{V}}}(D,\psi)$.

By definition, the ring ${\mathcal{O}}({\widetilde{\mathcal{V}}}(D,\psi))$ may
be identified with the subring of formal series $\hat{\alpha}$ in
$\mathbb{Z}[[X]]$ such that $\hat{\alpha}\circ\psi^{-1}$ is the Taylor
expansion in $0$ of an analytic function on some open neighborhood of $D$.

Similarly, the field ${\mathcal{M}}({\widetilde{\mathcal{V}}}(D,\psi))$ may be
identified with the field of Laurent series $\widehat{\alpha}$ in
$\mathbb{Q}((X))$ that belong to its subfield
$\mathrm{Frac}\,\mathbb{Z}[[T]]$, such that $\widehat{\alpha}\circ\psi^{-1}$
is the Taylor expansion in $0$ of an meromorphic function on some open
neighborhood of $D$.

######

For instance, for any $r\in\mathbb{R}_{+}^{\ast}$, we may define the _Borel
f.-a. arithmetic surface of radius $r$_, denoted by
$\widetilde{{\mathcal{B}}}(r),$ by the above construction applied to
$D:=\overline{D}(0;r)$, the closed disk of radius $r$, and to the “identity”
series $\psi=X$:

$\widetilde{{\mathcal{B}}}(r):={\widetilde{\mathcal{V}}}(\overline{D}(0;r),X).$

The equilibrium potential for $P_{\mathbb{C}}=0$ in $\overline{D}(0;r)$ is the
function:

$g_{\overline{D}(0;r),0}=(z\mapsto\log^{+}|z/r|),$

and therefore the associated capacitary metric
$\|.\|_{\overline{D}(0;r),0}^{\mathrm{cap}}$ satisfies:

$\|\partial/\partial z\|_{\overline{D}(0;r),0}^{\mathrm{cap}}=r^{-1}.$

Consequently the Hermitian line bundle
${\overline{N}}_{P}\widetilde{{\mathcal{B}}}(r)$ over ${\rm Spec\,}\mathbb{Z}$
may be identified with:

$(\mathbb{Z}\;\partial/\partial X,\|\partial/\partial X\|=r^{-1}),$

and therefore satisfies:

(6.4.2) $\widehat{\rm deg}\,N_{P}\widetilde{{\mathcal{B}}}(r)=\log r.$

The “size” of the ring ${\mathcal{O}}(\widetilde{{\mathcal{B}}}(r))$ clearly
depends on the sign of (6.4.2). Indeed when

$\widehat{\rm deg}\,N_{P}\widetilde{{\mathcal{B}}}(r)\geq 0,$

or equivalently when $r\geq 1$, it is easily seen to be reduced to the ring of
polynomials:

${\mathcal{O}}(\widetilde{{\mathcal{B}}}(r))=\mathbb{Z}[X].$

In contrast, when $\widehat{\rm deg}\,N_{P}\widetilde{{\mathcal{B}}}(r)<0$,
that is when $r<1$, the ring ${\mathcal{O}}(\widetilde{{\mathcal{B}}}(r))$ is
“very large”: it contains all formal series in $\mathbb{Z}[[X]]$ with bounded
coefficients and therefore has the cardinality of the continuum.

######

More generally, to any formal series $\psi$ in $\mathbb{R}[[X]]$ satisfying
the conditions (6.4.1), we may attach the f.-a. arithmetic surfaces
${\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)$. When $\psi=X/r$, it is
isomorphic to the Borel f.-a. arithmetic surface
$\widetilde{{\mathcal{B}}}(r)$.

The Hermitian line bundle
$N_{{\mathcal{P}}}{\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)$ may be
identified with $(\mathbb{Z}\,\partial/\partial X,\|.\|_{\psi})$ where the
metric $\|.\|_{\psi}$ satisfies:

$\|\psi^{\prime}(0)^{-1}\partial/\partial X\|_{\psi}=1.$

Consequently:

(6.4.3) $\widehat{\rm
deg}\,N_{P}{\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)=\log|\psi^{\prime}(0)|^{-1}.$

According to the uniformization theorem, every simply connected pointed
compact Riemann surface with boundary $(V^{+},P)$ is isomorphic to
$(\overline{D}(0,1)^{+},0)$. This implies that a smooth f.-a. arithmetic
surface ${\widetilde{\mathcal{V}}}$ over ${\rm Spec\,}\mathbb{Z}$ is
isomorphic to a f.-a. surface
${\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)$ for some formal series
$\psi$ in the group:

$G_{\mathrm{for}}(\mathbb{R}):=\mathbb{R}^{\ast}X+X^{2}\mathbb{R}[[X]]$

if and only if the underlying compact Riemann with boundary is simply
connected.

#####  6.4.2. The arithmetic surfaces
${\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi)$: the pseudoconvex case

In this subsection, we want to prove that, when the arithmetic degree (6.4.3)
is negative, the ring of regular functions
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi))$ is large.
This will extend the discussion above concerning the ring
${\mathcal{O}}(\widetilde{{\mathcal{B}}}(r))$ associated to the Borel f.-a.
arithmetic surface $\widetilde{{\mathcal{B}}}(r)$ when $r<1$.

To achieve this, we will rely on the following elementary proposition:

######  Proposition 6.4.1.

Let $\psi$ be a formal series in $\mathbb{R}[[X]]$ such that:

$\psi(0)=0\quad\mbox{and}\quad\lambda:=\psi^{\prime}(0)\neq 0.$

For every integer $e\geq 1,$ there exists a formal series in
$\mathbb{Z}[[X]]$:

$\widehat{\alpha}=X^{e}+\sum_{n\geq e+1}\alpha_{n}X^{n}$

such that the coefficients of the formal series in $\mathbb{R}[[T]]$:

$\widehat{\alpha}\circ\psi^{-1}(T)=:\lambda^{-e}\;T^{e}+\sum_{n\geq
e+1}a_{n}T^{n}$

satisfy the estimates:

(6.4.4) $|a_{n}|\leq|\lambda|^{-n}/2\quad\mbox{for every $n\geq e+1$.}$

###### Proof.

For every $n\geq e+1$, the $n$-th coefficients $a_{n}$ of the formal series
$\widehat{\alpha}\circ\psi^{-1}$ may be written:

$a_{n}=\lambda^{-n}\alpha_{n}+P_{n}(\alpha_{e+1},\dots,\alpha_{n-1}),$

where $P_{n}$ denotes some polynomial with real coefficients in $n-e-1$
variables, depending of $e$ and $\psi$; in particular $P_{e+1}$ is a constant
depending only on $e$ and $\psi$. We may construct the coefficients
$(\alpha_{n})_{n\geq e+1}$ inductively by choosing $\alpha_{n}$ as an integer
such that:

$|\alpha_{n}+\lambda^{n}\,P_{n}(a_{e+1},\dots,a_{n-1},\psi_{1},\dots,\psi_{n})|\leq
1/2.$

This immediately implies the estimates (6.4.4) ∎

We will use the following straightforward consequence of the estimates
(6.4.4):

######  Corollary 6.4.2.

The radius of convergence of the series $\widehat{\alpha}\circ\psi^{-1}(T)$ is
at least $|\lambda|$. When $|\lambda|>1,$ the series
$\widehat{\alpha}\circ\psi^{-1}$ defines a complex analytic function on some
open neighborhood of $\overline{D}(0;1)$; moreover:

(6.4.5)
$\max_{z\in\overline{D}(0;1)}|\widehat{\alpha}\circ\psi^{-1}(z)|\leq|\lambda|^{-e}+\sum_{n\geq
e+1}|\lambda|^{-n}/2=\frac{1-|\lambda|^{-1}/2}{1-|\lambda|^{-1}}\,|\lambda|^{-e}.$

Corollary 6.4.2 shows that, when $|\lambda|>1$, the series $\hat{\alpha}$
defines a non-constant element $\alpha:=(\widehat{\alpha},\alpha^{{\rm an}})$
in the ring
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0;1),\psi))$. If $e$ is
chosen large enough, the image

$\alpha^{{\rm
an}}(\overline{D}(0;1)):=\widehat{\alpha}\circ\psi^{-1}(\overline{D}(0;1))$

is a compact subset of the open disc $\mathring{D}(0;1)$.

The previous construction therefore establishes the following proposition:

######  Proposition 6.4.3.

Let ${\widetilde{\mathcal{V}}}:=(\widehat{{\mathcal{V}}},(V,P,i))$ be a smooth
f.-a. arithmetic surface over $\mathbb{Z}$ such that $V$ is simply
connected. If it satisfies the condition:

(6.4.6) $\widehat{\rm deg}\,{\overline{N}}_{P}{\widetilde{\mathcal{V}}}<0,$

then the algebra ${\mathcal{O}}({\widetilde{\mathcal{V}}})$ contains a non-
constant element $\alpha:=(\widehat{\alpha},\alpha^{{\rm an}})$ such that
$\widehat{\alpha}(P)=0$ and $\alpha^{{\rm an}}(V)$ is contained in
$\mathring{D}(0;1)$.

Observe that the conclusion of Proposition 6.4.3 implies that the ring
${\mathcal{O}}({\widetilde{\mathcal{V}}})$ is large. Indeed, for $\alpha$ as
in the conclusion of Proposition 6.4.3, for every $r\in(0,1)$ such that:

$\alpha^{{\rm an}}(V)\subseteq\overline{D}(0;r),$

we may define an injective morphism of rings:

${\mathcal{O}}(\widetilde{{\mathcal{B}}}(r))\longrightarrow{\mathcal{O}}({\widetilde{\mathcal{V}}}),\quad\beta:=(\widehat{\beta},\beta^{\rm
an})\longmapsto\beta\circ\alpha:=(\widehat{\beta}\circ\widehat{\alpha},\beta^{\rm
an}\circ\alpha^{\rm an}).$

It is likely that Proposition 6.4.3 still holds for every smooth f.-a.
arithmetic surface over ${\rm Spec\,}{{\mathcal{O}}_{K}}$ — where $K$ denotes
an arbitrary number field — that satisfies the pseudoconcavity condition
(6.4.6). This would constitute an arithmetic analogue of the theorem of
Grauert concerning pseudoconvex neighborhoods of projective curves discussed
in 1.4.3 above.

#####  6.4.3. The arithmetic surfaces
${\widetilde{\mathcal{V}}}(\overline{D}(0,1),\psi)$: the pseudoconcave case

In this subsection and in the next section, we are going to show that, if
$\lambda\in\mathbb{R}$ satisfies:

$0<|\lambda|<1,$

then for most choices of the formal series $\psi$ in $\lambda
X+X^{2}\mathbb{R}[[X]],$ the algebra
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0,1),\psi)$ is as small
as possible, namely is reduced to $\mathbb{Z}$.

To formulate our results, we need to introduce some notation.

For any commutative ring $A,$ we may consider the subset:

$D(A):=X+X^{2}A[[X]]$

of the algebra of formal series $A[[X]]$. Endowed with the composition $\circ$
of formal series with vanishing constant terms, it defines a group
$(D(A),\circ)$. For every $n\in\mathbb{N}$, we may consider its normal
subgroup:

$H_{n}(A):=X+X^{n+2}A[[X]].$

The quotient group:

$D_{n}(A):=D(A)/H_{n}(A)$

may be identified with the group $X+X^{2}A[[X]]/(X^{n+2})$ of formal series
truncated at order $n+1,$ endowed with the composition. The groups $D_{n}(A)$
are nilpotent, and $D(A)$ may be identified with the limit:

$D(A)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\varprojlim_{n}D_{n}(A)$

of the following diagram of surjective morphisms of groups:

$D_{0}(A)=\\{e\\}\longleftarrow D_{1}(A)\longleftarrow\cdots\longleftarrow
D_{n}(A)\longleftarrow D_{n+1}(A)\longleftarrow\cdots.$

For every $n\in\mathbb{N},$ $D_{n}(\mathbb{R})$ is a nilpotent Lie group of
dimension $n$. We shall denote by $\mu_{n}$ the measure on $D_{n}(\mathbb{R})$
image of the Lebesgue measure by the diffeomorphism:

$\iota_{n}:\mathbb{R}^{n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}D_{n}(\mathbb{R}),\quad(a_{2},\dots,a_{n+1})\longmapsto
X+\sum_{i=2}^{n+1}a_{i}X^{i}\mod X^{n+2}.$

It is easily seen to be a left and right Haar measure on the Lie group
$D_{n}(\mathbb{R})$.

For every $n\in\mathbb{N},$ the discrete subgroup $D_{n}(\mathbb{Z})$ of
$D_{n}(\mathbb{R})$ is cocompact, and the map

$[0,1)^{n}\longrightarrow
D_{n}(\mathbb{R})/D_{n}(\mathbb{Z}),\quad(a_{2},\dots,a_{n+1})\longmapsto[X+\sum_{i=2}^{n+1}a_{i}X^{i}]$

is bijective. This shows that the measure $\bar{\mu}_{n}$ on
$D_{n}(\mathbb{R})/D_{n}(\mathbb{Z})$ deduced from $\mu_{n}$ satisfies:

$\bar{\mu}_{n}(D_{n}(\mathbb{R})/D_{n}(\mathbb{Z}))=1.$

The measure $\bar{\mu}_{n}$ is the unique probability measure on the compact
space $D_{n}(\mathbb{R})/D_{n}(\mathbb{Z})$ invariant under the left action of
$D_{n}(\mathbb{R})$.

The group:

$D(\mathbb{R})\simeq\varprojlim_{n}D_{n}(\mathbb{R}),$

as the projective limit of a countable projective system of Lie groups, is a
Polish topological group. This topology coincide with the topology induces by
the natural topology of Fréchet space on
$\mathbb{R}[[X]]\simeq\mathbb{R}^{\mathbb{N}}.$ The topological group
$D(\mathbb{R})$ contains:

$D(\mathbb{Z})\simeq\varprojlim_{n}D_{n}(\mathbb{Z})$

as a closed subgroup, and the quotient space:

$D(\mathbb{R})/D(\mathbb{Z})\simeq\varprojlim_{n}D_{n}(\mathbb{R})/D_{n}(\mathbb{Z})$

is compact. We shall denote by $\bar{\mu}$ the unique probability measure on
the compact space $D(\mathbb{R})/D(\mathbb{Z})$ whose direct image by the
projection:

$\bar{p}_{n}:D(\mathbb{R})/D(\mathbb{Z})\longrightarrow
D_{n}(\mathbb{R})/D_{n}(\mathbb{Z})$

coincides with $\bar{\mu}_{n}$ for every $n\geq 1$. The measure $\bar{\mu}$ is
easily seen to be the unique probability measure on
$D(\mathbb{R})/D(\mathbb{Z})$ that is invariant under the left action of
$D(\mathbb{R})$. This notably implies that the mesure $\bar{\mu}(V)$ of any
non-empty open subset $V$ of $D(\mathbb{R})/D(\mathbb{Z})$ is positive.

We shall say that a Borel subset $B$ of $D(\mathbb{R})$, invariant under the
right action of $D(\mathbb{Z})$ _has full measure in $D(\mathbb{R})$_ when:

$\bar{\mu}(q(B))=1,$

where $q:D(\mathbb{R})\rightarrow D(\mathbb{R})/D(\mathbb{Z})$ denotes the
quotient map.

For every $\lambda\in\mathbb{R}^{\ast},$ we shall denote:

$[\lambda]:=\lambda X\quad(\in\mathbb{R}[[X]]).$

The map $(g\mapsto[\lambda]\circ g)$ establishes a bijection from
$D(\mathbb{R})$ onto $\lambda X+X^{2}\mathbb{R}[[X]].$

Our result concerning the generic triviality of the algebra
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0,1),\psi))$ in the
pseudoconcave case reads as follows:

######  Theorem 6.4.4.

For every $\lambda\in\mathbb{R}$ such that $|\lambda|>1,$ the set:

${\mathcal{T}}_{\lambda}:=\left\\{g\in
D(\mathbb{R})\mid{\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0,1),[\lambda]\circ
g))=\mathbb{Z}\right\\}$

is a $G_{\delta}$ subset, invariant under the right action of $D(\mathbb{Z}),$
dense and of full measure in $D(\mathbb{R})$.

In other words, when $|\lambda|>1,$ the algebra
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0,1),\psi)$ contains
only constant functions for “almost all” choice of $\psi$ with
$\psi^{\prime}(0)=\lambda$, both in the sense of measure theory and of Baire
category.

The proof of Theorem 6.4.4 is the object of the next section. The techniques
used in this proof play no role in the next chapters, and this section could
be skipped at first reading.

####  6.5. Proof of the generic triviality of
${\mathcal{O}}({\widetilde{\mathcal{V}}}(\overline{D}(0,1),\psi))$ in the
pseudo-concave case

##### 6.5.1.

In order to prove Theorem 6.4.4, we need to introduce some further notation.

For every commutative ring $A$, we define a left action of the group $D(A)$ on
the algebra $A[[X]]$ by:

(6.5.1) $g.\varphi:=\varphi\circ g^{-1},$

for $g\in D(A)$ and $\varphi\in A[[X]]$. For every $e\in\mathbb{N}_{>0}$ and
$a\in A\setminus\\{0\\}$, we also define:

$O(e,a):=aX^{e}+X^{e+1}A[[X]].$

The sets $O(e,a)$ are invariant under the action of $D(A),$ and we have:

(6.5.2) $XA[[X]]\setminus\\{0\\}=\coprod_{e\in\mathbb{N}_{>0},a\in
A\setminus\\{0\\}}O(e,a).$

Observe that, when $k$ is a field of characteristic zero, the action of $D(k)$
on $Xk[[X]]\setminus\\{0\\}$ is free, and the sets $O(e,a)$ for
$(e,a)\in\mathbb{N}_{>0}\times k^{\times}$ are precisely the orbits of this
action; see 6.5.2 (b) below when $k=\mathbb{R}$.

For every $\rho\in\mathbb{R}_{+},$ we define a pseudo-norm $\|.\|_{\rho}$ on
$\mathbb{R}[[X]]$ by the formula:

$\big{\|}\sum_{i\in\mathbb{N}}a_{i}X^{i}\big{\|}_{\rho}:=\sup_{i\in\mathbb{N}}|a_{i}|\rho^{i}\quad(\in[0,+\infty]).$

Moreover, for every $R\in\mathbb{R}_{+}$, we let:

$B_{\rho}(R):=\left\\{\varphi\in\mathbb{R}[[X]]\mid\|\varphi\|_{\rho}\leq
R\right\\}.$

Observe that a formal series $\varphi\in\mathbb{R}[[X]]$ defines a function
analytic on some open neighborhood of $\overline{D}(0,|\lambda|^{-1})$ if and
only if there exists $\rho$ in $(|\lambda|^{-1},+\infty)$ such that:

$\|\varphi\|_{\rho}<+\infty.$

For $e$ in $\mathbb{N}_{>0},$ $a$ in $\mathbb{Z}\setminus\\{0\\},$ and $\rho$
and $R$ in $\mathbb{R}_{+},$ let us consider the following subset of
$D(\mathbb{R})$:

$F(e,a,\rho,R):=\left\\{g\in D(\mathbb{R})\mid g.O(e,a)(\mathbb{Z})\cap
B_{\rho}(R)\neq\emptyset\right\\}.$

Our previous observation concerning the analyticity of elements of
$\mathbb{R}[[X]]$ and the decomposition (6.5.2) when $A=\mathbb{Z}$ imply the
following description of the complement of ${\mathcal{T}}_{\lambda}$:

$\displaystyle D(\mathbb{R})\setminus{\mathcal{T}}_{\lambda}$
$\displaystyle=\bigcup_{\rho>|\lambda|^{-1}}\left\\{g\in
D(\mathbb{R})\mid\exists\,\alpha\in
g(X\mathbb{Z}[[X]]\setminus\\{0\\}),\|\alpha\|_{\rho}<+\infty\right\\}$
$\displaystyle=\bigcup_{\rho>|\lambda|^{-1},R>0}\left\\{g\in D(\mathbb{R})\mid
g(X\mathbb{Z}[[X]]\setminus\\{0\\})\cap B_{\rho}(R)\neq\emptyset\right\\}$
$\displaystyle=\bigcup_{(e,a,\rho,R)\in{\mathcal{E}}(\lambda)}F(e,a,\rho,R),$

where:

${\mathcal{E}}(\lambda):=\mathbb{N}_{>0}\times(\mathbb{Z}\setminus\\{0\\})\times(|\lambda|^{-1},+\infty)\times\mathbb{R}_{+}^{\ast}.$

In turn, this immediately shows that
$D(\mathbb{R})\setminus{\mathcal{T}}_{\lambda}$ may be written as a countable
union:

(6.5.3)
$D(\mathbb{R})\setminus{\mathcal{T}}_{\lambda}=\bigcup_{(e,a,\rho,R)\in{\mathcal{E}}^{\prime}(\lambda)}F(e,a,\rho,R),$

where:

${\mathcal{E}}^{\prime}(\lambda):=\mathbb{N}_{>0}\times(\mathbb{Z}\setminus\\{0\\})\times\left\\{|\lambda|^{-1}+1/i;i\in\mathbb{N}_{>0}\right\\}\times\mathbb{N}.$

This decomposition of $D(\mathbb{R})\setminus{\mathcal{T}}_{\lambda}$ shows
that Theorem 6.4.4 is a consequence of the following proposition and of
Baire’s theorem:

######  Proposition 6.5.1.

(a) For every $(e,a,\rho,R)$ in
$\mathbb{N}_{>0}\times(\mathbb{Z}\setminus\\{0\\})\times\mathbb{R}_{+}^{2}$,
$F(e,a,\rho,R)$ is a closed subset of $D(\mathbb{R})$, invariant under the
right action of $D(\mathbb{Z}).$

(b) If moreover $\rho>1,$ then:

$\bar{\mu}(q(F(e,a,\rho,R)))=0,$

and $F(e,a,\rho,R)$ has an empty interior in $D(\mathbb{R})$.

#####  6.5.2. Proof of Proposition 6.5.1

######

_Proof of (a)._ The invariance of $F(e,a,\rho,R)$ under the right action of
$D(\mathbb{Z})$ follows from its definition.

To prove that $F(e,a,\rho,R)$ is closed in $D(\mathbb{R}),$ observe that the
left action of $D(\mathbb{R})$ on $\mathbb{R}[[X]]$, defined above by (6.5.1),
is continuous and that, for every
$(\rho,R)\in\mathbb{R}_{+}^{\ast}\times\mathbb{R}_{+},$ $B_{\rho}(R)$ is a
compact subset of the Fréchet space $\mathbb{R}[[X]]$.

Let us consider a sequence $(g_{n})$ in $F(e,a,\rho,R)$ that admits a limit
$g$ in $D(\mathbb{R})$. There exists a sequence $(\varphi_{n})$ in
$O(e,a)(\mathbb{Z})$ such that the formal series
$g_{n}.\varphi_{n}:=\varphi_{n}\circ g_{n}^{-1}$ belong to $B_{\rho}(R)$.
After possibly passing to a subsequence, we may assume that $(\varphi_{n}\circ
g_{n}^{-1})$ admits a limit $l$ in $B_{\rho}(R)$. Then
$(\varphi_{n})=((\varphi_{n}\circ g_{n}^{-1})\circ g_{n})$ converges to
$\varphi:=l\circ g$ in $\mathbb{R}[[X]]$. Since $O(e,a)(\mathbb{Z})$ is closed
in $\mathbb{R}[[X]],$ this limit $\varphi$ belongs to $O(e,a)(\mathbb{Z})$.
Finally,

$g.\varphi:=\varphi\circ g^{-1}=l,$

belongs to $g.O(e,a)(\mathbb{Z})\cap B_{\rho}(R)$, and therefore $g$ belongs
to $F(e,a,\rho,R)$.

######

_Proof of (b)._ Observe that, for every $(e,a)$ in
$\mathbb{N}_{>0}\times(\mathbb{R}\setminus\\{0\\}),$ the map:

$I_{e,a}:D(\mathbb{R}):=X+X^{2}\mathbb{R}[[X]]\longrightarrow
O(e,a)(\mathbb{R}),\quad\psi\longmapsto a\psi^{e}$

is bijective. This already implies that the action by composition of
$D(\mathbb{R})$ on $O(e,a)(\mathbb{R})$ is free and transitive.

By truncation, for every $n\in\mathbb{N},$ the action of $D(\mathbb{R})$ on
$O(e,a)(\mathbb{R})$ defines an action of $D_{n}(\mathbb{R})$ on

$O(e,a)_{n}(\mathbb{R}):=\left(aX^{e}+X^{e+1}\mathbb{R}[[X]]\right)\\!/(X^{n+e+1}).$

Moreover, by truncation, for every $n\in\mathbb{N},$ the map $I_{e,a}$ induces
a bijection:

$I^{n}_{e,a}:D_{n}(\mathbb{R})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}O(e,a)_{n}(\mathbb{R}),$

which is clearly compatible with the action of $D_{n}(\mathbb{R})$ by
composition on $O(e,a)_{n}(\mathbb{R})$, which is also free and transitive.

We shall denote by $\mu_{n}^{e,a}$ the measure on $O(e,a)_{n}(\mathbb{R})$
image of the Lebesgue measure on $\mathbb{R}^{n}$ by the diffeomorphism:

$\iota_{n}^{e,a}:\mathbb{R}^{n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}O(e,a)_{n}(\mathbb{R}),\quad(a_{e+1},\dots,a_{e+n})\longmapsto
aX^{e}+\sum_{i=e+1}^{e+n}a_{i}X^{i}\mod X^{n+e+1}.$

######  Lemma 6.5.2.

For every $(e,a)$ in $\mathbb{N}_{>0}\times(\mathbb{R}\setminus\\{0\\})$ and
every $\varphi\in O(e,a)_{n}(\mathbb{R})$, the image of the Haar measure
$\mu_{n}$ on $D_{n}(\mathbb{R})$ by the diffeomorphism

$\bullet.\varphi:D_{n}(\mathbb{R})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}O(e,a)_{n}(\mathbb{R}),\quad
g\longmapsto g.\varphi$

is the measure $(e|a|)^{-n}\mu_{n}^{e,a}$.

###### Proof.

Since $D_{n}(\mathbb{R})$ is unimodular, this is equivalent to the fact that
the image of $\mu_{n}$ by the diffeomorphism:

$\bullet^{-1}.\varphi:D_{n}(\mathbb{R})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}O(e,a)_{n}(\mathbb{R}),\quad
g\longmapsto\varphi\circ g$

is the measure $(e|a|)^{-n}\mu_{n}^{e,a}$. In turn, this is equivalent to the
fact that the direct image by the diffeomorphism:

$J(e,a,n,\varphi):=(\iota_{n}^{e,a})^{-1}\circ(\bullet^{-1}.\varphi)\circ\iota_{n}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$

of the $n$-dimensional Lebesgue measure is $(e|a|)^{-n}$ times the
$n$-dimensional Lebesgue measure. To establish this, we shall show that the
Jacobian of $J(e,a,n,\varphi)$, namely $\det DJ(e,a,n,\varphi)$, is constant
on $\mathbb{R}^{n}$ and satisfies:

(6.5.4) $\det DJ(e,a,n,\varphi)=(ea)^{n}.$

According to the change-of-variables formula for the Lebesgue measure, this
will complete the proof.

To compute the differential of the diffeomorphism $\bullet^{-1}.\varphi$, we
may identify $D_{n}(\mathbb{R})$ and $O(e,a)_{n}(\mathbb{R})$ to
$X+X^{2}\mathbb{R}[X]_{<n}$ and to $aX^{e}+X^{e+1}\mathbb{R}[X]_{<n}$
respectively, were $\mathbb{R}[X]_{<n}$ denotes the $\mathbb{R}$-vector space
of polynomial of degree $<n$. Then the differential of $\bullet^{-1}.\varphi$
at some point $g$ of $D_{n}(\mathbb{R})$ maps an element $\delta g$ in
$X^{2}\mathbb{R}[X]_{<n}$ to:

$D(\bullet^{-1}.\varphi)(g)\delta g=(\varphi^{\prime}\circ g)\,\delta
g\in\left(X^{e+1}\mathbb{R}[[X]]\right)/(X^{n+e+1})\simeq
X^{e+1}\mathbb{R}[X]_{<n}.$

This shows that, after identifying $\mathbb{R}^{n}$ and $\mathbb{R}[X]_{<n}$
by means of the standard basis:

$(1,X,\dots,X^{n-1}),$

the differential $DJ(e,a,n,\varphi)$ at the point $\iota_{n}^{-1}(g)$ is given
by the mutiplication:

$\mathbb{R}[[X]]_{<n}\longrightarrow\mathbb{R}[[X]]_{<n},\quad
P\longmapsto\lfloor X^{-e+1}(\varphi^{\prime}\circ g)P\rfloor,$

where $\lfloor X^{-e+1}(\varphi^{\prime}\circ g)P\rfloor$ denotes the unique
representative in $\mathbb{R}[[X]]_{<n}$ of the class $\mod X^{n}$ of
$\mathbb{R}[[X]]_{<n}$. Since $X^{-e+1}(\varphi^{\prime}\circ g)$ is an
element of $ae+X\mathbb{R}[[X]],$ the determinant of this map is $(ea)^{n}$. ∎

For every $(e,a)$ in $\mathbb{N}_{>0}\times(\mathbb{Z}\setminus\\{0\\}),$ we
may introduce the following subset of $O(e,a)(\mathbb{Z})$:

$\Delta(e,a):=\Big{\\{}aX^{e}+\sum_{i\geq
e+1}a_{i}X^{i}\in\mathbb{Z}[[X]]\mid\forall i\in\mathbb{N}_{\geq e+1},0\leq
a_{i}<e|a|\Big{\\}}.$

For every $n\in\mathbb{N}$, we shall also denote by $\Delta(e,a)_{n}$ the
image of $\Delta(e,a)$ in $O(e,a)_{n}(\mathbb{R})$ by the truncation morphism.
In other words:

$\Delta(e,a)_{n}:=\Big{\\{}\Big{[}aX^{e}+\sum_{i=e+1}^{e+n}a_{i}X^{i}\Big{]}\in\mathbb{Z}[[X]]/(X^{n+e+1})\mid\forall
i\in\\{e+1,\dots,e+n\\},0\leq a_{i}<e|a|\Big{\\}}.$

######  Lemma 6.5.3.

With the previous notation, we have:

$D(\mathbb{Z}).\Delta(e,a):=\bigcup_{\gamma\in
D(\mathbb{Z})}\gamma.\Delta(a,e)=O(e,a).$

###### Proof.

For every element:

$\varphi:=aX^{e}+\sum_{i\geq e+1}a_{i}X^{i}$

in $O(e,a)(\mathbb{Z})$ and for every integer $k\geq 2$ and every
$b\in\mathbb{Z},$ the image of $\varphi$ under the action of the element
$X+bX^{k}$ of $D(\mathbb{Z})$ is:

$\displaystyle(X+bX^{k}).\varphi$
$\displaystyle:=\varphi\circ(X+bX^{k})^{\circ(-1)}$
$\displaystyle=\varphi\circ(X-bX^{k})\mod X^{e+k}$
$\displaystyle=aX^{e}+\sum_{e+1\leq i\leq
e+k-2}a_{i}X^{i}+(a_{e+k-1}-aeb)X^{e+k-1}\mod X^{e+k}.$

This identity allows one to produce an element $\gamma$ in $D(\mathbb{Z})$
such that $\varphi$ is contained in $\gamma.\Delta(a,e)$ by defining
$\gamma^{-1}$ as the limit of elements of the form

$(X+b_{k}X^{k})\circ\dots\circ(X+b_{2}X^{2}),$

with $b_{2},\dots,b_{k}$ in $\mathbb{Z}$, when $k\geq 2$ goes to infinity. ∎

According to Lemma 6.5.3, we have:

(6.5.5) $q(F(e,a,\rho,R))=q(\widetilde{F}(e,a,\rho,R)),$

where:

$\widetilde{F}(e,a,\rho,R)):=\left\\{g\in D(\mathbb{R})\mid
g.\Delta(e,a)(\mathbb{Z})\cap B_{\rho}(R)\neq\emptyset\right\\}.$

For every $n\in\mathbb{N}$, we shall denote by:

$p_{n}:D(\mathbb{R})\longrightarrow
D_{n}(\mathbb{R}):=D_{n}(\mathbb{R})/H_{n}(\mathbb{R})$

the quotient (or truncation) morphism. It fits into the following commutative
diagram:

$\textstyle{D(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n}}$$\scriptstyle{q\quad}$$\textstyle{D(\mathbb{R})/D(\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{p}_{n}}$$\textstyle{D_{n}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{n}\quad}$$\textstyle{D_{n}(\mathbb{R})/D_{n}(\mathbb{Z}).}$

Therefore the following inequalities hold:

(6.5.6)
$\bar{\mu}(q(\widetilde{F}(e,a,\rho,R)))\leq\bar{\mu}(\bar{p}_{n}^{-1}(\bar{p}_{n}\circ
q(\widetilde{F}(e,a,\rho,R))))=\bar{\mu}_{n}(\bar{p}_{n}\circ
q(\widetilde{F}(e,a,\rho,R)))\\\ =\bar{\mu}_{n}(q_{n}\circ
p_{n}(\widetilde{F}(e,a,\rho,R)))\leq\mu_{n}(p_{n}(\widetilde{F}(e,a,\rho,R))).$

Moreover the image of $O(e,a)(\mathbb{R})\cap B_{\rho}(R)$ in
$O(e,a)_{n}(\mathbb{R})$ by the truncation map is contained in:

$\iota_{n}^{e,a}\Big{(}\prod_{i=e+1}^{e+n}[-R\rho^{-i},R\rho^{-i}]\Big{)}.$

Consequently we have:

$p_{n}(\widetilde{F}(e,a,\rho,R))\subseteq\bigcup_{\varphi\in\Delta(e,a)_{n}(\mathbb{Z})}\Big{\\{}g\in
D_{n}(\mathbb{R})\mid
g.\varphi\in\iota_{n}^{e,a}(\prod_{i=e+1}^{e+n}[-R\rho^{-i},R\rho^{-i}])\Big{\\}},$

and therefore:

$\mu_{n}(p_{n}(\widetilde{F}(e,a,\rho,R)))\leq\sum_{\varphi\in\Delta(e,a)_{n}(\mathbb{Z})}\mu_{n}\Big{(}\big{\\{}g\in
D_{n}(\mathbb{R})\mid
g.\varphi\in\iota_{n}^{e,a}(\prod_{i=e+1}^{e+n}[-R\rho^{-i},R\rho^{-i}])\Big{\\}}\Big{)}.$

Using Lemma 6.5.2, we finally obtain:

(6.5.7)
$\begin{split}\mu_{n}(p_{n}(\widetilde{F}(e,a,\rho,R)))&\leq\big{|}\Delta(e,a)_{n}(\mathbb{Z})\big{|}\,\,(e|a|)^{-n}\,\prod_{i=e+1}^{e+n}(2R\rho^{-i})\\\
&\quad\quad=(2R)^{n}\rho^{-\sum_{i=e+1}^{e+n}i}=(2R)^{n}\rho^{-(n+2e+2)(n-1)/2}.\end{split}$

If $\rho>1,$ the right-hand side of (6.5.7) goes to $0$ when $n$ goes to
infinity. Together with (6.5.5) and (6.5.6), this establishes the vanishing of
$\bar{\mu}(q(F(e,a,\rho,R)))$ when $\rho>1$. In turn this vanishing implies
that the interior $\mu(q(F(e,a,\rho,R)))$ in $D(\mathbb{R})/D(\mathbb{Z})$ is
empty, as observed above in 6.4.3.

###  Chapter 7 Maps from formal-analytic arithmetic surfaces to arithmetic
schemes

In this chapter, we define morphisms from a smooth f.-a. surface
${\widetilde{\mathcal{V}}}$ to an arithmetic scheme111that is, a separated
scheme of finite type over ${\rm Spec\,}\mathbb{Z}$. $X$ over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$, for $K$ some number field, and we discuss some
basic constructions involving them.

When $X$ is a normal arithmetic surface, we investigate the relations of the
rudimentary arithmetic intersection theory on ${\widetilde{\mathcal{V}}}$
introduced in Section 6.2 with the more classical arithmetic intersection
theory on quasi-projective arithmetic surfaces, as presented in Chapters 3 and
4. These relations involve the Archimedean overflow
$\mathrm{Ex}(\alpha:(V,P)\rightarrow N)$ studied in Chapter 5 and its
counterpart “at finite places”
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$.

Finally we extend these results to the situation where the morphisms from
${\widetilde{\mathcal{V}}}$ to $X$ are replaced by suitably defined
“meromorphic maps.”

We denote by $K$ a number field, and by ${{\mathcal{O}}_{K}}$ its ring of
integers.

####  7.1. Morphisms from formal-analytic arithmetic surfaces to
${{\mathcal{O}}_{K}}$-schemes

#####  7.1.1. Definitions and basic properties

######

If
${\widetilde{\mathcal{V}}}:=({\widehat{\mathcal{V}}},(V_{\sigma},O_{\sigma},\iota_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$
denotes a smooth f.-a. arithmetic surface over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$ as defined in 6.1.1, and if $X$ is a separated
scheme of finite type over ${{\mathcal{O}}_{K}}$, we may define a _morphism_

$f:{\widetilde{\mathcal{V}}}\longrightarrow X$

_over_ ${\rm Spec\,}{{\mathcal{O}}_{K}}$ as a pair:

$f:=(\widehat{f},(f_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

where:

$\widehat{f}:{\widehat{\mathcal{V}}}\longrightarrow X$

is a morphism of formal schemes over ${{\mathcal{O}}_{K}}$ and, for every
complex embedding $\sigma$ of $K$,

$f_{\sigma}:V_{\sigma}\longrightarrow X_{\sigma}(\mathbb{C})$

is a complex analytic map.

These data are moreover assumed to be compatible with the gluing data
$(\iota_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}$. Namely, for every
embedding $\sigma:K\hookrightarrow\mathbb{C},$ the morphisms from the smooth
formal curve ${\widehat{\mathcal{V}}}_{\sigma}$ to the complex scheme
$X_{\sigma}$ deduced from $\widehat{f}$ (by the base change
$\sigma:{{\mathcal{O}}_{K}}\rightarrow\mathbb{C}$) and from $f_{\sigma}$ (by
considering its “formal germ” at $P_{\sigma}$) are required to coincide; see
[Bos20, 10.6.3].

As a consequence, these data are compatible with complex conjugation.

We shall use the notation:

$f^{\rm an}:V^{+}_{\mathbb{C}}\longrightarrow X(\mathbb{C})$

for the complex analytic map from:

$V^{+}_{\mathbb{C}}:=\coprod_{\sigma:K\hookrightarrow\mathbb{C}}V^{+}_{\sigma}$

to:

$X(\mathbb{C})=\coprod_{\sigma:K\hookrightarrow\mathbb{C}}X_{\sigma}(\mathbb{C})$

defined by the maps $f_{\sigma}$.

If $E$ (resp. ${\overline{E}}:=(E,\|.\|)$) is a vector bundle over $X$ (resp.
a Hermitian vector bundle over $X$, supposed to have a reduced generic fiber
$X_{K}$), we may form its pull-back $f^{\ast}E$ (resp.
$f^{\ast}{\overline{E}}$), defined by the “formal” pull-back
$\widehat{f}^{\ast}E$ on ${\widehat{\mathcal{V}}}$ and the “complex analytic”
pull-back $f_{\sigma}^{\ast}E_{\sigma}$ (resp.
$f_{\sigma}^{\ast}{\overline{E}}_{\sigma}$) on $V_{\sigma}^{+}$, and some
canonical gluing data; see [Bos20, 10.6.3 ].

######

To every morphism $f:{\widetilde{\mathcal{V}}}\longrightarrow X$ as in 7.1.1,
we may attach the Zariski closures of the images of the morphisms of ringed
spaces:

$\widehat{f}:{\widehat{\mathcal{V}}}\longrightarrow
X,\quad\widehat{f}_{K}:{\widehat{\mathcal{V}}}_{K}\longrightarrow
X_{K},\quad\widehat{f}_{\sigma}:{\widehat{\mathcal{V}}}_{\sigma}\longrightarrow
X_{\sigma},\quad\mbox{and}\quad f_{\sigma}:V_{\sigma}^{+}\longrightarrow
X_{\sigma},$

which we shall denote by $\overline{\mathrm{im}\widehat{f}}$,
$\overline{\mathrm{im}\widehat{f}_{K}}$,
$\overline{\mathrm{im}\widehat{f}_{\sigma}}$ and
$\overline{\mathrm{im}f_{\sigma}}$. Each of these is defined as the smallest
closed subscheme of the range of the morphism through which the morphism
factors; see [Bos20, 10.6.4]. These schemes are integral. Moreover,
$\overline{\mathrm{im}\widehat{f}}$ is flat over ${{\mathcal{O}}_{K}}$,
$\overline{\mathrm{im}\widehat{f}_{K}}$ is geometrically irreducible over $K$,
and the following relations hold:

$\overline{\mathrm{im}\widehat{f}_{K}}=\big{(}\overline{\mathrm{im}\widehat{f}}\big{)}_{K}\quad\mbox{and}\quad\overline{\mathrm{im}f_{\sigma}}=\overline{\mathrm{im}\widehat{f}_{\sigma}}=\big{(}\overline{\mathrm{im}\widehat{f}}\big{)}_{\sigma}.$

We say that $f$ is a _constant_ morphism when the following equivalent
conditions are satisfied:

$\dim\overline{\mathrm{im}\widehat{f}}=1,\quad\dim\overline{\mathrm{im}\widehat{f}_{K}}=0,\quad\mbox{or}\quad\dim\overline{\mathrm{im}f_{\sigma}}=0.$

This holds precisely when $f$ factors through the morphism:

$\pi_{{\widetilde{\mathcal{V}}}}:{\widetilde{\mathcal{V}}}\longrightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}},$

or equivalently, when $f$ may written as
$f=Q\circ\pi_{{\widetilde{\mathcal{V}}}}$ for some ${{\mathcal{O}}_{K}}$-point
$Q$ of $X$. In concrete terms, this holds if and only if the morphism:

$\widehat{f}_{K}:{\widetilde{\mathcal{V}}}_{K}\simeq\mathrm{Spf}K[[T]]\longrightarrow
X_{K}$

is constant in the obvious sense.

The ${{\mathcal{O}}_{K}}$-morphism from ${\widetilde{\mathcal{V}}}$ to the
affine line $\mathbb{A}^{1}_{{{\mathcal{O}}_{K}}}$ may be identified with the
elements of the ${{\mathcal{O}}_{K}}$-algebra
$\mathcal{O}({\widetilde{\mathcal{V}}})$, and the morphism:

$f:{\widetilde{\mathcal{V}}}\longrightarrow\mathbb{A}^{1}_{{{\mathcal{O}}_{K}}}$

defined by an element $f$ of $\mathcal{O}({\widetilde{\mathcal{V}}})$ is
constant if and only if $f$ belongs to the subring ${{\mathcal{O}}_{K}}$ of
$\mathcal{O}({\widetilde{\mathcal{V}}}).$

Finally, we say that the image of an ${{\mathcal{O}}_{K}}$-morphism:

$f:{\widetilde{\mathcal{V}}}\longrightarrow X$

is _algebraic_ when the following equivalent conditions hold:

$\dim\overline{\mathrm{im}\widehat{f}}\leq
2,\quad\dim\overline{\mathrm{im}\widehat{f}_{K}}\leq
1,\quad\mbox{or}\quad\dim\overline{\mathrm{im}f_{\sigma}}\leq 1.$

#####  7.1.2. Morphisms and sections of vector bundles

######

As in 7.1.1, let us consider a scheme $X$ separated and of finite type over
${{\mathcal{O}}_{K}}$. Let us also assume that $X$ is reduced and that the
structure morphism $\pi_{X}:X\rightarrow{\rm Spec\,}{{\mathcal{O}}_{K}}$ is
proper and flat.

For every Hermitian vector bundle $\overline{E}=(E,\|.\|)$ over $X$ — defined
by a vector bundle $E$ on $X$ and a continuous metric $\|.\|$ on the complex
analytic vector bundle $E^{\mathrm{an}}_{\mathbb{C}}$ on:

$X(\mathbb{C})=\coprod_{\sigma:K\hookrightarrow\mathbb{C}}X_{\sigma}(\mathbb{C})$

— the direct image $\pi_{X*}E$ is a vector bundle over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$. Moreover, for every embedding
$\sigma:K\hookrightarrow\mathbb{C}$, the complex vector space:

$(\pi_{X*}E)_{\sigma}\simeq\Gamma(X_{\sigma},E_{\sigma})$

may be endowed with the norm $\|.\|_{\infty,\sigma}$ defined by:

$\|s\|_{\infty,\sigma}:=\sup_{x\in X_{\sigma}(\mathbb{C})}\|s(x)\|.$

In general, the norm $\|.\|_{\infty,\sigma}$ is not a Hermitian norm. However
we may consider the _John norm_ $\|.\|_{J,\sigma}$ on
$\Gamma(X_{\sigma},E_{\sigma})$ attached to $\|.\|_{\infty,\sigma}$, namely,
the smallest Hermitian norm such that

$\|.\|_{\infty,\sigma}\leq\|.\|_{J,\sigma};$

see [Bos20, Appendix F]. This construction defines a Hermitian vector bundle
over ${\rm Spec\,}{{\mathcal{O}}_{K}}$:

$\pi^{J}_{X\ast}\widetilde{\overline{E}}:=(\pi_{X*}E,(\|.\|_{J,\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}).$

The following proposition is a straightforward consequence of the definitions:

######  Proposition 7.1.1 (compare to [Bos20, 10.8.1]).

Let ${\widetilde{\mathcal{V}}}$ be a smooth f.-a. arithmetic surface over
${{\mathcal{O}}_{K}}$, let $\mu$ be a ${\mathcal{C}}^{\infty}$ positive volume
form on $V_{\mathbb{C}}$, invariant under complex conjugation, and let:

$f:{\widetilde{\mathcal{V}}}\longrightarrow X$

be a morphism over ${{\mathcal{O}}_{K}}$.

For every Hermitian vector bundle $\overline{E}$ over $X$, the pull-back of
sections of $E$ by $f$ defines a morphism:

$\varphi_{E}:\pi^{J}_{X\ast}\widetilde{\overline{E}}\longrightarrow\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}f^{*}\overline{E}$

of (pro-)Hermitian vector bundles over ${\rm Spec\,}{{\mathcal{O}}_{K}}$. If
moreover:

$\int_{V_{\sigma}}\mu\leq 1$

for every embedding $\sigma:K\hookrightarrow\mathbb{C}$, then the Archimedean
norms of $\varphi_{E}$ are bounded above by $1$.

######

If $\alpha:X\rightarrow Y$ is morphism between two separated
${{\mathcal{O}}_{K}}$-schemes of finite type, and if

$f:=(\widehat{f},(f_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\longrightarrow
X$

is a morphism over ${{\mathcal{O}}_{K}}$ from a smooth f.-a. arithmetic
surface ${\widetilde{\mathcal{V}}}$ to $X$, then we may form the composition
$\alpha\circ f$ from ${\widetilde{\mathcal{V}}}$ to $Y$:

$\alpha\circ f:=(\alpha\circ\widehat{f},(\alpha_{\sigma}\circ
f_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\longrightarrow
Y.$

This construction satisfies obvious compatibilities with the pull-back of
vector bundle, and with the pull-back of sections of vector bundles discussed
in 7.1.2.

####  7.2. Morphisms to arithmetic surfaces, arithmetic intersection numbers,
and overflow

In this section, we consider a smooth f.-a. arithmetic surface
${\widetilde{\mathcal{V}}}$ over ${{\mathcal{O}}_{K}}$ as defined in 6.1.1
above, an integral normal arithmetic surface:

$\pi_{X}:X\longrightarrow{\rm Spec\,}{{\mathcal{O}}_{K}},$

and a morphism over ${\rm Spec\,}{{\mathcal{O}}_{K}}$:

$\alpha:=(\widehat{\alpha},(\alpha_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\longrightarrow
X,$

as defined in 7.1.1.

We shall denote by:

$P:{\rm Spec\,}{{\mathcal{O}}_{K}}\longrightarrow{\widehat{\mathcal{V}}}$

the canonical section of the structural morphism of ${\widehat{\mathcal{V}}}$:

$\pi_{{\widehat{\mathcal{V}}}}:{\widehat{\mathcal{V}}}\longrightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}},$

and by:

$Q:=\widehat{\alpha}\circ P:{\rm Spec\,}{{\mathcal{O}}_{K}}\longrightarrow X$

the ${{\mathcal{O}}_{K}}$-point of $X$ defined as the image of $P$ by
$\widehat{\alpha}$. The existence of this ${{\mathcal{O}}_{K}}$-point
guarantees that $X_{K}$ is geometrically irreducible over $K$, and that the
fibers of $\pi_{X}$ are geometrically connected when moreover $X$ is
projective.

We shall also assume that $\alpha$ is non-constant, and we shall denote by
$e(\alpha)$ the ramification index of the morphism of smooth (formal) curves
over $K$:

$\widehat{\alpha}_{K}:{\widehat{\mathcal{V}}}_{K}\longrightarrow X_{K}.$

#####  7.2.1. The map
$\alpha_{\ast}:{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})\rightarrow{\overline{Z}}^{1}_{c}(X)$

To any Arakelov divisor
$(D,g)=(D,(g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}}),$ we may attach its _direct
image by the morphism_ $\alpha:{\widetilde{\mathcal{V}}}\rightarrow X$,
defined as:

$\alpha_{\ast}(D,g):=({\widehat{\alpha}}_{\ast}D,(\alpha_{\sigma\ast}g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}).$

It is straightforward that it is an Arakelov divisor in
${\overline{Z}}_{c}^{1}(X)$, and that this construction attaches a morphism of
$\mathbb{Z}$-modules:

$\alpha_{\ast}:{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})\longrightarrow{\overline{Z}}^{1}_{c}(X)$

to any non-constant morphism $\alpha:{\widetilde{\mathcal{V}}}\rightarrow X$
over ${\rm Spec\,}{{\mathcal{O}}_{K}}$.

Observe that the “arithmetic part” of this construction — namely the map
${\widehat{\alpha}}_{\ast}$ that maps a divisor supported by
$|{\widehat{\mathcal{V}}}|$ to its direct image in $X$ — is nothing more than
the map that sends a multiple $nP$ of the section $P$, for some
$n\in\mathbb{Z}$, to the multiple $nQ$ of the ${{\mathcal{O}}_{K}}$-point
$Q:=\alpha^{\rm an}(P)$ of $X$ image of $P$ by $\alpha^{\rm an}$.

This construction satisfies the following compatibility with the direct image
functoriality for the Arakelov divisors on quasi-projective arithmetic
surfaces:

######  Proposition 7.2.1.

For every ${{\mathcal{O}}_{K}}$-morphisms $f:X\rightarrow X^{\prime},$ from
$X$ to an integral normal arithmetic surface $X^{\prime}$ over ${\rm
Spec\,}{{\mathcal{O}}_{K}},$ and every Arakelov divisor $(D,g)$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, the following equality
holds in ${\overline{Z}}^{1}_{c}(X^{\prime})$:

(7.2.1) $(f\circ\alpha)_{\ast}(D,g)=f_{\ast}\alpha_{\ast}(D,g).$

The arithmetic intersection theory on the f.-a. arithmetic surface
${\widetilde{\mathcal{V}}}$ introduced in Subsection 6.2.3 and the “classical”
arithmetic intersection theory on the normal quasi-projective arithmetic
surface $X$ developed in Chapters 3 and 4 are related by the following
projection formula:

######  Proposition 7.2.2.

Let $\overline{L}:=(L,(\|.\|_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$
be a Hermitian line bundle on $X$, defined by Hermitian metrics
$\|.\|_{\sigma})$ of regularity $\mathcal{C}^{{\mathrm{b}\Delta}}$. For every
Arakelov divisor $(D,g)$ in
${\overline{Z}}_{c}^{1}({\widetilde{\mathcal{V}}}),$ the following equality of
arithmetic intersection numbers holds:

(7.2.2)
$\alpha^{*}{\overline{L}}\cdot(P,g)={\overline{L}}\cdot\alpha_{*}(P,g).$

Propositions 7.2.1 and 7.2.2 are straightforward consequences of the
definitions and of the basic properties of direct images of (Green) functions.

#####  7.2.2. The invariant
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$

######

In this paragraph, we assume that $X$ is regular. Then
$Q:=\widehat{\alpha}(P)$ is an effective Cartier divisor in $X$, and its
inverse image by $\widehat{\alpha}$ defines an effective (Cartier) divisor in
${\widehat{\mathcal{V}}}$:

$\widehat{\alpha}^{\ast}(Q)=\widehat{\alpha}^{\ast}(\widehat{\alpha}(P)).$

The divisor $P$ appears with multiplicity $e(\alpha)$ in this divisor, and we
may write:

(7.2.3) $\widehat{\alpha}^{*}(Q)=e(\alpha)P+R,$

where $R$ is an effective Cartier divisor in ${\widehat{\mathcal{V}}}$ that
intersects $P$ properly.

In particular, the intersection $R\cdot P$ is a well-defined effective
$0$-cycle with support on the arithmetic curve $P$; namely:

(7.2.4) $R\cdot P:=\sum_{x\in P_{0}}n_{x}x,$

where $P_{0}$ denotes the set of closed points of the scheme $P$ and $n_{x}$
the length of the $\mathcal{O}_{{\widehat{\mathcal{V}}},x}$-module
$\mathcal{O}_{R\cap P}$. The multiplicity $n_{x}$ is positive if and only if
the closed point $x$ belongs to the support $|R|$ of $R$.

The $0$-cycle $R\cdot P$ has a well-defined arithmetic degree:

(7.2.5) $\widehat{\rm deg}\,R\cdot P=\sum_{x\in
P_{0}}n_{x}\log|\kappa(x)|=\log|\mathcal{O}(R\cap P)|,$

where $\kappa(x)$ denotes the residue field of $x$ and $\mathcal{O}(R\cap P)$
is the ring of regular functions on the $0$-dimensional subscheme $R\cap P$ of
$P$. The arithmetic degree $\widehat{\rm deg}\,R\cdot P$ is non-negative and
vanishes if and only if $R=0$.

######  Example 7.2.3.

Assume that ${\widehat{\mathcal{V}}}=\mathrm{Spf\,}{{\mathcal{O}}_{K}}[[T]]$
and $X=\mathbb{A}^{1}_{{{\mathcal{O}}_{K}}}$. Then $\widehat{\alpha}$
identifies with a formal series in ${{\mathcal{O}}_{K}}[[T]]$, which may be
written:

$\widehat{\alpha}=a_{0}+\sum_{i\geq e}a_{i}T^{i}$

with $e\in\mathbb{N}_{>0}$ and $a_{e}\neq 0.$ Then $Q$ is the divisor
$(X-a_{0}=0)$ in $\mathbb{A}^{1}_{{{\mathcal{O}}_{K}}}$, the ramification
degree of $\widehat{\alpha}_{K}$ is $e$, and $R$ is the divisor of:

$T^{-e}(\widehat{\alpha}-a_{0})=\sum_{k\geq 0}a_{k+e}T^{k}$

in $\mathrm{Spf\,}{{\mathcal{O}}_{K}}[[T]]$. The cycle $R\cdot P$ coincides
with the divisor of $a_{e}$ in $P={\rm Spec\,}{{\mathcal{O}}_{K}}$, and
therefore:

$\widehat{\rm deg}\,R\cdot P=\log|N_{K/\mathbb{Q}}\,a_{e}|.$

The $0$-cycle $R\cdot P$, or equivalently the arithmetic degree $\widehat{\rm
deg}\,R\cdot P$, vanishes if and only if $a_{e}$ is a unit in
${{\mathcal{O}}_{K}}$.

######

When the arithmetic surface $X$ is no longer assumed to be regular, but only
normal, the previous construction still makes sense with the following
modifications.

The divisor $Q$ in $X$ may not be a Cartier divisor, but it is always
$\mathbb{Q}$-Cartier; see 0.6.3. If $N$ denotes a positive integer such that
$NQ$ is a Cartier divisor in $X,$ then we may define
${\widehat{\alpha}}^{\ast}(Q)$ as the effective $\mathbb{Q}$-divisor:

$\widehat{\alpha}^{\ast}(Q):=\frac{1}{N}\widehat{\alpha}^{\ast}(NQ).$

Then we may define $R$ as an effective $\mathbb{Q}$-divisor in
${\widehat{\mathcal{V}}}$ by (7.2.3), and $R\cdot P$ as an effective $0$-cycle
with $\mathbb{Q}$-coefficients supported by $P$. We may finally define
$\widehat{\rm deg}\,R\cdot P$ by (7.2.4) and by the first equality in (7.2.5).

######  Definition 7.2.4.

With the above notation, we define:

(7.2.6)
${\mathrm{Ex}}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}:=\widehat{\rm deg}\,R\cdot P.$

In other words, we have:

(7.2.7)
${\mathrm{Ex}}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}=\widehat{\rm
deg}\,\big{(}\widehat{\alpha}^{\ast}(\widehat{\alpha}(P))-e(\alpha)P\big{)}\cdot
P.$

The invariant
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$
constitutes an arithmetic counterpart of the Archimedean overflow invariant
${\mathrm{Ex}}(\alpha:V\rightarrow X)$ introduced in Chapter 5; compare for
instance (7.2.7) and the expression (5.2.3) for
${\mathrm{Ex}}(\alpha:V\rightarrow X)$ in terms of $\ast$-product.

The invariant
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$ is a
non-negative real number in $\mathbb{Q}^{\ast}_{+}\log\mathbb{Z}_{>0}$.
However, there are instances where we may guarantee that the invariant
$\mathrm{Ex}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$ is of the
form:

$\mathrm{Ex}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)=\log n$

for some positive integer $n$ even when $X$ is not assumed to be regular. This
is an immediate consequence of the following statement.

######  Proposition 7.2.5.

Assume that $\widehat{\alpha}$ is quasi-finite, namely, that the fibers of
$\widehat{\alpha}$ over closed points of $X$ are finite. Let $D$ be an
effective Weil divisor on $X$ and let $N$ be a positive integer such that $ND$
is Cartier. Then there exists a positive integer $n$ such that

$N^{-1}\widehat{\rm deg}\,\widehat{\alpha}^{*}(ND)\cdot P=\log n.$

In particular, if $N^{-1}\widehat{\rm deg}\,\widehat{\alpha}^{*}(ND)\cdot
P\neq 0,$ then $N^{-1}\widehat{\rm deg}\,\widehat{\alpha}^{*}(ND)\cdot
P\geq\log 2.$

To prove Proposition 7.2.5, it is enough to show that the effective
$\mathbb{Q}$-divisor $N^{-1}\widehat{\alpha}^{*}(ND)$ is actually a divisor,
namely, that it has integral coefficients. In turn, this is equivalent to
showing that if $E$ is an irreducible Cartier divisor on
${\widehat{\mathcal{V}}}$, that appears as a component of the effective
Cartier divisor $\widehat{\alpha}^{*}(ND)$, then the multiplicity of $E$ in
$\widehat{\alpha}^{*}(ND)$ is a multiple of $N$. If $E=P,$ the result is
clear, so we may assume that $E$ is not $P$.

In order to simplify the argument we will only prove that the multiplicity of
$E$ in $\widehat{\alpha}^{*}(NE)$ is larger than $N$. This still proves that
the intersection number $N^{-1}\widehat{\rm
deg}\,\widehat{\alpha}^{*}(ND)\cdot P$ is at least equal to $\log 2$ when it
is nonzero, which is the only part of the statement that we will use in this
monograph – see 9.1.3 below. We leave the derivation of the more precise
statement to the interested reader.

###### Proof.

Let $\widehat{Z}$ be the preimage of $D$ in ${\widehat{\mathcal{V}}}$ and let
$D^{\prime}$ be the associated effective Cartier divisor in
${\widehat{\mathcal{V}}}$ obtained by discarding the lower-dimensional
components and the embedded points of $\widehat{Z}$. By construction, the
schematic image of $D^{\prime}$ is contained in $D$ and $E$ is a component of
$D^{\prime}$. The schematic image of $E$ in $X$ is purely one-dimensional in
$X$ since $\widehat{\alpha}$ is quasi-finite. In particular, the schematic
image of $NE$ is contained in $ND$, so that $NE$ is a component of
$\widehat{\alpha}^{*}(ND)$. ∎

We shall call
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X)$ the
_overflow_ or _excess_ of the morphism
$\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow X$. It clearly depends
only on the morphism from ${\widehat{\mathcal{V}}}$ to the formal completion
$\widehat{X}_{Q}$ of $X$ along its section $Q=\widehat{\alpha}(P)$ defined by
$\widehat{\alpha}$, and accordingly we will also denote it by
${\mathrm{Ex}}(\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow\widehat{X}_{Q}).$

#####  7.2.3. The self-intersections of $\alpha_{\ast}(P,g)$ and
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$

######

Let $g:=(g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}$ be a family of Green
functions for the points $P_{\sigma}$ in the Riemann surfaces $V^{+}_{\sigma}$
such that $(P,g)$ is an Arakelov divisor in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$. In other words, for every
field embedding $\sigma:K\hookrightarrow\mathbb{C},$ $g_{\sigma}$ is a Green
function with ${\mathcal{C}}^{{\mathrm{b}\Delta}}$ regularity that vanishes on
$V^{+}_{\sigma}\setminus\mathring{V}_{\sigma}$, and the family
$(g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}$ is invariant under complex
conjugation.

To each of the Green functions $g_{\sigma}$ is associated a capacitary metric
$\|.\|_{g_{\sigma}}^{\mathrm{cap}}$ on the complex line :

$T_{P_{\sigma}}V_{\sigma}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(N_{P}{\widehat{\mathcal{V}}})_{\sigma},$

as defined in 3.2.1, and we may consider the Hermitian line bundle

${\overline{N}}_{P,g}{\widehat{\mathcal{V}}}:=(N_{P}{\widehat{\mathcal{V}}},(\|.\|_{g_{\sigma}}^{\mathrm{cap}})_{\sigma:K\hookrightarrow\mathbb{C}})$

over $P\simeq|{\widehat{\mathcal{V}}}|.$

The following proposition expresses the self-intersection of the direct image
$\alpha_{\ast}(P,g)$ in ${\overline{Z}}^{1}_{c}(X)$ in terms of the Arakelov
degree of the metrized normal bundle
${\overline{N}}_{P,g}{\widehat{\mathcal{V}}}$ and of the invariants
${\mathrm{Ex}}({\widehat{\alpha}}:{\widehat{\mathcal{V}}}\rightarrow X)$ and
${\mathrm{Ex}}(\alpha_{\sigma},g_{\sigma})$ attached to the formal and
Archimedean components ${\widehat{\alpha}}$ and $\alpha_{\sigma}$ of the
morphism $\alpha$.

######  Proposition 7.2.6.

With the notation above, the following equality holds:

(7.2.8)
$\alpha_{\ast}(P,g)\cdot\alpha_{\ast}(P,g)=e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P,g}{\widehat{\mathcal{V}}}+\mathrm{Ex}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\mathrm{Ex}(\alpha_{\sigma},g_{\sigma}).$

Proposition 7.2.6 will be a consequence of the following lemma:

######  Lemma 7.2.7.

Let $Q:={\widehat{\alpha}}(P)$ be the ${{\mathcal{O}}_{K}}$-point of $X$ image
of $P$ by ${\widehat{\alpha}}$, and let $N$ be a positive integer such the
divisor $NQ$ in $X$ is Cartier. Then the following equality holds:

(7.2.9) $N^{-1}\widehat{\rm
deg}\,Q^{\ast}\overline{\mathcal{O}}(NQ,N\alpha^{\rm
an}_{\ast}g)=e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P,g}{\widehat{\mathcal{V}}}+\mathrm{Ex}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}\\\
+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\int_{V_{\sigma}}g_{\sigma}\,\delta_{\alpha^{\mathrm{an}}_{\sigma*}(Q_{\sigma})-eP_{\sigma}}.$

###### Proof of Lemma 7.2.7.

With the notation introduced in 7.2.2, we have the equality of Cartier
divisors in ${\widehat{\mathcal{V}}}$:

(7.2.10) ${\widehat{\alpha}}^{\ast}(NQ)=e(\alpha)NP+NR,$

and therefore an isomorphism of line bundles over ${\widehat{\mathcal{V}}}$:

(7.2.11)
$I:{\widehat{\alpha}}^{\ast}{\mathcal{O}}_{X}(NQ)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{{\widehat{\mathcal{V}}}}(e(\alpha)NP)\otimes{\mathcal{O}}_{{\widehat{\mathcal{V}}}}(NR).$

Recall that $NR$ is an effective Cartier divisor in ${\widehat{\mathcal{V}}}$
such that $|NR|\cap|{\widehat{\mathcal{V}}}|$ is a finite set of closed
points. In particular $P\cap NR$ is a well-defined effective divisor in $P$.

By restriction to the section $P,$ the isomorphism $I$ this defines an
isomorphism of line bundles over ${\rm Spec\,}{{\mathcal{O}}_{K}}$:

(7.2.12) $I_{\mid
P}:Q^{\ast}{\mathcal{O}}_{X}(NQ)=P^{\ast}{\widehat{\alpha}}^{\ast}{\mathcal{O}}_{{\widehat{\mathcal{V}}}}(NQ)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}P^{\ast}{\mathcal{O}}_{\widehat{\mathcal{V}}}\big{(}e(\alpha)NP\big{)}\otimes
P^{\ast}{\mathcal{O}}_{\widehat{\mathcal{V}}}(NR)\\\
=P^{\ast}N_{P}{\widehat{\mathcal{V}}}^{\otimes e(\alpha)N}\otimes
P^{\ast}{\mathcal{O}}_{P}(P\cap NR).$

For every field embedding $\sigma:K\hookrightarrow\mathbb{C},$ the isomorphism
$I_{\mid P}$ induces an isomorphism of complex lines:

$I_{\mid P,\sigma}:\big{(}T_{Q_{\sigma}}X_{\sigma}\big{)}^{\otimes
N}\simeq{\mathcal{O}}_{X_{\sigma}}(NQ_{\sigma})_{\mid
Q_{\sigma}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(T_{Q_{\sigma}}X_{\sigma})^{\otimes
e(\alpha)N}.$

Its norm, with respect to the Hermitian metrics
$\|.\|_{\alpha^{\mathrm{an}}_{\sigma,*}g_{\sigma}}^{\otimes N}$ and
$(\|.\|^{\mathrm{cap}}_{g_{\sigma}})^{\otimes e(\alpha)N}$ is readily seen to
be:

$\|I_{\mid
P,\sigma}\|=\exp\Big{(}-N\int_{V_{\sigma}}g_{\sigma}\,\delta_{\alpha^{\mathrm{an}}_{\sigma,*}(Q_{\sigma})-eP_{\sigma}}\Big{)}.$

This implies the following relation between Arakelov degrees:

$\displaystyle\widehat{\rm
deg}\,Q^{\ast}\overline{\mathcal{O}}(NQ,N\alpha^{\rm an}_{\ast}g)$
$\displaystyle=\widehat{\rm
deg}\,(P^{*}\overline{N}_{P,g}{\widehat{\mathcal{V}}})^{\otimes
e(\alpha)N}+\widehat{\rm deg}\,(NR)\cdot
P-\sum_{\sigma:K\hookrightarrow\mathbb{C}}\log\|I_{\mid P,\sigma}\|$
$\displaystyle=\widehat{\rm
deg}\,(P^{*}\overline{N}_{P,g}{\widehat{\mathcal{V}}})^{\otimes
e(\alpha)N}+\widehat{\rm deg}\,(NR)\cdot
P+N\sum_{\sigma:K\hookrightarrow\mathbb{C}}\int_{V_{\sigma}}g_{\sigma}\delta_{\alpha^{\mathrm{an}}_{\sigma,*}(Q_{\sigma})-eP_{\sigma}}.$

Using the additivity of the map: $\widehat{\rm deg}\,:\overline{{\rm
Pic}}({\rm Spec\,}{{\mathcal{O}}_{K}})\rightarrow\mathbb{R}$ and dividing by
$N$, this becomes the equality (7.2.9). ∎

###### Proof of Proposition 7.2.6.

According to the definition (3.3.4) of the Arakelov intersection pairing on
$X$, we have:

$\displaystyle\alpha_{\ast}(P,g)\cdot\alpha_{\ast}(P,g)$
$\displaystyle=N^{-1}(NQ,N\alpha_{\ast}^{\rm an}g)\cdot(Q,\alpha_{\ast}^{\rm
an}g))$ $\displaystyle=N^{-1}\widehat{\rm
deg}\,Q^{\ast}\overline{\mathcal{O}}(NQ,N\alpha^{\rm
an}_{\ast}g)+N^{-1}\int_{X(\mathbb{C})}\alpha^{\rm
an}_{\ast}g\,\,\omega(N\alpha^{\rm an}_{\ast}g)$
$\displaystyle=N^{-1}\widehat{\rm
deg}\,Q^{\ast}\overline{\mathcal{O}}(NQ,N\alpha^{\rm
an}_{\ast}g)+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\int_{X_{\sigma}(\mathbb{C})}\alpha_{\sigma\ast}g_{\sigma}\,\alpha_{\sigma\ast}\omega(g_{\sigma}).$

Together with the equality (7.2.9) and the definition (5.1.1) of the
invariants $\mathrm{Ex}(\alpha_{\sigma},g_{\sigma})$, this establishes
(7.2.8). ∎

######

When $g=g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}},$ that is when the Green
functions $g_{\sigma}$ are the equilibrium potentials
$g_{V_{\sigma},P_{\sigma}}$ associated to the points $P_{\sigma}$ of the
compact Riemann surface with boundary $V_{\sigma}$, then the Hermitian line
bundle ${\overline{N}}_{P,g}{\widehat{\mathcal{V}}}$ is the Hermitian line
bundle $\overline{N}_{P}{\widetilde{\mathcal{V}}}$ introduced in (6.2.3), and
the invariant $\mathrm{Ex}(\alpha_{\sigma},g_{\sigma})$ is the overflow
${\mathrm{Ex}}(\alpha_{\sigma}:(V_{\sigma},P_{\sigma})\rightarrow
X_{\sigma})$. Consequently, applied to
$g=g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}},$ Proposition 7.2.6 becomes the
following result, which will play a key role in the remainder of this memoir.

######  Corollary 7.2.8.

For every non-constant morphism over ${\rm Spec\,}{{\mathcal{O}}_{K}}$,

$\alpha:=(\widehat{\alpha},(\alpha_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\longrightarrow
X,$

from a smooth f.-a. arithmetic surface ${\widetilde{\mathcal{V}}}$ to an
integral normal arithmetic surface $X$ over ${\rm Spec\,}{{\mathcal{O}}_{K}}$,
the following equality holds:

(7.2.13)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P}{\widetilde{\mathcal{V}}}+\mathrm{Ex}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}\\\
+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\mathrm{Ex}\big{(}\alpha_{\sigma}:(V_{\sigma},P_{\sigma})\rightarrow
X_{\sigma})\big{)}.$

In turn, the non-negativity of the overflow invariants implies:

######  Corollary 7.2.9.

With the notation above, we have:

(7.2.14)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\geq
e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P}{\widetilde{\mathcal{V}}}.$

#####  7.2.4. Self-intersection of
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$ and Green
function for the diagonal

######

We may combine Corollary 7.2.8 and the expression in Theorem 5.4.1 for the
archimedean overflow in terms of a Green function of the diagonal. To
formulate the resulting expression for the self-intersection of
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$, we need to
introduce some notation.

For every embedding $\sigma:K\hookrightarrow\mathbb{C},$ we assume that the
connected Riemann surface $X_{\sigma}(\mathbb{C})$ is endowed with a real
$2$-form $\beta_{\sigma}$ of class ${\mathcal{C}}^{\infty}$, and that:

$g_{X_{\sigma}}:X_{\sigma}(\mathbb{C})\times
X_{\sigma}(\mathbb{C})\longrightarrow(-\infty,+\infty]$

is a Green function for the diagonal of $X_{\sigma}(\mathbb{C})$ associated to
$\beta_{\sigma},$ as defined in 5.3.1. We assume that the family
$(g_{X_{\sigma}})_{\sigma:K\hookrightarrow\mathbb{C}},$ and therefore the
family $(\beta_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}$ is invariant
under complex conjugation.

Recall that we may attach the capacitary metric
$\|.\|_{g_{\sigma}}^{\mathrm{cap}}$ on the complex tangent bundle
$T_{X_{\sigma}}$ to the Green function $g_{\sigma}$; see 5.3.1. The family of
Hermitian metrics
$(\|.\|_{g_{\sigma}}^{\mathrm{cap}})_{\sigma:K\hookrightarrow\mathbb{C}}$ is
clearly invariant under complex conjugation.

We shall also assume that the image of _the ${{\mathcal{O}}_{K}}$-point
$Q:={\widehat{\alpha}}(P)$ of $X$ lies in the regular locus of_ $X$. Then the
normal bundle $N_{Q}X$ is a well-defined line bundle over $Q$, and equipped
with the capacitary metrics $\|.\|_{g_{\sigma},Q_{\sigma}}^{\mathrm{cap}}$ on
the complex lines:

$(N_{Q}X)_{\sigma}\simeq T_{X_{\sigma},Q_{\sigma}},$

it defines a Hermitian line bundle over $Q$:

${\overline{N}}_{Q}^{\mathrm{cap}}X:=\big{(}N_{Q}X,(\|.\|_{g_{\sigma},Q_{\sigma}}^{\mathrm{cap}})_{\sigma:K\hookrightarrow\mathbb{C}}\big{)}.$

######  Proposition 7.2.10.

With the notation above, the following formula holds:

(7.2.15)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=\widehat{\rm
deg}\,Q^{\ast}{\overline{N}}_{Q}^{\mathrm{cap}}X\\\
+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\Big{(}2\int_{V_{\sigma}}g_{V_{\sigma},P_{\sigma}}\,\alpha_{\sigma}^{*}\beta_{\sigma}-\int_{(\partial
V_{\sigma})^{2}}g_{X_{\sigma}}(\alpha_{\sigma}(z_{1}),\alpha_{\sigma}(z_{2}))\,d\mu_{V_{\sigma},P_{\sigma}}(z_{1})\,d\mu_{V_{\sigma},P_{\sigma}}(z_{2})\Big{)}.$

Observe that the image of the section $Q$ actually lies in the open subscheme
$X_{\rm sm}$ of $X$ where the structure morphisme $\pi_{X}:X\rightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}}$ is smooth. Endowed with the Hermitian metrics
$\big{(}\|.\|^{\rm
cap}_{g_{\sigma}}\big{)}_{\sigma:K\hookrightarrow\mathbb{C}}$ on the Riemann
surfaces $X_{{\rm sm},\sigma}(\mathbb{C})=X_{\sigma}(\mathbb{C}),$ the
relative tangent bundle $T_{\pi_{X}}$ over $X_{\rm sm}$ defines a Hermitian
line bundle ${\overline{T}}^{\rm cap}_{\pi_{X}}$ over $X_{\rm sm}$. The
Hermitian line bundle $Q^{\ast}{\overline{N}}_{Q}^{\mathrm{cap}}X$ may be
identified with $Q^{\ast}{\overline{T}}^{\rm cap}_{\pi_{X}}$, and its Arakelov
degree coincides with the height of $Q$ with respect to ${\overline{T}}^{\rm
cap}_{\pi_{X}}$:

$\widehat{\rm deg}\,Q^{\ast}{\overline{N}}_{Q}^{\mathrm{cap}}X=\widehat{\rm
deg}\,Q^{\ast}{\overline{T}}^{\rm cap}_{\pi_{X}}=:0pt_{{\overline{T}}^{\rm
cap}_{\pi_{X}}}(Q).$

###### Proof.

Let us write $e:=e(\alpha)$. The $e$-jet ${\widehat{\alpha}}_{K}^{[e]}(P_{K})$
of ${\widehat{\alpha}}_{K}:{\widehat{\mathcal{V}}}_{K}\rightarrow X_{K}$ at
$P_{K}$ defines an isomorphism of $K$-lines:

${\widehat{\alpha}}_{K}^{[e]}:\big{(}N_{P}{\widehat{\mathcal{V}}}\big{)}_{K}^{\otimes
e}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{Q_{K}}X_{K}.$

For every embedding $\sigma:K\hookrightarrow\mathbb{C}$, by means of the
identification:

$(T\iota_{\sigma})^{\otimes
e}:\big{(}N_{P}{\widehat{\mathcal{V}}})_{\sigma}^{\otimes
e}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\big{(}T_{P_{\sigma}}V_{\sigma}\big{)}^{\otimes
e},$

the base change of ${\widehat{\alpha}}_{K}^{[e]}$ by $\sigma$ becomes the
isomorphism of $\mathbb{C}$-lines:

$\alpha_{\sigma}^{[e]}(P_{\sigma}):\big{(}N_{P}{\widehat{\mathcal{V}}})_{\sigma}^{\otimes
e}{\simeq}\big{(}T_{P_{\sigma}}V_{\sigma}\big{)}^{\otimes
e}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{Q_{\sigma}}X_{\sigma},$

already considered, in a more general context, in Subsection 5.4.1. Its norm
with respect to the metric $\|.\|_{V_{\sigma},P_{\sigma}}^{\mathrm{cap}\otimes
e}$ on $\big{(}T_{P_{\sigma}}V_{\sigma}\big{)}^{\otimes e}$ and
$\|.\|^{\mathrm{cap}}_{g_{X_{\sigma}}}$ on $T_{Q_{\sigma}}X_{\sigma}$ will be
denoted by $\|\alpha_{\sigma}^{[e]}(P_{\sigma})\|^{\mathrm{cap}}_{e}$. With
this notation, according to Theorem 5.4.1, the following equality holds for
every embedding $\sigma:K\hookrightarrow\mathbb{C}$:

(7.2.16) $\mathrm{Ex}\big{(}\alpha_{\sigma}:(V_{\sigma},P_{\sigma})\rightarrow
X_{\sigma}\big{)}=2\int_{V_{\sigma}}g_{V_{\sigma},P_{\sigma}}\,\alpha_{\sigma}^{*}\beta_{\sigma}\\\
-\int_{(\partial
V_{\sigma})^{2}}g_{X_{\sigma}}(\alpha_{\sigma}(z_{1}),\alpha_{\sigma}(z_{2}))\,d\mu_{V_{\sigma},P_{\sigma}}(z_{1})\,d\mu_{V_{\sigma},P_{\sigma}}(z_{2})-\log\|\alpha_{\sigma}^{[e]}(P_{\sigma})\|^{\mathrm{cap}}_{e}.$

Observe that the isomorphism ${\widehat{\alpha}}_{K}^{[e]}$ is the inverse of
the restriction to ${\rm Spec\,}K$ of the isomorphism $I_{\mid P}$ in the
proof of Lemma 7.2.7 in the special case $N=1$. Therefore
${\widehat{\alpha}}_{K}^{[e]}$ extends to an isomorphism of line bundles over
${\rm Spec\,}{{\mathcal{O}}_{K}}$:

${\widehat{\alpha}}^{[e]}:P^{\ast}N_{P}{\widehat{\mathcal{V}}}^{\otimes
e}\otimes P^{\ast}{\mathcal{O}}_{P}(P\cap
R)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Q^{\ast}N_{Q}X,$

where $R$ is defined as in 7.2.2. This implies the following relation between
Arakelov degrees:

(7.2.17) $\widehat{\rm deg}\,Q^{\ast}\overline{N}_{Q}X=e\,\widehat{\rm
deg}\,P^{\ast}\overline{N}_{P}{\widetilde{\mathcal{V}}}+\widehat{\rm
deg}\,P\cdot
R-\sum_{\sigma:K\hookrightarrow\mathbb{C}}\log\|\alpha_{\sigma}^{[e]}(P_{\sigma})\|^{\mathrm{cap}}_{e}.$

The equality (7.2.15) follows from the expression (7.2.13) for the self-
intersection of $\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$,
combined with (7.2.16), (7.2.17), and the definition (7.2.6) of
${\mathrm{Ex}}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\rightarrow
X\big{)}$. ∎

######

Let $\psi$ a formal series in:

$G_{\mathrm{for}}(\mathbb{R}):=\mathbb{R}^{\ast}X+X^{2}\mathbb{R}[[X]],$

and let:

$\alpha:{\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)\longrightarrow\mathbb{A}^{1}_{\mathbb{Z}}$

be a morphism from the f.-a. arithmetic surface
${\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)$ over ${\rm
Spec\,}\mathbb{Z}$ attached to $\psi$ defined in 6.4.1.

Equivalently, $\alpha$ is an element of the ring
${\mathcal{O}}({\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)$ of regular
functions over ${\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)$. It is
defined as a pair:

$\alpha:=({\widehat{\alpha}},\alpha^{{\rm an}}),$

where ${\widehat{\alpha}}$ is a morphism from ${\rm Spf\,}\mathbb{Z}[[T]]$ to
$\mathbb{A}^{1}_{\mathbb{Z}}$ — that is an element of $\mathbb{Z}[[T]]$ — and
$\alpha^{{\rm an}}$ is an analytic function:

$\alpha^{{\rm
an}}:{\overline{D}}(0,1)^{+}\longrightarrow\mathbb{A}^{1}_{\mathbb{Z}}(\mathbb{C})=\mathbb{C},$

that satisfy the gluing relation:

(7.2.18) ${\widehat{\alpha}}=\alpha^{\rm an}\circ\psi,$

when ${\widehat{\alpha}},$ $\alpha^{\rm an}$, and $\psi$ are seen as formal
series in $\mathbb{C}[[X]]$. The morphism $\alpha$ is non-constant if and only
if $\alpha^{\rm an}$ belongs to $\mathbb{Z}[[T]]\setminus\mathbb{Z}$, or
equivalently if and only if $\alpha^{{\rm an}}$ is not a constant analytic
function.

######  Corollary 7.2.11.

For every $\psi\in G_{\mathrm{for}}(\mathbb{R})$, and every non-constant
morphism:

$\alpha:=({\widehat{\alpha}},\alpha^{{\rm
an}}):{\widetilde{\mathcal{V}}}:={\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)\longrightarrow\mathbb{A}^{1}_{\mathbb{Z}},$

the following equality holds:

(7.2.19)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=2\int_{0}^{1}\int_{0}^{1}\log\left|\alpha^{\rm
an}(e^{2\pi it_{1}})-\alpha^{\rm an}(e^{2\pi it_{2}})\right|\,dt_{1}\,dt_{2}.$

###### Proof.

The function:

$g_{\mathbb{C}}:\mathbb{C}\times\mathbb{C}\longrightarrow(-\infty,+\infty],\quad(z_{1},z_{2})\longrightarrow\log|z_{1}-z_{2}|^{-1}$

is a Green function for the diagonal of
$\mathbb{C}=\mathbb{A}^{1}_{\mathbb{Z}}(\mathbb{C})$, associated to the
$2$-form $\beta=0.$ The associated capacitary metric on
$T_{\mathbb{A}^{1}_{\mathbb{C}}}$ satisfies:

$\|\partial/\partial z\|^{\mathrm{cap}}_{g_{\mathbb{C}}}=1.$

Moreover the equilibrium measure $\mu_{{\overline{D}}(0,1),0}$ associated to
the equilibrium potential:
$g_{{\overline{D}}(0,1),0}=(z\mapsto\log^{+}|z|^{-1})$ for the origin in the
unit disk, is the normalized Haar measure on:

$\partial{\overline{D}}(0,1)=\\{z\in\mathbb{C}\mid|z|=1\\}=U(1).$

Using these observations, it is readily seen that, applied to ${\rm
Spec\,}{{\mathcal{O}}_{K}}={\rm Spec\,}\mathbb{Z}$, to the f.-a. arithmetic
surface
${\widetilde{\mathcal{V}}}={\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)$
and to $X=\mathbb{A}^{1}_{\mathbb{Z}}$ equipped with the Green function
$g_{\mathbb{C}}$, the expression (7.2.15) for the self-intersection of
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$ becomes
(7.2.19). ∎

######

Corollary 7.2.11 admits a variant, concerning a morphism $\alpha$ from
${\widetilde{\mathcal{V}}}({\overline{D}}(0,1),\psi)$ to
$\mathbb{P}^{1}_{\mathbb{Z}},$ obtained by using the Green function
$g_{\mathbb{P}^{1}(\mathbb{C})}$ for the diagonal of
$\mathbb{P}^{1}(\mathbb{C})$ introduced in 5.3.2 (2) in the expression
(7.2.15) for the self-intersection of
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})$. As already
observed in 5.4.2, the first integral in the right-hand side of (7.2.15) is
the value at 1 of the characteristic function $T_{\alpha}$ of Nevanlinna-
Ahlfors-Shimizu associated to $\alpha$.

The following formulation of this special instance of Proposition 5.4.2
emphasizes this relation to Nevalinna theory, and facilitates the comparison
of our results in Section 8.3 below with the ones in [CDT21].

Consider the following data:

  * •

a formal series $\psi$ in $G_{\mathrm{for}}(\mathbb{R})$;

  * •

a morphism of (formal) schemes:

${\widehat{\alpha}}:{\rm
Spf\,}\mathbb{Z}[[T]]\longrightarrow\mathbb{P}^{1}_{\mathbb{Z}};$

  * •

a meromorphic function, defined on the open disk $D(0,R)$ of radius
$R\in(0,+\infty]$:

$\alpha^{{\rm an}}:D(0,R)\longrightarrow\mathbb{P}^{1}(\mathbb{C});$

and assume that ${\widehat{\alpha}}$ and $\alpha^{\rm an}$ are not constant
and satisfy the gluing relation (7.2.18).

Let us denote by:

$0pt:\mathbb{P}^{1}(\mathbb{Q})\longrightarrow\mathbb{R}_{+}$

the usual height, defined by:

(7.2.20) $0pt(x_{0}:x_{1}):=\log\left(x_{0}^{2}+x_{1}^{2}\right)^{1/2}$

for every pair $(x_{0},x_{1})$ of integers that are prime together.

For every $r\in(0,+\infty),$ we may consider the following f.-a. surface
over ${\rm Spec\,}\mathbb{Z}$:

${\widetilde{\mathcal{V}}}_{r}:={\widetilde{\mathcal{V}}}(\overline{D}(0,r),\psi),$

and the morphism:

$\alpha_{r}:=\big{(}{\widehat{\alpha}},\alpha^{\rm
an}_{\mid\overline{D}(0,r)^{+}}\big{)}:{\widetilde{\mathcal{V}}}_{r}\longrightarrow\mathbb{P}^{1}_{\mathbb{Z}}.$

######  Corollary 7.2.12.

With the notation above, for every $r\in(0,R),$ the following equality holds:

(7.2.21)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{r,\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{r,\mathbb{C}}})=2\,0pt(\alpha(0))+2\,T_{\alpha^{\rm
an}}(r)\\\
-\int_{0}^{1}\int_{0}^{1}g_{\mathbb{P}^{1}(\mathbb{C})}\big{(}\alpha^{\rm
an}(r\,e^{2\pi it_{1}}),\alpha^{\rm an}(r\,e^{2\pi
it_{2}})\big{)}\,dt_{1}\,dt_{2}.$

In the right-hand side of (7.2.21), we denote by $\alpha(0)$ the point
${\widehat{\alpha}}(0)=\alpha^{\rm an}(0)$ of $\mathbb{P}^{1}(\mathbb{Q}),$ by
$T_{\alpha^{\rm an}}$ the characteristic function of ${\alpha^{\rm an}}$ as
defined in (5.4.10), and by $g_{\mathbb{P}^{1}(\mathbb{C})}$ the Green
function for the diagonal of $\mathbb{P}^{1}(\mathbb{C})$ defined by (5.3.2).

To derive (7.2.21) from the expression (7.2.15) applied to the morphism
$\alpha:=\alpha_{r}$ from
${\widetilde{\mathcal{V}}}:={\widetilde{\mathcal{V}}}_{r}$ to
$X:=\mathbb{P}^{1}_{\mathbb{Z}}$ and to the Green function
$g:=g_{\mathbb{P}^{1}(\mathbb{C})},$ observe that the Hermitian line bundle
${\overline{T}}^{\rm cap}_{\pi_{X}}$ is then isomorphic to the second tensor
power of the Hermitian line bundle
$\overline{\mathcal{O}}_{\mathbb{P}^{1}}(1)$ over
$\mathbb{P}^{1}_{\mathbb{Z}}$ that defines the usual height (7.2.20), and that
the equilibrium measure $\mu_{{\overline{D}}(0,1),r}$ is the rotation
invariant probability measure on $\partial{\overline{D}}(0,r).$

Observe that, since $g_{\mathbb{P}^{1}(\mathbb{C})}$ is non-negative, (7.2.21)
implies the following upper bound:

(7.2.22)
$\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{r,\mathbb{C}}})\cdot\alpha_{\ast}(P,g_{{\widetilde{\mathcal{V}}}_{r,\mathbb{C}}})\leq
2\,0pt(\alpha(0))+2\,T_{\alpha^{\rm an}}(r).$

####  7.3. Meromorphic maps from f.-a. arithmetic surfaces to proper
arithmetic schemes

The results of the previous sections may be extended to the situation where,
instead of a morphism $f:{\widetilde{\mathcal{V}}}\rightarrow X$ from a smooth
f.-a. surface ${\widetilde{\mathcal{V}}}$ to a quasi-projective scheme or an
arithmetic surface $X$ over ${{\mathcal{O}}_{K}},$ we consider a meromorphic
map:

(7.3.1) $f:{\widetilde{\mathcal{V}}}\dashrightarrow X$

from ${\widetilde{\mathcal{V}}}$ to some projective scheme over
${\mathcal{O}_{K}}$.

For lack of suitable available references concerning rational maps or
morphisms defined on general Noetherian formal schemes, we will follow a
pedestrian approach to define the meromorphic maps (7.3.1), by taking
advantage of the two-dimensional nature of the formal scheme
${\widehat{\mathcal{V}}}$ underlying the f.-a. surface
${\widetilde{\mathcal{V}}}$.

In this section, we denote by:

${\widetilde{\mathcal{V}}}:=\big{(}{\widehat{\mathcal{V}}},(V_{\sigma},P_{\sigma},\iota_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\big{)}$

a smooth f.-a. arithmetic surface over ${\mathcal{O}_{K}}$.

#####  7.3.1. Regular modifications of ${\widehat{\mathcal{V}}}$ and
meromorphic maps from ${\widehat{\mathcal{V}}}$ to projective
${{\mathcal{O}}_{K}}$-schemes

######

We define a _regular modification_ of ${\widehat{\mathcal{V}}}$ as a morphism
of formal schemes:

$\nu:{\widehat{\mathcal{V}}}^{\prime}\longrightarrow{\widehat{\mathcal{V}}}$

that may be written as a composition:

$\nu:=\nu_{1}\circ\dots\circ\nu_{n}:{\widehat{\mathcal{V}}}^{\prime}={\widehat{\mathcal{V}}}_{n}\stackrel{{\scriptstyle\nu_{n}}}{{\longrightarrow}}{\widehat{\mathcal{V}}}_{n-1}\stackrel{{\scriptstyle\nu_{n-1}}}{{\longrightarrow}}\dots\stackrel{{\scriptstyle\nu_{2}}}{{\longrightarrow}}{\widehat{\mathcal{V}}}_{1}\stackrel{{\scriptstyle\nu_{1}}}{{\longrightarrow}}{\widehat{\mathcal{V}}}_{0}={\widehat{\mathcal{V}}}$

where $\nu_{i}$ is the blowing up of a finite set of closed points of the
definition scheme $|{\widehat{\mathcal{V}}}_{i}|$ of
${\widehat{\mathcal{V}}}_{i}$.

By construction, ${\widehat{\mathcal{V}}}^{\prime}$ is a “regular formal
arithmetic surface”, whose reduced scheme of definition
$|{\widehat{\mathcal{V}}}^{\prime}|$ is purely one-dimensional and proper over
${\rm Spec\,}{{\mathcal{O}}_{K}}$.

The scheme $|{\widehat{\mathcal{V}}}^{\prime}|$ admits a unique irreducible
component that is finite over ${\rm Spec\,}{{\mathcal{O}}_{K}}$. We shall
denote it by $|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}$; the
restriction of $\nu$ defines an isomorphism:

$\nu_{\mid|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}}:|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}|{\widehat{\mathcal{V}}}|.$

The other components of $|{\widehat{\mathcal{V}}}|$ are (isomorphic to)
projective lines over some finite fields. We shall denote the union of these
vertical components by $|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{vert}}$.

We shall denote by ${\mathrm{Ind}}(\nu^{-1})$ the indeterminacy locus of
$\nu^{-1}$, namely the finite set of closed points of
$|{\widehat{\mathcal{V}}}|$ in the image of the centers of the morphisms
$\nu_{i}$:

${\mathrm{Ind}}(\nu^{-1}):=\bigcup_{1\leq i\leq
n}\nu_{1}\circ\dots\circ\nu_{i-1}(C_{i}),$

where $C_{i}$ denotes the center of the blowing up $\nu_{i}$.

The inverse image by $\nu$ of the closed subscheme ${\mathrm{Ind}}(\nu^{-1})$
of $|{\widehat{\mathcal{V}}}|$, hence of ${\widehat{\mathcal{V}}}$, is an
effective divisor on ${\widehat{\mathcal{V}}}^{\prime}$, of support
$|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{vert}}$.

######

A regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$ as
above is an adic morphism of formal schemes. For every coherent
${\mathcal{O}}_{{\widehat{\mathcal{V}}}^{\prime}}$-module ${\mathcal{F}}$, the
direct image $\nu_{\ast}{\mathcal{F}}$ is a coherent
${\mathcal{O}}_{{\widehat{\mathcal{V}}}}$-module. When ${\mathcal{F}}$ is
locally free, its direct image $\nu_{\ast}{\mathcal{F}}$ is not always locally
free. However it is torsion free, its bidual
$(\nu_{\ast}{\mathcal{F}})^{\vee\vee}$ is a locally free
${\mathcal{O}}_{\widehat{\mathcal{V}}}$-module, the tautological morphism:

$\tau:\nu_{\ast}{\mathcal{F}}\longrightarrow(\nu_{\ast}{\mathcal{F}})^{\vee\vee}$

is injective, and its cokernel ${\rm coker\,}\tau$ is an
${\mathcal{O}}_{{\widehat{\mathcal{V}}}}$-module of finite length, supported
by ${\mathrm{Ind}}(\nu^{-1}).$

By applying these observations to a line bundle $L^{\prime}$ over
${\widehat{\mathcal{V}}}^{\prime},$ one obtains:

######  Proposition 7.3.1.

For every regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$ as
above and every line bundle $L^{\prime}$ over
${\widehat{\mathcal{V}}}^{\prime}$, there exists a line bundle $L$ over
${\widehat{\mathcal{V}}}$, a divisor $W$ in ${\widehat{\mathcal{V}}}^{\prime}$
supported by $|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{vert}}$, and an
isomorphism of lines bundles over ${\widehat{\mathcal{V}}}^{\prime}$:

$\iota:L^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\nu^{\ast}L\otimes{\mathcal{O}}_{{\widehat{\mathcal{V}}}^{\prime}}(W).$

When this holds, $L$ is canonically isomorphic to
$(\nu_{\ast}L^{\prime})^{\vee\vee}$.

######

We may introduce the _category of regular modifications_ of
${\widehat{\mathcal{V}}}$ by defining a morphism for a regular modification
$\nu_{1}:{\widehat{\mathcal{V}}}_{1}\rightarrow{\widehat{\mathcal{V}}}$ to a
regular modification
$\nu_{2}:{\widehat{\mathcal{V}}}_{2}\rightarrow{\widehat{\mathcal{V}}}$ as a
morphism of formal schemes
$\varphi:{\widehat{\mathcal{V}}}_{1}\rightarrow{\widehat{\mathcal{V}}}_{2}$
such that $\nu_{2}\circ\varphi=\nu_{1}$.

Actually, if such a morphism exists, it is unique and may be written as a
composition of blowing-ups of closed points. Moreover, for any pair of regular
modifications:

$\nu_{i}:{\widehat{\mathcal{V}}}_{i}\longrightarrow{\widehat{\mathcal{V}}},\quad
i=1,2,$

there exists a regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$ and
two morphisms $\varphi_{1}$ and $\varphi_{2}$ from $\nu$ to $\nu_{1}$ and
$\nu_{2}$ respectively. This notably implies that the category of regular
modifications is equivalent to a directed set.

######

Let $X$ be a projective scheme over ${\rm Spec\,}{{\mathcal{O}}_{K}}$. We
define a _meromorphic map_ :

$\widehat{f}:{\widehat{\mathcal{V}}}\dashrightarrow X$

over ${\rm Spec\,}{{\mathcal{O}}_{K}}$ as a morphism
$\widehat{f}_{K}:\widehat{\mathcal{V}}_{K}\rightarrow X_{K}$ of (formal)
schemes over $K$ such that there exists a regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$ and a
morphism ${\widehat{f}}^{\prime}:\widehat{\mathcal{V}}^{\prime}\rightarrow X$
of (formal) schemes over ${\rm Spec\,}{{\mathcal{O}}_{K}}$: such that:

${\widehat{f}}^{\prime}_{K}=\widehat{f}_{K}\circ\nu_{K}.$

When this holds, we will say that the regular modification $\nu$ is _adapted_
to $\widehat{f}$.

The set of meromorphic maps from ${\widehat{\mathcal{V}}}$ to $X$ over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$ may be identified with the limit:

$\lim_{\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}}\mathrm{Mor}_{{{\mathcal{O}}_{K}}}({\widehat{\mathcal{V}}}^{\prime},X)$

of the sets
$\mathrm{Mor}_{{{\mathcal{O}}_{K}}}({\widehat{\mathcal{V}}}^{\prime},X)$ of
morphisms over ${\rm Spec\,}{{\mathcal{O}}_{K}}$, taken over the directed set
of formal modifications
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$.

The elements of the field ${\mathcal{M}}({\widehat{\mathcal{V}}})$ of formal
meromorphic functions on ${\widehat{\mathcal{V}}}$, as defined in [HM68], may
be identified with the meromorphic maps:

$\widehat{f}:{\widehat{\mathcal{V}}}\dashrightarrow\mathbb{P}^{1}_{{{\mathcal{O}}_{K}}}$

over ${\rm Spec\,}{{\mathcal{O}}_{K}}$ that are distinct from the “constant”
morphism $\infty$. This follows from the desingularization _à la_ M. Noether
of divisors in the regular formal scheme ${\widehat{\mathcal{V}}}$; see for
instance [Abh56] and [Sha66, pages 38-44].

More generally, consider a morphism
$\widehat{f}_{K}:\widehat{\mathcal{V}}_{K}\rightarrow X_{K}$ of (formal)
schemes over $K$, an affine open neighborhood $U$ of $\widehat{f}_{K}(P_{K})$
in $X_{K}$, and an embedding of $K$-schemes:

$(x_{1},\dots,x_{n}):U{\lhook\joinrel\longrightarrow}\mathbb{A}^{N}_{K}.$

Then $\widehat{f}_{K}$ defines a meromorphic map
$\widehat{f}:{\widehat{\mathcal{V}}}\dashrightarrow X$ if and only if the
elements $x_{1}\circ\widehat{f}_{K},\dots,x_{N}\circ\widehat{f}_{K}$ of
${\mathcal{O}}({\widehat{\mathcal{V}}}_{K})$ are elements of the subfield
${\mathcal{M}}({\widehat{\mathcal{V}}}):=\mathrm{Frac}\,{\mathcal{O}}({\widehat{\mathcal{V}}})$
of
${\mathcal{M}}({\widehat{\mathcal{V}}}_{K}):=\mathrm{Frac}\,{\mathcal{O}}({\widehat{\mathcal{V}}}_{K}).$

#####  7.3.2. Meromorphic maps from ${\widetilde{\mathcal{V}}}$ to projective
${{\mathcal{O}}_{K}}$-schemes

######

A _meromorphic map_ :

${f}:{\widetilde{\mathcal{V}}}\dashrightarrow X$

_over ${\rm Spec\,}{{\mathcal{O}}_{K}}$_ from the smooth f.-a. surface
${\widetilde{\mathcal{V}}}$ to a projective ${{\mathcal{O}}_{K}}$-scheme $X$
is defined by a generalization of the definition of morphisms in 7.1.1, namely
as a pair:

$f:=(\widehat{f},(f_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

where:

$\widehat{f}:{\widehat{\mathcal{V}}}\dashrightarrow X$

is a meromorphic map over ${{\mathcal{O}}_{K}}$ and where, for every complex
embedding $\sigma$ of $K$,

$f_{\sigma}:V_{\sigma}\longrightarrow X_{\sigma}(\mathbb{C})$

is a complex analytic map. Moreover, for every embedding
$\sigma:K\hookrightarrow\mathbb{C},$ the morphisms from the smooth formal
curve ${\widehat{\mathcal{V}}}_{\sigma}$ to the complex scheme $X_{\sigma}$
deduced from $\widehat{f}_{K}$ (by the base change $\sigma$) and from
$f_{\sigma}$ (by considering its “formal germ” at $P_{\sigma}$) are required
to coincide.

The discussion in 7.1.1 of the relations between the diverse “Zariski closures
of the image” associated to a morphism $f:{\widetilde{\mathcal{V}}}\rightarrow
X$ easily extends to the meromorphic maps
$f:{\widetilde{\mathcal{V}}}\dashrightarrow X$ so defined. In particular, it
makes sense to say that $f:{\widetilde{\mathcal{V}}}\dashrightarrow X$ is
constant, or that its image is algebraic.

######

The construction of the pro-Hermitian vector bundle
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}\widetilde{\overline{E}}$
over ${\rm Spec\,}{{\mathcal{O}}_{K}}$ associated to a Hermitian vector bundle
$\widetilde{\overline{E}}$ on ${\widetilde{\mathcal{V}}}$ discussed in 6.1.2,
and the construction of the morphism of (pro-)Hermitian vector bundles :

$\varphi_{E}:\pi^{J}_{X\ast}{\overline{E}}\longrightarrow\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}f^{*}\overline{E}$

associated to a morphism $f$ from ${\widetilde{\mathcal{V}}}$ to some
projective ${{\mathcal{O}}_{K}}$-scheme $X$ equipped with a Hermitian vector
bundle ${\overline{E}}$ that we discussed in 7.1.2 admit the following
generalizations involving regular modifications.

Consider a regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$ as
above. The pair:

${\widetilde{\mathcal{V}}}^{\prime}:=({\widehat{\mathcal{V}}}^{\prime},(V_{\sigma},O_{\sigma},\iota_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

defines a generalized smooth formal-analytic surface. There is an obvious
notion of a vector bundle:

$\widetilde{E}:=(\widehat{E},(E_{\sigma},\varphi_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}),$

and of a Hermitian vector bundle:

$\widetilde{\overline{E}}:=(\widehat{E},(E_{\sigma},\varphi_{\sigma},\|.\|_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}})$

over ${\widehat{\mathcal{V}}}^{\prime}$, defined as in 6.1.2, where
$\widehat{E}$ is now a vector bundle over ${\widehat{\mathcal{V}}}^{\prime}$.

Let $\mu$ be a positive volume form on $V_{\mathbb{C}}$, invariant under
complex conjugation, as in 6.1.2. Then to a Hermitian vector bundle
$\widetilde{\overline{E}}$ over ${\widetilde{\mathcal{V}}}^{\prime}$, we may
attach:

(7.3.2)
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}\widetilde{\overline{E}}:=\Big{(}\Gamma({\widehat{\mathcal{V}}}^{\prime},\widehat{E}),(\Gamma_{L^{2}}(V_{\sigma},\mu_{\sigma};E_{\sigma},\|.\|_{\sigma}),\widehat{\eta}_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\Big{)},$

where $\widehat{\eta}_{\sigma}$ is defined as in 6.1.2, with obvious minor
modifications. An extension of the proof in [Bos20, 10.6.2] shows that it is a
pro-Hermitian vector bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}}$, as in the
case where ${\widetilde{\mathcal{V}}}^{\prime}$ is ${\widetilde{\mathcal{V}}}$
considered in 6.1.2 above.

The construction (7.3.2) of the direct image on ${\rm
Spec\,}{{\mathcal{O}}_{K}}$ of some Hermitian vector bundle over
${\widetilde{\mathcal{V}}}^{\prime}$ is related as follows to the construction
of the direct image of Hermitian torsion free coherent sheaves and vector
bundles over ${\widetilde{\mathcal{V}}}$.

Consider a Hermitian vector bundle:

${\overline{E}}^{\prime}:=\big{(}\widehat{E}^{\prime},(E_{\sigma},\varphi_{\sigma},\|.\|_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\big{)}$

over ${\widetilde{\mathcal{V}}}^{\prime}$. The direct image
$\nu_{\ast}{\widehat{E}}^{\prime}$ of the underlying vector bundle
${\widehat{E}}^{\prime}$ over ${\widehat{\mathcal{V}}}$ by the modification
$\nu$ is a torsion free coherent sheaf over ${\widehat{\mathcal{V}}}$.
Moreover its “restriction” $(\nu_{\ast}{\widehat{E}}^{\prime})_{K}$ to
${\widehat{\mathcal{V}}}^{\prime}_{K}\simeq{\widehat{\mathcal{V}}}_{K}$ may be
identified with ${\widehat{E}}^{\prime}_{K}$. We therefore define a Hermitian
torsion free coherent sheaf over ${\widetilde{\mathcal{V}}}$ by letting:

$\nu_{\ast}{\overline{E}}^{\prime}:=\big{(}\nu_{\ast}\widehat{E}^{\prime},(E_{\sigma},\varphi_{\sigma},\|.\|_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}\big{)}.$

The identification of coherent sheaves of
${\mathcal{O}}_{{\widehat{\mathcal{V}}}}$-modules:

$\pi_{{\widehat{\mathcal{V}}}\ast}\nu_{\ast}{\widehat{E}}^{\prime}\simeq\pi_{{\widehat{\mathcal{V}}}^{\prime}\ast}{\widehat{E}}^{\prime}$

extends to an identification:

(7.3.3)
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}(\nu_{\ast}{\overline{E}}^{\prime})\simeq\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}{\overline{E}}^{\prime}.$

As explained in 6.1.2, the left-hand side of (7.3.3) is a pro-Hermitian vector
bundle over ${\rm Spec\,}{{\mathcal{O}}_{K}}$, and therefore
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}{\overline{E}}^{\prime}$
also is. Moreover, as shown by Proposition 6.1.1, the properties of the pro-
Hermitian vector bundle
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}{\overline{E}}^{\prime}$
— for instance its $\theta$-finiteness — are closely related to the properties
of $\pi^{L^{2}}_{({\widetilde{\mathcal{V}}},\mu)\ast}{\overline{E}}$, where:

${\overline{E}}:=(\nu_{\ast}{\overline{E}}^{\prime})^{\vee\vee}$

is the Hermitian vector bundle over ${\widetilde{\mathcal{V}}}$ deduced from
$\nu_{\ast}{\overline{E}}^{\prime}$ by biduality.

######

Once the direct image (7.3.2) has been introduced, Proposition 7.1.1
immediately extends to the situation where $f$ is a meromorphic map:

$f:=(\widehat{f},f^{\rm an}):{\widetilde{\mathcal{V}}}\dashrightarrow X$

from $X$ to a reduced flat projective ${{\mathcal{O}}_{K}}$-scheme $X$
equipped with some Hermitian vector bundle ${\overline{E}}$.

Indeed, if
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$
denotes a regular modification of ${\widehat{\mathcal{V}}}$ adapted to the
meromorphic map:

$\widehat{f}:{\widehat{\mathcal{V}}}\dashrightarrow X$

and if:

$f^{\prime}:=(\widehat{f}^{\prime},f^{\rm
an}):{\widetilde{\mathcal{V}}}^{\prime}\longrightarrow X$

denotes the associated ${{\mathcal{O}}_{K}}$-morphism, then we may consider
the Hermitian vector bundle $f^{\prime\ast}{\overline{E}}$ over
${\widetilde{\mathcal{V}}}^{\prime}$ and its direct image
$\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}f^{\prime\ast}{\overline{E}}$
over ${\rm Spec\,}{{\mathcal{O}}_{K}}$. It is readily seen to be “independent”
of the choice of the adapted modification $\nu$. Moreover the pull-back of
sections of $E$ by $f^{\prime}$ defines a morphism:

$\varphi_{E}:\pi^{J}_{X\ast}\widetilde{\overline{E}}\longrightarrow\pi^{L^{2}}_{({\widetilde{\mathcal{V}}}^{\prime},\mu)\ast}f^{\prime*}\overline{E}$

of (pro-)Hermitian vector bundles over ${\rm Spec\,}{{\mathcal{O}}_{K}}$, and
the last assertion of Proposition 7.1.1 still holds.

#####  7.3.3. Meromorphic maps from ${\widetilde{\mathcal{V}}}$ to projective
arithmetic surfaces and arithmetic intersections numbers

subsection

Assume that $X$ is an integral normal projective arithmetic surface over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$, and consider a non-constant meromorphic map:

$\alpha:({\widehat{\alpha}},(\alpha_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\dashrightarrow
X.$

As before, we shall denote by $\alpha^{\rm an}$ the complex analytic map from
$V_{\mathbb{C}}$ to $X(\mathbb{C})$ defined by the maps $\alpha_{\sigma}$, and
by $e(\alpha)$ the ramification index of the morphism
${\widehat{\alpha}}_{K}:{\widehat{\mathcal{V}}}_{K}\rightarrow X_{K}$.

In this subsection, we discuss the extension to this framework of the
constructions and results of Section 7.2.

######

Let us choose a regular modification
$\nu:{\widehat{\mathcal{V}}}^{\prime}\rightarrow{\widehat{\mathcal{V}}}$
adapted to ${\widehat{\alpha}}$, and let us denote by:

${\widehat{\alpha}}^{\prime}:{\widehat{\mathcal{V}}}^{\prime}\longrightarrow
X$

the morphism of (formal) schemes over ${\rm Spec\,}{{\mathcal{O}}_{K}}$ that
defines ${\widehat{\alpha}}$.

Let $(D,g)$ be an Arakelov divisor in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$. The inverse image
$\nu^{\ast}D$ is a divisor in ${\widehat{\mathcal{V}}}$ supported by
$|{\widehat{\mathcal{V}}}^{\prime}|,$ and its direct image by the proper map
${\widehat{\alpha}}^{\prime}:|{\widehat{\mathcal{V}}}^{\prime}|\rightarrow X$
defines a divisor:

$\Gamma_{{\widehat{\alpha}}\ast}D:={\widehat{\alpha}}^{\prime}_{\ast}\nu^{\ast}D$

in $Z^{1}(X)$. It is readily seen to be independent of the choice of the
adapted regular modification $\nu$, and we may define:

$\Gamma_{\alpha\ast}(D,g):=(\Gamma_{{\widehat{\alpha}}\ast}D,\alpha^{\rm
an}_{\ast}g)=(\Gamma_{{\widehat{\alpha}}\ast}D,(\alpha_{\sigma\ast}g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}).$

It is an Arakelov divisor in ${\overline{Z}}^{1}(X)$, and this construction
defines a morphism of $\mathbb{Z}$-modules:

$\Gamma_{\alpha\ast}:{\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})\longrightarrow{\overline{Z}}^{1}(X),$

which is compatible with the positivity of Arakelov divisors.

If $f:X\longrightarrow X^{\prime}$ is a dominant, or equivalently surjective,
morphism of integral normal projective arithmetic surfaces over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$, then:

$f\circ\alpha:=(f\circ{\widehat{\alpha}},(f_{\sigma}\circ\alpha_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}):{\widetilde{\mathcal{V}}}\dashrightarrow
X^{\prime}$

is a non-constant meromorphic map, and for every Arakelov divisor $(D,g)$ in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, the following equality
holds in ${\overline{Z}}^{1}(X^{\prime})$:

(7.3.4) $(f\circ\alpha)_{\ast}(D,g)=f_{\ast}\alpha_{\ast}(D,g).$

The proof is straighforward.

######

We shall denote by $P^{\prime}$ the section of the structure morphism:

$\pi_{{\widehat{\mathcal{V}}}^{\prime}}:{\widehat{\mathcal{V}}}^{\prime}\longrightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}}$

defined as the proper transform of $P$. It is the inverse of the isomorphism:

$\pi_{{\widehat{\mathcal{V}}}^{\prime}\mid|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}}:|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Spec\,}{{\mathcal{O}}_{K}}.$

The divisor $\nu^{\ast}P$ in ${\widehat{\mathcal{V}}}^{\prime}$ may be
written:

(7.3.5) $\nu^{\ast}P=P^{\prime}+V^{\prime},$

where $V^{\prime}$ is an effective divisor supported by
$|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{vert}}$.

As before, we denote by $Q$ the section ${\widehat{\alpha}}^{\prime}\circ P$
of the structure morphism $\pi_{X}:X\rightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}}$. The inverse image
${\widehat{\alpha}}^{\prime\ast}(Q)$ is a $\mathbb{Q}$-divisor in
${\widehat{\mathcal{V}}}$, and may be written:

(7.3.6) ${\widehat{\alpha}}^{\prime\ast}(Q)=e(\alpha)P^{\prime}+R^{\prime},$

where $R^{\prime}$ is an effective $\mathbb{Q}$-divisor in
${\widehat{\mathcal{V}}}^{\prime}$ such that $|R^{\prime}|\cap
P^{\prime}=|R^{\prime}|\cap|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}$
is a finite set of closed points, and therefore
$|R^{\prime}|\cap|\nu^{\ast}P|$ is proper over ${\rm
Spec\,}{{\mathcal{O}}_{K}}$.

The direct image
$\pi_{|{\widehat{\mathcal{V}}}|\ast}(R^{\prime}\cdot\nu^{\ast}P)$ by the
structure morphism
$\pi_{|{\widehat{\mathcal{V}}}|}:|{\widehat{\mathcal{V}}}|\rightarrow{\rm
Spec\,}{{\mathcal{O}}_{K}}$ of the $0$-dimensional intersection cycle of
$R^{\prime}$ and $\nu^{\ast}P$, which is defined up to vertical linear
equivalence, is a well defined $0$-dimensional $\mathbb{Q}$-cycle in ${\rm
Spec\,}{{\mathcal{O}}_{K}}$. It is readily seen to be independent of the
choice of the adapted regular modification $\nu$, and consequently its
Arakelov degree:

(7.3.7) $\widehat{\rm deg}\,R^{\prime}\cdot\nu^{\ast}P$

also is.

Observe that the Arakelov degree (7.3.7) is non-negative. Actually, for any
effective $\mathbb{Q}$-divisor $D$ in ${\widehat{\mathcal{V}}}^{\prime}$ that
does not contain $P^{\prime}$, the Arakelov degree:

$\widehat{\rm deg}\,D\cdot\nu^{\ast}P$

is well-defined and non-negative. This non-negativity indeed follows from the
fact that, for every component $W$ of
$|{\widehat{\mathcal{V}}}^{\prime}|_{\mathrm{hor}}$, the intersection number
$\widehat{\rm deg}\,W\cdot\nu^{\ast}P$ vanishes.

The direct image ${\widehat{\alpha}}_{\ast}^{\prime}V^{\prime}$ is a vertical
effective divisor in $X$, and its inverse image
${\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}_{\ast}^{\prime}V^{\prime}$
is an effective $\mathbb{Q}$-divisor in ${\widehat{\mathcal{V}}}^{\prime}$.
Here again
$|{\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}_{\ast}^{\prime}V|\cap
P^{\prime}$ is a finite set of closed points, the direct image
$\pi_{|{\widehat{\mathcal{V}}}|\ast}({\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}_{\ast}^{\prime}V^{\prime}\cdot\nu^{\ast}P^{\prime})$
is a well-defined 0-dimensional $\mathbb{Q}$-cycle in ${\rm
Spec\,}{{\mathcal{O}}_{K}}$ which does not depend of the choice of $\nu,$ and
its Arakelov degree:

$\widehat{\rm
deg}\,{\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}_{\ast}^{\prime}V^{\prime}\cdot\nu^{\ast}P^{\prime}$

also is non-negative.

######  Definition 7.3.2.

With the above notation, we define:

(7.3.8)
${\mathrm{Ex}}({\widehat{\alpha}}:{\widehat{\mathcal{V}}}\dashrightarrow
X):=\widehat{\rm
deg}\,(R^{\prime}+{\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}^{\prime}_{\ast}V^{\prime})\cdot\nu^{\ast}P.$

One easily checks that the invariant
${\mathrm{Ex}}({\widehat{\alpha}}:{\widehat{\mathcal{V}}}\dashrightarrow X)$
also admits the following expressions:

(7.3.9)
$\displaystyle{\mathrm{Ex}}({\widehat{\alpha}}:{\widehat{\mathcal{V}}}\dashrightarrow
X)$ $\displaystyle=\widehat{\rm
deg}\,\big{(}{\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}^{\prime}_{\ast}(\nu^{\ast}(P)\big{)}-e(\alpha)\,P^{\prime})\cdot\nu^{\ast}P$
(7.3.10) $\displaystyle=\widehat{\rm
deg}\,\big{(}{\widehat{\alpha}}^{\prime\ast}{\widehat{\alpha}}^{\prime}_{\ast}(\nu^{\ast}P)-e(\alpha)\,\nu^{\ast}P\big{)}\cdot\nu^{\ast}P,$

which extend the equality (7.2.7) to the situation, and are formally similar
to (5.2.3).

Observe that (7.3.9) makes clear that
${\mathrm{Ex}}({\widehat{\alpha}}:{\widehat{\mathcal{V}}}\dashrightarrow X)$
is non-negative, and (7.3.10) that it is independent of the choice of the
regular modification $\nu$ adapted to ${\widehat{\alpha}}$.

######

As in 7.2.3, let us consider a family
$g:=(g_{\sigma})_{\sigma:K\hookrightarrow\mathbb{C}}$ of Green functions for
the points $P_{\sigma}$ in the Riemann surfaces $V^{+}_{\sigma}$ such that
$(P,g)$ is an Arakelov divisor in
${\overline{Z}}^{1}_{c}({\widetilde{\mathcal{V}}})$, and the associated
Hermitian line bundle:

${\overline{N}}_{P,g}{\widehat{\mathcal{V}}}:=(N_{P}{\widehat{\mathcal{V}}},(\|.\|_{g_{\sigma}}^{\mathrm{cap}})_{\sigma:K\hookrightarrow\mathbb{C}})$

defined by the associated capacitary metrics
$\|.\|_{g_{\sigma}}^{\mathrm{cap}}$.

The following proposition is the generalization of Proposition 7.2.6 to
meromorphic maps.

######  Proposition 7.3.3.

With the notation above, the following equality holds:

(7.3.11)
$\Gamma_{\alpha\ast}(P,g)\cdot\Gamma_{\alpha\ast}(P,g)=e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P,g}{\widehat{\mathcal{V}}}+\mathrm{Ex}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\dashrightarrow
X\big{)}+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\mathrm{Ex}(\alpha_{\sigma},g_{\sigma}).$

When the Green functions $g_{\sigma}$ are the equilibrium potentials
$g_{V_{\sigma},P_{\sigma}}$, the equality (7.3.11) becomes the following
generalization of (7.2.13):

(7.3.12)
$\Gamma_{\alpha\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})\cdot\Gamma_{\alpha\ast}(P,g_{{\widetilde{\mathcal{V}}}_{\mathbb{C}}})=e(\alpha)\,\widehat{\deg}\,P^{*}\overline{N}_{P}{\widetilde{\mathcal{V}}}+\mathrm{Ex}\big{(}\widehat{\alpha}:{\widehat{\mathcal{V}}}\dashrightarrow
X\big{)}\\\
+\sum_{\sigma:K\hookrightarrow\mathbb{C}}\mathrm{Ex}\big{(}\alpha_{\sigma}:(V_{\sigma},P_{\sigma})\rightarrow
X_{\sigma})\big{)}.$

###### Proof of Proposition 7.3.3.

We may choose a positive integer $N$ such that the divisor $NQ$ and
$N{\widehat{\alpha}}^{\prime}_{\ast}V^{\prime}$ are Cartier. Then we may
define a Hermitian line bundle ${\overline{L}}$ over $X$ by:

(7.3.13)
${\overline{L}}:=\overline{\mathcal{O}}(N\Gamma_{\alpha*}P,N\alpha^{\rm
an}_{\ast}g)=\overline{\mathcal{O}}(NQ+N{\widehat{\alpha}}^{\prime}_{\ast}V^{\prime},N\alpha^{\rm
an}_{\ast}g).$

We want to show that $N^{-2}{\overline{L}}\cdot{\overline{L}}$ equals the
right-hand side of (7.3.11).

In the present context, the relations (7.2.10), (7.2.11), and (7.2.12) in the
proof of Lemma 7.2.7 are replaced by:

${\widehat{\alpha}}^{\prime\ast}(NQ)=e(\alpha)NP^{\prime}+NR^{\prime},$