## 1. Introduction

Let $(A,\lambda )$ be a principally polarized abelian variety of positive dimension over the field of algebraic numbers $\bar {\mathbb {Q}}$. Let $\Theta$ be an effective symmetric ample divisor on $A$ that defines the principal polarization $\lambda$ and put $L=\mathcal {O}_A(\Theta )$.

We are interested in the *Néron–Tate height* $\mathrm {h}'_L(\Theta )$ of the cycle $\Theta$. The Néron–Tate height of higher-dimensional cycles was first constructed by Philippon [Reference PhilipponPhi91] and soon afterwards re-obtained using different methods by, among others, Gubler [Reference GublerGub94], Bost *et al.* [Reference BostBos96b, Reference Bost, Gillet and SouléBGS94] and Zhang [Reference ZhangZha95]. The Néron–Tate height $\mathrm {h}'_L(\Theta )$ is non-negative and is an invariant of the pair $(A,\lambda )$.

Another natural invariant of $(A,\lambda )$ is the *stable Faltings height* $\mathrm {h}_F(A)$ of $A$ introduced by Faltings in [Reference FaltingsFal83] as a key tool in his proof of the Mordell conjecture. It is natural to ask how $\mathrm {h}'_L(\Theta )$ and $\mathrm {h}_F(A)$ are related.

Let $k \subset \bar {\mathbb {Q}}$ be a number field and assume that the pair $(A,L)$ is defined over $k$. In [Reference AutissierAut06, Reference HindryHin93], Hindry and Autissier proved an identity relating $\mathrm {h}'_L(\Theta )$ and $\mathrm {h}_F(A)$ under the assumption that $A$ has everywhere good reduction over $k$. In order to state their result, we introduce some notation.

Let $s$ be a non-zero global section of $L$. Let $M(k)_\infty$ denote the set of complex embeddings of $k$. For each $v \in M(k)_\infty$, we put the standard euclidean metric on $\bar {k}_v\cong \mathbb {C}$. Let $\|\cdot \|_v$ be a canonical metric on $L_v=L \otimes \bar {k}_v$ (i.e. a smooth hermitian metric with a translation-invariant curvature form). We then consider the local archimedean invariants

where $\mu _{H,v}$ denotes the Haar measure on the complex torus $A(\bar {k}_v)$, normalized to give $A(\bar {k}_v)$ unit volume, and where

is the $L^2$-norm of $s$. Here and below we denote by $\log$ the natural logarithm. The real number $I(A_v,\lambda _v)$ does not depend on the choice of $\Theta$ or $s$ or $\|\cdot \|_v$ (we verify this in § 12.4). We have $I(A_v,\lambda _v) > 0$ by the Jensen inequality.

In [Reference AutissierAut06, Théorème 3.1] and [Reference HindryHin93, formula (A.17)], we find the following result. Assume that $A$ has everywhere good reduction over $k$. Write $\kappa _0=\log (\pi \sqrt {2})$ and let $g=\dim (A)$. Then the equality

holds in $\mathbb {R}$. Formula (1.3) is obtained in both [Reference AutissierAut06, Reference HindryHin93] as the result of a calculation in Gillet–Soulé's arithmetic intersection theory, combined with Moret-Bailly's celebrated *key formula* for abelian schemes [Reference Moret-BaillyMor85].

In [Reference AutissierAut06, Question], it is asked whether an extension of (1.3) might hold for arbitrary principally polarized abelian varieties over $\bar {\mathbb {Q}}$ of the following shape. Assume that the abelian variety $A$ has *semistable reduction* over $k$. Let $M(k)_0$ denote the set of non-archimedean places of $k$ and, for $v \in M(k)_0$, denote by $Nv$ the cardinality of the residue field at $v$. Then, for each $v \in M(k)_0$, there should exist a natural local invariant $\alpha _v \in \mathbb {Q}_{\geqslant 0}$ of $(A,\lambda )$ at $v$ such that the equality

holds in $\mathbb {R}$. The local invariant $\alpha _v$ should vanish if and only if $A$ has good reduction at $v$.

In [Reference AutissierAut06], Autissier established the identity (1.4) for elliptic curves, for Jacobians of genus two curves and for arbitrary products of these. In [Reference de JongdJ18, Theorem 1.6], the first-named author exhibited natural $\alpha _v \in \mathbb {Q}_{\geqslant 0}$, and established (1.4), for all Jacobians and for arbitrary products of these. In both [Reference AutissierAut06, Reference de JongdJ18], the local non-archimedean invariants $\alpha _v$ are expressed in terms of the combinatorics of the dual graph of the underlying semistable curve at $v$.

### 1.1 Main result

The goal of this paper is to give a complete affirmative answer to [Reference AutissierAut06, Question]. This is established by combining Theorems A and B below.

Let $v \in M(k)_0$ be a non-archimedean place of $k$. Let $A^\mathrm {an}_v$ be the *Berkovich analytification* of $A$ over the completion $\mathbb {C}_v$ of the algebraic closure of the completion $k_v$ of $k$ at $v$. Similar to the archimedean setting, the analytification $L_v^\mathrm {an}$ of $L$ at $v$ can be endowed with a canonical metric $\|\cdot \|_{v}$; we refer to § 5 for a review of the construction and main properties of such canonical metrics.

Analogous to (1.1), we define

where

is the supremum norm of $s$ and where $\mu _{H,v}$ is the pushforward into $A_v^\mathrm {an}$ of the Haar measure of unit volume on the *canonical skeleton* of $A_v^\mathrm {an}$. The canonical skeleton of $A_v^\mathrm {an}$ is a natural real torus contained in $A_v^\mathrm {an}$ and to which $A_v^\mathrm {an}$ has a natural deformation retraction.

The invariant $I(A_v,\lambda _v)$ is independent of the choice of $L$, the choice of canonical metric $\|\cdot \|_v$ and the choice of global section $s$ (we verify this in § 7.2). It follows from the definition that $I(A_v,\lambda _v) \geqslant 0$ and equality is obtained if $A$ has good reduction at $v$ (we verify this in § 10.1).

Our first result is as follows.

Theorem A Let $(A,\lambda )$ be a principally polarized abelian variety over $\bar {\mathbb {Q}}$. Assume that $A$ has semistable reduction over the number field $k$ and set $g=\dim (A)$. Let $\Theta$ be a symmetric effective ample divisor on $A$ that defines the polarization $\lambda$ and put $L=\mathcal {O}_A(\Theta )$. Then the equality

holds in $\mathbb {R}$.

Note that the sum over $M(k)_0$ is indeed finite since $A$ has good reduction at almost all $v \in M(k)_0$. An important ingredient in our proof of Theorem A is Moret-Bailly's well-known *key formula* [Reference Moret-BaillyMor85] or more precisely Bost's version [Reference BostBos96b] of the key formula in the number field setting that expresses the stable Faltings height of a polarized abelian variety in terms of a so-called *Moret-Bailly model* of it. We review the notion of Moret-Bailly models, and state Bost's version of the key formula, in § 13.

### 1.2 Tropical moments

As we discuss next, for all non-archimedean places $v$ the term $I(A_v,\lambda _v)$ can be expressed as a ‘tropical moment’ of a principally polarized tropical abelian variety canonically associated to $A$ at $v$. This gives a concrete interpretation of the terms $I(A_v,\lambda _v)$ and makes it possible to calculate the terms $I(A_v,\lambda _v)$ explicitly.

A *principally polarized tropical abelian variety* is a tuple $(X,Y,\Phi,b)$, where $X, Y$ are finitely generated free abelian groups, $\Phi \colon Y \xrightarrow {\sim } X$ is an isomorphism and $b \colon Y \times X \to \mathbb {R}$ is a bilinear map such that $b(\cdot,\Phi (\cdot ))$ is positive definite. When $(X,Y,\Phi,b)$ is a principally polarized tropical abelian variety, we have a natural associated inclusion $Y \hookrightarrow X_\mathbb {R}^*$, where we write $X_\mathbb {R}^* = \operatorname {Hom}(X,\mathbb {R})$. The cokernel $\Sigma = X_\mathbb {R}^*/Y$ is a real torus and the bilinear map $b$ naturally induces a norm $\|\cdot \|$ on $X_\mathbb {R}^*$. We usually simply write $\Sigma$ for a principally polarized tropical abelian variety if the underlying data $(X,Y,\Phi,b)$ are understood.

Associated to the principally polarized tropical abelian variety $\Sigma$, we have its *Voronoi polytope* centered at the origin

The *tropical moment* of $\Sigma$ is defined to be the value of the integral

Here $\mu _L$ denotes the Lebesgue measure on $X_\mathbb {R}^*$, normalized to give $\textrm {Vor}(0)$ unit volume.

Let $v \in M(k)_0$ be a non-archimedean place of $k$. By Raynaud's classical theory of *non-archimedean uniformization*, see for example [Reference Baker and RabinoffBR15, Reference Bosch and LütkebohmertBL91, Reference Foster, Rabinoff, Shokrieh and SotoFRSS18] or §§ 6–7, the canonical skeleton $\Sigma _v$ of the Berkovich analytic space $A_v^\mathrm {an}$ is naturally equipped with a structure of a principally polarized tropical abelian variety.

Theorem B Let $v \in M(k)_0$. The following statements hold.

(a) The local non-archimedean term $I(A_v,\lambda _v)$ is equal to half the tropical moment of the canonical skeleton $\Sigma _v$.

(b) The term $I(A_v,\lambda _v)$ vanishes if and only if $A$ has good reduction at $v$.

(c) $I(A_v,\lambda _v)$ is a rational number.

Parts (b) and (c) of the theorem follow easily from part (a). Indeed, it is clear that the tropical moment of $\Sigma _v$ vanishes if and only if $\Sigma _v$ is a point. The latter holds if and only if $A$ has good reduction at $v$.

Next, the underlying bilinear form of the principally polarized tropical abelian variety $\Sigma _v$ is defined in terms of the discrete valuation corresponding to $v$ and is, in particular, $\mathbb {Z}$-valued. This implies that the Voronoi polytope associated to $\Sigma _v$ is a rational polyhedron and this gives that the tropical moment of $\Sigma _v$ is a rational number.

In the following we discuss a few applications of Theorems A and B.

### 1.3 Lower bounds for the stable Faltings height

First, as $\mathrm {h}'_L(\Theta )$ is non-negative, Theorem A immediately implies the following lower bound for the stable Faltings height of $A$.

Corollary Assume that $A$ has semistable reduction over the number field $k$ and set $g=\dim (A)$. Then the lower bound

holds.

With the terms $I(A_v,\lambda _v)$ interpreted as half the tropical moments of the canonical skeleta $\Sigma _v$, Wagener obtained the lower bound (1.7) in his 2016 PhD thesis [Reference WagenerWag16, Théorème A]. As for each $v \in M(k)_0$ we have $I(A_v,\lambda _v) \geqslant 0$, the lower bound (1.7) improves upon the well-known lower bound

for $\mathrm {h}_F(A)$ due to Bost [Reference BostBos96a, Reference Gaudron and RémondGR14]. We recall that (1.8) can in turn be used to obtain refinements of Masser's ‘matrix lemma’ [Reference MasserMas85]. See, for example, [Reference AutissierAut13] and the references therein.

### 1.4 Elliptic curves

Let $\Sigma$ be a circle of circumference $\sqrt {\ell }$. A small calculation yields that the tropical moment of $\Sigma$ is equal to $\frac {1}{12}\ell$. This has the following application. Let $(A,\lambda )$ be an elliptic curve with semistable reduction over the number field $k$. Let $\Theta$ be an effective symmetric divisor on $A$ that defines the principal polarization $\lambda$ and put $L=\mathcal {O}_A(\Theta )$. Then $\Theta$ is a two-torsion point of $A$, which gives $\mathrm {h}'_L(\Theta )=0$. If $A$ has bad reduction at $v \in M(k)_0$, then $\Sigma _v$ is a circle of circumference $\sqrt {\operatorname {ord}_v \Delta _v}$, where $\Delta _v$ is the minimal discriminant of $A$ at $v$. We conclude using Theorem B that $I(A_v,\lambda _v)=\frac {1}{24}\operatorname {ord}_v \Delta _v$ if $v \in M(k)_0$.

Next, let $v \in M(k)_\infty$ and write $A(\bar {k}_v) = \mathbb {C}/(\mathbb {Z}+\tau _v \mathbb {Z})$, where $\tau _v \in \mathbb {H}$ with $\mathbb {H}$ the Siegel upper half plane. For $\tau \in \mathbb {H}$, we set $q={e}^{2\pi \sqrt {-1}\tau }$ and let $\Delta (\tau )=q \prod _{n=1}^\infty (1-q^n)^{24}$ be the usual discriminant modular form. By [Reference AutissierAut06, Proposition 2.1], we have $I(A_v,\lambda _v) = -\frac {1}{24} \log (|\Delta (\tau _v)|(2 \,\mathrm {Im}\, \tau _v)^6)$.

Let $\mathrm {h}_F(A)$ be the stable Faltings height of $A$. Applying Theorem A, we find that

This recovers the well-known Faltings–Silverman formula for the stable Faltings height of an elliptic curve; cf. [Reference DeligneDel85, Exemple 1.4 and Remarque 1.5] or [Reference FaltingsFal84, Theorem 7] or [Reference SilvermanSil84, Proposition 1.1].

### 1.5 Jacobians

Let $\Gamma$ be a compact connected metric graph. Let $r(p,q)$ denote the *effective resistance* between points $p, q \in \Gamma$. Fix $q \in \Gamma$ and set $f(x) = \frac {1}{2}r(x,q)$. Following [Reference Baker and FaberBF06, Reference Baker and RumelyBR07], we set

The real number $\tau (\Gamma )$ is independent of the choice of $q$. Let $\operatorname {Jac}(\Gamma )$ denote the *tropical Jacobian* of $\Gamma$ as in [Reference Mikhalkin and ZharkovMZ08]. Then $\operatorname {Jac}(\Gamma )$ is a principally polarized tropical abelian variety canonically associated to $\Gamma$. We have shown in [Reference de Jong and ShokriehdJS18, Theorem B] that its tropical moment is equal to

where $\ell (\Gamma )$ is the total length of $\Gamma$ and $\tau (\Gamma )$ is the tau invariant of $\Gamma$ as in (1.9).

This leads to the following application. Let $C$ be a smooth projective geometrically connected curve of genus $g \geqslant 2$ with semistable reduction over $k$ and let $(J,\lambda )$ be its Jacobian. Let $v \in M(k)_0$. Let $\Gamma _v$ be the dual graph of the geometric special fiber $\mathcal {C}_{\bar {v}}$ of the minimal regular model $\mathcal {C}$ of $C$ at $v$, endowed with its canonical metric structure as in [Reference ZhangZha93]. In particular, $\Gamma _v$ is a compact connected metric graph and its total length $\ell (\Gamma _v)$ equals the number of singular points of $\mathcal {C}_{\bar {v}}$. By [Reference Faltings and ChaiFC90, Theorem III.8.3], the tropical Jacobian $\operatorname {Jac}(\Gamma _v)$ of $\Gamma _v$ is isometric with the principally polarized tropical abelian variety $\Sigma _v$ determined by $(J,\lambda )$ at $v$.

Using (1.10) and Theorem B, the formula from Theorem A specializes into the formula

for the stable Faltings height of $J$. This recovers [Reference de JongdJ18, Theorem 1.6].

Note that $\mathrm {h}_F(J)$ is equal to the stable Faltings height $\mathrm {h}_F(C)$ of the curve $C$ itself; cf. [Reference PazukiPaz19, Proposition 6.5]. In the case that $g=2$, the formula in (1.11) reproves [Reference AutissierAut06, Théorème 5.1] and gives a uniform explanation for the entries in the table that follows directly upon [Reference AutissierAut06, Théorème 5.1]. We refer to [Reference de Jong and ShokriehdJS18, § 9] for more details about the $g=2$ case.

### 1.6 The function field case

A slight variation of our arguments yields the following counterpart in the function field setting.

Let $S$ be a smooth projective connected curve over an algebraically closed field and let $F$ denote the function field of $S$. Let $(A,\lambda )$ be a principally polarized abelian variety of dimension $g$ with semistable reduction over $F$ and let $\pi \colon G \to S$ denote the connected component of the Néron model of $A$ over $S$ with zero section $e \colon S \to G$. Let $\mathrm {h}(A) = \deg e^* \Omega ^g_{G/S}$ in $\mathbb {Z}$ denote the *modular degree* of $A$.

Let $\Theta$ be a symmetric effective ample divisor on $A$ defining the principal polarization $\lambda$ and denote by $\mathrm {h}'_L(\Theta )$ the Néron–Tate height of $\Theta$ with respect to the line bundle $L=\mathcal {O}_A(\Theta )$. Then the equality

holds. Here $|S|$ denotes the set of closed points of $S$ and $I(A_v,\lambda _v)$ is half the tropical moment of the canonical skeleton of the Berkovich analytification of $A$ at $v$.

From the fact that $I(A_v,\lambda _v) \in \mathbb {Q}$, we obtain $\mathrm {h}'_L(\Theta ) \in \mathbb {Q}$, a fact that seems not clear *a priori*. Moreover, as the right-hand side of (1.12) is clearly non-negative, we obtain another proof of the well-known fact that $\mathrm {h}(A) \geqslant 0$; cf. [Reference Faltings and ChaiFC90, Proposition V.2.2] or [Reference Moret-BaillyMor85, Chapitre XI, 4.5]. We leave the details of the proof of (1.12) to the interested reader.

### 1.7 Structure of the paper

Sections 2–9 are mostly preliminary. In these sections we review basic notions and results concerning semistable models, cubical structures, Berkovich analytification, model metrics, admissible metrics, Green's functions, canonical metrics, Raynaud extensions, non-archimedean uniformization, non-archimedean theta functions and tropicalization of abelian varieties.

In § 10 we investigate the relationship between canonical metrics and cubical line bundles on semistable models and in § 11 we prove Theorem B. In § 12 we introduce the main relevant structures on the archimedean side needed for our proof of Theorem A and in § 13 we recall the stable Faltings height and state Bost's key formula for it.

In § 14 we review the notion of Néron–Tate heights of cycles on abelian varieties over number fields and prove a local decomposition formula for the height of a theta divisor. In § 15 we finally give our proof of Theorem A.

### Notation and terminology

When $R$ is a discrete valuation ring with fraction field $F$, we denote its maximal ideal by $\mathfrak {m}_R$, its residue field by $\tilde {F}$ and we let $\varpi$ denote a generator of $\mathfrak {m}_R$. Unless mentioned otherwise, we endow $F$ with the unique non-archimedean absolute value $|\cdot | \colon F \to \mathbb {R}$ whose valuation ring is $R$ and that is normalized such that $|\varpi |={e}^{-1}$.

When $M$ is a free rank-one $R$-module and $s$ is a non-zero element of $M\otimes _R F$, we write $\operatorname {ord}(s)$ for the *multiplicity* of $s$, by which we mean the largest integer $e$ such that $s$ is contained in $M \otimes \mathfrak {m}_R^e$.

When $\mathcal {X}$ is an integral noetherian scheme, and $\mathcal {L}$ is a line bundle on $\mathcal {X}$, we call a *rational section* of $\mathcal {L}$ any element of the stalk $\mathcal {L}_\eta$. Here $\eta$ is the generic point of $\mathcal {X}$. Let $\xi \in \mathcal {X}$ and let $K=\mathcal {O}_{\mathcal {X},\eta }$ be the function field of $\mathcal {X}$. The fraction field of the local ring $\mathcal {O}_{\mathcal {X},\xi }$ of $\mathcal {X}$ at $\xi$ coincides with $K$ and the natural map $\mathcal {L}_\xi \otimes _{\mathcal {O}_{\mathcal {X},\xi }} K \to \mathcal {L}_\eta$ is an isomorphism. When $\mathcal {O}_{\mathcal {X},\xi }$ is a discrete valuation ring and $s$ is a non-zero rational section of $\mathcal {L}$, we obtain via the natural isomorphism $\mathcal {L}_\xi \otimes _{\mathcal {O}_{\mathcal {X},\xi }} K \xrightarrow {\sim } \mathcal {L}_\eta$ a well-defined *multiplicity* $\operatorname {ord}_{\xi,\mathcal {L}} (s) \in \mathbb {Z}$ of $s$ at $\xi$.

When $V$ is a scheme over $\mathbb {C}$ or over an algebraically closed non-archimedean valued field, we denote by $V^\mathrm {an}$ the associated complex or Berkovich analytic space.

When $k$ is a number field, we denote by $M(k)_0$ the set of non-archimedean places of $k$, by $M(k)_\infty$ the set of complex embeddings of $k$ and we set $M(k) = M(k)_0 \sqcup M(k)_\infty$.

## 2. Semistable group schemes

Let $S$ be a locally noetherian scheme. Let $\pi \colon \mathcal {A} \to S$ be a smooth commutative group scheme of finite type over $S$ with zero section $e \colon S \to \mathcal {A}$. We call the *identity component* of $\mathcal {A}$ the open subscheme of $\mathcal {A}$ formed by taking the union of all fiberwise identity components. The group scheme $\mathcal {A}$ is called *semistable* if the identity component of $\mathcal {A}$ is a semiabelian group scheme.

Let $\mathcal {L}$ be a line bundle on $\mathcal {A}$. A *rigidification* of $\mathcal {L}$ is an isomorphism of line bundles $\mathcal {O}_S \xrightarrow {\sim } e^*\mathcal {L}$. For each $I \subset \{1,2,3\}$, let $m_I \colon \mathcal {A} \times _S \mathcal {A} \times _S \mathcal {A} \to \mathcal {A}$ be the morphism given functorially on points by sending $(x_1,x_2, x_3)$ to $\sum _{i \in I} x_i$. Write $\mathcal {D}_3(\mathcal {L})$ for the line bundle $\mathcal {D}_3(\mathcal {L}) = \bigotimes _{\varnothing \neq I \subset \{1,2,3\}} m_I^*\mathcal {L}^{\otimes (-1)^{\sharp I}}$ on $\mathcal {A} \times _S \mathcal {A} \times _S \mathcal {A}$. A *cubical structure* on $\mathcal {L}$ is an isomorphism $\mathcal {O}_{\mathcal {A}^3}\xrightarrow {\sim }\mathcal {D}_3(\mathcal {L})$ satisfying suitable symmetry and cocycle conditions as described in [Reference Moret-BaillyMor85, Définition I.2.4.5]. A line bundle $\mathcal {L}$ on $\mathcal {A}$ endowed with a cubical structure is called a *cubical line bundle*. A cubical line bundle is canonically rigidified.

By the theorem of the cube, each rigidified line bundle on an abelian variety over a field has a unique cubical structure.

### 2.1 Cubical extensions

Assume that $S$ is the spectrum of a discrete valuation ring $R$ and assume that the generic fiber of $\mathcal {A}$ is an abelian variety $A$. Let $L$ be a cubical (that is, rigidified) line bundle on $A$. A cubical line bundle $\mathcal {L}$ on $\mathcal {A}$ extending the cubical line bundle $L$ is unique up to isomorphism, once one exists, by [Reference Moret-BaillyMor85, Théorème II.1.1]. We have the following two important existence results for cubical extensions. Let $\Phi _\mathcal {A}$ denote the group of connected components of the special fiber of $\mathcal {A}$ and let $n \in \mathbb {Z}_{>0}$ be such that $n \cdot \Phi _\mathcal {A}=0$.

Lemma 2.1 The cubical line bundle $L^{\otimes 2n}$ extends as a cubical line bundle over $\mathcal {A}$.

Proof. This is [Reference Moret-BaillyMor85, Proposition II.1.2.1].

Lemma 2.2 Let $R \to R'$ be a finite extension and let $F'$ be the fraction field of $R'$. Let $e$ denote the ramification index of $R \to R'$. Assume that $2n | e$ if $n$ is even and $n|e$ if $n$ is odd. Then the cubical line bundle $L_{F'}$ on $A_{F'}$ extends as a cubical line bundle over the group scheme $\mathcal {A} \times _R R'$.

Proof. This is [Reference Moret-BaillyMor85, Proposition II.1.2.2].

## 3. Berkovich analytification

The purpose of this section is to set terminology and recall some basic notions concerning Berkovich spaces. We use [Reference BerkovichBer90, Reference Chambert-LoirCha11, Reference Chambert-LoirCha06, Reference Chambert-Loir and ThuillierCT09, Reference GublerGub10, Reference GublerGub07] as our main references.

### 3.1 Berkovich analytic spaces

Let $R$ be a complete discrete valuation ring, with fraction field $F$, and let $\mathbb {F}$ be the completion of an algebraic closure of $F$, endowed with the unique extension of $|\cdot |$ as an absolute value on $\mathbb {F}$. We write $\mathbb {F}^\circ$ for the valuation ring of $\mathbb {F}$ and $\tilde {\mathbb {F}}$ for the residue field of $\mathbb {F}^\circ$. When $V$ is a separated scheme of finite type over $F$, we are interested in the Berkovich analytification of the $\mathbb {F}$-scheme $V_\mathbb {F}$, denoted by $V^\mathrm {an}$. The step of passing to $\mathbb {F}$ first is natural for our purposes and moreover some of the references that we use only consider Berkovich analytic spaces over algebraically closed fields.

We recall that the underlying set of $V^\mathrm {an}$ consists of pairs $x=(y,|\cdot |)$, where $y$ is a point of $V_\mathbb {F}$ and where $|\cdot | \colon \kappa (y) \to \mathbb {R}$ is an absolute value on the residue field at $y$ that extends the given absolute value on $\mathbb {F}$. The point $y$ is called the *center* of $x$. The space $V^\mathrm {an}$ contains the set $V^\mathrm {alg}=V(\mathbb {F})$ of algebraic points of $V$ naturally as a dense subset. The underlying topological space of $V^\mathrm {an}$ is Hausdorff, locally compact, locally contractible and path-connected if $V_\mathbb {F}$ is connected. The construction $V \mapsto V^\mathrm {an}$ is functorial; for example, when $L$ is a line bundle on $V$, analytification produces a line bundle $L^\mathrm {an}$ on $V^\mathrm {an}$.

Assume that $V$ is geometrically integral and let $\mathbb {F}(V)$ denote the function field of $V_\mathbb {F}$. Let $y \in V_\mathbb {F}$ and let $\mathcal {O}_{V_\mathbb {F},y}$ denote the local ring of $V_\mathbb {F}$ at $y$. For each $x \in V^\mathrm {an}$ with center $y$, pullback along the canonical map $\mathcal {O}_{V_\mathbb {F},y} \to \kappa (y)$ gives rise to a multiplicative seminorm on $\mathbb {F}(V) = \operatorname {Frac} \mathcal {O}_{V_\mathbb {F},y}$. We denote this seminorm by $|\cdot |_x$. For $f \in \mathbb {F}(V)$, we sometimes write $|f(x)|$ instead of $|f|_x$.

### 3.2 Reduction map

Let $V$ be a geometrically integral and projective $F$-scheme. Write $S = \operatorname {Spec} R$. One way of obtaining $V^\mathrm {an}$ as an analytic space is as follows [Reference GublerGub07, § 2.7]. Let $\mathcal {V}$ be an integral scheme and let $\mathcal {V} \to S$ be a projective and flat morphism with generic fiber isomorphic to $V$. By base change and $\varpi$-adic completion, one obtains from $\mathcal {V}$ an admissible formal scheme $\mathfrak {V}$ over $\mathbb {F}^\circ$. The analytic space $V^\mathrm {an}$ is naturally identified with the generic fiber of $\mathfrak {V}$. The special fiber of $\mathfrak {V}$ is naturally identified with the $\tilde {\mathbb {F}}$-scheme $\mathcal {V}_{\tilde {\mathbb {F}}}$. By virtue of these identifications, we obtain by [Reference BerkovichBer90, § 2.4] a canonical *reduction map* $\operatorname {red}_\mathcal {V} \colon V^\mathrm {an} \to \mathcal {V}_{\tilde {\mathbb {F}}}$. The reduction map $\operatorname {red}_\mathcal {V}$ is surjective and, if $\xi$ is a generic point of $\mathcal {V}_{\tilde {\mathbb {F}}}$, there exists a *unique* $x \in V^\mathrm {an}$ such that $\operatorname {red}_\mathcal {V}(x) = \xi$; see [Reference BerkovichBer90, Proposition 2.4.4]. We call this point the *Shilov point* corresponding to $\xi$, denoted by $x_\xi$.

## 4. Metrics and Green's functions

We continue with the setting of § 3 and review the notions of metrics and Green's functions.

### 4.1 Model metrics

Let $L$ be a line bundle on $V$ and let $L^\mathrm {an}$ be its analytification over $\mathbb {F}$. One has a natural notion of continuous metrics on $L^\mathrm {an}$. An important class of continuous metrics on $L^\mathrm {an}$ is provided by models of (tensor powers of) $L$, as follows: let $\mathcal {V} \to S$ be an integral, projective and flat model of $V$ and let $\mathcal {L}$ be a line bundle on $\mathcal {V}$ whose restriction to $V$ is equal to $L$. There exists a continuous metric $\|\cdot \|_{\mathcal {L}}$ on $L^\mathrm {an}$ uniquely determined by the following property. Let $s$ be a non-zero rational section of $L$ and view $s$ as a rational section of $\mathcal {L}$ on $\mathcal {V}$. Let $x \in V^\mathrm {an}$ and write $\xi = \operatorname {red}_\mathcal {V}(x)$, viewed as a point on $\mathcal {V}$. Then $\|s(x)\|_\mathcal {L}$ is given by the following prescription. Let $U$ be an open neighborhood of $\xi$ in $\mathcal {V}$ such that $\mathcal {L}$ is trivialized on $U$. Let $t$ be a trivializing element of $\mathcal {L}(U)$ and let $f \in F(V)$ be the unique rational function on $U$ satisfying $s = f \cdot t$ on $U$. Then we put $\|s(x)\|_\mathcal {L} = |f(x)|$.

A small verification shows that the assignment $(s,x) \mapsto \|s(x)\|_\mathcal {L}$ is well defined and, in particular, is independent of the choice of $U$ and of the trivializing section $t$. One calls $\|\cdot \|_{\mathcal {L}}$ the *model metric* on $L^\mathrm {an}$ determined by the model $(\mathcal {V},\mathcal {L})$ of $(V,L)$. More generally, a model metric on $L^\mathrm {an}$ is any metric that is obtained by taking $e$th roots of a model metric determined by some model of $L^{\otimes e}$ for some $e \in \mathbb {Z}_{>0}$. The notion of model metrics can be extended to the setting of formal models of $V$, but our assumption that $V$ is projective ensures that for our purposes we do not need them.

### 4.2 Semipositive and admissible metrics

Let $\|\cdot \|$ be a continuous metric on $L^\mathrm {an}$. We call the metric $\|\cdot \|$ *semipositive* if $\|\cdot \|$ is obtained as a uniform limit of model metrics on $L^\mathrm {an}$ associated to pairs $(\mathcal {V},\mathcal {L})$ consisting of an integral projective flat model $\mathcal {V}$ of $V$ and a model $\mathcal {L}$ of some tensor power $L^{\otimes e}$ as above, such that for each $(\mathcal {V},\mathcal {L})$, the first Chern class $c_1(\mathcal {L})$ has non-negative intersection with all complete curves in the special fiber of $\mathcal {V}$. We call the metric $\|\cdot \|$ *admissible* if $\|\cdot \|$ can be written as a quotient of two semipositive metrics. We call the metric $\|\cdot \|$ *bounded continuous* if there exists a pair $(\mathcal {V},\mathcal {L})$, where $\mathcal {L}$ extends $L$ such that the quotient $\| \cdot \|/\| \cdot \|_{\mathcal {L}}$ is a bounded and continuous function on $V^\mathrm {an}$. An admissible metric is bounded continuous.

We refer to [Reference Chambert-LoirCha11, Reference Chambert-LoirCha06, Reference Chambert-Loir and ThuillierCT09, Reference ZhangZha95] for more precise definitions and extensive discussions. The definitions of semipositive and admissible metrics given in [Reference GublerGub10, Reference GublerGub07] are more involved, and work more generally for proper schemes $V$, but coincide with the current definitions since we are assuming that $V$ is projective. An important class of admissible metrics is given by the canonical metrics on a rigidified symmetric ample line bundle on an abelian variety over $F$. We discuss these canonical metrics in § 5.

### 4.3 Green's functions

Let $D$ be an effective Cartier divisor on $V$. Following [Reference Chambert-Loir and ThuillierCT09, § 2], a *Green's function* with respect to $D$ is any continuous function $g_D \colon V^\mathrm {an} \setminus \operatorname {Supp}(D) \to \mathbb {R}$ obtained as follows: put $L=\mathcal {O}_V(D)$ and let $\|\cdot \|$ be any admissible metric on $L^\mathrm {an}$. The divisor $D$ determines a canonical global section $s_D$ of $L$. Then, for each $x \in V^\mathrm {an} \setminus \operatorname {Supp}(D)$, we put $g_D(x) = -\log \! \|s_D(x)\|$. The notion of a Green's function readily generalizes to arbitrary Cartier divisors on $V$. When $g_D \colon V^\mathrm {an} \setminus \operatorname {Supp}(D) \to \mathbb {R}$ is a Green's function on $V^\mathrm {an}$ with respect to the Cartier divisor $D$, the restriction of $g_D$ to $V^\mathrm {alg} \setminus \operatorname {Supp}(D)$ is a *Weil function* on $V^\mathrm {alg}$ with respect to $D$ in the sense of [Reference LangLan83, § 10.2].

## 5. Canonical metrics

Let $A$ be an abelian variety over $F$ and let $L$ be a rigidified ample line bundle on $A$ that we assume moreover to be *symmetric*. We thus have a unique isomorphism of rigidified line bundles $[-1]^*L \xrightarrow {\sim } L$. Gubler constructed in [Reference GublerGub07, § 3.3] a *canonical metric* $\|\cdot \|_L$ on $L^\mathrm {an}$ using formal model metrics and taking uniform limits. The discussion in [Reference ZhangZha95, § 2] shows that $\|\cdot \|_L$ can alternatively be obtained by working with model metrics obtained from integral projective flat models of $A$ and taking uniform limits of such.

### 5.1 Axiomatic characterization

This leads to the following characterization of $\|\cdot \|_L$; cf. [Reference ZhangZha95, Theorem 2.2]. Let $m \in \mathbb {Z}_{>0}$ and denote by $[m] \colon A \to A$ the multiplication-by-$m$. We have a unique isomorphism $\varphi _m \colon [m]^*L \xrightarrow {\sim } L^{ \otimes m^2}$ of rigidified line bundles on $A$ (see, for instance, [Reference Moret-BaillyMor85, Proposition I.5.5]). The canonical metric $\|\cdot \|_L$ is the unique bounded continuous metric on $L^\mathrm {an}$ that has the property that the isomorphism $\varphi _m$ is an isometry with respect to the canonically induced metrics on $[m]^*L$ and $L^{\otimes m^2}$.

The canonical metric $\|\cdot \|_L$ is independent of the choice of $m$ and is invariant under extensions of the base field $F$. As can be verified immediately, if the given rigidification of $L$ is multiplied by a scalar $\lambda \in F^\times$, then the canonical metric on $L$ associated to the new rigidification is obtained by multiplying $\|\cdot \|_L$ by $|\lambda |$. Furthermore, for each $n \in \mathbb {Z}_{>0}$, the canonical metric on the symmetric rigidified ample line bundle $L^{\otimes n}$ is given by $\|\cdot \|_L^{\otimes n}$.

In § 10, we see how $\|\cdot \|_L$ is connected to the Néron model of $A$ over the valuation ring of $F$. We discuss here a special case. If $A$ has good reduction over the valuation ring $R$ of $F$, then, by [Reference Moret-BaillyMor85, II.3.5 and VI.2.1], the line bundle $L$ extends uniquely as a cubical symmetric ample line bundle $\mathcal {L}$ over the Néron model of $A$ over $R$, which now is an abelian scheme over $R$. In this case $\|\cdot \|_L$ is just the model metric associated to $\mathcal {L}$. The metric $\|\cdot \|_L$ is in general not a model metric, but it is always an admissible metric [Reference Chambert-LoirCha11, Reference GublerGub07, Reference ZhangZha95].

### 5.2 Translation by a two-torsion point

We continue to assume that $L$ is symmetric, ample and rigidified. As our considerations are analytic in nature, we work over the field $\mathbb {F}$. Let $y \in A[2]$ be a two-torsion point of $A$ and write $T_y \colon A \xrightarrow {\sim } A$ for translation along $y$. We have that $T_y^*L$ is a symmetric ample line bundle on $A$.

Lemma 5.1 Let $\|\cdot \|_L$ be the canonical metric on $L^\mathrm {an}$. Then the pullback metric $T_y^*\|\cdot \|_L$ is a canonical metric on $T_y^*L^\mathrm {an}$.

Proof. It suffices to show that $T_y^* \|\cdot \|_L^{\otimes 4}$ is a canonical metric on $(T_y^*L^\mathrm {an})^{\otimes 4}$. As the metric $T_y^* \|\cdot \|_L^{\otimes 4}$ is bounded continuous, it suffices to show that there exists an isomorphism $[2]^* (T_y^*L)^{\otimes 4} \xrightarrow {\sim } (T_y^*L)^{\otimes 16}$ that is an isometry for the metrics induced from $\|\cdot \|_L$. We may construct such an isomorphism as follows. Starting from the isomorphism $\varphi _2 \colon [2]^*L \xrightarrow {\sim } L^{\otimes 4}$, we obtain by pullback along $T_y$ an isomorphism $[2]^*L = T_y^*[2]^*L \xrightarrow {\sim } (T_y^*L)^{\otimes 4}$. From this, we obtain by pullback along $[2]$ an isomorphism $[4]^*L \xrightarrow {\sim } [2]^*(T_y^*L)^{\otimes 4}$. On the other hand, we have, again starting from the isomorphism $\varphi _2 \colon [2]^*L \xrightarrow {\sim } L^{\otimes 4}$, by pullback along $[2]$, an isomorphism $\varphi _4 \colon [4]^*L \xrightarrow {\sim } L^{\otimes 16}$ and then by pullback along $T_y$ an isomorphism $[4]^*L = T_y^*[4]^*L \xrightarrow {\sim } (T_y^*L)^{\otimes 16}$. Combining the results of applying $[2]^* \circ T_y^*$, respectively $T_y^* \circ [2]^*$, to $\varphi _2$, we find an isomorphism $[2]^* (T_y^*L)^{\otimes 4} \xrightarrow {\sim } (T_y^*L)^{\otimes 16}$. This isomorphism is by construction an isometry for the induced metrics from $\|\cdot \|_L$.

### 5.3 Néron functions

Let $s$ be a non-zero rational section of the rigidified symmetric ample line bundle $L$ and write $D=\operatorname {div}_L s$. Following [Reference LangLan83, § 11.1], we call a *Néron function* with respect to $D$ any Weil function $\Lambda \colon A^\mathrm {alg} \setminus \operatorname {Supp}(D) \to \mathbb {R}$ with respect to $D$ such that there exists a rational function $h$ on $A$ whose divisor is equal to $-[2]^*D+4D$ and such that, away from the support of $\operatorname {div} h$, the identity

is satisfied. For the notion of Weil function, we refer to [Reference LangLan83, § 10.2].

We note that the isomorphism $\varphi _2 \colon [2]^*L \xrightarrow {\sim } L^{\otimes 4}$ of line bundles from § 5.1 allows us to view the rational section $s^{\otimes 4} \otimes [2]^*s^{\otimes -1}$ of the rigidified trivial line bundle $L^{\otimes 4} \otimes [2]^*L^{\otimes -1}$ as a rational function $h$ on $A$ whose divisor is equal to $-[2]^*D+4D$. The fact that $\varphi _2$ is an isometry for the canonical metrics translates into the identity

on $A^\mathrm {an}$ wherever each of the three terms is defined.

Write $g_D$ for the Green's function $-\log \|s\|_L$ on $A^\mathrm {an}$ (cf. § 4.3). The restriction of $g_D$ to $A^\mathrm {alg} \setminus \operatorname {Supp}(D)$ is a Weil function with respect to $D$. We conclude from (5.2) that the restriction of $g_D$ to $A^\mathrm {alg} \setminus \operatorname {Supp}(D)$ is in fact a Néron function with respect to $D$. It is shown in [Reference LangLan83, § 11.1] that a Néron function with respect to $D$ is unique up to an additive constant.

## 6. Raynaud extensions

Let $R$ be a complete discrete valuation ring with fraction field $F$. We briefly discuss the theory of Raynaud extensions for polarized abelian varieties over $F$, following [Reference Faltings and ChaiFC90, Chapter II].

Assume that we are given an abelian variety $A$ over $F$ with split semistable reduction, and a rigidified ample line bundle $L$ on $A$, determining a polarization $\lambda _{A,L} \colon A \to A^t$ of $A$. Write $S = \operatorname {Spec} R$ and let $v$ denote the closed point of $S$. Let $G$ denote the identity component of the Néron model of $A$ over $S$. By our assumptions, the scheme $G$ is a semiabelian scheme over $S$. By [Reference Moret-BaillyMor85, II.3.5 and VI.2.1], the group scheme $G$ is endowed with a unique cubical ample extension $\mathcal {L}_G$ of $L$. The Raynaud extension construction [Reference Faltings and ChaiFC90, II.1–2] can be applied to the pair $(G,\mathcal {L}_G)$ to yield a canonical short exact sequence, for the fppf topology, of commutative group schemes

over $S$. Here $T$ is a split torus and $\mathcal {B}$ an abelian scheme. The Raynaud construction produces an ample cubical line bundle $\tilde {\mathcal {L}}_{\tilde {G}}$ on $\tilde {G}$ and an isomorphism $\tilde {G}_v \xrightarrow {\sim } G_v$ of special fibers. In particular, the formal completions of $G$ and $\tilde {G}$ are identified.

Let $G^t$ denote the identity component of the Néron model of the dual abelian variety $A^t$. We similarly have, associated to $G^t$, a Raynaud extension

over $S$ with $T^t$ a split torus. Let $\underline {X} = \mathcal {H}om(T,\mathbb {G}_m)$ and $\underline {Y} = \mathcal {H}om(T^t,\mathbb {G}_m)$, both viewed as étale group schemes over $S$. For $u \in \underline {X}(S)$, we usually denote by $\chi ^u \colon T \to \mathbb {G}_m$ the corresponding character. From extension (6.1), we obtain an associated pushout diagram

in the category of commutative $S$-group schemes with the fppf topology. The pushout construction gives rise to a morphism $\Omega \colon \underline {X} \to \mathcal {B}^t$ of group schemes by sending $u$ to the class of the algebraically trivial line bundle on $\mathcal {B}$ determined by the $\mathbb {G}_m$-torsor $\tilde {G}_u$. Similarly, the extension (

6.2) determines an assignment $u' \mapsto e_{u'}$ and a morphism of group schemes $\Omega ' \colon \underline {Y} \to \mathcal {B}$.

The polarization $\lambda _{A,L} \colon A \to A^t$ associated to $L$ extends canonically into an isogeny $\lambda _{G,L} \colon G \to G^t$. Functoriality of the Raynaud extension gives an isogeny $\lambda _{T,L} \colon T \to T^t$, an isogeny $\lambda _{\tilde {G},L} \colon \tilde {G} \to \tilde {G}^t$ and a polarization $\lambda _{\mathcal {B},L} \colon \mathcal {B} \to \mathcal {B}^t$. These morphisms fit together into a morphism of short exact sequences of commutative group schemes

over $S$. The morphism $\lambda _{T,L}$ induces by pullback a morphism $\Phi \colon \underline {Y} \to \underline {X}$ of group schemes and the diagram

commutes. If $L$ defines a *principal* polarization, then $\Phi \colon \underline {Y} \to \underline {X}$ and each of the maps $\lambda$ in (

) is an isomorphism.

In addition to the above canonical data associated to $(A,L)$, we may and do pick some non-canonical further data as follows. For these extra data, a finite extension of the field $F$ may be needed, but this is harmless for our purposes. Denote the generic fibers of $\tilde {G}$ and $\mathcal {B}$ by $E$ and $B$, respectively. Let $q \colon E \to B$ be the map induced by $\mathfrak {q} \colon \tilde {G} \to \mathcal {B}$. Set $X=\underline {X}(F)$, $Y=\underline {Y}(F)$. First of all, we may and do pick an injective lift

of the map $\Omega '_\eta$. Similarly, we may pick an injective lift $\upsilon ^t \colon X \to E^t(F)$ of the quotient map $E^t(F) \to B^t(F)$. We view $Y$ as a subgroup of $E(F)$ and $X$ as a subgroup of $E^t(F)$ via the maps $\upsilon$, $\upsilon ^t$. We can arrange that $e_u(u')=e_{u'}(u)$ for $u \in X$, $u' \in Y$.

Let $\mathcal {P}$ denote the Poincaré bundle on $\mathcal {B} \times _S \mathcal {B}^t$, endowed with its canonical rigidification, and let $P$ be its generic fiber. We may further suppose that the map

defines a trivialization of the invertible sheaf $(\Omega '_\eta \times \Omega _\eta )^*P$ on $Y \times X$.

Next, we may and do pick an ample cubical line bundle $\mathcal {M}$ on $\mathcal {B}$ such that $\mathcal {M}$ determines the polarization $\lambda _{\mathcal {B},L} \colon \mathcal {B} \to \mathcal {B}^t$ and such that $\mathfrak {q}^* \mathcal {M}$ is identified with $\tilde {\mathcal {L}}_{\tilde {G}}$ as cubical line bundles. Denote by $M$ the generic fiber of $\mathcal {M}$, which is thus a rigidified ample line bundle on $B$. If $L$ determines a principal polarization, then so does $M$.

Let $m_\mathcal {B} \colon \mathcal {B} \times _S \mathcal {B} \to \mathcal {B}$ denote the additively written group operation of $\mathcal {B}$ and denote by $p_1, p_2 \colon \mathcal {B} \times _S \mathcal {B} \to \mathcal {B}$ the projections onto the first, respectively the second, factor. We have a canonical identification

of rigidified line bundles on $\mathcal {B} \times _S \mathcal {B}$.

Finally, we may and do, pick, given our choices of $\upsilon, \upsilon '$ and $\mathcal {M}$, a trivialization $c \colon Y \to M$ of the rigidified line bundle $(\Omega '_\eta )^*M$ on $Y$ such that via the restriction of the canonical identification given by (6.6) to the generic fiber, the trivialization $c$ satisfies the relation

for all $u', v' \in Y$.

We call the data $(M,\Phi,c)$ a *triple* associated to the rigidified ample line bundle $L$. For $u \in X$, we denote by $E_u$ the rigidified line bundle determined by the generic fiber of $\tilde {G}_u$. By [Reference Foster, Rabinoff, Shokrieh and SotoFRSS18, Theorem 3.6], two triples $(M_1,\Phi _1,c_1)$ and $(M_2,\Phi _2,c_2)$ define the same rigidified line bundle if and only if $\Phi _1= \Phi _2$ and there exists $u \in X$ such that $M_1 \otimes M_2^{-1} \cong E_u$ and $c_1 \otimes c_2^{-1} \cong \varepsilon _u$. Here, for $u \in X$, we denote by $\varepsilon _u \colon Y \to E_u(F)$ the composite of the inclusion $\upsilon \colon Y \to E(F)$ and the map $e_u \colon E(F) \to E_u(F)$. It is straightforward to extend the notion of associated triple to the setting of rigidified ample line bundles defined over $\mathbb {F}$.

Denote by $\|\cdot \|_M$ the model metric on $M^\mathrm {an}$ derived from $\mathcal {M}$ and by $\|\cdot \|_P$ the model metric on $P^\mathrm {an}$ derived from $\mathcal {P}$. By construction, the rigidification of $M$ is an isometry for the metric $\|\cdot \|_M$ and the canonical rigidification of $P$ is an isometry for the metric $\|\cdot \|_P$. For $u' \in Y$, $v \in X$, we put

Then $b$ is a $\mathbb {Z}$-valued bilinear map on $Y \times X$ and $c_\mathrm {trop}$ is a $\mathbb {Z}$-valued function on $Y$. From (6.6) and (6.7), we derive the fundamental identity

for all $u', v' \in Y$. The assumption that $L$ is ample implies that the map $Y \times Y \to \mathbb {Z}$ given by sending $(u',v') \in Y \times Y$ to $b(u',\Phi (v'))$ is *positive definite*.

The tuple $(X,Y,\Phi,b)$ constitutes a *polarized tropical abelian variety*. Let $X^*=\operatorname {Hom}(X,\mathbb {Z})$ and $X^*_\mathbb {R} = X^* \otimes \mathbb {R} = \operatorname {Hom}(X,\mathbb {R})$. The bilinear map $b$ realizes $Y$ as a subgroup of $X^*$ of finite index. We write $\Sigma$ for the real torus $X_\mathbb {R}^*/Y$. We note that $\Sigma$ is a point if and only if $A$ has good reduction over $R$.

## 7. Non-archimedean uniformization of abelian varieties

In [Reference BerkovichBer90, § 6.5], the classical rigid analytic uniformization of abelian varieties (see [Reference Bosch and LütkebohmertBL91] for a thorough treatment) is established in the context of Berkovich analytic spaces. We discuss the matter here briefly. Our main references are [Reference Foster, Rabinoff, Shokrieh and SotoFRSS18, Reference GublerGub10]. We continue with the notation and assumptions from § 6. The map $\upsilon$ from (6.5) induces, upon analytification, an exact sequence

of analytic groups. We refer to the map $p \colon E^\mathrm {an} \to A^\mathrm {an}$ as the *non-archimedean uniformization* of $A$ and we call the group $Y$ the group of *periods* of $A^\mathrm {an}$. We have a canonical isomorphism $p^*L^\mathrm {an} \xrightarrow {\sim } q^*M^\mathrm {an}$ of rigidified analytic line bundles on $E^\mathrm {an}$.

### 7.1 Tropicalization

Let $\langle \cdot,\cdot \rangle \colon X \times X_\mathbb {R}^* \to \mathbb {R}$ denote the natural evaluation pairing. The *tropicalization map* $\mathrm {trop} \colon T^\mathrm {an} \to X_\mathbb {R}^*$ is given by the rule

The tropicalization map is a surjective homomorphism and extends in a natural way to a surjective homomorphism $\mathrm {trop} \colon E^\mathrm {an} \to X_\mathbb {R}^*$ by setting

Here $\| \cdot \|_{E_u}$ is the model metric on $E_u^\mathrm {an}$ determined by the $\mathbb {G}_m$-torsor $\tilde {G}_u$ on $\mathcal {B}$.

Write $\Sigma = X_\mathbb {R}^*/Y$. The homomorphism $\mathrm {trop} \colon E^\mathrm {an} \to X_\mathbb {R}^*$ gives rise to a morphism of short exact sequences

The map $\tau \colon A^\mathrm {an} \to \Sigma$ turns out to be a deformation retraction. Following [Reference Foster, Rabinoff, Shokrieh and SotoFRSS18, § 4] and [Reference GublerGub10, Example 7.2], there exists a natural section $\sigma \colon X_\mathbb {R}^* \to E^\mathrm {an}$ of $\mathrm {trop}$. We denote by $\iota \colon \Sigma \to A^\mathrm {an}$ the resulting section of $\tau$. We usually view $\Sigma =X_\mathbb {R}^*/Y$ as a subspace of $A^\mathrm {an}$ via the map $\iota$. When viewed as a subspace of $A^\mathrm {an}$ via $\iota$, we call $\Sigma$ the *canonical skeleton* of $A^\mathrm {an}$.

Lemma 7.1 The restriction of the retraction map $\tau \colon A^\mathrm {an} \to \Sigma =X_\mathbb {R}^*/Y$ to $A(F)$ induces an isomorphism of groups $A(F)/G(R) \xrightarrow {\sim } X^*/Y$.

Proof. We have canonical identifications (cf. [Reference Faltings and ChaiFC90, p. 78])

and

The restriction of the tropicalization map $\mathrm {trop} \colon E^\mathrm {an} \to X_\mathbb {R}^*$ to $E(F)=\tilde {G}(F)$ induces an isomorphism of groups $\tilde {G}(F)/\tilde {G}(R)=\operatorname {Hom}(X,F^*/R^*) \xrightarrow {\sim } X^*$. This descends to an isomorphism $\tilde {G}(F)/ \upsilon (Y) \cdot \tilde {G}(R) \xrightarrow {\sim } X^*/Y$. Since, by construction, the retraction map $\tau \colon A^\mathrm {an} \to \Sigma$ descends from $\mathrm {trop}$, we see that $\tau$ sends $A(F)$ onto $X^*/Y$ with kernel $G(R)$.

Let $\mathcal {N}$ be the Néron model of $A$ over $S = \operatorname {Spec} R$, let $\Phi _\mathcal {N}$ be the group of components of its special fiber and let $\mathrm {sp} \colon \mathcal {N}(R) \to \Phi _\mathcal {N}$ denote the specialization map. We note that $\mathrm {sp}$ induces a group isomorphism $\mathcal {N}(R)/G(R) \xrightarrow {\sim } \Phi _\mathcal {N}$. As $\mathcal {N}(R)=A(F)$, we immediately deduce from Lemma 7.1 the following.

Corollary 7.2 The map $\Phi _\mathcal {N} \to X^*/Y$ that sends $\mathrm {sp}(x)$ for $x \in \mathcal {N}(R)$ to $\tau (x)$ is a group isomorphism.

Compare with [Reference Faltings and ChaiFC90, Corollary III.8.2].

This has the following consequence. Let $\mathcal {V}$ be a projective integral model of $A$ containing $\mathcal {N}$ as an open subscheme. In particular, we have an open immersion $\mathcal {N}_{\tilde {\mathbb {F}}} \hookrightarrow \mathcal {V}_{\tilde {\mathbb {F}}}$ of special fibers. Let $\xi \in \Phi _\mathcal {N}$ and let $x_\xi \in A^\mathrm {an}$ denote the Shilov point determined by the generic point of the irreducible component corresponding to $\xi$ in $\mathcal {V}_{\tilde {\mathbb {F}}}$. Then $x_\xi$ is an element of the canonical skeleton $\Sigma$ of $A^\mathrm {an}$.

### 7.2 The invariant $I(A,\lambda )$

Let $\lambda \colon A \xrightarrow {\sim } A^t$ be a principal polarization of $A$ and let $L$ be any rigidified symmetric ample line bundle on $A$ determining $\lambda$. Let $s$ be a non-zero global section of $L$. Let $\|\cdot \|_L$ denote the canonical metric on $L^\mathrm {an}$.

We define

where

is the supremum norm of $s$ and where $\mu _{H}$ is the pushforward, along the inclusion $\iota \colon \Sigma \hookrightarrow A^\mathrm {an}$, of the Haar measure of unit volume on the canonical skeleton $\Sigma$ of $A^\mathrm {an}$.

Lemma 7.3 The quantity $I(A,\lambda )$ is independent of the choice of symmetric ample line bundle $L$, of section $s$ and of rigidification of $L$ and hence defines an invariant of the principally polarized abelian variety $(A,\lambda )$.

Proof. Choose one symmetric ample line bundle $L$ on $A$ determining $\lambda$. A change of rigidification results in a replacement of $\|\cdot \|_L$ by a scalar multiple of