Faltings height and Néron–Tate height of a theta divisor

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 $n \cdot \Phi _\mathcal {A}=0$.

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 $x_\xi$.

4.1 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].

5.1 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 $A$.

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

(5.1)\begin{equation} 4\,\Lambda(x) - \Lambda(2x) ={-}\log|h(x)| \end{equation}

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

(5.2)\begin{equation} {-}4 \log\! \|s(x)\|_L + \log\! \|s(2x)\|_L ={-}\log|h(x)| \end{equation}

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.

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

(7.1)\begin{equation} \langle u, \mathrm{trop}(z) \rangle ={-}\log|\chi^u(z)| , \quad u \in X , \ z \in T^\mathrm{an}. \end{equation}

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

(7.2)\begin{equation} \langle u, \mathrm{trop}(z) \rangle ={-}\log\! \|e_u(z) \|_{E_u} , \quad u \in X , \ z \in E^\mathrm{an}. \end{equation}

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

(7.3)

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 $A^\mathrm {an}$.

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

\begin{align*} A(F)/G(R) & = E(F)/\upsilon(Y) \cdot G(R) \\ & = \tilde{G}(F)/\upsilon(Y) \cdot G(R) \\ & = \tilde{G}(F)/ \upsilon(Y) \cdot \tilde{G}(R) \end{align*}

and

\begin{align*} \tilde{G}(F)/\tilde{G}(R) & = T(F) / T(R) \\ & = \operatorname{Hom}(X, F^*/R^*). \end{align*}

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.

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

(7.4)\begin{equation} I(A,\lambda) = \log\! \|s\|_{L,\sup} -\int_{A^\mathrm{an}} \log\! \|s\|_{L} \, d \mu_{H}, \end{equation}

where

(7.5)\begin{equation} \|s\|_{L,\sup} = \sup_{x \in A^\mathrm{an}} \|s(x)\|_L \end{equation}

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 $A^\mathrm {an}$.