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# Specialization of canonical heights on abelian varieties

Alexander Carney

###### Abstract.

Given a family of abelian varieties over a quasiprojective smooth curve
$T^{0}$ over a global field and a point $P$ on the generic fiber, we show that
the Néron-Tate canonical height $h_{X_{t}}(P_{t})$ of $P_{t}$ along each fiber
is exactly equal to a Weil height $h_{\overline{M}}(t)$ given by an adelic
metrized line bundle $\overline{M}$ on the unique smooth projective curve $T$
containing $T^{0}$. As a consequence, we show that a conjecture of Zhang on
the finiteness of small-height specializations of $P$ is equivalent to
$\overline{M}$ being big.

##  1\. Introduction

Let $K$ be a number field or a transcendence degree one function field of any
characteristic, and let $T^{0}$ be a smooth, quasiprojective curve over $K$.
Suppose we have an abelian scheme $A\to T^{0}$, so that every fiber $A_{t}$ is
an abelian variety. By fixing a line bundle on $A$ which restricts to a
symmetric and ample line bundle on each fiber, we may specify the Néron-Tate
height $h_{A_{t}}$ on each fiber in a consistent way.

Fix a section $P:T^{0}\to A$ with specializations $P_{t}$. If we let
$F=K(T^{0})$ be the function field of $T$, this corresponds to an $F$ point on
the generic fiber $A_{\eta}$. Following a line of study begun by Silverman and
Tate, we ask how the canonical height $h_{A_{t}}(P_{t})$ varies as we vary the
parameter $t$ along $T^{0}(\overline{K})$. Let $T$ be the unique smooth
projective curve containing $T^{0}$, and let $h_{T}(t)$ be any height on $T$
corresponding to a degree one line bundle on $T$. The general formulation of
several theorems and conjectures is that

$h_{A_{t}}(P_{t})=h_{A}(P)h_{T}(t)+(\text{error term}),$

Where $h_{A}$ is the Néron-Tate canonical height on $A_{\eta}$.

Silverman conjectured [Sil82], and Tate proved [Tat83, Main Theorem] that if
$A$ is a family of elliptic curves, a divisor $D_{P}$ can be found on $T$ so
that replacing $h_{T}$ with $h_{D_{P}}$ reduces the error term to $O(1)$. Call
[Cal89] and [Gre89] prove similar results in higher dimension. A series of
subsequent results of Silverman culminating in [Sil94] further classify the
nature of the bounded error function for families of elliptic curves, and work
of Biesel, Holmes, and de Jong [BHdJ17] makes similar refinements in higher
dimension. More recently, DeMarco and Mavraki [DM20, Theorem 1.1] showed how
to eliminate the error term entirely for families of elliptic curves by
replacing $h_{T}$ with a specific height constructed using adelic metrized
line bundles. Here, we generalize their result to any dimension, resolving
this question in full for abelian varieties.

######  Theorem 1.1.

Let $P\in A(F)$ such that $h_{A}(P)\neq 0$. The function

$t\mapsto h_{A_{t}}(P_{t}),$

defined for all but finitely many $t\in T$, can be extended to a height
function $h_{\overline{M}}$ on all of $T$ which is given by a nef adelic line
bundle $\overline{M}$ on $T$ whose underlying line bundle $M$ is ample.

This is proved by a completely different method from that of DeMarco and
Mavraki. We use Yuan-Zhang’s theory of adelic line bundles and vector heights
[YZ21] to construct a canonical adelic line bundle, essentially extending
Tate’s limiting method from $\mathbb{R}$-valued height functions to the
underlying geometry. Once this theory is established, the existence of a Néron
model for $A$ provides the needed global geometric setting on which to
complete the argument.

######  Remark 1.2.

By the Lang-Néron Theorem (see [Con06]), if $A_{\eta}$ has trivial
$F/K$-trace, the condition that $h_{A}(P)\neq 0$ is equivalent to $P$ being
non-torsion. In general, it’s equivalent to $P$ being outside a torsion coset
of Tr${}_{F/K}(A_{\eta})$.

With a metrized line bundle giving $h_{A_{t}}(P_{t})$ exactly, one could use
equidistribution arguments to show, for example, that if $P,Q\in A(F)$ are two
points whose heights are both non-zero and there exists a sequence
$\\{t_{n}\\}\subset T(\overline{\mathbb{Q}})$ such that both
$h_{A_{t_{n}}}(P_{t_{n}})$ and $h_{A_{t_{n}}}(Q_{t_{n}})$ tend to zero as
$n\to\infty$, then $P$ and $Q$ specialize to torsion points at the exact same
places. This is likely vacuous, however, due to the following conjecture of
Zhang.

######  Conjecture 1.3.

([Zha98]) Suppose $A\to T^{0}$ is a non-isotrivial family of abelian varieties
as above, with $K$ a number field, and that $A_{\eta}$ is simple of dimension
at least two. For each non-torsion section $P:T^{0}\to A$ defined over
$\overline{\mathbb{Q}}$, there exists an $\epsilon>0$ such that

$\\{t\in T^{0}(\overline{\mathbb{Q}}):h_{A_{t}}(P_{t})\leq\epsilon\\}$

is finite.

Instead, using Theorem 1.1, we can restate Zhang’s conjecture as a positivity
statement for self-intersections.

######  Corollary 1.4.

Let $\overline{M}$ be the metrized line bundle defined in Theorem 1.1 so that
$h_{A_{t}}(P_{t})=h_{\overline{M}}(t)$. Then the conclusion of Zhang’s
conjecture is equivalent to

$\overline{M}^{2}>0.$

The author is extremely grateful to Xinyi Yuan, Laura DeMarco, Niki Myrto
Mavraki, and Tom Tucker for several ideas and insightful discussion while
preparing this paper.

##  2\. Vector-valued heights

We begin with an overview of adelic line bundles and vector-valued height
functions, in so much as is needed here. For a more complete treatment, see
[YZ21, Ch. 2] and [Car20, Sec. 2] for more development of the function field
setting.

We will work in two parallel settings The first is for varieties over a field,
in which all constructions are purely geometric, and which we will call the
_geometric case_. The second is for varieties over
$\operatorname{Spec}\mathbb{Z}$, which we will call the _arithmetic case_.
Note that when $K$ is a number field, we will require both settings, as the
height on $A_{\eta}$ is geometric, while that on each fiber is arithmetic.

Geometric case: Fix any field $k$, and let $\mathcal{U}$ be a quasi-projective
$k$ variety. A _projective model_ for $\mathcal{U}$ is a projective variety
$\mathcal{X}$ along with an open immersion
$\mathcal{U}\hookrightarrow\mathcal{X}$ defined over $k$. We write
$\operatorname{Div}(\mathcal{X})$ for the group of Cartier divisors on
$\mathcal{X}$ and $\operatorname{Pr}(\mathcal{X})$ for the subgroup of
principal Cartier divisors.

Arithmetic case: Let $\mathcal{U}\to\operatorname{Spec}\mathbb{Z}$ be a quasi-
projective arithmetic variety. In this setting, a _projective model_ for
$\mathcal{U}$ is a projective arithmetic variety
$\mathcal{X}\to\operatorname{Spec}\mathbb{Z}$ with an open embedding
$\mathcal{U}\to\mathcal{X}$ over $\operatorname{Spec}\mathbb{Z}$. Write
$\widehat{\operatorname{Div}}(\mathcal{X})$ for the group of arithmetic
divisors on $\mathcal{X}$, and $\widehat{\operatorname{Pr}}(\mathcal{X})$ for
the subgroup of principal arithmetic divisors.

In both cases the projective models for $\mathcal{U}$ form an inverse system
via dominating morphisms. Using pullbacks on that system, we define the
following, first in the geometric case. If
$\mathcal{U}\to\operatorname{Spec}k$ is a quasi-projective variety,

$\widehat{\operatorname{Div}}(\mathcal{U}/k)_{\operatorname{mod}}:=\lim_{\mathcal{U}\hookrightarrow\mathcal{X}}\operatorname{Div}(\mathcal{X})_{\mathbb{Q}}\quad\text{and}\quad\widehat{\operatorname{Pr}}(\mathcal{U}/k)_{\operatorname{mod}}:=\lim_{\mathcal{U}\hookrightarrow\mathcal{X}}\operatorname{Pr}(\mathcal{X})_{\mathbb{Q}}.$

An element of
$\widehat{\operatorname{Div}}(\mathcal{U}/k)_{\operatorname{mod}}$ is called
_effective_ if it comes from an effective divisor in some
$\operatorname{Div}(\mathcal{X})_{\mathbb{Q}}$. If
$\mathcal{U}\to\operatorname{Spec}\mathbb{Z}$ is a quasi-projective arithmetic
variety, we define

$\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}:=\lim_{\mathcal{U}\hookrightarrow\mathcal{X}}\widehat{\operatorname{Div}}(\mathcal{X})_{\mathbb{Q}}\quad\text{and}\quad\widehat{\operatorname{Pr}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}:=\lim_{\mathcal{U}\hookrightarrow\mathcal{X}}\widehat{\operatorname{Pr}}(\mathcal{X})_{\mathbb{Q}}.$

An element of
$\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}$ is
called _effective_ if it comes from an effective arithmetic divisor in some
$\widehat{\operatorname{Div}}(\mathcal{X})_{\mathbb{Q}}$, meaning that the
finite part of the divisor is effective, and the Green’s function is non-
negative.

Next we define a topology on both of these groups, stemming from effectivity.
Since the argument is identical in both the arithmetic and geometric cases we
provide it only for the arithmetic case.

Let $\mathcal{U}$ be a quasi-projective arithmetic variety. Effectivity
provides a partial ordering on
$\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}$.
Fix some projective model $\mathcal{X}_{0}$ and a strictly effective
arithmetic divisor $\overline{\mathcal{D}}_{0}$ on $\mathcal{X}_{0}$ such that
the support of $\mathcal{D}_{0}$ is equal to
$\mathcal{X}_{0}\backslash\mathcal{U}$. We call such $\mathcal{D}_{0}$ a
_boundary divisor_. For $\epsilon\in\mathbb{Q}_{>0}$, define a basis of
epsilon balls around $0$ by

$B(\epsilon,0):=\left\\{\overline{\mathcal{E}}\in\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}:\epsilon\overline{\mathcal{D}}_{0}\pm\overline{\mathcal{E}}\text{
are both effective}\right\\}.$

Via translation, this defines a topology on all of
$\widehat{\operatorname{Div}}(\mathcal{U})_{\operatorname{mod}}$, and it is
easy to check that this topology does not depend on the choice of
$\overline{\mathcal{D}}_{0}$.

We now define $\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})$ to be the
completion of
$\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}}$
with respect to this topology. $\widehat{\operatorname{Div}}(\mathcal{U}/k)$
is defined identically, except without a Green’s function attached to each
divisor. Finally, define the groups of _adelic line bundles_ in each setting:

$\widehat{\operatorname{Pic}}(\mathcal{U}/\mathbb{Z}):=\widehat{\operatorname{Div}}(\mathcal{U}/\mathbb{Z})/\widehat{\operatorname{Pr}}(\mathcal{U}/\mathbb{Z})_{\operatorname{mod}},$
$\widehat{\operatorname{Pic}}(\mathcal{U}/k):=\widehat{\operatorname{Div}}(\mathcal{U}/k)/\widehat{\operatorname{Pr}}(\mathcal{U}/k)_{\operatorname{mod}}.$

To further justify notating these as Picard groups, elements of
$\widehat{\operatorname{Pic}}(\mathcal{U}/k)$ (resp.
$\widehat{\operatorname{Pic}}(\mathcal{U}/\mathbb{Z})$) can be represented by
sequences $\\{\mathcal{X}_{i},\psi_{i},\mathcal{L}_{i},\ell_{i}\\}_{i\geq 1}$
(resp.
$\\{\mathcal{X}_{i},\psi_{i},\overline{\mathcal{L}}_{i},\ell_{i}\\}_{i\geq
1}$) where $\mathcal{X}_{i}$ is a projective model for $\mathcal{U}$ with a
morphism $\psi_{i}:\mathcal{X}_{i}\to\mathcal{X}_{1}$, where $\mathcal{L}_{i}$
(resp. $\overline{\mathcal{L}}_{i}$) is a $\mathbb{Q}$-line bundle (resp.
Hermitian $\mathbb{Q}$-line bundle) on $\mathcal{X}$, and $\ell_{i}$ is a
rational section of $\mathcal{L}_{i}\otimes\psi^{*}\mathcal{L}_{1}^{-1}$ with
support contained in $\mathcal{X}_{i}\backslash\mathcal{U}$. The equality of
this representation with the definitions above is shown in [YZ21, Lemma
2.5.1].

We will typically just write
$\overline{\mathcal{L}}=\\{\mathcal{X}_{i},\mathcal{L}_{i}\\}\in\widehat{\operatorname{Pic}}(\mathcal{U}/k)$
or
$\overline{\mathcal{L}}=\\{\mathcal{X}_{i},\overline{\mathcal{L}}_{i}\\}\in\widehat{\operatorname{Pic}}(\mathcal{U}/\mathbb{Z})$.
From the conditions on $\ell_{i}$, we get a well-defined restriction map
$\overline{\mathcal{L}}\mapsto\mathcal{L}\in\operatorname{Pic}(\mathcal{U})_{\mathbb{Q}}$.

Such a sequence is _Cauchy_ provided that $\\{\text{div}(\ell_{i})\\}$ (resp.
$\\{\widehat{\text{div}}(\ell_{i})\\}$) is Cauchy under the topology defined
above. A sequence converges to zero if there exists a sequence of rational
sections $s_{i}$ of $\mathcal{L}_{i}$ such that
$\ell_{i}=s_{i}\otimes\psi_{i}^{*}s_{1}^{-1}$, and such that
$\\{\text{div}(s_{i})\\}$ (resp. $\\{\widehat{\text{div}}(s_{i})\\}$) is
itself Cauchy.

We call an adelic line bundle _nef_ if it is isomorphic to a sequence where
every (Hermitian) line bundle is nef, and we call an adelic line bundle
_integrable_ if it can be written as the difference of two nef ones. Denote
the cones of nef elements

$\widehat{\operatorname{Pic}}(\mathcal{U}/k)_{\operatorname{nef}}\quad\text{
and
}\quad\widehat{\operatorname{Pic}}(\mathcal{U}/\mathbb{Z})_{\operatorname{nef}}$

and the subgroups of integrable elements

$\widehat{\operatorname{Pic}}(\mathcal{U}/k)_{\operatorname{int}}\quad\text{
and
}\quad\widehat{\operatorname{Pic}}(\mathcal{U}/\mathbb{Z})_{\operatorname{int}}.$

###  2.1. The relative setting

From here on out, we unify the notation for the base $k$ and base $\mathbb{Z}$
settings. Let $b$ be either $\operatorname{Spec}\mathbb{Z}$ or
$\operatorname{Spec}k$. Let $K$ be a field which is finitely generated over
$\mathbb{Q}$, if $b=\operatorname{Spec}\mathbb{Z}$, or finitely generated over
$k$, when $b=\operatorname{Spec}k$. An _open model_ for $K$ is a quasi-
projective $b$-variety $\mathcal{V}$ with function field $K$.

Next, let $X$ be a quasi-projective variety over $K$. An open model for $X/K$
consists of an open model $\mathcal{V}$ for $K$, together with a quasi-
projective and flat morphism $\mathcal{U}\to\mathcal{V}$ whose generic fiber
is $X\to\operatorname{Spec}K$.

The open models for $X$ form an inverse system via inclusion. Taking the limit
over this system, we define

$\widehat{\operatorname{Pic}}(X/b):=\lim_{\mathcal{U}\to\mathcal{V}}\widehat{\operatorname{Pic}}(\mathcal{U}/b).$

When $X=\operatorname{Spec}K$, we will simply write
$\widehat{\operatorname{Pic}}(K/b)$. Restricting these limits to the nef and
integrable elements, we define the nef cone and integrable subgroup

$\widehat{\operatorname{Pic}}(X/b)_{\operatorname{nef}}\subset\widehat{\operatorname{Pic}}(X/b)_{\operatorname{int}}\subset\widehat{\operatorname{Pic}}(X/b).$

We will write elements as $\overline{L}\in\widehat{\operatorname{Pic}}(X/b)$.
Given such
$\overline{L}=\\{\mathcal{X}_{i},\overline{\mathcal{L}}_{i}\\}_{i\geq
0}\in\widehat{\operatorname{Pic}}(X/b)$, we can restrict via base change to

$\overline{L}_{K}:=\\{\mathcal{X}_{i,K},\mathcal{L}_{i,K}\\}\in\widehat{\operatorname{Pic}}(X/K),$

as each $\mathcal{X}_{i,K}$ is a projective model for $X$. Since, by
construction, all $\mathcal{L}_{i,K}$ agree on $X\subset\mathcal{X}_{i,K}$, we
have a well defined restriction to $X$ which we denote simply $L\in
Pic(X)_{\mathbb{Q}}$.

If $X^{0}$ is a smooth, quasi-projective $K$ curve, since $X^{0}$ has a unique
projective model $X$ over $K$,

(2.1)
$\widehat{\operatorname{Pic}}(X^{0}/K)=\operatorname{Pic}(X)_{\mathbb{R}}:=\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}.$

When $K$ is a number field or a function field of transcendence degree one
over $k$ and $X$ is a projective $K$-variety,
$\widehat{\operatorname{Pic}}(X/b)$ corresponds to the group of adelic
metrized line bundles, as defined by Zhang [Zha95b]. There also exists an
analytic theory of adelic line bundles over larger fields, for example [YZ21,
Sec. 3], [Car20, Sec. 2], but we will not require it here.

###  2.2. Pullbacks

Given a map $f:Y\to X$, since a priori the models defining
$\widehat{\operatorname{Pic}}(Y/b)$ and $\widehat{\operatorname{Pic}}(X/b)$
may not be compatible, it is not immediately apparent that we can define
pullbacks of adelic line bundles. We rectify that.

######  Lemma 2.2.

(C.f. [YZ21, Sec. 2.5.5]) Let $f:Y\to X$ a morphism of quasi-projective
varieties which is flat over $b$. Then there exists a well-defined pullback
morphism

$f^{*}:\widehat{\operatorname{Pic}}(X/b)\to\widehat{\operatorname{Pic}}(Y/b).$

###### Proof.

Assume for simplicity of notation that $b=\operatorname{Spec}\mathbb{Z}$; the
only difference when $b=\operatorname{Spec}k$ is the lack of Hermitian
metrics. Let $\overline{L}\in\widehat{\operatorname{Pic}}(X/b)$ be represented
by a Cauchy sequence $\\{\mathcal{X}_{i},\overline{\mathcal{L}}_{i}\\}$.

By moving to smaller open subvarieties as needed, we may extend $f$ to a
morphism $f:\mathcal{W}\to\mathcal{U}$ of open models for $X$ and $Y$, and it
suffices to prove that there exists a well defined pullback
$\widehat{\operatorname{Pic}}(\mathcal{U}/b)\to\widehat{\operatorname{Pic}}(\mathcal{W}/b)$.

Let $\mathcal{Y}$ be a projective model for $\mathcal{W}$. By Raynaud’s
flattening theorem [RG71], by blowing up along
$\mathcal{Y}\backslash\mathcal{W}$, we may assume that
$f:\mathcal{W}\to\mathcal{X}_{i}$ extends to a flat morphism
$f_{i}:\mathcal{Y}_{i}\to\mathcal{X}_{i}$. Replacing $\mathcal{Y}_{i}$ with a
morphism which dominates it, we may further assume that there is a morphism
$\psi_{i}:\mathcal{Y}_{i}\to\mathcal{Y}_{0}$ for every $i$, which is the
identity on $\mathcal{W}$.

We now have a compatible system of models on which to pull back
$\overline{L}$. Define

$f^{*}\overline{L}:=\left\\{\mathcal{Y}_{i},\psi_{i},f_{i}^{*}\overline{\mathcal{L}}_{i},f^{*}\ell_{i}\right\\}_{i\geq
0}.$

Since effective Cartier divisors pull back to effective Cartier divisors under
both flat and dominant maps, and the topology on
$\widehat{\operatorname{Pic}}(\mathcal{W}/b)$ does not depend on the choice of
boundary divisor, the topology on
$\widehat{\operatorname{Pic}}(\mathcal{U}/b)$ induces that on
$\widehat{\operatorname{Pic}}(\mathcal{W}/b)$ via pullback of any boundary
divisor on $\mathcal{X}_{0}$, and the sequence defining $f^{*}\overline{L}$ is
Cauchy with respect to this topology.

∎

##  3\. Canonical heights

######  Definition 3.1.

Let $X$ be a quasi-projective $K$ variety, with $K$ and $b$ as in the previous
section. Let
$\overline{L}\in\widehat{\operatorname{Pic}}(X/b)_{\operatorname{int}}$, and
suppose that $L\in\operatorname{Pic}(X)$ is ample. Let $P\in X(K)$. We define
the _vector valued $\overline{L}$-height_ of $P$ to be

$\mathfrak{h}_{\overline{L}}(P):=P^{*}\overline{L}\in\widehat{\operatorname{Pic}}(K/b)_{\operatorname{int}}.$

If $P$ is defined over some finite extension $K^{\prime}$, we compose with the
map
$\widehat{\operatorname{Pic}}(K^{\prime}/b)_{\operatorname{nef}}\to\widehat{\operatorname{Pic}}(K/b)_{\operatorname{nef}}$
induced by the norm functor $N_{K^{\prime}/K}$ and scale the result by
$1/[K^{\prime}:K]$, so that this is a relative height which does not depend on
the choice of $K^{\prime}$. For simplicity, we will usually omit this
formalism and assume without loss of generality that $P$ is defined over $K$.

When $K$ is a number field or a transcendence degree one function field over
$k$, taking the limit of the arithmetic degrees produces a Weil height
[Zha95b], which we notate

$h_{\overline{L}}(P):=\widehat{\deg}\left(\mathfrak{h}_{\overline{L}}(P)\right),$

and thus vector-valued heights extend the theory of $\mathbb{R}$-valued
heights.

######  Lemma 3.2.

(C.f. [YZ21, Sec. 6.1.1], [Car20, Sec. 4.1]) Let $f:X\to X$ be an endomorphism
of projective varieties over $K$ with a _polarization_ , i.e. an ample
$\mathbb{Q}$-line bundle $L\in\operatorname{Pic}(X)_{\mathbb{Q}}$ and a
rational number $q>1$ such that $f^{*}L=qL$. Then there exists a nef adelic
line bundle
$\overline{L}_{f}\in\widehat{\operatorname{Pic}}(X/b)_{\operatorname{nef}}$
extending $L$ such that

$f^{*}\overline{L}_{f}=q\overline{L}_{f}.$

In particular, there exists a canonical height
$\mathfrak{h}_{f}:=\mathfrak{h}_{\overline{L}_{f}}$ such that for all $P\in
X(\overline{K})$,

$\mathfrak{h}_{f}(f(P))=q\mathfrak{h}_{f}(P).$

###### Proof.

We again prove this only in the base $\mathbb{Z}$ case, as the base $k$ case
is identical besides the lack of Hermitian metrics. To start, we can find an
open model $\mathcal{U}\to\mathcal{V}$ over $b$ for $X\to\operatorname{Spec}K$
on which $f$ extends to a $\mathcal{V}$-morphism
$f:\mathcal{U}\to\mathcal{U}$, and over which there exists an extension
$\mathcal{L}\in\operatorname{Pic}(\mathcal{U})_{\mathbb{Q}}$ of $L$ such that
$f^{*}\mathcal{L}=q\mathcal{L}$. Fix a projective model
$\mathcal{X}_{0}\to\mathcal{B}$ for $\mathcal{U}\to\mathcal{V}$, and a nef
extension
$\overline{\mathcal{L}}_{0}\in\widehat{\operatorname{Pic}}(\mathcal{X}_{0})_{\mathbb{Q}}$
of $\mathcal{L}$. By shrinking $\mathcal{V}$ if needed, we may assume
$\mathcal{B}\backslash\mathcal{V}$ is an effective Cartier divisor.

Write $f^{m}$ to mean the $m$-th iterate of $f$. For each $m\geq 1$, define
$f_{m}:\mathcal{X}_{m}\to\mathcal{X}_{0}$ to be the normalization of the
composition $X\xrightarrow{f^{m}}X\hookrightarrow\mathcal{X}_{0}$, and define

$\overline{\mathcal{L}}_{m}:=\frac{1}{q^{m}}f_{m}^{*}\overline{\mathcal{L}}_{0}.$

For each $m\geq 0$, let $\mathcal{X}_{m}^{\prime}$ be a projective model for
$\mathcal{U}$ which dominates both $\mathcal{X}_{m}$ and $\mathcal{X}_{m+1}$
via $\phi_{m}:\mathcal{X}_{m}^{\prime}\to\mathcal{X}_{m}$ and
$\rho_{m}:\mathcal{X}_{m}^{\prime}\to\mathcal{X}_{m+1}$, both of which
restrict to the identity on $\mathcal{U}$. Further assume that for $m\geq 1$,
there exists a morphism
$\tau_{m}:\mathcal{X}_{m}^{\prime}\to\mathcal{X}_{0}^{\prime}$ which commutes
with $\phi_{m},$ $\rho_{m}$ and $f_{m}$. By construction $\tau_{m}$ extends
$f^{m}$ on $\mathcal{U}$. Since
$\phi_{0}^{*}\overline{\mathcal{L}}_{0}-\rho_{0}^{*}\overline{\mathcal{L}}_{1}$
is trivial on $\mathcal{U}$, we can find a boundary divisor
$\overline{\mathcal{D}}_{0}$ supported on $\mathcal{B}\backslash\mathcal{V}$
such that for the topology defined by $\pi^{*}\overline{\mathcal{D}}_{0}$, we
have
$\phi_{0}^{*}\overline{\mathcal{L}}_{0}-\rho_{0}^{*}\overline{\mathcal{L}}_{1}\in
B(1,0)$. Now

$\phi_{m}^{*}\overline{\mathcal{L}}_{m}-\rho_{m}^{*}\overline{\mathcal{L}}_{m+1}=q^{-m}\tau_{m}^{*}\left(\phi_{0}^{*}\overline{\mathcal{L}}_{0}-\rho_{0}^{*}\overline{\mathcal{L}}_{1}\right).$

and thus

$\phi_{m}^{*}\overline{\mathcal{L}}_{m}-\rho_{m}^{*}\overline{\mathcal{L}}_{m+1}\in
B\left(q^{-m},0\right).$

We get a Cauchy sequence

$\overline{L}_{f}:=\left\\{\mathcal{X}_{m},\overline{\mathcal{L}}_{m}\right\\}_{m\geq
0}\in\widehat{\operatorname{Pic}}(X/b)_{\operatorname{nef}},$

which is easily seen to satisfy the desired conditions. ∎

##  4\. Proof of Theorem 1.1

Let $K$ be a number field and $b=\operatorname{Spec}\mathbb{Z}$, or let $K$ be
a transcendence degree one function field with constant field $k$, and
$b=\operatorname{Spec}k$. Let $T^{0}$ be a smooth quasiprojective curve over
$K$, and let $F=K(T^{0})$ be its function field. Let $A$ be a family of
abelian varieties over $T^{0}$, and let $T$ be the unique smooth projective
curve with function field $F$.

Write $\mathfrak{h}_{A}=\mathfrak{h}_{[2]}$ for the
$\widehat{\operatorname{Pic}}(F/K)_{\operatorname{int}}$-valued canonical
height on the generic fiber $A_{\eta}$, and on each fiber $A_{t}$ over $t\in
T^{0}(\overline{K})$, write $\mathfrak{h}_{A_{t}}$ for the
$\widehat{\operatorname{Pic}}(K/b)_{\operatorname{int}}-$valued canonical
height on $A_{t}$. As defined in the previous section, the usual
$\mathbb{R}$-valued Néron-Tate canonical heights are

$h_{A}(P)=\widehat{\deg}\left(\mathfrak{h}_{A}(P)\right),\qquad
h_{A_{t}}(P_{t})=\widehat{\deg}\left(\mathfrak{h}_{A_{t}}(P_{t})\right)$

We show that if $h_{A}(P)\neq 0$, there exists a nef adelic line bundle
$\overline{M}\in\widehat{\operatorname{Pic}}(K/b)_{\operatorname{nef}}$ such
that on the $\overline{K}$-points of $T$ over which $A$ has good reduction,
the function

$t\mapsto\mathfrak{h}_{A_{t}}(P_{t})$

is equal to a vector-valued height function $\mathfrak{h}_{\overline{M}}$,
defined on all of $T$.

###### Proof.

Let $\mathcal{A}\to T$ be the Néron model for $A_{\eta}$. This is a
quasiprojective $K$-variety. Replacing $F$, and thus $T$, with a finite
extension will not alter the result, as the heights involved are invariant
under finite extensions. Thus, we may assume that $\mathcal{A}\to T$ has
semiabelian reduction [BLR90, Ch. 7.4, Thm 1]. We also write
$P:T\to\mathcal{A}$ for the unique section extending $P\in A(F)$, by the Néron
mapping property.

Since $\mathcal{A}$ has abelian reduction at all but finitely many fibers, and
the bad fibers have finitely many components, by replacing the point $P$ with
$nP$ if needed, we may assume that the section $P$ is contained in the
connected component of the identity, which we denote $\mathcal{A}^{\circ}$.
This replacement scales the canonical height on both $A$ and on each abelian
fiber $A_{t}=\mathcal{A}_{t}$ by $n^{2}$, and thus does not affect the result.

Let $\mathcal{X}\to b$ be a projective model for $\mathcal{A}^{\circ}$ over
$b$. Fix a symmetric and ample line bundle $L$ on $A_{\eta}$, and extend this
to a nef line bundle $\mathcal{L}$ on $\mathcal{X}$. By applying the
normalization construction from the proof of Theorem 3.2, we can find a second
projective model $\mathcal{X}^{\prime}\to b$ with a morphism

$[-1]:\mathcal{X}^{\prime}\to\mathcal{X}$

extending the automorphism $[-1]$ on $\mathcal{A}^{\circ}$. Fix a third model
$\mathcal{X}_{0}$ dominating both $\mathcal{X}$ and $\mathcal{X}^{\prime}$. We
can then define a nef line bundle

$\mathcal{L}_{0}=\frac{1}{2}\left(\mathcal{L}+[-1]^{*}\mathcal{L}\right)\in\operatorname{Pic}(\mathcal{X}_{0})_{\mathbb{Q}},$

where we use the implied pullbacks from $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ to $\mathcal{X}_{0}$.

We adapt an argument of Green [Gre89]. Consider the difference
$[n]^{*}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}-n^{2}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}$.
By the theorem of the cube, this becomes trivial when restricted to
$A_{\eta}=\mathcal{A}_{\eta}$. Then, since $\mathcal{A}^{\circ}$ is smooth and
has integral fibers over $T$, this difference is equal to the pullback under
$\mathcal{A}^{\circ}\to T$ of a line bundle on $T$, by [Gro67, Cor. 21.4.13 of
Ch. 4, Errata et Addenda]. Finally, pulling back by the identity section
$E:T\to\mathcal{A}^{\circ}$, we see that
$[n]^{*}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}-n^{2}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}$
must be trivial. Thus,

$[n]^{*}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}=n^{2}\mathcal{L}_{0}|_{\mathcal{A}^{\circ}}.$

######  Remark 4.1.

For a different version of the above argument extending a polarization to
$\mathcal{A}^{\circ}$, one can use Moret-Bailly’s extension of cubical
structure, [MB85, Thm. 3.5], as in [K9̈8, Sec. 4].

Next, apply Lemma 3.2 to extend $L$ to a nef adelic line bundle
$\overline{L}=\\{\mathcal{X}_{m},\overline{\mathcal{L}}_{m}\\}\in\widehat{\operatorname{Pic}}(A/b)_{\operatorname{nef}}$
such that

$[2]^{*}\overline{L}=4\overline{L}.$

In general, the model adelic line bundles $\overline{\mathcal{L}}_{m}$ may all
differ along a boundary divisor. Using the above construction extending the
polarization to $\mathcal{A}^{\circ}$, however, the line bundles
$\mathcal{L}_{m,K}$ can be made to all agree on $\mathcal{A}^{\circ}$. Since
the section $P$ lands fully within $\mathcal{A}^{\circ}$, we have a well-
defined pullback

$\overline{M}=P^{*}\overline{L}\in\widehat{\operatorname{Pic}}(T/b)_{\operatorname{int}},$

since each $P^{*}(\mathcal{L}_{m,K})$ is the same line bundle on $T$.

By construction,
$\mathfrak{h}_{\overline{M}}(t)=t^{*}\overline{M}=P_{t}^{*}\overline{L}_{t}=\mathfrak{h}_{A_{t}}(P_{t})$
for all $t\in T^{0}(\overline{K})$. Since $T$ is a curve over $K$, the
$\mathbb{R}$-valued Néron-Tate height of $P$ is simply
$\deg(M)=\deg((P^{*}\overline{L})_{K})\in\mathbb{Q}$. Since $P$ does not have
height zero, we conclude that $M$ is ample, and that $\overline{M}$ is nef, as
the pullback of a nef adelic line bundle. ∎

######  Remark 4.2.

Without the Néron model, which exists for abelian varieties, but not in
general for dynamical systems (see for example [Hsi96]), the above
construction would instead produce only
$\overline{M}\in\widehat{\operatorname{Pic}}(T^{0}/b)$. Since $T^{0}$ has a
unique projective closure, this gives an $\mathbb{R}$-line bundle
$M\in\operatorname{Pic}(T)_{\mathbb{R}}$, but it remains an open question
whether the adelic structure extends over all of $T$, i.e. whether the metrics
defined by each $\overline{\mathcal{L}}_{m}$ converge to an adelic metric on
$T$.

###  4.1. Proof of Corollary 1.4

Since $\overline{M}$ is nef, we know that $\overline{M}^{2}\geq 0$. Define

$e_{1}:=\sup_{\text{open }U\subset T}\inf_{t\in
U(\overline{K})}h_{\overline{M}}(t).$

It can easily be seen that $e_{1}\geq 0$, and is positive if and only if
Zhang’s conjecture holds.

By Zhang’s essential inequalities [Zha95a, Thm 5.2],

$e_{1}\geq\frac{\overline{M}^{2}}{\deg M}\geq\frac{e_{1}}{2}.$

From this, the equivalence of the restatement of the conjecture in terms of
the bigness of $\overline{M}$ is immediate.

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