†† _2020 Mathematics Subject Classification_. 11G10, 11G35, 14K12.†† _Key
words and phrases_. Unlikely intersections, sub-schemes of abelian schemes,
reduction modulo a prime, Galois orbits of torsion points, lifting abelian
subvarieties.

# Manin–Mumford in arithmetic pencils

Gregorio Baldi, Rodolphe Richard, and Emmanuel Ullmo

###### Abstract.

We obtain a refinement of Manin–Mumford (Raynaud’s Theorem) for abelian
schemes over some ring of integers. Torsion points are replaced by _special
0-cycles_ , that is reductions modulo some, possibly varying, prime of Galois
orbits of torsion points. There is a flat/horizontal part and a vertical one.
The irreducible components of the flat part are given by the Zariski closure,
over the integers, of torsion cosets of the generic fibre of the abelian
scheme. The vertical components are given by translates of abelian
subvarieties, which “come from characteristic zero”.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Main results
    3. 1.3 Strategy of the proof
      1. 1.3.1 Horizontal and weak vertical
      2. 1.3.2 Vertical and lifting
      3. 1.3.3 Points of $p$-torsion in characteristic $p$
    4. 1.4 Outline of the paper
  2. 2 Preliminaries and Hindry’s method
    1. 2.1 Lang group and adelic Tate module
    2. 2.2 Degree, stabiliser subgroup, and a criterion for specialness
  3. 3 Horizontal and weak vertical, Theorem 1.2.1
    1. 3.1 Statement of the main proposition
    2. 3.2 Proof of Proposition 3.1.2
      1. 3.2.1 First lemma
      2. 3.2.2 Second lemma
      3. 3.2.3 Proof of Proposition 3.1.1
      4. 3.2.4 Effective estimates for Proposition 3.1.1
    3. 3.3 Proof of Theorem 2.1.4, and Theorem 1.2.1
  4. 4 Strong vertical, Theorem 1.2.3
    1. 4.0.1 Proof of Theorem 1.2.3
    2. 4.1 Variants of the Tate conjecture (after Faltings et. al.)
    3. 4.2 Prelimaniry results
    4. 4.3 Proof of Theorem 4.0.1
      1. 4.3.1 Proof of Theorem 4.0.1
      2. 4.3.2 Combinatorial result
  5. 5 Purely $p$-adic phenomena, Theorem 1.2.6
    1. 5.1 Proof of Theorem 5.0.2
      1. 5.1.1 Variant of Lemma 4.2.2
    2. 5.2 Proof of Theorem 1.2.6
    3. 5.3 Some $p$-adic Hodge theory (for abelian varieties with good reduction)
      1. 5.3.1 Notation
      2. 5.3.2 Purpose
      3. 5.3.3 Statement of the main result
      4. 5.3.4 Witt vectors notations
    4. 5.4 Crystalline-étale comparison theorem
    5. 5.5 The Connected-étale exact sequence
    6. 5.6 Unit root eigenspaces
      1. 5.6.1 Dieudonné-Manin semilinear eigenspaces
      2. 5.6.2 Unit root isosplitting of étale quotient
      3. 5.6.3 Spectral projector $\pi$ and functorial Frobenius
      4. 5.6.4 Comparison Theorem for $p$-divisible groups
    7. 5.7 Assembling everything together

##  1\. Introduction

###  1.1. Motivation

The following is often referred to as the _Manin–Mumford conjecture_. It was
first proved by Raynaud [26, 27], reproved then by Hindry [17] and more
recently by Pila–Zannier [24], among others.

######  Theorem 1.1.1 (Raynaud).

Let $A$ be an abelian variety over a field of characteristic $0$ and $E$ be a
subset of torsion points. Then the Zariski closure of $E$ is _special_. That
is, it is a finite union of torsion cosets $a+B$ for some abelian subvariety
$B\subseteq A$ and a torsion point $a\in A$.

Theorem 1.1.1, together with its multiplicative analogue (as originally
appeared in [21]), can be thought as the first instances of a more general
philosophy, now referred to as _Unlikely Intersections_ , and proposed by
Zilber and Pink [25], see also the book [16]. Recently Richard [28] has
introduced a new net of conjectures aiming to understand unlikely
intersections phenomena _over the integers_. This includes new problems
regarding the arithmetic of (models of) both abelian and Shimura varieties.
The role of torsion points is replaced by the reduction to some residue field
of positive characteristic of the Zariski closure, over the integers, of
either a torsion or a CM-point. In an arithmetic family there are two points
of view:

  * •

_Horizontal_ unlikely intersections. This case corresponds to an intersection
with infinitely many vertical fibres;

  * •

_Vertical_ unlikely intersections. When the vertical fibres form a finite set.

Even formulating the vertical part for Shimura varieties is challenging.
Consider $Y_{0}(1)/\mathbb{Z}\cong\mathbb{A}^{1}_{\mathbb{Z}}$, the modular
curve parametrising elliptic curves. Under the generalised Riemann hypothesis
for quadratic inmaginary fields, Edixhoven and Richard [8] proved the vertical
André–Oort for $Y_{0}(1)\times Y_{0}(1)/\mathbb{Z}$ and later,
unconditionally, Richard [28] proved the horizontal André–Oort for
$Y_{0}(1)/\mathbb{Z}$. The above conjectures contain, as special cases, new
statements for subvarieties of abelian varieties that can be thought as an
arithmetic counterpart of Theorem 1.1.1. The aim of this paper is to prove
both a horizontal and vertical Manin–Mumford, whose statement we explain in
the next section.

###  1.2. Main results

Let $K$ be a number field (with a fixed algebraic closure $\overline{K}$), $S$
a finite set of finite places of $K$ and $\mathcal{O}_{K,S}$ be the ring of
$S$-integers of $K$. Write $\operatorname{Gal}(\overline{K}/K)$ for the
absolute Galois group of $K$. Let $\mathcal{A}$ be an abelian scheme over (the
spectrum of) $\mathcal{O}_{K,S}$. We denote by $A=A_{K}$ its generic fibre,
and by $\mathcal{A}_{\mathfrak{p}}$ its special fibre at a closed point
$\mathfrak{p}\in\operatorname{Spec}(\mathcal{O}_{K,S})$. The latter is an
abelian variety defined at the residue field $\kappa(\mathfrak{p})$. In
$\mathcal{A}$ we consider _special 0-cycles_ , that is the reduction to
$\kappa(\mathfrak{p})$, of positive characteristic, of the
$\operatorname{Gal}(\overline{K}/K)$-orbit of a torsion point $a\in
A(\overline{K})$, for some prime ideal
$\mathfrak{p}\in\operatorname{Spec}(\mathcal{O}_{K,S})$. Our main result
describes the Zariski closure of a set of special 0-cycles in $\mathcal{A}$.
To do so we introduce two classes of subvarieties of $\mathcal{A}$:

  * •

_Flat cycles_. Those obtained as Zariski closure in
$\mathcal{A}/\mathcal{O}_{K,S}$ of special subvarieties of $A_{\overline{K}}$
(in the sense of Theorem 1.1.1), and

  * •

_Torsion cycles_. That is torsion coset of abelian subvarieties of some
vertical fibre.

Intersections between flat cycles and vertical fibres are particular cases of
finite unions of torsion cycles, but in general torsion cycles are given by an
abelian subvariety of $\mathcal{A}_{\mathfrak{p}}$ which is not, a priori,
coming from a flat subvariety of $\mathcal{A}$. _Special subvarieties_ of
$\mathcal{A}$ are the ones obtained as a finite union of flat and torsion
cycles. Moreover flat cycles can be globally (additively) torsion.

######  Theorem 1.2.1.

Let $E_{n}$, for $n\geq 0$, be a sequence of special 0-cycles in
$\mathcal{A}$. The Zariski closure of

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}$

is a special subvariety of $\mathcal{A}$, that is it can be written as a
finite union of flat and torsion cycles.

It is natural to try to understand when the torsion cycles given by Theorem
1.2.1 happen to be intersections between flat cycles and vertical fibres, but
there is a _$p$ -adic obstruction_ to this, as the next example shows.

######  Example 1.2.2.

Let $A/K$ be geometrically simple abelian surface, and $\mathfrak{p}$ a prime
of residue characteristic $p>0$, such that the reduction of $A$ mod
$\mathfrak{p}$, denoted by $\mathcal{A}_{\mathfrak{p}}$, is isogenous to
$\mathfrak{B}\times\mathfrak{B}^{\prime}/\kappa(\mathfrak{p})$, with
$\mathfrak{B}$ ordinary and $\mathfrak{B}^{\prime}$ super-singular, that is
$\mathfrak{B}^{\prime}(\overline{\kappa(\mathfrak{p}}))[p^{\infty}]\cong 0$.
Let $(a_{n})_{n}\subset A_{\text{tors}}$ be a sequence of points such that
$\mathrm{ord}(a_{n})=p^{n}$. Since $\mathfrak{B}^{\prime}[p^{\infty}]=0$, the
reduction to $\overline{\kappa(\mathfrak{p})}$ of the
$\operatorname{Gal}(\overline{K}/K)$-orbit of the $a_{n}$’s is contained in
$\mathfrak{B}\times\\{0_{\mathfrak{B}^{\prime}}\\}$. We can choose the
$a_{n}$’s in such a way that Zariski closure of the union of special cycles
$E_{n}$ is
$\mathfrak{B}\times\\{0_{\mathfrak{B}^{\prime}}\\}\subset\mathcal{A}_{\mathfrak{p}}$
(as predicted by Theorem 1.2.1), but
$\mathfrak{B}\times\\{0_{\mathfrak{B}^{\prime}}\\}$ can not come from
characteristic zero, being $A$ simple.

The following is the second main result of the paper and considers only points
of order prime to the residue characteristics.

######  Theorem 1.2.3.

Let $E_{n}$, for $n\geq 0$, be a sequence of special 0-cycles in $\mathcal{A}$
of order coprime with the residue characteristic. Then the Zariski closure of
$\bigcup_{n\geq 0}E_{n}$ is a finite union of flat cycles and intersections
between flat cycles and vertical fibres.

Each special $0$-cycle $E_{n}\subset\mathcal{A}$, by definition, can be
written as the image of $\operatorname{Gal}(\overline{K}/K)\cdot a_{n}$, for a
family of points $a_{n}\in A_{\text{tors}}(\overline{K})$ along a111Even if
there is no canonical map
$\pi_{\mathfrak{p}_{n}}:A(\overline{K})\to\mathcal{A}(\overline{\kappa(\mathfrak{p}_{n})})$,
the image of a Galois orbit in
$\mathcal{A}(\overline{\kappa(\mathfrak{p}_{n})})$ is well defined (i.e. does
not depend on a particular choice of $\pi_{\mathfrak{p}_{n}}$). Indeed it is
just the intersection between the vertical fibre $\mathcal{A}_{\mathfrak{p}}$
and the Zariski closure over $\mathcal{O}_{K,S}$ of $a_{n}$. map

$A(\overline{K})\to\mathcal{A}(\overline{\kappa(\mathfrak{p}_{n})})$

to a family of fields of positive characteristic $\kappa(\mathfrak{p}_{n})$.
Notice that the result is purely vertical when the set
$\\{\mathfrak{p}_{n}\\}_{n\geq 0}$ is finite. It may be helpful to give a
reformulation of Theorem 1.2.3, assuming that all special $0$-cycles lie in a
fixed vertical fibre $\mathcal{A}_{\mathfrak{p}}$. The following is a special
case of Theorem 1.2.3, in the purely vertical result part, and may be of
independent interest.

######  Theorem 1.2.4.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$ of residue characteristic
$p$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme,
$(a_{n})_{n}\subset A_{\operatorname{tors}}$ be a sequence of torsion points
of order coprime with $p$, and write
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$, for
a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Then every component of $\mathcal{V}$, the Zariski closure of

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}_{\mathfrak{p}}\subset\mathcal{A}$

is a component of the intersection between a flat cycle
$\mathcal{B}\subset\mathcal{A}$ and $\mathcal{A}_{\mathfrak{p}}$.

An interesting and simple corollary of Theorem 1.2.4 is the following.

######  Corollary 1.2.5.

With the notations and the assumptions of Theorem 1.2.4, if the sequence of
subsets $(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})_{n}$ is Zariski dense
in $A_{K}$, then the special 0-cycles $E_{n}$ are Zariski dense in
$\mathcal{A}_{\mathfrak{p}}$.

The final main result of the paper shows that the “$p$-rank zero factor”
explained in Example 1.2.2 is indeed the only obstruction to the lifting
problem.

######  Theorem 1.2.6.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$ of residue characteristic
$p$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme,
$(a_{n})_{n}\subset A_{\operatorname{tors}}$ be a sequence of torsion points
(possibly of order divisible by $p$), and write
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$, for
a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Then every component $\alpha+\mathfrak{B}$ of the Zariski closure
$\mathcal{V}$

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}_{\mathfrak{p}}\subset\mathcal{A}$

is such that there exists $B\subseteq A$ an abelian subvariety with the
following property: $B_{\mathfrak{p}}$ contains $\mathfrak{B}$, and the
quotient $B_{\mathfrak{p}}/\mathfrak{B}$ is of $p$-rank zero.

######  Remark 1.2.7.

It may be interesting to notice that both Theorem 1.2.1 and 1.2.3 are
refinements of the classical Manin–Mumford (at least after it is reduced to
$\overline{\mathbb{Q}}$). To see how they imply Theorem 1.1.1 we argue as
follows. Let $V\subset A_{\overline{\mathbb{Q}}}$ be a component of the
Zariski closure of a sequence of torsions points $a_{n}\in A$. Consider the
special zero cycles $E_{m,n}$ given by reducing modulo $\mathfrak{p}_{m}$ the
Galois orbit of $a_{n}$. By the horizontal Manin–Mumford, we have a torsion
coset $B\subset A_{\overline{\mathbb{Q}}}$ such that

$E_{m,n}\subseteq B\subseteq V\subseteq A_{\overline{\mathbb{Q}}}.$

Repeating this on $A/B$ concludes the argument. The same proof works with the
strong vertical Manin–Mumford established in Theorem 1.2.3, choosing carefully
the reduction place (in both cases we need a characteristic zero abelian
subvariety of $\mathcal{A}$).

Hindry’s approach to the Manin–Mumford conjecture turned out to provide a
fruitful strategy for the _André–Oort conjecture_ , as first envisioned by
Edixhoven [7]. We refer to [20] and [16] for an overview of recent results and
progress. We hope that our strategy can be applied to obtain new evidences in
favour of Richard’s arithmetic conjectures. Finally the generality of Hindry’s
article is not the focus of the present paper. Probably one could treat, if
not all the semi-abelian varieties he considers, the product case of a torus
and an abelian variety. Unsurprisingly the base field $K$ may be replaced by
any finitely generated field of characteristic zero (and probably also of
finite characteristic).

###  1.3. Strategy of the proof

The proof of the horizontal part of Theorem 1.2.1 is inspired by the approach
of Hindry [17] to his proof of Theorem 1.1.1, and it can be made effective (as
we do in Section 3.2, see especially 3.2.4). The strategy for Theorem 1.2.3
and Theorem 1.2.6 is more complicate, and requires different ideas and tools.

####  1.3.1. Horizontal and weak vertical

As for Hindry’s proof, the only “Galois input” we need to know is a comparison
with the action of the _homotheties_. More precisely let $a\in
A(\overline{K})_{\operatorname{tors}}$ be a point of order $d$, and let $l$ be
a positive integer prime to $d$. As recalled in Section 2.1, there exists
$\sigma\in\operatorname{Gal}(\overline{K}/K)$ such that

$\sigma\cdot a=[l^{c}](a),$

where $[l^{c}]$ denotes the multiplication by $l^{c}$ on $A$ (for some
positive integer $c$, depending only on $K$ and $A$). We denote by $L$ the
group of $c$-th powers of homoteties. We actually prove a stronger form of
Theorem 1.2.1.

######  Theorem 1.3.1.

Let $\mathcal{V}\subseteq\mathcal{A}$ be a subvariety of
$\mathcal{A}/\mathcal{O}_{K,S}$ which admits a Zariski dense set of torsion
points $a$ such that $L\cdot a\subseteq\mathcal{V}$. Then $\mathcal{V}_{K}$ is
special in $A_{K}$, that is, each of its geometric irreducible components are
cosets $\alpha+B$ of an abelian subvariety $B\subset A/\overline{K}$ through a
torsion point $\alpha\in A$.

Let $V$ be a subvariety of $A$ of degree $D$ and dimension $\Delta$, and
assume that $L\cdot a\subseteq V$. In Proposition 3.1.1 we show that there
exist $\alpha\in A_{\operatorname{tors}}$ and $B\subseteq A$ an abelian
subvariety of $A$ such that

$L\cdot a\subset L\cdot\alpha+B\subseteq V,$

where $\mathrm{ord}(\alpha+B)\leq\delta$, for some constant $\delta$ depending
only on $D$, $\Delta$, and $c$, and $\dim(B)>0$ if $\mathrm{ord}(a)$ is bigger
than a function depending on $D$ and $\Delta$. The idea is to replace $V$ by a
special subvariety $Z$, still containing $L\cdot a$, of degree as small as
possible. In this case we may show that the subgroup generated by $a$ in $A$
is fully contained in $Z$ (rather than only $L\cdot a$). Once this is proven,
Theorems 1.3.1 and 1.2.1 easily follow.

######  Remark 1.3.2.

If a torsion coset $a+A^{\prime}$ of $A$ is contained in a subvariety
$V\subseteq A$ and in $[q](V)$, for some $q>1$, then

$a+A^{\prime}\subseteq V\cap[q](V)$

and in particular there is an irreducible component $V^{\prime}$ of $V$ such
that $a+A^{\prime}\subseteq V^{\prime}$. Compared with Hindry’s approach,
where everything is defined over a number field $K$, the main difference is
that in this situation

$\operatorname{Gal}(\overline{K}/K)\cdot a+A^{\prime}\subset V^{\prime}.$

In an arithmetic family, this point becomes much more subtle, since the
standard habit of reducing to irreducible components is usually incompatible
with the problem.

####  1.3.2. Vertical and lifting

Refining Theorem 1.2.1, to get Theorem 1.2.3, requires different ideas and
more sophisticated tools. It turns out that, using various forms of the Tate
conjecture (Faltings theorem, see Section 4.1), the only abelian subvarieties
we have to consider are the ones coming from characteristic zero. This is
interesting because there are many more abelian subvarieties of
$\mathcal{A}_{\mathfrak{p}}$, than the one coming from characteristic zero.
The analogy with Shimura varieties is less clear: what is a _special
subvariety_ of the reduction mod $p$ of a Shimura variety, which does not come
from characteristic zero (that is from reducing mod $p$ a Shimura sub-
variety)?

We briefly explain our strategy for Theorems 1.2.3 and 1.2.4 (and use the same
notation Theorem 1.2.4, so we let $p$ be the characteristic of the residue
field). By looking at the order of the $a_{n}$, there are essentially two
possibilities to keep in mind:

  1. (1)

$\mathrm{ord}(a_{n})=\ell^{n}$, for all $n$ (where $\ell$ is a prime different
from $p$);

  2. (2)

$\mathrm{ord}(a_{n})$ is the $n$-th prime number, for all $n$.

The main protagonists are (considered as
$\operatorname{Gal}(\overline{K}/K)$-representation):

  * •

The $\ell$-adic (rational) Tate module of $A$ (for $\ell\neq p$);

  * •

The adelic Tate module of $A$ (or quotients thereof).

Each object plays a similar role, depending only on the order of the torsion
points $a_{n}$: the $\ell$-adic Tate module in (1), and the adelic one in (2).
The idea in both cases is similar, even if it may require different tools and
can be summarised as follows. Let $T$ be one of the above _semisimple_
$\operatorname{Gal}(\overline{K}/K)$-representations. The Galois orbits of the
points $a_{n}$ determine a sub-representation $W\subset T$, and write
$T=W\oplus W^{\prime}$. Faltings’s theorem asserts that such decomposition is
induced by an element $u\in\operatorname{End}(A)\otimes E_{T}$, where
$E_{T}\in\\{\mathbb{Q}_{\ell},\text{ a quotient of }\widehat{\mathbb{Z}}\\}$,
depending on $T$. Knowing from Theorem 1.2.1 that such decomposition is,
“after reducing mod $\mathfrak{p}$”, induced by an element of
$\operatorname{End}(A_{\mathfrak{p}})$, we prove that $u$ can be found in
$\operatorname{End}(A)\otimes\mathbb{Q}$. It is not hard to see that this
corresponds to lifting a torsion cycle to a flat one.

####  1.3.3. Points of $p$-torsion in characteristic $p$

The proof of Theorem 1.2.6 is inspired by the $\ell$-adic argument presented
above, but it is more subtle. As Example 1.2.2 shows, when the torsion points
$a_{n}$ are such that $\mathrm{ord}(a_{n})=p^{n}$, for all $n$, and are
reduced modulo a prime $\mathfrak{p}$ of residue characteristic $p$, we have
to take care of the $p$-rank zero factors of the special fiber. To emulate the
argument of Section 1.3.2, we need to find a vector space where both
$\operatorname{End}(A)\otimes\mathbb{Q}_{p}$ and
$\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes\mathbb{Q}_{p}$ act
faithfully and compatibly. This is done using some comparison theorem from
$p$-adic Hodge theory, but it requires to extend the coefficients from
$\mathbb{Q}_{p}$ to one of Fontaine’s period rings. Such results are presented
in a self contained way in Section 5.3, the reader fluent in $p$-adic Hodge
theory can simply read Proposition 5.3.1, and its proof given in the Section
5.7.

###  1.4. Outline of the paper

In Section 2 we describe the Lang group and recall some degree formulae of
Hindry that will be used for the horizontal and weak vertical part of our
Manin–Mumford. In Section 3 we prove the horizontal part and the weak vertical
part. To prove Theorem 1.2.1 it suffices to work with a smaller part of the
Galois orbit, and indeed we consider only Lang orbits. The main technical
result is stated in Section 3.1. In Section 4, we prove Theorem 1.2.3 by
showing that torsion cycles are components of the intersection of a flat cycle
with some vertical fibre. In this section we use the full Galois group, mainly
to invoke the Tate conjecture (Faltings theorem). Extra care is required when
the order or the torsion points is divisible by the characteristic of the
residue field, and the final Section 5 is devoted precisely to the proof of
Theorem 1.2.6. Finally Corollary 5.2.2 shows how to obtain the most general
result in the case where torsion points are allowed to have order divisible by
$p$ (combining Theorems 1.2.3 and 1.2.6 together).

### Acknowledgements

The second author would like to thank the I.H.E.S. for a research visit in
September and October 2020, during which most of the work regarding the
horizontal Manin–Mumford was done. Rodolphe Richard was supported by the
Leverhulme grant RPG-2019-180.

##  2\. Preliminaries and Hindry’s method

We first recall some results which appear in Hindry’s method for Manin–Mumford
[17], and reduce Theorem 1.2.1 to a more geometric statement, where we replace
the Galois orbits of torsion points by the orbit under the _Lang group_. In
the process we check that Hindry’s results hold true also over fields of
arbitrary characteristics.

### Notations and conventions

Some notations and conventions we freely use in the text are presented below.

  * •

Given a torsion coset $a+B$ of an abelian variety $A$ we denote by
$\mathrm{ord}(a+B)$ its order in $A/B$. We may always assume that $a$ is also
of order $\mathrm{ord}(a+B)$;

  * •

By _subvariety_ we mean a closed subvariety;

  * •

By _component_ we mean irreducible component;

  * •

We write $<$ to mean strictly less than, and $\leq$ to mean less, but possibly
equal, than;

  * •

We say that $q$ _is a positive integer_ to mean that $q\in\mathbb{Z}$ and
$q>0$;

  * •

Let $A$ be an abelian variety and $n$ an positive integer, we denote by
$[n]:A\to A$ the multiplication by $n$, and by $A[n]$ its kernel;

  * •

Let $A_{\operatorname{tors}}$ denote the torsion points of $A$ (defined over
$\overline{\mathbb{Q}}$);

  * •

We usually write $\ell$, $p$ for finite places of $\mathbb{Q}$ (usually
different), and $\mathbb{Q}_{\ell},\mathbb{Q}_{p}$ for the respective local
fields;

  * •

We write $\\#$ for cardinality (of finite sets);

  * •

$a|b$ for saying that $a$ divides $b$, and $a\nmid b$ to say that $a$ does not
divide $b$;

  * •

We write $a\wedge b=1$ to mean that $a$ and $b$ are coprime;

###  2.1. Lang group and adelic Tate module

Let $K$ be a number field, and $A/K$ be a $g$-dimensional abelian variety (for
some $g>0$). Denote by $T(A)$ the $\operatorname{Gal}(\overline{K}/K)$-module
obtained by taking the inverse limit of the $A[n]$’s, that is the (integral)
_adelic Tate module_ of $A$. We write $\widehat{\mathbb{Z}}$ for the profinite
completion of the ring of integers $\mathbb{Z}$, and notice that
$T(A)\cong\widehat{\mathbb{Z}}^{2g}$. The proof of the following can be found
for example in [5, Theorem B] (which is stated also for finitely generated
field extensions of $\mathbb{Q}$). See also [3, Corollary 1, page 702], [17,
Section 2 (a)], [32], and references therein.

######  Theorem 2.1.1 (Bogomolov–Serre).

There exists a positive integer $c=c(A,K)$ such that the image of the natural
map describing the action of $\operatorname{Gal}(\overline{K}/K)$ on the
torsion points,

$\rho_{A}:\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}(T(A)),$

contains the $c$-th powers of the _homoteties_. That is the group

(2.1)
$L=L(c):=\ker\left(\widehat{\mathbb{Z}}^{*}\to\widehat{\mathbb{Z}}^{*}\otimes\mathbb{Z}/c\mathbb{Z}\right)$

is a subgroup of $\operatorname{Im}(\rho_{A})$.

By definition, for any $c$, $L=L(c)$ acts on the torsion points of $A$. We
refer to the group $L$ as the _Lang group_ , implicitly fixing the constant
$c$.

######  Remark 2.1.2.

As a corollary of Theorem 2.1.1, we have the following. Let $a\in
A_{\operatorname{tors}}$ be a point of order $d$, and let $l$ be a positive
integer prime to $d$. Then there exists
$\sigma\in\operatorname{Gal}(\overline{K}/K)$ such that

$\sigma\cdot a=\sigma(a)=[l^{c}](a).$

######  Remark 2.1.3.

For completness we remark that the constant $c$ of Theorem 2.1.1 can be made
effective. See for example [15, 35]. In particular in the former reference,
applications to explicit bounds in Manin–Mumford are described, and similar
remarks apply also here.

In Section 3.3 we actually prove the following, which, in virtue of Theorem
2.1.1, implies Theorem 1.2.1. Such statement appeared in the Introduction as
Theorem 1.3.1. In this formulation it becomes less arithmetic and more of
geometric nature, there is indeed no mention of the absolute Galois group of
the base field.

######  Theorem 2.1.4.

Let $\mathcal{V}\subseteq\mathcal{A}$ be a sub-scheme of
$\mathcal{A}/\mathcal{O}_{K,S}$ which admits a Zariski dense set of torsion
points $a$ such that $L\cdot a\subseteq\mathcal{V}$. Then $\mathcal{V}_{K}$ is
special in $A_{K}$. That is, the geometric irreducible components of
$\mathcal{V}_{K}$ are each a coset $\alpha+B$ of an abelian subvariety
$B\subset A/\overline{K}$ through a torsion point $\alpha\in A$.

The case where the points $a$ are all of residue characteristic $0$ follows
from Hindry’s method for Manin–Mumford [17]. The novelty of Section 3.3
consists in proving Theorem 2.1.4 when the $a$’s have positive, and possibly
varying, characteristic.

###  2.2. Degree, stabiliser subgroup, and a criterion for specialness

We discuss Hindry’s results of Bézout type in abelian varieties (see [17,
Section 1]). The first use of a Bézout theorem/intersection theory to approach
Lang conjecture, can be dated back to Tate’s proof [21, Section 2 (page 231)]
of a case of the multiplicative Manin–Mumford. See also the discussion about
abelian varieties in Section 3 of _op. cit._.

Let $k$ be an arbitrary field of characteristic $p\geq 0$, and $A/k$ be an
abelian variety (of dimension $>1$). Given a (positive dimensional, strict)
subvariety $V$ of $A$, we speak of its _degree_ as in [17, 1.b), page 578],
notice that this makes sense regardless the characteristic of the base field.

######  Lemma 2.2.1.

Let $V$ be an irreducible subvariety of $A$, and let $B\subset A$ be its
stabiliser. Let $q$ be a positive integer not divisible by $p$. We have

(2.2) $\deg([q](V))=q^{2\dim(V)}\deg(V)\cdot(\\#B[q])^{-1}.$

In [17] the stabiliser of $V$ in $A$ is denoted by $G_{V}$, and $G$ denotes a
commutative group scheme which is not necessarly assumed to be an abelian
variety. For simplicity, we present only the case of abelian varieties (and so
$G=A$).

###### Proof.

The proof presented in [17, Lemma 6] applies also to cover the above
statement, when $p$ does not divide $q$. ∎

For simplicity, form now, we simply write that $q$ is prime to $p$, to mean
that $q$ is non zero and not divisible by $p$ (which makes sense both when
$p=0$ and $p>0$).

######  Corollary 2.2.2.

Let $A,V$, and $B$ as in Lemma 2.2.1. Let $q,q^{\prime}$ be positive integers,
prime to each other and to $p$, such that

$[q](V)=[q^{\prime}](V).$

Then

$\\#B[q]=q^{2\dim(V)},\text{ }\\#B[q^{\prime}]=q^{\prime 2\dim(V)},\text{ and
}\\#B[qq^{\prime}]=(qq^{\prime})^{2\dim(V)}.$

###### Proof.

From Lemma 2.2.1 we have that

$q^{2\dim V}\\#B[q]^{-1}=q^{\prime 2\dim V}\\#B[q^{\prime}]^{-1}.$

Since $q$ and $q^{\prime}$ are coprime, the only possibility is that
$\\#B[q]=q^{2\dim(V)}$ and $\\#B[q^{\prime}]=q^{\prime 2\dim(V)}$. Again by
the fact that $q$ and $q^{\prime}$ are coprime, we have

$B[qq^{\prime}]=B[q]\times B[q^{\prime}],$

concluding the proof. ∎

######  Lemma 2.2.3.

Let $A,V$, and $B$ as in Lemma 2.2.1. Denote by $B^{+}$ the neutral component
of $B$, and the group of components of $B$ by $\pi_{0}(B)=B/B^{+}$. We have

$\\#\pi_{0}(B)\leq\deg(B)=\\#\pi_{0}(B)\cdot\deg(B^{+})\leq\deg(V),$

and

$\\#B[q]=\\#\pi_{0}(B)[q]\cdot\\#B^{+}[q]\leq\\#\pi_{0}(B)\cdot q^{2\dim(B)}.$

Assume now that $\\#B[q]=q^{2\dim(V)}$. Then

$q^{2(\dim(V)-\dim(B))}\leq\\#\pi_{0}(B)\leq\deg(V).$

If moreover

$\deg(V)<q^{2},$

then $V$ is special.

###### Proof.

The first formula follows from [17, Lemma 8]. For the second formula, just
notice that $\\#B^{+}[q](\overline{k})=q^{2\dim B^{+}}$, as long as $q$ is
prime to $p$. The third formula is a direct consequence of the previous two.

Regarding the last conclusion, we necessarily have $\dim(V)-\dim(B)=0$, since
otherwise

$q^{2(\dim(V)-\dim(B))}\leq\\#\pi_{0}(B)\leq\deg(V)<q^{2}.$

Choose a point $v\in V$, and notice that necessarly $v+B\subset V$. As
$\dim(v+B)=\dim(B)=\dim(V)$, and $V$ is irreducible, we have that $V=v+B$ is
special. ∎

######  Proposition 2.2.4.

Let $V$ be an irreducible subvariety of $A$, $q$ and $q^{\prime}$ be prime to
each other and to $p$ be such that

$[q](V)=[q^{\prime}](V),$

and

$\deg(V)^{1/2}<q\cdot q^{\prime}.$

Then $V$ is a special subvariety of $A$, and so is $[q](V)$.

Notice that the condition $\deg(V)^{1/2}<q\cdot q^{\prime}$ is actually
implied by $\deg(V)^{1/4}\leq q<q^{\prime}$. We only use the latter condition.

###### Proof.

Corollary 2.2.2 implies that

$\\#B[qq^{\prime}]=(qq^{\prime})^{2\dim(V)}.$

Therefore the assumption on $\\#B[qq^{\prime}]$ (with $qq^{\prime}$ for $q$)
in Lemma 2.2.3 is satisfied and so $V$ is special. ∎

Finally we check for reference that Hindry’s criterion [17, Lemma 15] stays
valid in any characteristics, and that it applies to subvarieties which are
not necessary irreducible. See also [3, Corollary on page 703].

######  Corollary 2.2.5 (Hindry).

Let $V$ be subvariety of $A$ such that $V=[q](V)$, for some $q>1$ prime to
$p$. Then each component of $V$ is special.

###### Proof.

Let $r\leq\deg(V)$ the number of components of $V$, and $r!$ be the
cardinality of the symmetric group acting on a set with $r$ elements. Since
$[q]$ permutes the components of $V$, every component $Z$ is stable under
$[q^{r!}]$, and similarly under $[m]$, for $m:=q^{\deg(V)!}$. Of course
$m^{2}>\deg(V)$, and Proposition 2.2.4 on $Z$, $m$ and $q^{\prime}=1$ applies
to conclude. ∎

##  3\. Horizontal and weak vertical, Theorem 1.2.1

In this section we prove Theorems 1.2.1 and 1.3.1. That is that the Zariski
closure of a sequence of special 0-cycles in $\mathcal{A}$ is a special
subvariety of $\mathcal{A}$. The statement contains both a horizontal and a
vertical part, but the proof is essentially the same. It is only to have the
refined conclusion of Theorem 1.2.3 (that is the lift to characteristic zero),
that we need more delicate arguments, which are in Sections 4 and 5.

###  3.1. Statement of the main proposition

Let $L=L(c)$ be the Lang group associated to some positive integer $c$, as
defined in Section 2.1 (equation (2.1)). The next result is the key for
proving Theorem 1.3.1, and it is proven in Section 3.2. Here we consider a
field $k$ (with a fixed algebraic closure $\overline{k}$) of characteristic
$p\geq 0$, and a $g$-dimensional abelian variety $A/k$ (for some $g>1$).

######  Proposition 3.1.1.

Let $V$ be a subvariety of $A$ (not necessarly irreducible) of degree
$D:=\deg(V)$ and dimension $\Delta:=\dim(V)<g$, and let $a$ be a torsion point
of $A(\overline{k})$ of order $d$. Assume that

(3.1) $L\cdot a\subseteq V.$

Then there exist $\alpha\in A_{\operatorname{tors}}$ and $B\subset A$ an
abelian subvariety of $A$ such that

$L\cdot a\subseteq L\cdot\alpha+B\subseteq V,$

and

$\mathrm{ord}(\alpha+B)\leq\delta.$

Where $\delta=\delta(D,\Delta,c)$ is a constant depending only on $D$,
$\Delta$, and $c$.

From the proof of Proposition 3.1.1, we will see that the constant $\delta$
can be made very explicit, see indeed Section 3.2.4. The strategy is to
replace $V$ by a special subvariety $Z\subset V$, still containing $L\cdot a$,
and of degree as small as possible. In this case one can hope that not only
$L\cdot a$, but most of the subgroup of $A$ generated by $a$, which we denote
by $<a>,$ is contained in $Z$ (and hence in $V$).

In the proof we will actually prove the following stronger statement, where we
work with arbitrary torsion cosets $a+A^{\prime}$ of $A$, rather than just
torsion points.

######  Proposition 3.1.2.

Let $A^{\prime}$ be an abelian subvariety of $A$, and $a+A^{\prime}$ be a
torsion coset of $A$ of order $d$. Let $V$ be a subvariety of $A$ of degree
$D$ and dimension $\Delta$, such that

(3.2) $L\cdot a+A^{\prime}\subseteq V,$

and that

(3.3) $\text{every component of $V$ contains a component of $L\cdot
a+A^{\prime}$}.$

There exist $\alpha\in A_{\operatorname{tors}}$ and $B\subset A$ an abelian
subvariety of $A$ such that

$\mathrm{ord}(\alpha+B)\leq\delta,$

and

$L\cdot a+A^{\prime}\subseteq L\cdot\alpha+B\subseteq V.$

Where $\delta=\delta(D,\Delta,c)$ is a constant depending only on $D$,
$\Delta$, and $c$.

###  3.2. Proof of Proposition 3.1.2

The proof of Proposition 3.1.2 consists of two lemmas, which are proven below.
To obtain explicit bounds we need some extra care in the arguments, see
especially Section 3.2.4. For this purpose we introduce the following
functions. Given a positive integer $d$, we write:

  * •

$p(d)$ for the $d$-th prime number;

  * •

$\omega(d)$ for the number of distinct prime divisors of $d$;

  * •

$g(d)$ for the _Jacobsthal’s function_. That is it represents the smallest
number $M$ such that every sequence of $M$ consecutive integers contains an
integer coprime to $d$.

The Jacobsthal’s function plays a role in providing sharp estimates. The
reader not interested in explicit estimates may replace this arguments by less
precise, but easier, estimates that will be explained during the proofs,
indeed we only need to know that $g(d)=d^{o(1)}$.

######  Proposition 3.2.1.

Let $d,n$ be positive integers, and $d^{\prime}$ be the square-free part of
$d/\gcd(d,n)$. Let $a$ be a positive integer such that

(3.4) $a+k\cdot n+d\mathbb{Z}\in\mathbb{Z}/(d)^{\times},$

for some $k\in\mathbb{Z}$. (Namely
$a+\operatorname{gcd}(n,d)\mathbb{Z}\in\mathbb{Z}/(\gcd(n,d))^{\times}$). Then
we can find $k\geq 0$ satisfying (3.4), and such that

$0\leq k<g(d^{\prime})\leq g(d)=d^{o(1)}.$

In case $0\leq a<n$, we have $a+k\cdot n\leq g(d^{\prime})\cdot n$.

Iwaniec [18] proved that

$g(d)\leq C\cdot\omega(d)^{2}\cdot\log\omega(d)^{2},$

for some unknown $C$ (this gives the best asymptotic upper bound on $g(d)$).
The best known explicit upper bounds, of

(3.5) $g(d)\leq 2^{\omega(d)},\text{ and}\qquad g(d)\leq
2\omega(d)^{2+2e\log\omega(d)},$

are due to Kanold [19] and Stevens [30] (with the second bound being stronger
for $\omega(d)\geq 260$).

###### Proof of Proposition 3.2.1.

By the Chinese Remainder theorem, we may replace $d$ by $d^{\prime}$ and
assume that $d^{\prime}$ is prime to $n$. We rewrite the question into finding
the smallest $k\in\mathbb{N}$ such that

$a/n+k\cdot d^{\prime}\mathbb{Z}\in\mathbb{Z}/(d^{\prime})^{\times},$

where by $a/n$ denotes $a\cdot n^{\ast}$ for $n^{\ast}+d^{\prime}\mathbb{Z}$
an inverse of $n+d^{\prime}\mathbb{Z}$ in $\mathbb{Z}/(d^{\prime})^{\times}$.
By definition of $g(d^{\prime})$, this happens for some $0\leq
k<g(d^{\prime})$. Finally the $d^{o(1)}$ estimate follows from the sharper
estimates reviewed above. ∎

######  Remark 3.2.2.

Using a more well known theorem of Linnik, about least prime in the arithmetic
progressions, it seems that we can even choose the $k$ in Proposition 3.2.1 to
be searched through the set for which $a+k\cdot n$ is a prime number $\ell$
with estimate, enough for our purpose, of the sort

$\ell\leq n^{Q}\cdot(\omega(d)+1),$

with a constant $Q$ which, up to date, has been worked out to be at most
$5.2$. For a complete discussion we refer to [33] and references therein.
Linnik proved that

$\ell\leq Cn^{Q},$

for some $C$ and $Q$ to be effectively computable constants.

####  3.2.1. First lemma

The first step towards Propositions 3.1.1 and 3.1.2 consists in finding a
subvariety $V^{\prime}\subset V$ of suitably bounded degree, depending on $d$
as a $d^{o(1)}$. From now on we denote by $o(1)$ some functions depending only
on $c$, and $\Delta$. To be more precise/explicit our $o(1)$ depends only on
$A$, via $g=\dim(A)$ and $h(A)$, the semistable (logarithmic) Faltings height
of $A/K$ (see indeed Remark 2.1.3, and references therein).

######  Lemma 3.2.3.

Let $V$ be a subvariety of $A$ of degree $D$ and dimension $\Delta$, such that
$L\cdot a+A^{\prime}\subseteq V$.

  * •

Either a component of maximal dimension $X$ of $V$ is a special subvariety,
say of the form $\alpha+B$, for some torsion point $\alpha\in A$ and $B$ an
abelian subvariety of $A$, such that for every $l\in L$, $l\alpha+B$ is a
component of $V$ and the order of each $l\alpha+B$ is bounded by
$D^{o(1)}\cdot d^{o(1)}$, and

$L\cdot a+A^{\prime}\subseteq L\cdot\alpha+B\subseteq V.$

Otherwise

  * •

There exists $V^{\prime}\subset V$ such that $\dim(V^{\prime})<\dim(V)$,
$V^{\prime}$ satisfies (3.2) and (3.3), and of degree bounded as follows:

(3.6) $\deg(V^{\prime})\leq D^{o(1)}\cdot d^{o(1)}.$

More precisely and explicitly we prove that

(3.7) $\deg(V^{\prime})\leq
f(D,d):=D^{2}\cdot\left(p(\left\lceil{\sqrt[4c]{D}}\right\rceil+D+\omega(d)+1)^{c}\cdot
g(d\cdot\max\\{1,p\\})\right)^{2\cdot c\cdot\Delta},$

where $\left\lceil{\sqrt[4c]{D}}\right\rceil$ denotes the smallest integer
bigger or equal than $\sqrt[4c]{D}$. Bounding $p(-)$ and $g(-)$ as reviewed in
the previous section, we have $f(D,d)\leq D^{o(1)}\cdot d^{o(1)}$.

First two easy observations, which follow directly from the assumptions
(3.2)-(3.3), and are also at the heart of Hindry’s method:

  * •

For every integer $m$ prime to $d=\mathrm{ord}(a+A^{\prime})$, we have
$V\supseteq L\cdot a+A^{\prime}=m^{c}\cdot L\cdot
a+A^{\prime}\subseteq[m^{c}](V)$. Hence

$L\cdot a+A^{\prime}\subseteq V\cap[m^{c}](V);$

  * •

For every component $X$ of $V$, we have

$\emptyset\neq X\cap(L\cdot a+A^{\prime})\subseteq X\cap[m^{c}](V).$

In Lemma 3.2.3 we did not assume that (3.3) holds, but the statements is still
true, since we may discard a finite number of components of $V$, to assume
that (3.3) holds.

###### Proof.

Let

(3.8) $x:=\left\lceil{\sqrt[4c]{D}}\right\rceil+D+\omega(d)+1,\text{ and
}n:=p(x).$

For simplicity one can also simply replace $x$ by $x_{D}:=2D+\omega(d)+1$,
just by noticing that $x\leq x_{D}$. This is indeed what we will do in the
second Lemma. Since $x$ is at least $4$, we can use some non-asymptotic bounds
on the prime-counting function and write

(3.9) $n=p(x)\leq x\log(x)(1+\log\log(x)),$

see for example [29, Theorem 3, page 69]. Finally we set

$N:=n^{c}\cdot g(d\cdot\max\\{1,p\\}),$

where $g(-)$ denotes the Jacobsthal’s function, as introduced in Section 3.2.
The reason why the factor $\max\\{1,p\\}$ appears, is that it allows to deal
simultaneously the case of zero and positive characteristic.

Let $X$ be a component of $V$ of maximal dimension, and set

$X^{\prime}=X^{\prime}(m):=X\cap[m^{c}](V),$

for some $m=m(X)$ chosen in the following set

(3.10) $\Sigma=\Sigma(d,D):=\\{m^{c}\text{ such that
}m\in\\{1,\ldots,N\\},m\wedge d=1,\text{ and }p\nmid m\\}.$

Assume now that there exists $m^{c}\in\Sigma$ such that
$\dim(X^{\prime})<\dim(X)$. Then the degree formula from Section 2.2, namely
(2.2), and the fact that $m^{c}\leq N^{c}=n^{c^{2}}\cdot
g(d\cdot\max\\{1,p\\})^{c}$, imply that

$\deg(X^{\prime})\leq\deg(X)\cdot\deg([m^{c}](V))\leq\deg(X)\cdot\deg(V)\cdot
N^{2\cdot c\cdot\Delta}.$

Assume now that for each component $X$, of maximal dimension, there exists
$m^{c}\in\Sigma$ such that $\dim(X^{\prime}=X^{\prime}(m))<\dim(X)$, and let
$V^{\prime}$ be the union the $X^{\prime}$s and the lower dimensional
components of $V$. In a similar way we can control the degree of $V^{\prime}$:

$\deg(V^{\prime})\leq\deg(V)^{2}\cdot N^{2\cdot c\cdot\Delta}.$

The above, combined with the explicit bounds for the Jacobsthal’s function
discussed in Section 3.2, give the second conclusion.

We are left to the case where the above assumption fails. That is the desired
$m\in\Sigma$ can not be found for some $\Delta$-dimensional component $X$ of
$V$. In other words

(3.11) $X\subseteq\bigcap_{m^{c}\in\Sigma}[m^{c}](V).$

We show that, if this is the case, then the first conclusion holds. Assuming
that $X$ satisfies (3.11), it is enough to prove that the following properties
hold true:

  1. (1)

$X$ is a special subvariety, say of the form $\alpha+B$, for some torsion
point $\alpha\in A$ and $B$ an abelian subvariety of $A$;

  2. (2)

The order of $X$ is bounded by $\mathrm{ord}(X)\leq n^{c}=d^{o(1)}$ (and the
same applies for the order of $l\alpha+B$, for $l\in L$)222The estimate
$n^{c}=d^{o(1)}$ follows from the more precise (3.9).;

  3. (3)

For every $l\in L$, $l\alpha+B$ is a component of $V$.

Letting $Z$ be the union of such components provides the first conclusion.

###### Proof of (1)..

From the set $\Sigma$, defined in (3.10), pick

$q_{1}=(p_{1})^{c}<q_{2}=(p_{2})^{c}<\ldots<q_{D+1}=(p_{D+1})^{c},$

such that

(3.12) $\sqrt{D}\leq q_{1}<q_{D+1}\leq n^{c},$

and $p_{1},\ldots,p_{D+1}$ relatively prime to each other. For simplicity we
can find these $p_{i}$’s among prime numbers, which are known to abound. We
can actually choose the $p_{i}$’s among

$\Sigma\cap\\{p(\left\lceil{\sqrt[4c]{D}}\right\rceil),\ldots,p(\left\lceil{\sqrt[4c]{D}}\right\rceil+D+\omega(d)+1)\\},$

to take care that we might have at most $\omega(d)$ primes dividing $d$, and
to exclude at most one more prime, accounting for the characteristic $p$ of
the base field $k$. The lower bound in (3.12) then holds, as well as the upper
bound, as $n$ was defined for this very purpose (the choice of $N$ will be
clear later).

For every $q\in\\{q_{1},\ldots,q_{D+1}\\}$ (or more generally in the set
$\Sigma$) by assumption (3.11), we have

$X\subseteq[q](V),$

and so there exists at least a component $Y(q)$ of $V$ such that

$X\subseteq[q](Y(q)).$

Since $X$ is a component of maximal dimension, it follows that $X=[q](Y(q))$.
As $q$ ranges through $\\{q_{1},\dots,q_{D+1}\\}$ the numbers of distinct
$Y(q_{i})$ cannot be more than $\deg(V)=D$. That is, a collision must occur.
We have

$Y(q_{i_{1}})=Y(q_{i_{2}}),$

with $1\leq i_{1}\neq i_{2}\leq D+1$ (this is why we have chosen to work with
$D+1$ numbers). Write $Y:=Y(q_{i_{1}})=Y(q_{i_{2}})$, we have

$\deg(Y)\leq\deg(V)=D\qquad\text{and}\qquad[q_{i_{1}}](Y)=X=[q_{i_{2}}](Y).$

As we guaranteed that $D<(q_{i_{1}}q_{i_{2}})^{2}$, Proposition 2.2.4 implies
that $Y$ is special. Hence so is $X$. ∎

This covers the first claim, and we are now ready to bound $\mathrm{ord}(X)$.

###### Proof of (2).

Thanks to (1), we can find an abelian subvariety $B\subset A$ such that

$Y=\alpha_{Y}+B,$

for some torsion point $\alpha_{Y}\in A$, and we may assume that
$\mathrm{ord}(Y)=\mathrm{ord}(\alpha_{Y})$ be the general assumptions. We then
have

$X=q_{i_{1}}\cdot\alpha_{Y}+B=q_{i_{2}}\cdot\alpha_{Y}+B.$

Necessarily we have the divisibility

$\mathrm{ord}(Y)|\left|{q_{i_{1}}-q_{i_{2}}}\right|\leq n^{c}.$

On the other hand we know, for some $l\in L$, that $la+A^{\prime}\subset Y$,
and so

$\mathrm{ord}(Y)|\mathrm{ord}(l\cdot
a+A^{\prime})=\mathrm{ord}(a+A^{\prime})=d.$

Hence each $q$ in the set $\Sigma$, being relatively prime to $d$, is
relatively prime to $\mathrm{ord}(Y)$. We deduce that

$\mathrm{ord}(X)=\mathrm{ord}(Y)\leq n^{c},$

concluding the proof of (2). ∎

Set $d^{\prime}:=\mathrm{ord}(X)$. We are left to prove that for every $l\in
L$, $l\alpha+B$ is a component of $V$. Namely, for any $1\leq j<d^{\prime}$
such that $j\wedge d^{\prime}=1$, we have

$[j^{c}](X)\subseteq V.$

To take care of the fact that such $j^{c}$, a priori, does not lie in the set
$\Sigma$ (since $j^{c}\wedge d$ could be different from $1$), we use some
results on the Jacobsthal’s function $g(d)$, as discussed in Section 3.2. It
is only in this proof that we need to argue with $N$, rather than $n$.

###### Proof of (3).

We choose an inverse $j^{\ast}+d^{\prime}\mathbb{Z}$ of
$j+d^{\prime}\mathbb{Z}$, represented by some $1\leq j^{\ast}<d^{\prime}$. By
Proposition 3.2.1 there is some $m$ such that $m\equiv
j^{\ast}\pmod{d^{\prime}}$, relatively prime to $d$ and not divisible by $p$,
satisfying

$m\leq d^{\prime}\cdot g((d\cdot\max\\{1,p\\})/d^{\prime})\leq n^{c}\cdot
g(d\cdot\max\\{1,p\\})=N,$

since, by (2), $d^{\prime}\leq n^{c}$ (this explains our definition of $N$).
Therefore $m^{c}$ belongs to the set $\Sigma=\Sigma(d,D)$, defined in (3.10).
In particular we have

$X=[m^{c}](Y(m^{c})),$

for the component $Y=Y(m^{c})$ of $V$. We know, for some $l\in L$, that
$a_{Y}:=l\cdot a$ belongs to $Y$ and we can write

$Y=a_{Y}+B,$

for some abelian subvariety $B$ of $A$. Therefore

$X=[m^{c}](Y)=m^{c}\cdot a_{Y}+B.$

Applying $[j^{c}]$, the above identity becomes

$[j^{c}](X)=[(jm)^{c}](a_{Y}+B)=(jm)^{c}\cdot a_{Y}+B.$

But $\mathrm{ord}(a_{Y}+B)=d^{\prime}$ and $jm\equiv 1\pmod{d^{\prime}}$, so

$[j^{c}](X)=Y$

is indeed a component of $V$. ∎

The proof of Lemma 3.2.3 is finally completed. ∎

####  3.2.2. Second lemma

The following refines the conclusion of Lemma 3.2.3, constructing a special
subvariety in $V$ of explicitly bounded degree.

######  Lemma 3.2.4.

Under the same assumptions and notations of Proposition 3.1.2, there is $Z$
such that

$L\cdot a+A^{\prime}\subseteq Z\subseteq V,$

and of the form $L\cdot\alpha+B$, for some abelian subvariety $B\subset A$,
and $\alpha\in A_{\operatorname{tors}}$, such that:

$\deg(Z)\leq D^{o(1)}\cdot d^{o(1)}.$

The proof of the above result is obtained by induction from Lemma 3.2.3.
Notice that, for every $l\in L$, we have

$\mathrm{ord}(l\alpha+B)\leq\deg(Z)\leq D^{o(1)}\cdot d^{o(1)},$

which is exactly what we need. During the proof we will actually prove that

$\deg(Z)\leq\max\\{D,(\omega(d)+1)\\}^{\delta(\Delta,c)}\left(2\cdot
g(d\cdot\max\\{1,p\\})\right)^{\delta^{\prime}(\Delta,c)},$

where $\delta(\Delta,c)$ and $\delta^{\prime}(\Delta,c)$ are explicit constant
depending only on $\Delta$ and $c$ (see (3.13) for their definition).

###### Proof.

We define inductively

$f_{0}(D)=D,\qquad f_{i+1}(D)=f(f_{i}(D),d),$

where $f(D,d)=D^{2}\cdot\left(n^{c}\cdot g(d\cdot\max\\{1,p\\})\right)^{2\cdot
c\cdot\Delta}$ was defined in (3.7), for $n=p(x)$, and
$x=x_{D}:=2D+\omega(d)+1$. For simplicity, from now on, we ignore the factor
$\max\\{1,p\\}$ in the argument of the Jacobsthal’s function. For example, we
have

$f_{2}(D,d)=(D^{2}n^{2c^{2}\Delta}g(d)^{2c\Delta})^{2}({n_{2}}^{c}g(d))^{2c\Delta}=D^{2}n^{2c^{2}\Delta}{n_{2}}^{2c^{2}\Delta}g(d)^{6c\Delta},$

where $n_{2}=p(x_{2})$, and

$x_{2}:=x_{f(D,d)}=2f(D,d)+\omega(d)+1=2D^{2}\cdot\left(n^{c}\cdot
g(d)\right)^{2\cdot c\cdot\Delta}+\omega(d)+1.$

Set also

$V_{0}=V,\qquad V_{i+1}={V_{i}}^{\prime},$

with ${V_{i}}^{\prime}$ to be $V^{\prime}$ constructed from Lemma 3.2.3
applied to the $V_{i}$. The latter is defined until $i_{0}$, for some $i_{0}$,
where the first point of Lemma 3.2.3 occurs, and in this case we obtain $Z$
with $\deg(Z)\leq\deg(V_{i_{0}})$ and of the form

$L\cdot\alpha+B$

for some abelian subvariety $B\subset A$, and $\alpha\in
A_{\operatorname{tors}}$. This has to happen for some $i_{0}\leq\Delta$, since
at each step the dimension is dropping by one, and each $V_{i}$ is not empty.

The functions $f_{i}(D)$ are increasing in both $D$ and $i$, and so we can
carry over upper bounds and rely on the computation below. For example, for
any $i$, we get

$\deg(V_{i})\leq f_{i}(D)\leq D^{\prime}:=f_{\Delta}(D),$

which will be bounded in an explicit way in (3.13) below.

Set $D_{\ast}:=\max\\{D,\omega(d)+1\\}$, and notice that

$4\leq x=2D+\omega(d)+1\leq 3D_{\ast}.$

In particular we can write

$n=p(x)\leq x\log(x)(1+\log\log(x))\leq
3D_{\ast}\log(3D_{\ast})(1+\log\log(3D_{\ast}))\leq{D_{\ast}}^{1+\epsilon},$

for some $\epsilon>0$, and

$f_{1}(D)=f(D,d)=D^{2}\cdot n^{2c^{2}\Delta}\cdot g(d)^{2\cdot
c\cdot\Delta}\leq{D_{\ast}}^{2(1+c^{2}\cdot\Delta\cdot(1+\epsilon))}\cdot
g(d)^{2\cdot c\cdot\Delta}.$

To bound the $f_{i}(D)$, for $1<i\leq\Delta$ we argue inductively. A direct
computation shows that

(3.13) $\displaystyle f_{\Delta}(D)=D^{\prime}$
$\displaystyle\leq{D_{\ast}}^{{2^{\Delta}}(1+\lambda)}\cdot(2\cdot
g(d))^{2^{(\Delta+1)}c\Delta}$ with
$\displaystyle\lambda=c^{2}\cdot\frac{\Delta^{2}-\Delta}{2}\cdot(1+\epsilon),$
(3.14) $\displaystyle=\max\\{D,(\omega(d)+1)\\}^{\delta}\cdot(2\cdot
g(d))^{\delta^{\prime}}$ with $\displaystyle\left\\{\begin{aligned}
\delta=\delta(\Delta,c)&={2^{\Delta}}(1+\lambda);\\\
\delta^{\prime}=\delta^{\prime}(\Delta,c)&=2^{\Delta}\lambda/c.\end{aligned}\right.$

where $\epsilon=0.5$ will always do. We have therefore proven the desired

$\deg(Z)\leq D^{o(1)}\cdot d^{o(1)},$

concluding the proof of Lemma 3.2.4. ∎

We are ready to prove Proposition 3.1.1, as was stated in Section 3.1. It
consists in a refinement of the conclusion of Lemma 3.2.4.

####  3.2.3. Proof of Proposition 3.1.1

Let $V$ be a subvariety of $A$ of degree $D$ and dimension $\Delta<g$, and
$a\in A_{\operatorname{tors}}$, of order $d$, such that $L\cdot a\subseteq V$,
where $L=L(c)$ is the Lang group as in (2.1). Consider the set $\Theta$ of
subvarieties of $V$ of the form

$Y=L\cdot a_{Y}+B_{Y},$

where $a_{Y}\in A_{\operatorname{tors}}$ and $B_{Y}$ is an abelian subvariety
of $A$, such that

$L\cdot a\subseteq Y.$

Notice that $a\in\Theta$, and in particular $\Theta\neq\emptyset$. We can
choose a $Y_{\min}=L\cdot{a}_{\min}+B_{\min}$ such that the order of the coset
$a_{\min}+B_{\min}$ is minimal among the elements in $\Theta$. Lemma 3.2.4,
applied to the pair $(a_{\min}+B_{\min}\in A_{\operatorname{tors}},V)$ asserts
that there exists $Z=L\alpha_{Z}+B_{Z}$, containing $Y_{\min}$, and of order
bounded by $d^{o(1)}$, and implicit constant depending only on
$D,\Delta,c$333Here is where we needed a version of Lemma 3.2.4 for torsion
cosets, rather than for just torsion points.. Let $\delta=\delta(D,\Delta,c)$
be the biggest number such that $d^{o(1)}<d$, for all $d\geq\delta$. My
minimality of the order of $a_{\min}$, it follows that

$\mathrm{ord}(a_{\min})\leq\min\\{\delta,d\\}$

and therefore $Y_{\min}=L\cdot{a}_{\min}+B_{\min}$ satisfies the desired
conclusion. Proposition 3.1.1 is finally proven.

######  Remark 3.2.5.

The same argument applies to prove also Proposition 3.1.2, but, in Section 3.3
we only need the version proved above. Notice however that it was important to
have a version of Lemma 3.2.4 for torsion cosets, rather than just for torsion
points.

####  3.2.4. Effective estimates for Proposition 3.1.1

For completeness we explain how to get explicit estimates for
$\delta(D,\Delta,c)$ (as in the proof of the previous lemma, we ignore the
factor $\max\\{1,p\\}$ in the argument of the Jacobsthal’s function). We break
the bound from Lemma 3.2.4, in a system of inequalities

$\begin{cases}d\geq(\omega(d)+1)^{\delta}\cdot(2\cdot
g(d))^{\delta^{\prime}},\\\ d\geq D^{\delta}\cdot(2\cdot
g(d))^{\delta^{\prime}}.\end{cases}$

Where $\delta=\delta(\Delta,c),\delta^{\prime}=\delta^{\prime}(\Delta,c)$ are
the explicit constants defined in (3.13). We bound $\omega(d)\leq\log(d)$ and
$g(d)\leq\alpha\cdot(\omega(d))^{\beta}$ for some explicit constants $\alpha$
and $\beta$ obtained from bounds on $g(d)$, or more precisely on
$g(d\cdot\max\\{1,p\\})$, as explained in Section 3.2.

The first inequality is independent of $D$ and it is satisfied as soon as (up
to replacing $d$ by $d+6$)

$d\geq\alpha^{\prime}\log(d)^{\beta^{\prime}}\geq\alpha^{\prime}\omega(d)^{\beta^{\prime}}\geq(\omega(d)+1)^{\delta}\cdot(2\cdot
g(d))^{\delta^{\prime}},$

where we set

$\alpha^{\prime}:=\alpha^{\delta^{\prime}},\text{ and
}\beta^{\prime}:=\delta^{\prime}\cdot\beta+\delta.$

This is satisfied whenever

$d\geq\alpha^{\prime}\cdot{\beta^{\prime}}^{\beta^{\prime}},$

which we assume in studying now the second inequality. Note that

$\log(d)\leq\alpha^{\prime(-1/\beta^{\prime})}d^{1/\beta^{\prime}}.$

The second inequality now becomes

$d\geq(\alpha^{\prime(-1/\beta^{\prime})}d^{1/\beta^{\prime}})^{\delta}\geq\log(d)^{\delta}D^{\delta^{\prime}},$

and it holds precisely when

$d^{1-1/\beta^{\prime}}\geq\alpha^{\prime(-1/\beta^{\prime})}D^{\gamma^{\prime}}.$

Finally we can set

(3.15)
$\delta(D,\Delta,c):=\max\\{\alpha^{\gamma}\cdot{\beta^{\prime}}^{\beta^{\prime}},\alpha^{\gamma\cdot\beta^{\prime}/(\beta^{\prime}-1)}D^{\gamma^{\prime}\beta^{\prime}/(1-\beta^{\prime})}\\}$

to be a sufficient bound. This bound can explicitly be written in terms of
$\delta=\delta(\Delta,c)$ and $\delta^{\prime}=\delta^{\prime}(\Delta,c)$ of
(3.13), and explicit bounds $\alpha,\beta$ on the Jacobsthal’s function, as
recalled in (3.5).

###  3.3. Proof of Theorem 2.1.4, and Theorem 1.2.1

We are finally ready to prove Theorem 2.1.4, which, as explained in Section
2.1, implies Theorem 1.2.1. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian
scheme. Our proof works both when the special 0-cycles are of characteristic
zero and when they are of arbitrary characteristics. Here
$A=A_{K}=\mathcal{A}\times_{\mathcal{O}_{K,S}}K$, and the implicit constant
$c=c(A)$ in the Lang group $L=L(c)$ is the one given by Theorem 2.1.1 applied
to $A/K$.

###### Proof of horizontal part of Theorem 2.1.4.

Let $(E_{n})_{n\geq 0}\subset\mathcal{A}$ be a sequence of special 0-cycles
associated to $a_{n}\in A_{\operatorname{tors}}$ and
$\mathfrak{p}_{n}\in\operatorname{Spec}(\mathcal{O}_{K,S})$. Consider the
image of the $a_{n}$ along the corresponding map

$\pi_{n}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}_{n}}(\overline{\kappa(\mathfrak{p}_{n})}).$

Denote by $d_{n}$ the order of $a_{n}\in A$, and by $d_{n}^{\prime}$ the order
of $\pi_{n}(a_{n})$ in $\mathcal{A}(\overline{\kappa(\mathfrak{p}_{n}}))$.

We study $\mathcal{V}$, the Zariski closure of $\bigcup_{n\geq 0}E_{n}$ in
$\mathcal{A}$. It is a finite union of vertical fibres and horizontal
components (i.e. flat over the base $\operatorname{Spec}(\mathcal{O}_{K,S})$).
We may assume that $d_{n}$ and $d_{n}^{\prime}$ tend to infinity, as $n$
grows, since otherwise $\mathcal{V}$ is finite and in particular special.
Notice that, for any $n\geq 0$, we have

$L\cdot\pi_{n}(a_{n})=\pi_{n}(L\cdot a_{n})\subseteq E_{n}.$

Moreover, up to extracting a subsequence, we may assume that
$|\mathfrak{p}_{n}|\to+\infty$.

We first look at the horizontal/flat part, even if the same argument applies
to the vertical case. Set $D$ for the maximum, over $n$, of
$\deg(V_{\mathfrak{p}_{n}})$, $\Delta:=\dim(V_{K})$, and let $\delta$ for the
maximum of $\delta(\deg(V_{\mathfrak{p}_{n}}),\Delta,c)$ be as in Proposition
3.1.1 applied to the pair $\pi_{n}(a_{n})$ and
$V_{\mathfrak{p}_{n}}\subset\mathcal{A}_{\mathfrak{p}_{n}}$ (it is a finite
set, since $\deg(V_{\mathfrak{p}_{n}})$ takes only finitely many values as $n$
varies). We can find a positive integer $q$ such that

  * •

$q$ is a $c$-th power;

  * •

$q$ is relatively prime to $\delta$;

  * •

$q$ is relatively prime to
$p_{n}:=\operatorname{char}(\kappa(\mathfrak{p}_{n}))$, for all $n$ bigger
than a given constant $M$.

Proposition 3.1.1 asserts that, for each torsion point
$\pi_{n}(a_{n})\in\mathcal{A}_{\mathfrak{p}_{n}}$, we have

$L\cdot\pi_{n}(a_{n})\subseteq\left(L\cdot\alpha_{n}+B_{n}\right)\subseteq\mathcal{V},$

for some $\alpha_{n}\in\mathcal{A}_{\mathfrak{p}_{n}}$, with
$\mathrm{ord}(\alpha_{n}+B_{n})\leq\delta$ (in particular the order of
$\alpha_{n}$ does not depend on $d_{n}^{\prime}$, nor on $p_{n}$), and
$B_{n}\subset\mathcal{A}_{\mathfrak{p}_{n}}$ an abelian subvariety. Notice
that

$L\cdot\alpha_{n}+B_{n}$

is stable under the multiplication by $[q]$ (thanks to Remark 2.1.2). In
particular $\mathcal{V}$ can be written as the Zariski closure of the set

$\left(\bigcup_{n\geq M}L\cdot\alpha_{n}+B_{n}\right)\subseteq\mathcal{V}.$

As explained above, the left hand side is stable under multiplication by
$[q]$, and so $\mathcal{V}$ itself is stable under $[q]$. Corollary 2.2.5,
applied to $\mathcal{V}_{K}\subseteq A/K$ shows that $\mathcal{V}_{K}$ is
special. This means, following the definitions introduced in Section 1.2, that
the horizontal part of $\mathcal{V}$ is a finite union of flat cycles. ∎

To conclude the proof of Theorem 2.1.4, we have to show that and the vertical
part of $\mathcal{V}$ is a finite union of torsion cycles, essentially by
adapting the argument just presented. The lifting argument is more delicate,
and presents some challenges depending on the order of the torsion points
$a_{n}$, this will be dealt with in the next section.

###### Proof of the Weak vertical part of Theorem 2.1.4.

We are left to considering the case where

$(E_{n})_{n\geq 0}\subset\mathcal{A}$

is a sequence of special 0-cycles associated to $(a_{n})_{n}\subset
A_{\operatorname{tors}}$ and a fixed
$\mathfrak{p}\in\operatorname{Spec}(\mathcal{O}_{K,S})$. Consider the image of
the $a_{n}$ along the corresponding map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Let $d_{n}^{\prime}$ be the order of
$\pi_{\mathfrak{p}}(a_{n})\in\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})})$.
If the $d_{n}^{\prime}$ are bounded as $n$ goes to infinity, then the set

$\bigcup_{n\geq 0}E_{n}$

is finite and so there is nothing to prove. We may therefore assume that the
$d_{n}^{\prime}$ are unbounded.

The argument described above implies that every component of $\mathcal{V}$ is
a translate of an abelian subvariety of $\mathcal{A}_{\mathfrak{p}}$. Indeed,
let $c$ be the constant associated to $A/K$ by Theorem 2.1.1, and let
$\delta=\delta(\deg(\mathcal{V}),\dim(\mathcal{V}),c)$ be the constant given
by Proposition 3.1.1 applied to $\mathcal{V}\subset\mathcal{A}_{\mathfrak{p}}$
(and any $\pi_{\mathfrak{p}}(a_{n})$). We can find a positive integer $q$ such
that $q$ is a $c$-th power, and $q\wedge\delta p=1$. Proposition 3.1.1 asserts
that, for each torsion point
$\pi_{\mathfrak{p}}(a_{n})\in\mathcal{A}_{\mathfrak{p}}$, we have

$L\cdot\pi_{\mathfrak{p}}(a_{n})\subseteq\pi_{\mathfrak{p}}(L\cdot
a_{n})\subseteq\left(L\cdot\alpha+\mathfrak{B}_{n}\right)\subseteq\mathcal{V},$

for some
$\alpha\in\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})})$, with
$\mathrm{ord}(\alpha+\mathfrak{B}_{n})\leq\delta$, for some $\delta$ depending
only on $D,\Delta$ and $c$ (where $\mathfrak{B}_{n}$ is some abelian
subvariety of $\mathcal{A}_{\mathfrak{p}}$). Notice that
$L\cdot\alpha+\mathfrak{B}_{n}$ is stable under the multiplication by $[q]$.
Since $\mathcal{V}$ is defined as the Zariski closure of the $E_{n}$, and from
the fact that $\delta$ does not depend on $d_{n}^{\prime}$, it follows that
$\mathcal{V}$ is stable under the multiplication by $[q]$. Proposition 2.2.4
then shows that $\mathcal{V}$ is special, that is a finite union of translates
by a (torsion) point of
$\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})})$ of abelian
subvarieties of $\mathcal{A}_{\mathfrak{p}}$. The proof of Theorems 1.2.1 and
2.1.4 is finally completed. ∎

The last statement will be referred to as the _weak vertical part_ of the
arithmetic Manin–Mumford. The _stronger_ part, that is the lift of the torsion
cosets of $\mathcal{A}_{\mathfrak{p}}$ to $K$, possibly up to $p$-rank zero
factors, is described in the next two sections.

##  4\. Strong vertical, Theorem 1.2.3

In this section we prove Theorem 1.2.3, which may be of independent interest
from the arguments presented so far. As in Section 1.2, $K$ denotes a number
field, $S$ a finite set of finite places of $K$, and $A=A_{K}$ the generic
fibre of an abelian scheme $\mathcal{A}/\mathcal{O}_{K,S}$. To explain the
method of the proof we first prove the following easier statement, which
appeared in the introduction as Theorem 1.2.4. It can be thought as the
analogue of the main theorem of Edixhoven and Richard [8] for abelian
varieties (rather than $\mathbb{A}^{1}\times_{\mathbb{Z}}\mathbb{A}^{1}$).

######  Theorem 4.0.1.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$ of residue characteristic
$p>0$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme,
$(a_{n})_{n}\subset A_{\operatorname{tors}}$ be a sequence of torsion points
of order coprime with $p$ and write
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$, for
a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Then every component of the Zariski closure $\mathcal{V}$ of

$\bigcup_{n\geq 0}E_{n}\subseteq\mathcal{A}_{\mathfrak{p}}\subset\mathcal{A},$

is a component of the intersection between a flat cycle
$\mathcal{B}\subseteq\mathcal{A}$ and $\mathcal{A}_{\mathfrak{p}}$.

The novelty of Theorem 4.0.1, compared to Theorem 1.2.1, is that the
components of $\mathcal{V}$ are not only given by torsion cycles of
$\mathcal{A}_{\mathfrak{p}}$, but they actually come from characteristic zero.
The case where torsion points are allowed to have order divisible by $p$, is
the main topic of Section 5. From now on, we let $p$ be the characteristic of
the residue field $\kappa(\mathfrak{p})$.

Before proving Theorem 4.0.1, we explain how it implies Theorem 1.2.3.

####  4.0.1. Proof of Theorem 1.2.3

Assuming Theorem 4.0.1, which will be proven in Section 4.3, we show how to
conclude the proof of the more general Theorem 1.2.3, as stated in the
introduction. Let $E_{n}$ be a sequence of special zero cycles in
$\mathcal{A}$, each one associated to $a_{n}\in A_{\operatorname{tors}}$ and
primes $\mathfrak{p}_{n}\subset\mathcal{O}_{K,S}$.

Let $\mathcal{V}\subseteq\mathcal{A}$ be the Zariski closure of

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}$

and write it as a finite union of flat and vertical components. From Theorem
1.2.1, which was proven in Section 3.3, we now that the flat part is a finite
union of flat cycles, and that each vertical component at
$\mathfrak{p}_{1},\dots,\mathfrak{p}_{m}\subset\mathcal{O}_{K,S}$ is a torsion
cycle of some $A_{\mathfrak{p}_{j}}$ for $j\in\\{1,\dots,m\\}$. Theorem 4.0.1,
applied at $\mathfrak{p}_{1},\dots,\mathfrak{p}_{m}$, now implies that each
torsion cycles is given as the intersection between a flat cycle and vertical
fibres. That is $\mathcal{V}$ can be written as a finite union of flat cycles
and intersections between flat cycles and vertical fibres. Theorem 1.2.3 is
finally proven.

###  4.1. Variants of the Tate conjecture (after Faltings et. al.)

We recall Faltings’s results from the landmark paper [9] known as the _semi-
simplicity of the Tate module_ and the _Isogeny Theorem_ (or often referred to
as the _Tate conjecture_). Some variants thereof are also discussed, and we
remark here that all results are known to hold true even for finitely
generated fields over $\mathbb{Q}$. This section is purely in characteristic
zero, and it is independent from the prime $p$ fixed in the previous one.

######  Theorem 4.1.1 (Faltings).

Let $K$ be a number field, and $A/K$ an abelian variety. Let $\ell$ be a
rational prime, and $V_{\ell}(A)$ the rational $\ell$-adic Tate module of $A$.
The canonical map

$\operatorname{End}_{K}(A)\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\to\operatorname{End}_{\operatorname{Gal}(\overline{K}/K)}(V_{\ell}(A))$

is an isomorphism. Moreover $V_{\ell}(A)$ is a semi-simple representation of
$\operatorname{Gal}(\overline{K}/K)$.

We recall also a version of the Tate conjecture with finite coefficients. For
its proof we refer to [34, Section 5.4], which builds again on Faltings’s
proof of the Shafarevich conjecture [9].

######  Theorem 4.1.2 (Faltings, Zarhin).

Let $K$ be a number field, and $A/K$ an abelian variety. For any $\ell$ large
enough and any $m$, the $\operatorname{Gal}(\overline{K}/K)$-module
$A[\ell^{m}]$ is semisimple, and the canonical map

$\operatorname{End}_{K}(A)\otimes\mathbb{Z}/\ell^{m}\mathbb{Z}\to\operatorname{End}_{\operatorname{Gal}(\overline{K}/K)}(A[\ell^{m}])$

is an isomorphism. Moreover $A[\ell^{m}]$ is a semi-simple representation of
$\operatorname{Gal}(\overline{K}/K)$.

For completeness we recall an effective version of Theorem 4.1.2. Compared to
Theorem 4.1.2, it will be important that the dependence on $K$ is only via its
degree over $\mathbb{Q}$. We write $h(A)$ for the semistable (logarithmic)
Faltings height of $A/K$. The proof of such statement, which builds on
previous work of the authors [22], can be found in [23].

######  Theorem 4.1.3 (Masser-Wüstholz).

For any positive integer $m$, the cokernel of the canonical map

$\operatorname{End}_{K}(A)\to\operatorname{End}_{\operatorname{Gal}(\overline{K}/K)}(A[m])$

is killed by $M$, for some positive integer $M$ such that

$M\leq c\left(\max\\{1,h(A)\\}\right)^{\tau},$

where $\tau$ is a constant depending only on $[K:\mathbb{Q}]$, and $c$ depends
only on the dimension of $A$ and the degree of $[K:\mathbb{Q}]$.

The last refinement of the Tate conjecture we need is the following. Let
$(\ell_{n})_{n\in\mathbb{N}}$ be a sequence of distinct prime numbers, and
consider the following quotient of $\widehat{\mathbb{Z}}$

$R=\prod_{n\in\mathbb{N}}\mathbb{F}_{\ell_{n}}.$

Since $R\otimes_{\mathbb{Z}}\mathbb{Q}$ is a non-zero $\mathbb{Q}$-algebra,
there exists a field $E$ (of characteristic zero) and a ring map
$R\otimes_{\mathbb{Z}}\mathbb{Q}\to E$ corresponding to some maximal ideal.
Given an abelian variety $A/K$, we let $V_{E}(A)$ be the $E$-Tate module of
$A$:

$V_{E}(A):=T(A)\otimes_{\widehat{\mathbb{Z}}}R\otimes_{\mathbb{Q}}E.$

Here the action of $\operatorname{Gal}(\overline{K}/K)$ on $V_{E}(A)$ is
trivial on $R\otimes_{\mathbb{Q}}E$.

######  Theorem 4.1.4.

The canonical map

$\operatorname{End}_{K}(A)\otimes_{\mathbb{Z}}E\to\operatorname{End}_{\operatorname{Gal}(\overline{K}/K)}(V_{E}(A))$

is an isomorphism, and $V_{E}(A)$ is semisimple
$\operatorname{Gal}(\overline{K}/K)$-representation.

###  4.2. Prelimaniry results

We first present the simplest result we use to lift torsion cycles, it
encompasses the power of Theorem 4.1.1, making it ready to be applied for our
purposes. Let $A/K$ be an abelian variety, $\mathcal{A}_{\mathfrak{p}}$ its
reduction mod $\mathfrak{p}$, a place of good reduction. Set
$p=\operatorname{char}(\kappa(\mathfrak{p}))>0$, and

$M^{0}:=\operatorname{End}_{K}(A)\otimes_{\mathbb{Z}}\mathbb{Q},\ \ \
N^{0}:=\operatorname{End}_{\kappa(\mathfrak{p})}(\mathcal{A}_{\mathfrak{p}})\otimes_{\mathbb{Z}}\mathbb{Q}.$

The reduction mod $\mathfrak{p}$ defines a canonical embedding

$M^{0}\hookrightarrow N^{0},$

and, for every $\ell\neq p$, we have

$M^{0}\otimes\mathbb{Q}_{\ell}\hookrightarrow
N^{0}\otimes\mathbb{Q}_{\ell}\hookrightarrow\operatorname{End}_{\mathbb{Q}_{\ell}}(V_{\ell}(A)),$

thanks to the identification $V_{\ell}(A)\cong
V_{\ell}(\mathcal{A}_{\mathfrak{p}})$ given again by the reduction mod
$\mathfrak{p}$.

######  Proposition 4.2.1.

Let $A/K$ be an abelian variety, $\mathcal{A}_{\mathfrak{p}}$ its reduction
mod $\mathfrak{p}$, a place of good reduction. Let $\ell$ be a rational prime
different from $p$. Let $W$ be a
$\operatorname{Gal}(\overline{K}/K)$-invariant subspace of $V_{\ell}(A)$ and
let $\mathfrak{B}\subseteq\mathcal{A}_{\mathfrak{p}}$ be an abelian subvariety
such that

$W\subseteq V_{\ell}(\mathfrak{B})\subseteq V_{\ell}(A).$

There exists an abelian subvariety $D$ of $A$, defined over $K$, such that

$W\subseteq V_{\ell}(D)\subseteq V_{\ell}(\mathfrak{B}).$

Notice that Proposition 4.2.1 can be rewritten in purely algebraic therms.
Here $M^{0}$ and $N^{0}$ are finite dimensional semisimple
$\mathbb{Q}$-algebras. Moreover
$\mathfrak{B}\subseteq\mathcal{A}_{\mathfrak{p}}$ corresponds to an idempotent
$u=u^{2}\in N^{0}$ such that

$W\subseteq u\cdot V_{\ell}(A),$

And Proposition 4.2.1 asserts that there exists an idempotent $v$ in $M^{0}$
such that

$W\subseteq v\cdot V_{\ell}(A)\subseteq u\cdot V_{\ell}(A).$

###### Proof of Proposition 4.2.1.

Being $V_{\ell}(A)$ a semisimple Galois representation, as established by
Theorem 4.1.1, we have that $W\subseteq V_{\ell}(A)$ is of the form $e\cdot
V_{\ell}(A)$ for some idempotent

$e\in\operatorname{End}_{\operatorname{Gal}(\overline{K}/K)}(V_{\ell}(A)).$

By the Tate conjecture, that is the first part of Theorem 4.1.1, we can
actually write

$W=w\cdot V_{\ell}(A)$

for some $w\in M^{0}\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}$ (and such $w$ is
unique by faithfulness). Lemma 4.2.2, applied with $E=\mathbb{Q}_{\ell}$,
shows how to get $v$ in $M^{0}$ (rather than only in
$M^{0}\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}$) such that the conclusion holds.
∎

######  Lemma 4.2.2.

Let $M,N$ be two semisimple (associative, non necessarily commutative, finite
dimensional) $\mathbb{Q}$-algebras, with $M$ a sub-algebra of $N$. Let
$E/\mathbb{Q}$ be a field extension (not necessarly of finite type), and

$i_{E}:N\otimes_{\mathbb{Q}}E\hookrightarrow\operatorname{End}_{E}(V)$

be a faithful representation on $V$, a finite dimensional $E$-linear vector
space. We let $u$ be an idempotent of $N$, and $w$ be an element of
$M\otimes_{\mathbb{Q}}E$ such that $i_{E}(w)\cdot V\subseteq i_{E}(u)\cdot V$
. Then there exists an idempotent $v\in M$, such that

$i_{E}(w)\cdot V\subseteq i_{E}(v)\cdot V\subseteq i_{E}(u)\cdot V.$

In the proof of Lemma 4.2.2 we use the following standard fact about
semisimple $\mathbb{Q}$-algebras. Let $M$ be a semisimple associative finite
dimensional $\mathbb{Q}$-algebra, then any $\mathbb{Q}$-linear subspace $X$ of
$M$ is a right ideal if and only if it is of the form $vM$ for some idempotent
$v$ of $M$ (possibly, of course, $v\in\\{0,1\\}$). In particular, the above
result applies to the field $E$ described in Theorem 4.1.4.

###### Proof of Lemma 4.2.2.

Consider $Y=uN$, as a right $N$ linear submodule of $N$. Let $X:=Y\cap M$ be
the associated right $M$-module. By the previous fact, there is at least one
idempotent $v\in M$ with the following property: for any $m$ in $M$ we have
$u\cdot m=m$ if and only if $v\cdot m=m$. Otherwise stated, we have $X=v\cdot
M$. The same property holds true with elements in $M\otimes E$. It is indeed
enough to notice, just by computing dimensions, that we have
$(Y\otimes_{\mathbb{Q}}E)\cap(M\otimes_{\mathbb{Q}}E)=X\otimes_{\mathbb{Q}}E$.

As $\operatorname{Im}(i_{E}(w))$ is a submodule of
$\operatorname{Im}(i_{E}(y))$, by assumption, and $i_{E}(u)$ is an idempotent,
we conclude that $i_{E}(u\cdot w)=i_{E}(w)$. By faithfulness of $i_{E}$, we
have also $u\cdot w=w\in N\otimes_{\mathbb{Q}}E$.

Let $v\in M$ be the idempotent associated to $u\in N$ as above. It follows
that

$i_{E}(v)\cdot i_{E}(w)=i_{E}(w),$

that is

$\operatorname{Im}(i_{E}(w))\subseteq\operatorname{Im}(i_{E}(v)).$

Finally, as $v\in vM\subset uN$, we get $u\cdot v=v$, and likewise

$\operatorname{Im}(i_{E}(v))\subseteq\operatorname{Im}(i_{E}(u)).$

In another writing

$i_{E}(w)\cdot V\subseteq i_{E}(v)\cdot V\subseteq i_{E}(u)\cdot V,$

as desired. This concludes the proof of the Lemma 4.2.2. ∎

Replacing in the proof of Proposition 4.2.1 the role of (the $\ell$-adic)
Faltings’s Theorem by the Tate conjecture with adelic (or $E$) coefficients
recalled in Theorem 4.1.4, we also have the following.

######  Proposition 4.2.3.

Let $A/K$ be an abelian variety, $\mathcal{A}_{\mathfrak{p}}$ its reduction
mod $\mathfrak{p}$, a place of good reduction. Let $(\ell_{n})$ be a sequence
of distinct primes different from $p$, and let $E$ be the field associated to
$\mathbb{Q}\otimes\prod_{n}\mathbb{F}_{\ell_{n}}$ as in Theorem 4.1.4. Let $W$
be a $\operatorname{Gal}(\overline{K}/K)$-invariant subspace of $V_{E}(A)$ and
let $\mathfrak{B}\subseteq\mathcal{A}_{\mathfrak{p}}$ be an abelian subvariety
such that

$W\subseteq V_{E}(\mathfrak{B})\subseteq V_{E}(A).$

There exists an abelian subvariety $D$ of $A$, defined over $K$, such that

$W\subseteq V_{E}(D)\subseteq V_{E}(\mathfrak{B}).$

We record here another application of Lemma 4.2.2 which will be useful in the
next subsection.

######  Lemma 4.2.4.

Let $A/K$ be an abelian variety, and $K^{\prime}/K$ be a finite field
extension. Let $E$ be $\mathbb{Q}_{\ell}$, for some prime $\ell$, (resp. a
field as in Proposition 4.2.3) and $W$ be a
$\operatorname{Gal}(\overline{K}/K)$-invariant subspace of $V_{\ell}(A)$
(resp. $V_{E}(A)$). Let
$u\in\operatorname{End}(A_{K^{\prime}})\otimes\mathbb{Q}$ be an idempotent,
and $w$ be an element of $\operatorname{End}(A_{K})\otimes_{\mathbb{Q}}E$ such
that $w\cdot V_{E}(A)=W$. Then there exists an idempotent
$v\in\operatorname{End}(A_{K})\otimes\mathbb{Q}$, such that

$w\cdot V_{E}(A)\subseteq v\cdot V_{E}(A)\subseteq u\cdot V_{E}(A).$

The above result, for example, can be applied to $K^{\prime}$ a finite field
extension such that all endomorphism of $A\times\overline{K}$ are defined over
$K^{\prime}$.

###  4.3. Proof of Theorem 4.0.1

As we show in Section 4.3.1, the proof of Theorem 4.0.1 follows from Theorem
1.2.1 (the weak vertical Manin–Mumford) and the following result.

######  Proposition 4.3.1.

Let $\mathfrak{p}\subset\mathcal{O}_{K,S}$ be a prime ideal of residue
characteristic $p>0$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian
scheme, and fix a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

For $i=1,\dots,C$, let $\mathfrak{B}_{i}\subset\mathcal{A}_{\mathfrak{p}}$ be
abelian subvarieties of $\mathcal{A}_{\mathfrak{p}}$. Let $D_{i}/K$ be the
biggest abelian subvariety of $A$ whose reduction mod $\mathfrak{p}$ is
contained in $\mathfrak{B}_{i}$. Let $(a_{n})_{n\geq 0}\subset
A_{\operatorname{tors}}$ be a sequence of torsion points of order coprime with
$p$ and, for every $n$, let
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$. If

$\bigcup_{n}E_{n}\subset\bigcup_{i=1}^{C}\beta_{i}+\mathfrak{B}_{i},$

then there exists a finite set $F\subset A$ such that

$a_{n}\in F+\bigcup_{i=1}^{C}D_{i},$

for all $n$.

###### Proof.

Heading for a contraction, we may pass to a subsequence of $(a_{n})_{n}$ for
which, for every $i=1,\dots,C$, we have

$\mathrm{ord}(a_{n}+D_{i}/D_{i})\to+\infty,$

as $n\to+\infty$. The strategy is to construct some $1\leq i\leq C$ and a non-
zero abelian subvariety $A^{\prime}\subset A/D_{i}$ such that
$A^{\prime}_{\mathfrak{p}}\subseteq\mathfrak{B}_{i}/D_{i,\mathfrak{p}}$. The
preimage of $A^{\prime}$ along the quotient map $A\to A/D_{i}$ would then
contradict the maximality of $D_{i}$.

For every $n$ there exists an $1\leq i\leq C$ such that the following
inequality of cardinalities holds true

$\\#\\{\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot
a_{n})\cap\mathfrak{B}_{i}\\}\geq\frac{1}{C}\\#\\{\operatorname{Gal}(\overline{K}/K)\cdot
a_{n}\\}.$

Upon extracting a subsequence, we may assume from now on that $i$ is fixed.
And for simplicity set $D=D_{i}$, $\mathfrak{B}=\mathfrak{B}_{i}$ and

$d_{n}=\mathrm{ord}(a_{n}),\text{ and
}d_{n}^{\prime}=\mathrm{ord}(a_{n}+D/D).$

By assumption we know that $d_{n}^{\prime}$ goes to infinity and, possibly
extracting a subsequence, at least one of the following holds true:

  1. (1)

There exists a rational prime number $\ell(\neq p)$, and a sequence
$(k_{n})\subset\mathbb{N}$ going to infinity, as $n\to+\infty$, such that

$\forall n,\ell^{k_{n}}\text{ divides }d_{n}^{\prime}.$

  2. (2)

There exists an infinite sequence of distinct prime numbers $\ell_{n}(\neq
p)$, such that

$\forall n,\ell_{n}\text{ divides }d_{n}^{\prime}.$

We argue each case separately and with different techniques, even if the
strategy is the same one.

###### Proof in Case (1).

We set

$a_{n}^{\prime}:=[d_{n}^{\prime}/\ell^{k_{n}}]a_{n}+D/D,$

which is an element of order exactly $\ell^{k_{n}}$ of $A/D$. We say that a
sequence $b_{n}\in\mathbb{Z}/\ell^{k_{n}}\mathbb{Z}$, of elements of order
$\ell^{k_{n}}$, converges to some $b_{\infty}\in T_{\ell}(A/D)$, the
$\mathbb{Z}_{\ell}$-Tate module of $A/D$, if there exist $\epsilon_{n}$
converging to $0\in T_{\ell}(A/D)$ such that $b_{n}$ is the reduction modulo
$\ell^{k_{n}}$ of $b_{\infty}+\epsilon_{n}$. Extracting if needed, we may
assume that $a^{\prime}_{n}$ converges to some non-zero
$a^{\prime}_{\infty}\in T_{\ell}(A/D)$.

Let $\mathcal{G}_{\ell}$ be the image of the natural continuous representation

$\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}(V_{\ell}(A/D)).$

For all $n$, we have444By abuse of notation we still denote by
$\pi_{\mathfrak{p}}$ the reduction mod $\mathfrak{p}$ map
$A/D(\overline{K})\to\mathcal{A}_{\mathfrak{p}}/D_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})})$,
induced by the map
$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\kappa(\mathfrak{p}))$.

(4.1) $\\#\\{(\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot
a_{n}^{\prime}))\cap\mathfrak{B}/D_{\mathfrak{p}}\\}\geq\frac{1}{C}\\#\\{\operatorname{Gal}(\overline{K}/K)\cdot
a^{\prime}_{n}\\}.$

We define the set

$\Sigma_{n}:=\\{\sigma\in\mathcal{G}_{\ell}:\pi_{\mathfrak{p}}(\sigma\cdot
a^{\prime}_{n})\in\mathfrak{B}/D_{\mathfrak{p}}\\}.$

For any sequence $(\sigma_{n})_{n}\in\mathcal{G}_{\ell}$ such that
$\sigma_{n}\in\Sigma_{n}$ for every $n$, for any limit of any subsequence (for
the $\ell$-adic topology) $\sigma_{\infty}$ we have, along such subsequence

$\sigma_{n}\cdot a^{\prime}_{n}\to\sigma_{\infty}\cdot a^{\prime}_{\infty}.$

As $\pi_{\mathfrak{p}}(\sigma\cdot a^{\prime}_{n})$ belongs to
$\mathfrak{B}/D_{\mathfrak{p}}$, we see that

$\sigma_{\infty}\cdot a^{\prime}_{\infty}\in
V_{\ell}(\mathfrak{B}/D_{\mathfrak{p}})\subseteq V_{\ell}(A/D),$

where we identify $V_{\ell}(A/D)$ with
$V_{\ell}(\mathcal{A}_{\mathfrak{p}}/D_{\mathfrak{p}})$ (being $p$ different
from $\ell$). We define $\Sigma_{\infty}$ to be the set of the
$\sigma_{\infty}\in\mathcal{G}_{\ell}$ obtained as above.

Let $\mu$ be the Haar probability measure on $\mathcal{G}_{\ell}$. From the
equation (4.1) we have $\mu(\Sigma_{n})$ has Haar measure at least $(1/C)$,
for every $n$. In particular, we have that $\Sigma_{\infty}$ has measure at
least $1/C$. Let $H_{\ell}/\mathbb{Q}_{\ell}$ be the Zariski closure of
$\mathcal{G}_{\ell}$ in $\operatorname{GL}(V_{\ell}(A/D))$ and $H_{\ell}^{0}$
its Zariski neutral component. The Zariski closure of $\Sigma_{\infty}$
contains one irreducible component $H_{\ell}^{0}\cdot\sigma_{0}$, for some
$\sigma_{0}\in\Sigma_{\infty}$. Let $U$ be the sub-vector space of
$V_{\ell}(A/D)$ given by $V_{\ell}(\mathfrak{B}/D_{\mathfrak{p}})$, and $W$ be
the $\mathbb{Q}_{\ell}$-subvector space generated by
$H_{\ell}^{0}(\mathbb{Q}_{\ell})\cdot\sigma_{0}\cdot a^{\prime}_{\infty}$.

By Zariski density, we have that
$H_{\ell}^{0}(\mathbb{Q}_{\ell})\cdot\sigma_{0}\cdot
a^{\prime}_{\infty}\subset U$. Notice that, by construction, $W$ is
$H_{\ell}^{0}$ invariant. Let $\mathcal{G}_{\ell}^{0}=\mathcal{G}_{\ell}\cap
H_{\ell}^{0}$. It is a finite index subgroup of $\mathcal{G}_{\ell}$ which
corresponds to a finite extension of fields $K_{\ell}/K$. We apply Proposition
4.2.1, with $E=\mathbb{Q}_{\ell}$-coefficients to $W$ and $U$. So we get an
abelian subvariety $A^{\prime\prime}/K_{\ell}\subseteq A/D\times K_{\ell}$
such that

(4.2) $V_{\ell}(A^{\prime\prime})\subseteq
V_{\ell}(\mathfrak{B}/D_{\mathfrak{p}}),$

and

$\operatorname{Gal}(\overline{K}/K_{\ell})\cdot\sigma\cdot
a_{\infty}^{\prime}\subset W\subseteq V_{\ell}(A^{\prime\prime}).$

Thanks to Lemma 4.2.4, we have that the same holds true for some abelian
subvariety $A^{\prime}/K\subseteq A/D$. By definition of $D$, thanks to (4.2),
we see that $A^{\prime}=0$, but since $0\neq a^{\prime}_{\infty}\in
V_{\ell}(A^{\prime})$, we must also have that $A^{\prime}\neq 0$. This is the
contradiction we were aiming for. ∎

###### Proof in Case (2).

In this case, we set

$a_{n}^{\prime}:=[d_{n}^{\prime}/\ell_{n}]a_{n}+D/D,$

which is an element of order exactly $\ell_{n}$ of $A/D$. Consider the ring

$R:=\prod_{n\geq 0}\mathbb{F}_{\ell_{n}}.$

The sequence $(a_{n}^{\prime})_{n}$ is naturally an element of
$T_{\widehat{\mathbb{Z}}}(A/D)\otimes R$, where we denote by
$T_{\widehat{\mathbb{Z}}}(A/D)$ the adelic Tate module of $A/D$ (which is
denoted from now on by $V_{E}$, the $E$-adic Tate module of $A/D$). By
assumption, we know that the order of $a_{n}^{\prime}$ is going to infinity,
as $n\to+\infty$, and in particular $(a_{n}^{\prime})_{n}$ is not a torsion
element. The image of $(a_{n}^{\prime})_{n}$ in

$T_{\widehat{\mathbb{Z}}}(A/D)\otimes
R\otimes\mathbb{Q}\cong(R\otimes\mathbb{Q})^{2\dim(A/D)}$

is non-zero. There exists a field $E$ and a ring map $R\otimes\mathbb{Q}\to E$
such that the image $a^{\prime}_{\infty}\in
T_{\widehat{\mathbb{Z}}}(A/D)\otimes E$ is non-zero.

Notice that, for every $n$,

$\\#\\{(\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot
a^{\prime}_{n})\cap\mathfrak{B}/D_{\mathfrak{p}})\\}\geq\frac{1}{C}\\#\\{\operatorname{Gal}(\overline{K}/K)\cdot
a_{n}^{\prime}\\}.$

Let $G_{\ell,n}$ be the image of $\operatorname{Gal}(\overline{K}/K)$ in
$\operatorname{GL}(A/D[\ell_{n}])$. We can apply the combinatorial result of
Section 4.3.2 (with $V=\mathfrak{B}/D_{\mathfrak{p}})\\}[\ell_{n}]$) to
guarantee the existence a subgroup $H_{\ell,n}$ of $G_{\ell,n}$, and a
$\sigma_{n}\in\operatorname{Gal}(\overline{K}/K)$, such that

$[G_{\ell,n}:H_{\ell,n}]\leq\psi(C)=3\cdot C^{4^{2\dim A/D}},$

and

$\pi_{\mathfrak{p}}(H_{\ell,n}\cdot\sigma_{n}\cdot
a_{n}^{\prime})\subset\mathfrak{B}/D_{\mathfrak{p}}.$

Let $K_{H_{\ell,n}}/K$ be the finite extension of fields corresponding to
$H_{\ell,n}$, and notice that $[K_{H}:\mathbb{Q}]\leq\psi(C)[K:\mathbb{Q}]$.
Let $M$ be the constant appearing in Theorem 4.1.3, for
$K_{H_{\ell,n}}$-abelian variety $A/D$, and notice that $M$ is bounded
independently of $n$. Upon extraction, we may assume that $\ell_{n}$ is
strictly bigger than $M$ for every $n$. Let $W_{n}$ be the
$\mathbb{F}_{\ell_{n}}$-subvector space of $A/D[\ell_{n}]$ generated by
$H_{\ell,n}\cdot\sigma_{n}\cdot a^{\prime}_{n}$. Notice that $W_{n}$ is
invariant under the action of
$\operatorname{Gal}(\overline{K}/K_{H_{\ell,n}})$, and so by Theorem 4.1.3,
there exists an element
$f\in\operatorname{End}(A)\otimes\mathbb{F}_{\ell_{n}}$, such that

$W_{n}=f(A[\ell_{n}]).$

Being $K$ such that all endomorphism of $A_{\overline{K}}$ are defined over
$K$, we see that $W_{n}$ is actually
$\operatorname{Gal}(\overline{K}/K)$-invariant. So we have, for every $n$,

$\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot\sigma_{n}\cdot
a_{n}^{\prime})\subset\mathfrak{B}/D_{\mathfrak{p}}.$

In particular we see that

$\operatorname{Gal}(\overline{K}/K)\cdot\sigma_{\infty}\cdot
a^{\prime}_{\infty}\subset V_{E}(\mathfrak{B}/D_{\mathfrak{p}})\subset
V_{E}(A/D).$

We now conclude as in the previous case. Let indeed $W$ be the $E$-subvector
space generated by $\operatorname{Gal}(\overline{K}/K)\cdot
a^{\prime}_{\infty}$. We apply Proposition 4.2.3, with $E$-coefficients to $W$
and
$u\in\operatorname{End}(\mathcal{A}_{\mathfrak{p}}/D_{\mathfrak{p}})\otimes\mathbb{Q}$
an idempotent corresponding to
$\mathfrak{B}/D_{\mathfrak{p}}\subset\mathcal{A}_{\mathfrak{p}}/D_{\mathfrak{p}}$.
So we get an abelian subvariety $A^{\prime}\subseteq A/D$ such that

(4.3) $V_{E}(A^{\prime})\subseteq V_{E}(\mathfrak{B}/D_{\mathfrak{p}}),$

and

$\operatorname{Gal}(\overline{K}/K)\cdot a_{\infty}^{\prime}\subset W\subseteq
V_{E}(A^{\prime}).$

By definition of $D$, thanks to (4.3), we see that $A^{\prime}=0$, but since
$0\neq a^{\prime}_{\infty}\in V_{E}(A^{\prime})$, we must also have that
$A^{\prime}\neq 0$. This is the contradiction we were aiming for. ∎

The proof of Proposition 4.3.1 is finally completed. ∎

####  4.3.1. Proof of Theorem 4.0.1

At least assuming the combinatorial result of Section 4.3.2, we are ready to
prove the main Theorem of the section, namely Theorem 4.0.1. Let $E_{n}$ be a
sequence of special zero cycles in $\mathcal{A}$, each one associated to some
torsion point $a_{n}$ of order coprime to $p$. We may assume that the
$a_{n}$’s are all distinct, and of order $>1$. In particular
$\mathrm{ord}(a_{n})$ is unbounded, and we will only consider the positive
dimensional components of the Zariski closure of the $E_{n}$’s. We can apply
the weaker Theorem 1.2.1 to the Zariski closure of the $0$-cycles $E_{n}$. It
is of the form

$\bigcup_{i=1}^{C}\alpha_{i}+\mathfrak{B}_{i},$

for some positive integer $C$ denoting the number of components, where
$\mathfrak{B}_{i}$ are abelian sub-varieties of $\mathcal{A}_{\mathfrak{p}}$,
and $\alpha_{i}$ torsion points. Let $A_{i}$ be the biggest special subvariety
of $A$ whose reduction mod $\mathfrak{p}$ is in $\mathfrak{B}_{i}$. Applying
Proposition 4.3.1 we have that each $a_{n}$ lies in the union of the $A_{i}$,
up to a finite set $F$. We have

$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot
a_{n})\subset
F+\bigcup_{i=1}^{C}A_{i,\mathfrak{p}}\subset\bigcup_{i=1}^{C}\alpha_{i}+\mathfrak{B}_{i}.$

Being $\bigcup_{i=1}^{C}\alpha_{i}+\mathfrak{B}_{i}$ the Zariski closure of
the $E_{n}$, we conclude. The Proof of Theorem 4.0.1 is finished and we
observe that

$F+\bigcup_{i=1}^{C}A_{i,\mathfrak{p}}=\bigcup_{i=1}^{C}\alpha_{i}+\mathfrak{B}_{i}.$

####  4.3.2. Combinatorial result

Let $\ell$ be a prime number, set $\mathbb{F}:=\mathbb{F}_{\ell}$ for the
field with $\ell$ elements, and let $d$ be a positive integer.

######  Proposition 4.3.2.

Let $\mathbb{F}^{d}$ be a $d$-dimensional $\mathbb{F}$-vector space and let
$a\in\mathbb{F}^{d}$. Let $G\leq\operatorname{GL}_{d}(\mathbb{F})$ be a
subgroup. If $V\subseteq\mathbb{F}^{d}$ is a sub-vector space such that, for
some $1\leq C\leq\infty$,

$\\#\\{G\cdot a\cap V\\}\geq\frac{1}{C}\\#\\{G\cdot a\\},$

then there exists a non trivial sub-vector space $W\subseteq V$ such that

(4.4) $[G:\operatorname{Stab}_{G}(W)]\leq 3\cdot C^{4^{\dim(V)}}.$

Moreover $G\cdot a\cap W\neq\emptyset$ and it generates $W$.

######  Corollary 4.3.3.

With the notation of Proposition 4.3.2, in particular $W$ and $V$ contain the
orbit $H\cdot g\cdot a$, for some $g\in G$, under the subgroup
$H=\operatorname{Stab}_{G}(W)\subseteq G$, and $[G:H]\leq 3\cdot C^{4^{\dim
V}}$.

The proof is inspired by the arguments of [4, Section 3].

###### Proof of Proposition 4.3.2.

Given a subset $S\subseteq G\cdot a$ and $W\subseteq\mathbb{F}^{d}$, we set

${\varepsilon}(S):=\frac{\\#S}{\\#G\cdot
a},\qquad{\varepsilon}(W):={\varepsilon}(G\cdot a\cap W)=\frac{\\#\\{G\cdot
a\cap W\\}}{\\#G\cdot a}.$

Write $\langle S\rangle=\sum_{s\in S}\mathbb{F}\cdot s$ for the sub-vector
space of $\mathbb{F}^{d}$ generated by $S$. Notice that
$0\leq{\varepsilon}(W)\leq 1$.

We argue considering subvector spaces $W\subseteq V$ chosen to be of minimal
dimension among all sub-vector spaces of $V$ and maximizing the quantity

${\varepsilon}(W)^{4^{(\dim(W)-\dim(V))}}.$

Such $W$ exists since there are only finitely many possibilities for
${\varepsilon}(W)$ and $\dim(W)$. In particular we have

(4.5)
${\varepsilon}(W)^{4^{(\dim(W)-\dim(V))}}\geq{\varepsilon}(V)^{4^{0}}>0,$

and

(4.6)
${(1/{\varepsilon}(W))}\leq{(1/{\varepsilon}(V))}^{4^{(\dim(V)-\dim(W))}}\leq{(1/{\varepsilon}(V))}^{4^{\dim(V)}}\leq
C^{4^{\dim(V)}}.$

Comparing $W$ to all its sub-vector spaces $W^{\prime}<W$, we also have

(4.7)
${\varepsilon}(W^{\prime})<{\varepsilon}(W)^{4^{(\dim(W)-\dim(W^{\prime}))}}\leq{\varepsilon}(W)^{4}\leq{\varepsilon}(W).$

We consider $S:=W\cap(G\cdot a)$, and notice that it generates $W$, since
otherwise $W_{S}=\langle S\rangle$ would contradict the inequality (4.7). The
following simple observations will be useful for the sequel:

  * •

If $gW=W$, then $gS=g({G}\cdot a\cap W)=g{G}\cdot a\cap gW={G}\cdot a\cap
W=S$;

  * •

Reciprocally, if $gS=S$ then $gW=g\langle S\rangle=\langle gS\rangle=\langle
S\rangle=W$.

Denote by $G(W)\subseteq G$ the stabiliser of $W$ in $G$ (which is also the
stabiliser of $S$). If $gS\neq S$, the vector space $W^{\prime}:=gW\cap W$
satisfies $W^{\prime}<W$, as well as

$gS\cap S\subseteq gW\cap W<W.$

The inequality (4.7) then gives

$\frac{\\#gS\cap S}{\\#G\cdot a}\leq\frac{\\#(G\cdot a\cap
W^{\prime})}{\\#G\cdot a}={\varepsilon}(W^{\prime})<{\varepsilon}(W)^{4}.$

More generally, given $g\cdot G(W)\neq g^{\prime}\cdot G(W)$, we similarly
find

(4.8) $\frac{\\#gS\cap g^{\prime}S}{\\#G\cdot
a}\leq\frac{\\#g^{\prime-1}gS\cap S}{\\#G\cdot a}<{{\varepsilon}(W)}^{4}.$

To ease the notation, from now on, we simply set
${\varepsilon}={\varepsilon}(W)$. Let $c=[G:G(W)]$ and choose some
representative $g_{1}\cdot G(W),\ldots,g_{c}\cdot G(W)$ of $G/G(W)$. We are
left to prove that $c\leq 3\cdot 1/{\varepsilon}$.

###### Proof of the first bound for $c$.

If $c>3\cdot 1/{\varepsilon}$, we set $\gamma=\lceil 1/{\varepsilon}+1\rceil$.
Recall that $0<{\varepsilon}\leq 1$, and the evident bounds

(4.9) $\displaystyle 1/{\varepsilon}+1\leq\gamma$
$\displaystyle<1/{{\varepsilon}}+2$ (4.10) $\displaystyle\gamma$
$\displaystyle\leq c$

(since $2\leq 2\cdot(1/{\varepsilon})$). We can also notice that

(4.11) $\displaystyle\gamma\cdot{\varepsilon}$
$\displaystyle\geq(1/{{\varepsilon}}+1)\cdot{{\varepsilon}}=1+{{\varepsilon}}$
(4.12) $\displaystyle\frac{\gamma(\gamma-1)}{2}\cdot{{\varepsilon}}^{4}$
$\displaystyle<\frac{(1/{{\varepsilon}}+1)\cdot
1/{{\varepsilon}}}{2}\cdot{{\varepsilon}}^{4}=\frac{{{\varepsilon}}^{2}+{{\varepsilon}}^{3}}{2}\leq{{\varepsilon}}^{2}.$

Thanks to (4.10), the classes $g_{1}\cdot G(W),\ldots,g_{\gamma}\cdot G(W)$
are well defined and all distinct. Thanks to inclusion-exclusion we estimate:

(4.13) $\\#G\cdot
a\geq\\#\bigcup_{i=1}^{\gamma}g_{i}S\geq\sum_{i=1}^{\gamma}\\#g_{i}S-\sum_{1\leq
i<j\leq\gamma}\\#(g_{i}S\cap g_{j}S).$

The number of terms of the last summation is $(\gamma(\gamma-1))/2$, and each
term is bounded, via (4.8), by ${{\varepsilon}}^{4}\cdot\\#G\cdot a$.
Therefore

(4.14) $\sum_{1\leq i<j\leq\gamma}\\#\\{g_{i}S\cap
g_{j}S\\}\leq\frac{\gamma(\gamma-1)}{2}\cdot{{\varepsilon}}^{4}\cdot\\#G\cdot
a$

For the second to last summation in (4.13), we notice that

(4.15) $\sum_{i=1}^{\gamma}\\#g_{i}S=\gamma\cdot{\varepsilon}\cdot\\#G\cdot
a\geq(1/{\varepsilon}+1)\cdot{\varepsilon}\cdot\\#G\cdot
a=(1+{\varepsilon})\cdot\\#G\cdot a.$

Substituting (4.15) and (4.14) in (4.13), and simplifying by $\\#G\cdot a>0$,
we obtain the inequality

$1>1+{\varepsilon}-{\varepsilon}^{2},$

which is the contradiction we were aiming for, since the right hand side
satisfies $1+{\varepsilon}-{\varepsilon}^{2}\geq 1$.∎

To conclude the proof, we are left to show that

$c=[G:G(W)]\leq 3/{\varepsilon}.$

Bounding the right hand side by three times the inequality (4.6), we obtain
the desired bound (4.4). We know that ${\varepsilon}(W)>0$, thanks (4.5), and
that there exists an element $g\cdot a$ of $G\cdot a$ in $W$. This concludes
the proof of Proposition 4.3.2. ∎

##  5\. Purely $p$-adic phenomena, Theorem 1.2.6

In this final section we prove Theorem 1.2.6, which, for the convenience of
the reader, is recalled below. The most general result which combines Theorem
1.2.3 and Theorem 1.2.6 is also proven here as Corollary 5.2.2.

######  Theorem 5.0.1.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$ of residue characteristic
$p$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme,
$(a_{n})_{n}\subset A_{\operatorname{tors}}$ be a sequence of torsion points
(possibly of order divisible by $p$), and write
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$, for
a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Then every component $\alpha+\mathfrak{B}$ of the Zariski closure
$\mathcal{V}$ of

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}_{\mathfrak{p}}\subset\mathcal{A}$

is such that there exists $B\subseteq A$ an abelian subvariety such that
$B_{\mathfrak{p}}$ contains $\mathfrak{B}$ and the quotient
$B_{\mathfrak{p}}/\mathfrak{B}$ is of $p$-rank zero.

The key input needed to adapt the strategy of the proof of Theorem 4.0.1 to
the above, is the following $p$-adic version of the arguments of Section 4.2.
Compared to the $\ell$-adic argument, the crux is to have some control on the
reduction map from $T_{p}(A)$ to $T_{p}(\mathcal{A}_{\mathfrak{p}})$ and to
find a faithful representation of the endomorphism algebra of $A$ and
$\mathcal{A}_{\mathfrak{p}}$. All this is summarised in the statement of
Proposition 5.3.1, which we first use as a black box, and then explain its
proof (in the very end of the section).

######  Theorem 5.0.2.

Let $W$ be a $\operatorname{Gal}(\overline{K}/K)$-invariant subspace of
$V_{p}(A)$ and denote by $W_{\mathfrak{p}}$ its image in
$V_{p}(\mathcal{A}_{\mathfrak{p}})$. If there exists
$u\in\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes\mathbb{Q}$ such
that

$W_{\mathfrak{p}}\subseteq u\cdot V_{p}(\mathcal{A}_{\mathfrak{p}}),$

then there is an idempotent $v$ in $\operatorname{End}(A)\otimes\mathbb{Q}$
such that

$W_{\mathfrak{p}}\subseteq v\cdot V_{p}(\mathcal{A}_{\mathfrak{p}})\subseteq
u\cdot V_{p}(\mathcal{A}_{\mathfrak{p}}).$

In particular, if the idempotent
$u\in\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes\mathbb{Q}$
corresponds to an abelian subvariety
$\mathfrak{B}\subset\mathcal{A}_{\mathfrak{p}}$, then $v$ corresponds to an
abelian subvariety $B\subseteq A$ with the following property:
$B_{\mathfrak{p}}$ contains $\mathfrak{B}$,
$B_{\mathfrak{p}}[p^{\infty}]=\mathfrak{B}[p^{\infty}]$, and the quotient
abelian variety $B_{\mathfrak{p}}/\mathfrak{B}$ is of $p$-rank zero.

###  5.1. Proof of Theorem 5.0.2

There exists a finite extension $K^{\prime}/K$ such that all endomorphisms of
$A\times\overline{K}$ are defined over $K^{\prime}$. In first approximation we
may replace $K$ by $K^{\prime}$ (and we will find an abelian subvariety of
$A_{K^{\prime}}$, rather than $A_{K}$). We explain at the end of the proof how
to get rid of the extension $K^{\prime}/K$, to find the desired $B$ in $A$,
using Lemma 4.2.4. The place $\mathfrak{p}$ is a place of good reduction for
$A$, and, we now have

$\operatorname{End}_{K^{\prime}}(A_{K^{\prime}})=\operatorname{End}_{K^{\prime}_{\mathfrak{p}}}(A_{K^{\prime}_{\mathfrak{p}}}).$

As a $\mathbb{Q}_{p}$-vector spaces of dimension $2g$, we may also naturally
identify $V_{p}(A_{K^{\prime}})$ with $V_{p}(A_{K^{\prime}_{\mathfrak{p}}})$,
via the map induced by
$\overline{K^{\prime}}=\overline{K}\to\overline{K^{\prime}_{\mathfrak{p}}}$.
We now use some facts that will be explained in details in Section 5.3 (where
we let $L$ to be $K^{\prime}_{\mathfrak{p}}$). Set $E/\mathbb{Q}$ be the field
extension of Proposition 5.3.1, and consider

$V:=V_{p}(A_{K^{\prime}})\otimes_{\mathbb{Q}_{p}}E,$

with its natural action of
$\operatorname{End}(A_{K^{\prime}})\otimes\mathbb{Q}_{p}$ and the
$\rho$-action of $\operatorname{End}(A_{k})\otimes\mathbb{Q}_{p}$ coming from
(1) of Proposition 5.3.1.

We can consider the element

(5.1)
$\pi\in\left(\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes\overline{\mathbb{Q}}\right)\cap\left(\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes\mathbb{Q}_{p}\right),$

defined in (2) of Proposition 5.3.1. By construction, $\pi$ is a central
element (it is a $(\mathbb{Q}_{p}\cap\overline{\mathbb{Q}})$-combination of
powers of the Frobenius endomorphism
$F_{A,\mathfrak{p}}\in\operatorname{End}(\mathcal{A}_{\mathfrak{p}})$). We
have that $W\otimes E=w\cdot V$, and that $\rho(u)\cdot V$ is related to the
abelian variety $\mathfrak{B}$. The key point of Proposition 5.3.1 is that

(5.2) $\rho(\pi)(V)=V^{r}\otimes E$

where $V^{r}$ denotes the kernel of the reduction map
$V_{p}(A_{K^{\prime}})\to V_{p}(A_{\kappa(\mathfrak{p})})$. By assumption $W$,
modulo $V^{r}$, is in the naïve Tate module of $\mathfrak{B}$, which is
$u\cdot V_{p}(A_{\kappa(\mathfrak{p})})$. Thanks to (3) of Proposition 5.3.1,
we may identify the $W\otimes E$, modulo $V^{r}\otimes E$, with $\rho(u)\cdot
V$.

Thanks to the definition of $\pi$ from (5.1), and (5.2), we have an inclusion,
modulo $V^{r}$:

(5.3) $\operatorname{Im}(w)\otimes E\subseteq\operatorname{Im}(\rho(u)).$

We can now apply Proposition 5.1.1 (which is proven below): there exists an
idempotent $v\in\operatorname{End}(A_{K^{\prime}})\otimes\mathbb{Q}$ such that

$w\cdot V+V^{r}\otimes E\subseteq v\cdot V+V^{r}\otimes E\subseteq\rho(u)\cdot
V+V^{r}\otimes E.$

The image of $v$ gives, up to a quasi-isogeny, the desired abelian subvariety
$B\subset A/K^{\prime}$: $B_{\mathfrak{p}}$ contains $\mathfrak{B}$, and by
construction $B_{\mathfrak{p}}[p^{\infty}]=\mathfrak{B}[p^{\infty}]$.
Equivalently the quotient abelian variety $B_{\mathfrak{p}}/\mathfrak{B}$ is
of $p$-rank zero.

To conclude the proof, we have to show how to get rid of the extension
$K^{\prime}/K$. Indeed assume that there is a finite extension $K^{\prime}/K$
(which we may assume to be Galois), a $K^{\prime}$ abelian subvariety
$B_{K^{\prime}}$ of $A_{K^{\prime}}$ such that
$\mathfrak{B}_{\kappa(\mathfrak{p^{\prime}})}$ contains
$B_{\mathfrak{p^{\prime}}}=B_{K^{\prime}}\otimes\kappa(\mathfrak{p^{\prime}})$
for some prime $\mathfrak{p^{\prime}}$ of $\mathcal{O}_{K^{\prime}}$ above
$\mathfrak{p}\subset\mathcal{O}_{K}$ and, up to a finite subgroup,

$\pi_{\mathfrak{p^{\prime}}}(\operatorname{Gal}(\overline{K}/K^{\prime}).a_{n})\subseteq
B_{\mathfrak{p^{\prime}}}\subseteq\mathfrak{B}\otimes_{\kappa(\mathfrak{p})}{\kappa(\mathfrak{p^{\prime}})}.$

Here we use that $E_{n}$ were actually
$\operatorname{Gal}(\overline{K}/K)$-orbits, to deduce the stronger version.
Along the map $K\to K^{\prime}$, we can identify, as $\mathbb{Q}_{p}$-vector
spaces $V_{p}(A)$ and $V_{p}(A_{K^{\prime}})$ for some prime $\ell$ (possible
even $\ell=p$). Consider

$\operatorname{End}(A_{K})\otimes\mathbb{Q}_{p}\subseteq\operatorname{End}(A_{K^{\prime}})\otimes\mathbb{Q}_{p}\subseteq\operatorname{End}_{\mathbb{Q}_{p}}(V_{p}(A)).$

Lemma 4.2.4, which is just a simple application of Lemma 4.2.2, produces the
desired conclusion (with $E=\mathbb{Q}_{p}$).

####  5.1.1. Variant of Lemma 4.2.2

The variant of Lemma 4.2.2 needed for the previous argument is the following.

######  Proposition 5.1.1.

Let $M,N$ be two semisimple finite dimensional $\mathbb{Q}$-algebras, with $M$
a sub-algebra of $N$. Let $E/\mathbb{Q}$ be a field extension, $\pi\in
N\otimes E$ a central idempotent, and

$\left(N\otimes_{\mathbb{Q}}E\right)\hookrightarrow\operatorname{End}_{E}(V)$

be a faithful representation in $V$ a finite dimensional $E$ linear vector
space. Let $u$ be an idempotent of $N$, and $w$ be an idempotent of
$M\otimes_{\mathbb{Q}}E$ such that

$w\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot V.$

Then there exists an idempotent $v\in M$, such that

$w\cdot V+\pi\cdot V\subseteq v\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot
V.$

######  Remark 5.1.2.

Let $B$ be a semisimple algebra, and $\pi$ be a _central_ idempotent. Then we
can write, as $\mathbb{Q}$-algebras

$B=\pi B\oplus(1_{B}-\pi)B.$

Let $I$ be a left (resp. right) ideal of $B$ containing $\pi$, we may write

$I=\pi B\oplus I^{\prime},$

for $I^{\prime}:=(1_{B}-\pi)I$. So we have a correspondence between left
(resp. right) ideals of $B$ containing $\pi$, and left (resp. right) ideals of
$(1_{B}-\pi)B$.

For the proof of Proposition 5.1.1, we need two lemmas (the first being a
simple fact in the theory of semisimple algebras).

######  Lemma 5.1.3.

Let $M\subset B$ be a semisimple algebras, $\pi$ a central idempotent such
that $B=M[\pi]$. Then $M$ surjects onto $B/(\pi)$, and every idempotent of
$B/(\pi)$, comes from an idempotent of $M$.

######  Lemma 5.1.4.

Let $M,B,\pi$ be as in Lemma 5.1.3, and $V$ be a $\mathbb{Q}$-vector space
with a faithful action of $B$. Let $u$ be an idempotent of $B$, and consider
the ideal of $B$

$I:=uB+\pi B\subset B.$

An element $b\in B$ belongs to $I$ if and only if

$b\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot V.$

###### Proof.

We need only to prove $\Leftarrow$, the other implication being evident. Let
$b\in B$ be such that $b\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot V$. We
write

$b=\pi b+(1-\pi)b.$

We also have an induced decomposition on $V$:

$V=\ker(\pi\cdot)\oplus\operatorname{Im}(\pi\cdot).$

As $u$ and $b$ commute with $\pi$, they are compatible with the above direct
sum decomposition, that is $u$ acts on $\ker(\pi\cdot)$ as $(1-\pi)u$ and on
$\operatorname{Im}(\pi\cdot)$ as $\pi u$, and similarly for $b$. We have

$\operatorname{Im}((1-\pi)b)\oplus\operatorname{Im}(\pi)=\operatorname{Im}(b)+\operatorname{Im}(\pi)\subseteq\operatorname{Im}(u)+\operatorname{Im}(\pi)=\operatorname{Im}((1-\pi)u)\oplus\operatorname{Im}(\pi).$

Hence $\operatorname{Im}((1-\pi)b)\subseteq\operatorname{Im}((1-\pi)u)$. But
$(1-\pi)u$ is unipotent, by Lemma 4.2.2, $(1-\pi)b$ belongs to $(1-\pi)uB$,
say

$(1-\pi)b=(1-\pi)ux,$

for some $x\in B$. Finally $b=\pi b+(1-\pi)b=\pi b+(1-\pi)ux=\pi
b+u(1-\pi)x\in I$. ∎

###### Proof of Proposition 5.1.1.

We first consider

$X_{\pi}:=(uN[\pi]+\pi N[\pi])\cap(M[\pi]),$

where $u\in N$ acts diagonally on $N[\pi]$. It is a right ideal of $M[\pi]$,
and as such it is generated by an idempotent $m=m^{2}$ of $M[\pi]$ (as
recalled before the proof of Lemma 4.2.2):

$X_{\pi}=mM[\pi].$

Notice that $X_{\pi}$ contains $(\pi)$, and so, by Lemma 5.1.3 (applied to
$B=M[\pi]$), we can write $X_{\pi}=vM[\pi]+\pi M[\pi]$, where $v$ is an
idempotent of $M$ such that

$v\mod(\pi)=m\mod(\pi),$

that is $v+(\pi)=m+(\pi)$. Hence $mM[\pi]=mM[\pi]+\pi mM[\pi]=vM[\pi]+\pi
M[\pi]$.

Notice that

$X_{\pi}\otimes E=((uN[\pi]+\pi N[\pi])\otimes E)\cap(M[\pi]\otimes E).$

We may apply Lemma 5.1.4 to $b=v\in I=uN[\pi]\otimes E+\pi N[\pi]\otimes E$,
to get

$v\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot V.$

Again applying Lemma 5.1.4 $w\in uN[\pi]\otimes E+\pi N[\pi]\otimes E$, since
we know that

$w\cdot V+\pi\cdot V\subseteq u\cdot V+\pi\cdot V.$

By construction $w\in M[\pi]\otimes E$, so $w\in X_{\pi}\otimes E$, and we can
write

$w\cdot V+\pi\cdot V\subseteq v\cdot V+\pi\cdot V.$

∎

###  5.2. Proof of Theorem 1.2.6

The following is a version of Proposition 4.3.1 that works also for points of
order divisible by $p$, the main difference being in the definition of the
$D_{i}$. Here we denote by $\operatorname{red}$ the map

$\operatorname{red}:T_{p}(A)\to T_{p}(\mathcal{A}_{\mathfrak{p}}).$

######  Proposition 5.2.1.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$, of residue characteristic
$p>0$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme, and fix a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

For $i=1,\dots,C$, let $\mathfrak{B}_{i}\subset\mathcal{A}_{\mathfrak{p}}$ be
abelian subvarieties of $\mathcal{A}_{\mathfrak{p}}$. Let $D_{i}/K$ be the
biggest abelian subvariety of $A$ with the following property:

$\operatorname{red}T_{p}(D_{i})\subseteq T_{p}(\mathfrak{B}_{i})$

Let $(a_{n})_{n}\subset A_{\text{tors}}$ be an arbitrary sequence of torsions
points, and write
$E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot a_{n})$. If

$E_{n}\subset\bigcup_{i=1}^{C}\mathfrak{B}_{i},$

then there exists a finite set $F\subset A$ such that for all $n$

$a_{n}\in F+\bigcup_{i=1}^{C}D_{i}.$

###### Proof.

Heading for a contraction, we may pass to a subsequence of $(a_{n})_{n}$ for
which, for every $i=1,\dots,C$, we have

$\mathrm{ord}(a_{n}+D_{i}/D_{i})\to+\infty,$

as $n\to+\infty$. We will construct some $i$ and a non zero abelian subvariety
$A^{\prime}\subset A/D_{i}$ such that
$A^{\prime}_{\mathfrak{p}}\subseteq\mathfrak{B}_{i}/D_{i,\mathfrak{p}}$. The
preimage of $A^{\prime}$ along the quotient map $A\to A/D_{i}$ would then
contradict the maximality of $D_{i}$.

For every $n$ there exists an $1\leq i\leq C$ such that

$\\#\\{\operatorname{Gal}(\overline{K}/K\cdot
a_{n})\cap\pi_{\mathfrak{p}}^{-1}(\mathfrak{B}_{i})\\}\geq\frac{1}{C}\\#\\{\operatorname{Gal}(\overline{K}/K)\cdot
a_{n}\\}.$

Upon extracting a subsequence, we may and do assume from now on that $i$ is
fixed. And for simplicity set $D=D_{i}$, $\mathfrak{B}=\mathfrak{B}_{i}$ and

$d_{n}=\mathrm{ord}(a_{n}),\text{ and
}d_{n}^{\prime}=\mathrm{ord}(a_{n}+D/D).$

By assumption we know that $d_{n}^{\prime}$ goes to infinity and, possibly
extracting a subsequence, at least one of the holds true:

  1. (1)

There exists a rational prime number $\ell(\neq p)$, and a sequence
$(x_{n})_{n}\subset\mathbb{N}$ going to infinity, as $n\to+\infty$, such that

$\forall n,\ell^{x_{n}}\text{ divides }d_{n}^{\prime}.$

  2. (2)

There exists an infinite sequence of distinct prime numbers $\ell_{n}(\neq
p)$, such that

$\forall n,\ell_{n}\text{ divides }d_{n}^{\prime}.$

  3. (3)

There exists a sequence $(k_{n})_{n}\subset\mathbb{N}$ going to infinity, as
$n\to+\infty$, such that

$\forall n,p^{k_{n}}\text{ divides }d_{n}^{\prime}.$

For the first two cases, arguing as in the proof of Proposition 4.3.1, we
obtain the contradiction we were aiming for (we actually have a stronger
conclusion, namely a true lift to characteristic zero, rather than a lift up
to $p$-rank zero factors). For the third case, we can argue in parallel with
the $\ell$-adic one. The only change needed is indeed to replace the reference
to Proposition 4.2.1 by Theorem 5.0.2 which was proven in the previous
section, and has the effect of producing a lift up to $p$-rank zero factors. ∎

As explained in Section 4.3.1, Theorem 1.2.6 easily follows by combining
Proposition 5.2.1 and the weak vertical Manin–Mumford (Theorem 1.2.1). The
arguments presented so far prove even the following stronger version.

######  Corollary 5.2.2.

Fix a prime $\mathfrak{p}\subset\mathcal{O}_{K,S}$ of residue characteristic
$p$. Let $\mathcal{A}/\mathcal{O}_{K,S}$ be an abelian scheme,
$(a_{n})_{n}\subset A_{\operatorname{tors}}$ be a sequence of torsion points,
and write $E_{n}=\pi_{\mathfrak{p}}(\operatorname{Gal}(\overline{K}/K)\cdot
a_{n})$, for a map

$\pi_{\mathfrak{p}}:A(\overline{K})\to\mathcal{A}_{\mathfrak{p}}(\overline{\kappa(\mathfrak{p})}).$

Then every component $\alpha+\mathfrak{B}$ of $\mathcal{V}$, the Zariski
closure of

$\bigcup_{n\geq 0}E_{n}\subset\mathcal{A}_{\mathfrak{p}}\subset\mathcal{A},$

is such that there exists $B\subseteq A$ an abelian subvariety with the
following property:

  * •

If the $p$ part of the order of the $a_{n}$ (or a subsequence thereof) is
bounded, then $B_{\mathfrak{p}}=\mathfrak{B}$;

  * •

Otherwise $B_{\mathfrak{p}}$ contains $\mathfrak{B}$, and the quotient
$B_{\mathfrak{p}}/\mathfrak{B}$ is of $p$-rank zero.

###  5.3. Some $p$-adic Hodge theory (for abelian varieties with good
reduction)

To conclude the paper we need to justify the steps of the proof of Theorem
5.0.2 that used some $p$-adic Hodge theory. The reader fluent in $p$-adic
Hodge Theory can simply read Proposition 5.3.1 and its proof given in Section
5.7. Here we try to make our argument as understandable as possible, by
recalling some standard facts in $p$-adic Hodge Theory.

####  5.3.1. Notation

Let $p$ be a rational prime number, $L$ be a finite extension of
$\mathbb{Q}_{p}$, $\mathcal{O}_{L}$ its ring of integers and
$k=\mathbb{F}_{q}=\mathbb{F}_{p^{f}}$ its residue field. Let
$\mathcal{A}/\mathcal{O}_{L}$ be an abelian scheme, of relative dimension
$g>0$, and denote its special and generic fibres respectively by $A_{k}$ and
$A_{L}$. We choose $\overline{L}$ an algebraically closed extension of $L$ and
denote $\overline{k}$ the corresponding residue field extension of $k$, so
that we can properly define the $\mathbb{Z}_{p}$-linear Tate modules:

(5.4)
$T_{p}(A_{k}):=\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},A_{k}[p^{\infty}](\overline{k}))\text{
and
}T_{p}(A_{L}):=\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},A_{L}[p^{\infty}](\overline{L}))$

and the reduction map of the statement below. (The former is the _naïve_ Tate
module accounting only for the $p$-_rank_ of $A$, whereas the second has rank
$2g$.)

We denote by $\overline{\mathbb{Q}}$ the algebraic closure of $\mathbb{Q}$
inside of $\overline{L}$, so that the field
$\overline{\mathbb{Q}}\cap\mathbb{Q}_{p}$ is properly defined.

We write the corresponding $\mathbb{Q}_{p}$-linear Tate modules as follows

$V_{p}(A_{k})=T_{p}(A_{k})\mathbin{\otimes}_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\simeq\operatorname{Hom}(\mathbb{Q}_{p},A_{k}[p^{\infty}](\overline{k})),$

and

$V_{p}(A_{L})=T_{p}(A_{L})\mathbin{\otimes}_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\simeq\operatorname{Hom}(\mathbb{Q}_{p},A_{L}[p^{\infty}](\overline{L})).$

We are interested in the reduction map $A_{L}(\overline{L})\to
A_{k}(\overline{k})$ induced by

$A_{L}(\overline{L})\xleftarrow{\sim}\mathcal{A}(\mathcal{O}_{\overline{L}})\xrightarrow{}A_{k}(\overline{k}).$

Passing to the Tate modules there is a reduction map

$T_{p}(A_{L})\to T_{p}(A_{k}),$

which is equivariant with respect to the covariant action of
$\operatorname{End}(A_{L})\subseteq\operatorname{End}(A_{\overline{L}})$
whereas

$\operatorname{End}(A_{k})\subseteq\operatorname{End}(A_{\overline{k}})$

acts a priori only on $T_{p}(A_{k})$.

Inside of $\operatorname{End}(A_{k})$ there is a specific element, the
Frobenius of $A_{k}$ relative to $k$,

(5.5) $\operatorname{Frob}_{A_{k}}:A_{k}\to A_{k}$

associated to the $q$-power map, where $q=\\#k$. On coordinates
$A(\overline{k})\to A(\overline{k})$ it acts with the Frobenius map

$\operatorname{Frob}_{\overline{k}/k}:\overline{k}\xrightarrow{x\mapsto
x^{q}}\overline{k}.$

(This is just the morphism of local ringed spaces given by the identity at the
topological level and the $|k|$-power at the level of structural sheaves.)

####  5.3.2. Purpose

In difference with the $\ell$-adic reduction map, for $\ell\neq p$, the
$p$-adic reduction map is never an isomorphism, unless $A=0$, which makes it
difficult to relate:

  * •

On the one hand our special $0$-cycles, which are related to abelian varieties
of the special fibres, and to the action of $\operatorname{End}(A_{k})$ on
$V_{p}(A_{k})$;

  * •

On the other hand the invariant subspaces of the global Galois group
$\operatorname{Gal}({\overline{\mathbb{Q}}/K})$ from which
$L=K_{\mathfrak{p}}$, which are related to the action of
$\operatorname{End}(A_{L})$ on $V_{p}(A_{L})$.

To top it all, we may have no possible action of $\operatorname{End}(A_{k})$
on $V_{p}(A_{L})$ and non-zero representation of
$\operatorname{Gal}({\overline{\mathbb{Q}}/K})$ on $V_{p}(A_{k})$.

####  5.3.3. Statement of the main result

The notations are those of Section 5.3.1, and set $V:=V_{p}(A_{L})$.

######  Proposition 5.3.1.

There exist a field extension $E$ of $L$, and a polynomial $Q(X)$ with
coefficients in $\overline{\mathbb{Q}}\cap\mathbb{Q}_{p}$ such that:

  1. (1)

The left action $\operatorname{End}(A_{L})$ on
$V\mathbin{\otimes}_{\mathbb{Q}_{p}}E$ extends to an $E$-linear left action

$\rho:\operatorname{End}(A_{k})\to\operatorname{End}_{E}(V\mathbin{\otimes}_{\mathbb{Q}_{p}}E);$

  2. (2)

The kernel of the reduction map, which we call _the ramified part of V_ , is
denoted by

$V^{r}:=\ker\left(V_{p}(A_{L})\to V_{p}(A_{k})\right),$

and the Frobenius endomorphism
$\operatorname{Frob}_{A_{k}}\in\operatorname{End}(A_{k})$ satisfy the identity
of $E$-linear subspaces

$\rho(\pi)(V\mathbin{\otimes}E)=V^{r}\mathbin{\otimes}E,$

where $\pi=Q(\rho(\operatorname{Frob}_{A_{k}}))$ is seen as an element of
$\operatorname{End}_{E}(V\mathbin{\otimes}_{\mathbb{Q}_{p}}E)$. Moreover $\pi$
is a central idempotent;

  3. (3)

Consider $V\otimes E$ modulo $V^{r}\otimes E$. Via $\rho$, it is an
$\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes E$-module, and it can
naturally be identified with the
$\operatorname{End}(\mathcal{A}_{\mathfrak{p}})\otimes E$-module
$V_{p}(\mathcal{A}_{\mathfrak{p}})\otimes E$ (where the action on the latter
is the natural one coming from the naïve $p$-adic Tate module of
$\mathcal{A}_{\mathfrak{p}}$, as an
$\operatorname{End}(\mathcal{A}_{\mathfrak{p}})$-module).

The proof appears in Section 5.7, after all required objects are introduced
and described (for example $E$ is one of Fontaine’s famous period rings).

####  5.3.4. Witt vectors notations

Two standard references for crystalline cohomology are [1] and [2]. We let
${}{\operatorname{W}(k)}{}$ denote the ring of Witt vectors with coefficients
in $k$ (and $L_{0}=\mathbb{Q}_{q}$ its fraction field), and
$F:=\operatorname{F}_{{}{\operatorname{W}(k)}{}},V:=\operatorname{V}_{{}{\operatorname{W}(k)}{}}\in\operatorname{End}_{W(\mathbb{F}_{p})}(W(k))$
the _Frobenius_ and _Verschiebung_ operators.

In practice: the first acts on the Witt coordinates by applying on each
coordinate the $p$-th power map

$\operatorname{Frob}_{k/\mathbb{F}_{p}}:k\xrightarrow{x\mapsto x^{p}}k;$

the second acts by shifting those coordinates.

Concretely the ring $\operatorname{W}(\mathbb{F}_{p})$ is $\mathbb{Z}_{p}$
defined in a schematic way, whereas the ring
$\operatorname{W}(\mathbb{F}_{q})$ is naturally isomorphic to
$\mathbb{Z}_{q}$, the ring of integers of an unramified extension
$\mathbb{Q}_{q}$ of $\mathbb{Q}_{p}$ with residue field $\mathbb{F}_{q}$. One
should know that the fraction field
$\operatorname{W}(k)\mathbin{\otimes}_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ is
naturally isomorphic to the maximal unramified extension $L_{0}$ of
$\mathbb{Q}_{p}$ contained in $L$.

Of relevance is the Dieudonné ring
${}{\operatorname{W}(k)}{}[\operatorname{F}_{{}{\operatorname{W}(k)}{}},\operatorname{V}_{{}{\operatorname{W}(k)}{}}]$
as the (non commutative) ${}{\operatorname{W}(k)}{}$-algebra of additive
endomorphisms of ${}{\operatorname{W}(k)}{}$ generated by
${}{\operatorname{W}(k)}{}$ acting by ring multiplication, and
$\operatorname{F}_{{}{\operatorname{W}(k)}{}}$ and
$\operatorname{V}_{{}{\operatorname{W}(k)}{}}$ just defined. One usually
recovers $V$ from $F\circ V=p=V\circ F$.

###  5.4. Crystalline-étale comparison theorem

On the category of abelian schemes $\mathcal{A}/\mathcal{O}_{L}$, we consider
the crystalline and $p$-adic étale cohomologies as contravariant functors

(5.6) ${\operatorname{H}}^{1}_{\operatorname{crys}}(A_{k})\text{ and
}{\operatorname{H}}^{1}_{\operatorname{\acute{e}t}}(A_{\overline{L}},\mathbb{Z}_{p}).$

The first is a free ${}{\operatorname{W}(k)}{}$-module of rank $2g$ functorial
in the special fibre $A_{k}$, and comes equipped with a
${}{\operatorname{W}(\mathbb{F}_{p})}{}$-linear endomorphism
$\phi\in\operatorname{End}_{{}{\operatorname{W}(\mathbb{F}_{p})}{}}({\operatorname{H}}^{1}_{\operatorname{crys}}(A_{k}))$
which is semi-linear with respect to
$\operatorname{F}_{{}{\operatorname{W}(k)}{}}$: mapping
$\operatorname{F}_{{}{\operatorname{W}(k)}{}}$ to $\phi$ makes
${\operatorname{H}}^{1}_{\operatorname{crys}}(A_{k})$ a
${}{\operatorname{W}(k)}{}[\operatorname{F}_{{}{\operatorname{W}(k)}{}},\operatorname{V}_{{}{\operatorname{W}(k)}{}}]$-module
which is free as a ${}{\operatorname{W}(k)}{}$-module.

The second is a free $\mathbb{Z}_{p}$-module of rank $2g$, and is functorial
in the generic fibre $A_{L}$, and depends upon a choice of algebraically
closed extension $\overline{L}/L$. It is in guise the usual Tate module, or
rather its dual.

The following comparison theorem holds for whichever of the Fontaine rings
$E=B_{\text{dR}}$ (and then generalises to varieties, according to Faltings)
or, sufficiently for abelian schemes, $E=B_{\operatorname{crys}}$. Those
constructions depend a priori on $L$ and $\overline{L}$, but the resulting
rings not, up to the relevant isomorphism. The construction of period rings,
with all details, can be found in [14]. We following is a specail case of the
so called $C_{\operatorname{crys}}$-theorem (for $H^{1}s$ and abelian
varieties with good reduction). Of course everything holds in a greater
generality, even if we limit our discuss to abelian varieties. The general
case is due to Faltings [10].

######  Theorem 5.4.1.

There is an field extension $E$ of the fraction field $L_{0}$ of $W(k)$ and a
_natural_ isomorphism

(5.7)
$\gamma_{\operatorname{crys}}(A):E\otimes_{W(k)}{\operatorname{H}}^{1}_{\operatorname{crys}}(A_{k})\xrightarrow{\sim}E\otimes_{\mathbb{Z}_{p}}{\operatorname{H}}^{1}_{\operatorname{\acute{e}t}}(A_{\overline{L}},\mathbb{Z}_{p}).$

_of functors_ from the category of smooth projective varieties over
$\mathcal{O}_{L}$ to that of vector spaces over $E$.

The next remark explains the reason why we need to extend to
$B_{\operatorname{crys}}$-coefficients rather than $\mathbb{C}_{p}$.

######  Remark 5.4.2.

Originally Tate [31] proved the so called Hodge-Tate decomposition for the
first étale cohomological group of abelian varieties. That is that the
existence of a canonical equivariant isomorphism

$H^{1}_{\text{et}}(A_{\overline{L}},\mathbb{Q}_{p})\otimes\mathbb{C}_{p}\cong
H^{1}(A,\mathcal{O}_{A})\otimes\mathbb{C}_{p}(-1)\oplus
H^{0}(A,\Omega^{1})\otimes\mathbb{C}_{p}.$

By the degeneration of the Hodge to de Rham spectral sequence in
characteristic zero, the RHS is isomorphic to $H^{1}_{dR}(A/\mathbb{C}_{p})$.
The problem is that such isomorphism is, in general, not canonical.

On the homology side, we get the covariant ${}{\operatorname{W}(k)}{}$-dual
and $\mathbb{Z}_{p}$-dual functors

${\operatorname{H}}_{1}^{\operatorname{crys}}(A_{k}):=\operatorname{Hom}_{\operatorname{W}(k)}({\operatorname{H}}^{1}_{\operatorname{crys}}(A_{k}),{}{\operatorname{W}(k)}{})\text{
and
}{\operatorname{H}}_{1}^{\operatorname{\acute{e}t}}(A_{\overline{L}}):=\operatorname{Hom}_{\mathbb{Z}_{p}}({\operatorname{H}}^{1}_{\operatorname{\acute{e}t}}(A_{\overline{L}},\mathbb{Z}_{p}))$

and, passing to the $E$-linear duals, a natural isomorphism

(5.8)
$\gamma^{\vee}_{\operatorname{crys}}(A):E\otimes_{W(k)}H_{1}^{\operatorname{crys}}(A_{k})\xleftarrow{\sim}E\otimes_{\mathbb{Z}_{p}}H_{1}^{\operatorname{et}}(A_{\overline{L}},\mathbb{Z}_{p}).$

###  5.5. The Connected-étale exact sequence

Let $\overline{k}$ be the residue field of $\overline{L}$. We consider the
reduction map

(5.9) $\operatorname{red}:A(\overline{L})[p^{\infty}]\to
A(\overline{k})[p^{\infty}],$

in terms of the connected-étale exact sequence.

This sequence involves $p$-divisible groups and to do so we recall some
notions from $p$-divisible groups and Dieudonné theory, following Demazure’s
classical reference [6] (both over $L$ and $k$). To the abelian scheme
$\mathcal{A}$, and the abelian varieties $A_{L}$ and $A_{k}$, Tate attached
_$p$ -divisible groups_

$\mathcal{G}_{\mathcal{A}}=\varinjlim_{n\in{\mathbb{Z}_{\geq
0}}}\mathcal{A}[p^{n}]\hookrightarrow\mathcal{A}[p^{n+1}],$

and its generic and special fibres

$\mathcal{G}_{A_{L}}=\varinjlim_{n\in{\mathbb{Z}_{\geq
0}}}A_{L}[p^{n}]\hookrightarrow A_{L}[p^{n+1}]\text{ and
}\mathcal{G}_{A_{k}}=\varinjlim_{n\in{\mathbb{Z}_{\geq
0}}}A_{k}[p^{n}]\hookrightarrow A_{k}[p^{n+1}].$

These are _ind-schemes_ of groups schemes: its functor of points is tested
against $\mathcal{O}_{L}$-algebras $R$ and is
$\mathcal{G}_{\mathcal{A}}(R)=\mathcal{A}(R)[p^{n}]$.

We denote the neutral connected components (with respect to the union of
Zariski topologies) as

$\mathcal{G}_{\mathcal{A}}^{0},\text{ as }\mathcal{G}_{A_{L}}^{0}\text{ and as
}\mathcal{G}^{0}_{A_{k}}.$

These first and third are actually the formal Lie groups attached to
$\mathcal{A}$ and $A_{k}$.

The corresponding quotients are well defined as $p$-divisible groups and there
are short exact sequences of $p$-divisible groups:

(5.10) $\displaystyle
0\xrightarrow{}\mathcal{G}_{\mathcal{A}}^{0}\xrightarrow{\iota}\mathcal{G}_{\mathcal{A}}\xrightarrow{\varpi}\mathcal{G}_{\mathcal{A}}^{\operatorname{\acute{e}t}}\xrightarrow{}0$
$\displaystyle\text{ over }\mathcal{O}_{L},$ (5.11) and $\displaystyle
0\xrightarrow{}\mathcal{G}_{A_{k}}^{0}\xrightarrow{\iota_{k}}\mathcal{G}_{A_{k}}\xrightarrow{\varpi_{k}}\mathcal{G}_{A_{k}}^{\operatorname{\acute{e}t}}\to
0$ $\displaystyle\text{ over }k.$

The (co)homologies we evoked in (5.6) factor as functor through the “passing
to the attached $p$-divisible groups over $\mathcal{O}_{L}$”, and
${\operatorname{H}}^{1}_{\operatorname{crys}}$, resp.
${\operatorname{H}}^{1}_{\operatorname{\acute{e}t}}$, factors even more
through “passing to the special (resp. generic) fibre”. We can moreover
identify respectively

${\operatorname{H}}_{1}^{\operatorname{crys}}(A_{k})\text{ and
}{\operatorname{H}}_{1}^{\operatorname{\acute{e}t}}(A_{\overline{L}};\mathbb{Z}_{p})$

with the _covariant Dieudonné module_ $M(\mathcal{G}_{A_{k}})$ of
$\mathcal{G}_{A_{k}}$, and its _Tate module_

$T(\mathcal{G}_{A_{\overline{L}}})=\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\mathcal{G}_{A_{L}}(\overline{L})).$

As Dieudonné modules description is fairly explicit by reduction to the finite
group schemes $A_{k}[p^{n}]$, this can serve as a working definition of the
${\operatorname{H}}^{1}_{\operatorname{crys}}$ and
${\operatorname{H}}^{1}_{\operatorname{\acute{e}t}}$ cohomologies applied to
abelian varieties.

Important to us is that the Dieudonné module, resp. Tate module still make
sense for any the $p$-divisible groups appearing in (5.11) and (5.10)
respectively. Even though these are not the full $p$-divisible group of an
abelian variety, or of an abelian scheme resp. We still get covariant functors

$\mathbf{M}()\text{ and }\mathbf{T}()$

on the category of $p$-divisible groups over $k$ and $L$ respectively, to that
of free ${}{\operatorname{W}(k)}{}$-modules and $\mathbb{Z}_{p}$-modules
respectively.

Applying the functors $\mathbf{T}()$ and $\mathbf{M}()$ to (5.10) and (5.11)
respectively gives us exact sequences that we relabel

(5.12) $\displaystyle
0\xrightarrow{}T^{0}\xrightarrow{}T\xrightarrow{}T^{\operatorname{\acute{e}t}}\xrightarrow{}0=$
$\displaystyle
0\xrightarrow{}\mathbf{T}(\mathcal{G}_{\mathcal{A}}^{0})\xrightarrow{\mathbf{T}(\iota)}\mathbf{T}(\mathcal{G}_{\mathcal{A}})\xrightarrow{\mathbf{T}(\varpi)}\mathbf{T}(\mathcal{G}_{\mathcal{A}}^{\operatorname{\acute{e}t}})\xrightarrow{}0$
(5.13) and $\displaystyle
0\xrightarrow{}M^{0}\xrightarrow{}M\xrightarrow{}M^{\operatorname{\acute{e}t}}\xrightarrow{}0=$
$\displaystyle
0\xrightarrow{}\mathbf{M}(\mathcal{G}_{A_{k}}^{0})\xrightarrow{\mathbf{M}(\iota_{k})}\mathbf{M}(\mathcal{G}_{A_{k}})\xrightarrow{\mathbf{M}(\varpi_{k})}\mathbf{M}(\mathcal{G}_{A_{k}}^{\operatorname{\acute{e}t}})\to
0.$

#### Fact

We will use the following explicit description of the Dieudonné module of an
étale $p$-divisible group $\mathcal{G}$ over $k$. We have, as $W(k)$-modules,
a natural identification

$\mathbf{T}(\mathcal{G})\mathbin{\otimes}_{\mathbb{Z}_{p}}W(k)\simeq\mathbf{M}(\mathcal{G}).$

We won’t need to explicit semi-linear action making it a _Dieudonné_ module,
but we stress above identification is functorial, and in particular receives a
left action of $\operatorname{End}(\mathcal{G})$.

The reduction map $\operatorname{red}$ factors through the above exact
sequences, by which we mean the commutativity of the square

${\mathcal{G}_{\mathcal{A}}[p^{\infty}](\mathcal{O}_{\overline{L}})}$${\mathcal{G}_{\mathcal{A}}^{\operatorname{\acute{e}t}}[p^{\infty}](\mathcal{O}_{\overline{L}})}$${\mathcal{G}_{A_{k}}[p^{\infty}](\overline{k})}$${\mathcal{G}_{A_{k}}^{\operatorname{\acute{e}t}}[p^{\infty}](\overline{k}).}$$\scriptstyle{\operatorname{red}}$$\scriptstyle{\varpi}$$\scriptstyle{\sim}$$\scriptstyle{\varpi_{k}}$

Applying the dualising $\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},-)$
to pass to the Tate modules, we get equivalently

${T}$${T^{\operatorname{\acute{e}t}}}$${T_{p}({A_{k}}[p^{\infty}](\overline{k}))}$${T_{p}({A_{k}}[p^{\infty}](\overline{k})).}$$\scriptstyle{\sim}$

where in the bottom row is the naïve Tate module of $A_{k}$: its rank as a
free $\mathbb{Z}_{p}$ module is the $p$-_rank_ of $A_{k}$.

In another formulation, we have the identity

(5.14) $\ker\bigl{(}\mathbf{T}(\varpi):T\to
T^{\operatorname{\acute{e}t}}\bigr{)}=\ker\bigl{(}T(\operatorname{red}):T\to
T_{p}({A_{k}}[p^{\infty}](\overline{k}))\bigr{)}$

###  5.6. Unit root eigenspaces

We now work with the connected-étale exact sequence of Dieudonné modules:

$0\xrightarrow{}M^{0}\xrightarrow{}M\xrightarrow{}M^{\operatorname{\acute{e}t}}\xrightarrow{}0.$

####  5.6.1. Dieudonné-Manin semilinear eigenspaces

We recall a notion of eigenspaces for semilinear actions, and detail it for
convenience. Let $M^{\prime}$ be any free Dieudonné module over $k$, namely a
finite rank free module over ${}{\operatorname{W}(k)}{}$ equipped with
${}{\operatorname{W}(\mathbb{F}_{p})}{}$-linear endomorphism $\phi$ which is
semi-linear with respect to ${}{\operatorname{W}(k)}{}$. We have the following
notion of eigenspace on the $L_{0}$-linear vector space
$D=M^{\prime}\mathbin{\otimes}_{{}{\operatorname{W}(k)}{}}L_{0}$. Pick any
${}{\operatorname{W}(k)}{}$ basis of $M^{\prime}$ over $L_{0}$, the
representative matrix $\Phi$ of $\phi$ applied to this basis has associated
(generalised) eigenspaces in $D\mathbin{\otimes}_{L_{0}}\overline{L}$, and
these are defined over $L_{0}$ when the corresponding eigenvalue belongs to
$L_{0}$. A change of basis of matrix $B$ will change
$\Phi^{\prime}=\Phi\mapsto\operatorname{F}_{{}{\operatorname{W}(k)}{}}(B)\Phi
B^{-1}$, each eigenvalue obtained from the $\lambda$ by some operation

$\lambda\mapsto\lambda^{\prime}=\operatorname{F}_{{}{\operatorname{W}(k)}{}}(c_{\lambda})/c_{\lambda}\cdot\lambda$

for some invertible $c(\lambda)$ in
${}{\operatorname{W}(k)}{}\subseteq{L_{0}}^{\text{nr}}$. As
$\operatorname{F}_{{}{\operatorname{W}(k)}{}}$ preserves the $p$-adic absolute
values $\left\lvert{}\right\rvert_{p}:L_{0}\mapsto\mathbb{R}_{\geq 0}$, we
will have the invariance

$\left\lvert{\lambda}\right\rvert_{p}=\left\lvert{\lambda^{\prime}}\right\rvert_{p}.$

For every real number $\mu$, we let $\overline{V}_{\mu}$ be the sum, in
$\overline{D}=D\mathbin{\otimes}_{L_{0}}\overline{L}$, of the generalised
eigenspaces for eigenvalues $\lambda$ having valuation
$\mu=-\log_{p}\left\lvert{\lambda}\right\rvert_{p}$. Then $\overline{D}_{\mu}$
does not depend on the choice of basis, and is defined over $L_{0}$.

This can be approached another way. Denote $f=[k:\mathbb{F}_{p}]$, so that
${}{\operatorname{W}(k)}{}^{f}$ will actually be the identity of
${}{\operatorname{W}(k)}{}$. But the power $\phi^{f}$, being
${}{\operatorname{W}(k)}{}^{f}$-semi-linear, will actually be
${}{\operatorname{W}(k)}{}$-linear. We let $P(X)$ be its characteristic
polynomial, and $P_{\mu}(X)$ be, in $\overline{L}[X]$, its maximal unitary
factor whose roots with roots $\lambda$ have valuation
$f\mu=-f\cdot\log\left\lvert{\lambda}\right\rvert_{p}$. Then $P_{\mu}(X)$ is
actually defined over $L_{0}$, and the kernel $D_{\mu}=\ker P_{\mu}(\phi)$
will produce
$\overline{D}_{\mu}=D_{\mu}\mathbin{\otimes}_{L_{0}}\overline{L}$.

These “Dieudonné modules” $M^{\prime}$ are also known as _crystals_ , the
corresponding $D$ as _iso_ crystals, and the subspace $D_{0}$ as a _unit root_
isocrystal.

####  5.6.2. Unit root isosplitting of étale quotient

It happens that the exact sequence

${0}$${M^{0}\mathbin{\otimes}_{W(k)}L_{0}}$${M\mathbin{\otimes}_{W(k)}L_{0}}$${M^{\operatorname{\acute{e}t}}\mathbin{\otimes}_{W(k)}L_{0}}$${0}$$\scriptstyle{t}$$\scriptstyle{\pi}$$\scriptstyle{s}$

is (uniquely) split as an exact sequence of
($\operatorname{F}_{{}{\operatorname{W}(k)}{}}$-semi-linear) modules, and that
a (unique semi-linear) splitting is given by the unit root subspace

$s(D^{\operatorname{\acute{e}t}})=D_{0}\text{ of
}D=M\mathbin{\otimes}_{W(k)}L_{0}\text{, letting