universal property of $\pi$ that $f:Y^{\prime}\to Y$ is a categorical quotient
in schemes over $B$ of the pre-equivalence relation

$R\to Y^{\prime}\times_{B}Y^{\prime}$

defined by the groupoid in algebraic spaces $(Y^{\prime},R,s,t,c)$ over $B$,
where $s,t:R\to Y^{\prime}$ correspond to the two projections and
$c:R\times_{Y^{\prime}}R\to R$ corresponds to the projection which comes from
the canonical equivalence $R\times_{Y^{\prime}}R\cong
Y^{\prime}\times_{\mathcal{M}_{\mathcal{P}}}Y^{\prime}\times_{\mathcal{M}_{\mathcal{P}}}Y^{\prime}$.

To describe the categorical quotient $f$ more explicitly, we may and do assume
that the Hilbert moduli stack $\mathcal{M}=\mathcal{M}^{I}$ is associated to
some $(L,I,I_{+})$. Let $\mathcal{O}$ be the ring of integers of $L$, and let
$G$ be the constant group scheme over $B$ defined by the finite group
${\textup{GL}}_{2}(\mathcal{O}/n\mathcal{O})$ which acts on
$\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}$ via (5.2) and (5.3). Then we
obtain an action of $G$ on the $B$-scheme $Y^{\prime}$ by transport of
structure using the equivalence
$Y^{\prime}\overset{\sim}{\to}\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}$ in
(5.1) and the inverse defined right after (5.1). We now show that $f$ is a
categorical quotient in schemes over $B$ for the induced morphism

$G\times_{B}Y^{\prime}\to Y^{\prime}\times_{B}Y^{\prime}.$ (5.12)

As $f$ is a categorical quotient in schemes over $B$ for $R\to
Y^{\prime}\times_{B}Y^{\prime}$, it suffices to prove for any morphism of
$B$-schemes $Y^{\prime}\to X$ that it is $R$-invariant if and only if it is
$G$-invariant. It follows from (1) below that this is equivalent to the
statement that for any geometric point $\textnormal{Spec}(k)\to B$ the map
$Y^{\prime}(k)\to X(k)$ is constant on weak $R$-orbits if and only if it is
constant on weak $G$-orbits. But this statement holds since the weak
$R$-orbits are precisely the weak $G$-orbits by (2) below. Hence we conclude
(5.12).

(1) Let $Y^{\prime}\to X$ be a morphism of schemes over $B$. Then
$Y^{\prime}\to X$ is $R$-invariant (resp. $G$-invariant) if and only if for
any geometric point $\textnormal{Spec}(k)\to B$ the map $Y^{\prime}(k)\to
X(k)$ is constant on weak $R$-orbits (resp. weak $G$-orbits). This follows for
example from the results in [Sta, 048M] which require that the schemes $R$ and
$G\times_{B}Y^{\prime}$ are reduced with $R\rightrightarrows Y^{\prime}$ and
$G\times_{B}Y^{\prime}\rightrightarrows Y^{\prime}$ locally of finite type.
These conditions are satisfied. Indeed, on using that
$Y^{\prime}\to\mathcal{M}_{\mathcal{P}}$ is a base change of the finite étale
$\mathcal{M}_{\mathcal{Q}}\to\mathcal{M}$ and that $G$ is finite étale over
$B$, we see that $R$ and $G\times_{B}Y^{\prime}$ are both finite étale over
$Y^{\prime}$ which is finite étale over $\mathcal{M}_{\mathcal{P}}$ and by
assumption $\mathcal{M}_{\mathcal{P}}$ is reduced and of finite type over $B$.

(2) For any geometric point $\textnormal{Spec}(k)\to B$, two $k$-points
$u,u^{\prime}\in Y^{\prime}(k)$ are weakly $R$-equivalent if and only if they
are weakly $G$-equivalent. This can be verified by a direct computation. Let
$\phi:Y^{\prime}\overset{\sim}{\to}\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}$
be the equivalence in (5.1), let $\psi$ be the inverse defined right after
(5.1), and write $\phi(u)=(x,\alpha,\beta)$ and
$\phi(u^{\prime})=(x^{\prime},\alpha^{\prime},\beta^{\prime})$. The following
computations crucially exploit that $\phi$ and $\psi$ are morphisms of
categories over $\mathcal{M}_{\mathcal{P}}$. We first suppose that $u$ is
weakly $R$-equivalent to $u^{\prime}$. Then there exists $r\in R(k)$ with
$s(r)=u$ and $t(r)=u^{\prime}$. Hence $r=(u,u^{\prime},\tau)$ for some
isomorphism $\tau:(x,\alpha)\overset{\sim}{\to}(x^{\prime},\alpha^{\prime})$
in $\mathcal{M}_{\mathcal{P}}(k)$ which induces an isomorphism
$(x,\alpha,\beta^{*})\overset{\sim}{\to}(x^{\prime},\alpha^{\prime},\beta^{\prime})$
where $\beta^{*}\in\mathcal{Q}(x)$. Then we compute that $g\cdot
u=\psi(x,\alpha,\beta^{*})=u^{\prime}$ where $g\in G(k)$ is defined by
$(\beta^{*})^{-1}\beta$. Hence $u$ is weakly $G$-equivalent to $u^{\prime}$ as
desired. To prove the converse, we now suppose that $u$ is weakly
$G$-equivalent to $u^{\prime}$. Then there exists $g\in G(k)$ such that
$g\cdot u=u^{\prime}$ and hence $(x,\alpha,g\cdot\beta)$ is isomorphic to
$(x^{\prime},\alpha^{\prime},\beta^{\prime})$ in
$\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}(k)$. Thus there is an isomorphism
$\tau:(x,\alpha)\overset{\sim}{\to}(x^{\prime},\alpha^{\prime})$ in
$\mathcal{M}_{\mathcal{P}}(k)$ such that $r=(u,u^{\prime},\tau)\in R(k)$
satisfies $s(r)=u$ and $t(r)=u^{\prime}$, which means that $u$ is weakly
$R$-equivalent to $u^{\prime}$ as desired. This completes the proof of (2).

##### 3.

Now we are ready to show that the morphism $f$ is étale over any point $y\in
Y$ with $|\textup{Aut}(y)|\leq 3$. For this purpose we may assume that $f$ is
the geometric quotient morphism $q:Y^{\prime}\to Y^{\prime}/G$ which sends a
geometric point to its $G$-orbit coming from the action of $G$ on the quasi-
projective $B$-scheme $Y^{\prime}$. Indeed (5.12) combined with the universal
property of categorical quotients gives a unique isomorphism
$\chi:Y\overset{\sim}{\to}Y^{\prime}/G$ of schemes over $B$ such that $q=\chi
f$. In particular $q$ is étale over $\chi(y)$ if and only if $f$ is étale over
$y$, and the separable rank $|\textup{Aut}(\chi(y))|$ defined with respect to
the initial morphism $\chi\pi:\mathcal{M}_{\mathcal{P}}\to Y^{\prime}/G$
satisfies $|\textup{Aut}(y)|=|\textup{Aut}(\chi(y))|$. In what follows we
assume that $f=q$.

Let $y\in Y$ be a point with $|\textup{Aut}(y)|\leq 3$. We consider a point
$y^{\prime}\in Y^{\prime}$ with $f(y^{\prime})=y$ and we denote by
$D\subseteq{\textup{GL}}_{2}(\mathcal{O}/n\mathcal{O})$ the stabilizer group
of $y^{\prime}$ with respect to the action of
${\textup{GL}}_{2}(\mathcal{O}/n\mathcal{O})$ on $Y^{\prime}$. Then
$f:Y^{\prime}\to Y^{\prime}/G$ is étale at $y^{\prime}$ if and only if the
geometric quotient morphism $Y^{\prime}\to Y^{\prime}/D$ is étale at
$y^{\prime}$. In particular $f$ is étale at $y^{\prime}$ in the situation when
$D$ acts trivially on all points of $Y^{\prime}$. We now show that we are in
this situation. To understand the action of $D$ on $Y^{\prime}$, we shall
construct in step 4. below a group $\textup{Aut}(y)(k)$ with the following
properties: Its cardinality equals $|\textup{Aut}(y)|$, it has at most one
element $\sigma$ of order two and there exist an isomorphism of groups

$\textup{Aut}(y)(k)\overset{\sim}{\to}D$ (5.13)

which sends this $\sigma$ to $-1$. Then we see that
$|\textup{Aut}(y)(k)|=|\textup{Aut}(y)|=2$, since by assumption each geometric
automorphism group of $\mathcal{M}_{\mathcal{P}}$ has even order and
$|\textup{Aut}(y)|\leq 3$. Thus the above mentioned properties of
$\textup{Aut}(y)(k)$ imply that this group is generated by $\sigma$. Hence we
obtain that $D=\\{1,-1\\}$ since the isomorphism (5.13) sends $\sigma$ to
$-1$. To understand the action of $-1\in D$ on the points of $Y^{\prime}$, we
consider a geometric point $\textnormal{Spec}(k^{\prime})\to B$. Let
$(x,\alpha)$ be in $\mathcal{M}_{\mathcal{P}}(k^{\prime})$. The forgetful
morphism $\mathcal{M}_{\mathcal{P}}\to\mathcal{M}$ identifies
$\textup{Aut}(x,\alpha)$ with a subgroup of $\textup{Aut}(x)$. Further, as the
inertia stack
$\mathcal{I}_{\mathcal{M}_{\mathcal{P}}}\to\mathcal{M}_{\mathcal{P}}$ is
finite, the group $\textup{Aut}(x,\alpha)$ is finite. Hence Cauchy’s theorem
and our assumption on the geometric automorphism groups of
$\mathcal{M}_{\mathcal{P}}$ imply that $\textup{Aut}(x,\alpha)$ contains an
element of order two. Then it follows from Lemma 5.3 and
$\textup{Aut}(x,\alpha)\subseteq\textup{Aut}(x)$ that this element is given by
multiplication with $[-1]$. Thus for each object $(x,\alpha,\beta)$ of
$\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}(k^{\prime})$ there exists an
isomorphism $(x,\alpha,\beta)\overset{\sim}{\to}(x,\alpha,-1\cdot\beta)$
defined by the element in $\textup{Aut}(x,\alpha)$ given by $[-1]$. This
implies that $-1$, and hence the whole group $D=\\{1,-1\\}$, acts trivially on
all points of $Y^{\prime}$. We conclude that $f$ is étale at each
$y^{\prime}\in Y^{\prime}$ with $f(y^{\prime})=y$, that is $f$ is étale over
$y$. This shows that $y$ does not lie in $B_{f}$ and then the equality
$B_{f}=B_{\mathcal{P}}$ obtained in step 1. assures that our point $y$ does
not lie in $B_{\mathcal{P}}$ as desired.

##### 4.

It remains to construct the group $\textup{Aut}(y)(k)$ in (5.13) with all the
desired properties. For this purpose we use again the equivalence
$\phi:Y^{\prime}\overset{\sim}{\to}\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}$.
Let $y^{\prime}:\textnormal{Spec}(k)\to Y^{\prime}$ be a geometric point over
$y^{\prime}\in Y^{\prime}$ and denote by $\phi(y^{\prime})=(x,\alpha,\beta)$
its image in $\mathcal{M}_{\mathcal{P}\times\mathcal{Q}}(k)$. Then
$(x,\alpha)$ lies in $\mathcal{M}_{\mathcal{P}}(k)$ and we consider the group
algebraic space over $k$ given by

$\textup{Aut}(y)=\textnormal{Isom}_{\mathcal{M}_{\mathcal{P}}}((x,\alpha),(x,\alpha)).$

Notice that the group $\textup{Aut}(y)(k)$ identifies with
$\textup{Aut}(x,\alpha)$. As $f(y^{\prime})=y$ and $\phi$ is a morphism over
$\mathcal{M}_{\mathcal{P}}$, we find that $\pi(x,\alpha)=y$. This implies that
$|\textup{Aut}(y)|=|\textup{Aut}(y)(k)|$ since the separable rank
$|\textup{Aut}(y)|$ is an invariant of the point $\pi^{-1}(y)$ of
$\mathcal{M}_{\mathcal{P}}$ and since $k$ is an algebraically closed field.
The forgetful morphism $\mathcal{M}_{\mathcal{P}}\to\mathcal{M}$ identifies
$\textup{Aut}(y)(k)=\textup{Aut}(x,\alpha)$ with a subgroup of
$\textup{Aut}(x)$, and it follows from Lemma 5.3 that any element of order two
in $\textup{Aut}(x)$ is given by multiplication with $[-1]:A\to A$ where
$x=(A,\iota,\varphi)$. Thus $\textup{Aut}(y)(k)$ has at most one element of
order two. Next, we construct an isomorphism
$\textup{Aut}(x,\alpha)\overset{\sim}{\to}D$. For each
$\sigma\in\textup{Aut}(x,\alpha)$ we denote by $\sigma_{n}$ the restriction to
$A_{n}$ of the automorphism of $A$ underlying $\sigma$. Then we obtain a group
morphism

$\textup{Aut}(x,\alpha)\overset{\sim}{\to}D,\quad\sigma\mapsto\beta^{-1}\sigma_{n}\beta.$

Indeed $g=\beta^{-1}\sigma_{n}\beta$ lies in the decomposition group $D$ of
$y^{\prime}$ since $\sigma$ defines an isomorphism
$(x,\alpha,\beta)\overset{\sim}{\to}(x,\alpha,g\cdot\beta)$ and hence $g\cdot
y^{\prime}=y^{\prime}$, where as before we identify an automorphism of the
constant group scheme $(\mathcal{O}/n\mathcal{O})^{2}_{k}$ over the
algebraically closed field $k$ with the corresponding element in
$G(k)={\textup{GL}}_{2}(\mathcal{O}/n\mathcal{O})$. To see that the displayed
morphism is injective, we take $\lambda\in I_{+}$ where $I_{+}$ are the
totally positive elements of the $\mathcal{O}$-submodule $I$ of $L$ which
appears in the definition of $\mathcal{M}$. Now we consider the automorphism
group $\textup{Aut}(A,\varphi(\lambda))$ of the polarized abelian variety
$(A,\varphi(\lambda))$ over $k$. The forgetful morphisms induce an embedding
$\textup{Aut}(x,\alpha)\hookrightarrow\textup{Aut}(A,\varphi(\lambda))$, and
the restriction morphism
$\textup{Aut}(A,\varphi(\lambda))\to\textup{Aut}(A_{n})$ is injective since
$n\geq 3$. Therefore the composition
$\textup{Aut}(x,\alpha)\to\textup{Aut}(A_{n})$ given by
$\sigma\mapsto\sigma_{n}$ is injective, which implies that the displayed
morphism is injective. On the other hand, the displayed morphism is surjective
since for any $g\in D$ there exists an isomorphism
$(x,\alpha,\beta)\overset{\sim}{\to}(x,\alpha,g\cdot\beta)$ which defines an
automorphism $\sigma$ of $(x,\alpha)$ with $g=\beta^{-1}\sigma_{n}\beta$.
Further, we notice that $\beta^{-1}[-1]\beta=-1$ and we recall that
$\textup{Aut}(y)(k)=\textup{Aut}(x,\alpha)$. Therefore we see that the
displayed group morphism, and hence the group $\textup{Aut}(y)(k)$, has all
the desired properties. This completes the proof of Proposition 5.5. ∎

We now briefly discuss where we used the assumptions appearing in Proposition
5.5.

######  Remark 5.7.

The assumption that $\mathcal{M}_{\mathcal{P}}\to\mathcal{M}$ is finite
assures in 2. that the scheme
$Y^{\prime}=\mathcal{M}_{\mathcal{P}}\times_{\mathcal{M}}\mathcal{M}_{\mathcal{Q}}$
is quasi-projective over $B$ which we use to view $Y^{\prime}\to Y$ as a
geometric quotient $Y^{\prime}\to Y^{\prime}/G$ induced by the action of
$G={\textup{GL}}_{2}(\mathcal{O}/n\mathcal{O})$. The assumption that
$\mathcal{M}_{\mathcal{P}}$ is reduced assures that $Y^{\prime}$ is reduced
which we use to relate $R$-invariance with $G$-invariance, see (1). Finally,
the assumption on the geometric automorphism groups assures that certain
decomposition groups $D\subseteq G$ act trivially on all points of
$Y^{\prime}$ and hence $Y^{\prime}\to Y^{\prime}/D$ is étale. This assumption
can be relaxed by studying the étale locus of $Y^{\prime}\to Y^{\prime}/D$ in
terms of the automorphism groups of $\mathcal{M}_{\mathcal{P}}$; such a study
could be the subject of a future work.

##  6 Morphims of abelian schemes over Dedekind schemes

Let $S$ be a connected Dedekind scheme. In this section we introduce some
notation and terminology, and we collect basic results for morphisms of
abelian schemes over $S$ which we shall use freely in what follows. We refer
to [vKK19, $\mathsection$5] for detailed proofs or precise references for
those statements in this section which are not further explained.

##### Basic properties.

Let $A$, $A^{\prime}$ and $A^{\prime\prime}$ be abelian schemes over $S$.
Composition induces a pairing $\circ$ from
$\textup{Hom}^{0}(A,A^{\prime})\times\textup{Hom}^{0}(A^{\prime},A^{\prime\prime})$
to $\textup{Hom}^{0}(A,A^{\prime\prime})$. Further $f\mapsto f\otimes 1$
embeds the finitely generated free abelian group $\textup{Hom}(A,A^{\prime})$
into $\textup{Hom}^{0}(A,A^{\prime})$. In what follows we shall often identify
a morphism $f:A\to A^{\prime}$ with its image $f\otimes 1$ in
$\textup{Hom}^{0}(A,A^{\prime})$. For each nonzero $n\in\mathbb{Z}$ and any
$f\in\textup{Hom}(A,A^{\prime})$ we denote by $f/n$ the element $f\otimes
n^{-1}$ of $\textup{Hom}^{0}(A,A^{\prime})$. Let $k$ be the function field of
the connected Dedekind scheme $S$. Base change from $S$ to the spectrum of $k$
induces canonical isomorphisms

$\textup{Hom}(A,A^{\prime})\overset{\sim}{\to}\textup{Hom}(A_{k},A_{k}^{\prime})\quad\textnormal{and}\quad\textup{End}(A)\overset{\sim}{\to}\textup{End}(A_{k}).$
(6.1)

A direct computation shows that any element in
$\textup{Hom}^{0}(A,A^{\prime})$ takes the form $f\otimes q$ with
$f\in\textup{Hom}(A,A^{\prime})$ and $q\in\mathbb{Q}$. In particular, we see
that the center of the $\mathbb{Q}$-algebra $\textup{End}^{0}(A)$ identifies
with $Z\otimes_{\mathbb{Z}}\mathbb{Q}$ where $Z$ denotes the center of the
ring $\textup{End}(A)$. We say that $A$ is simple if $A_{k}$ is simple.
Further we say that $A^{\prime}$ is a factor of $A$ if there exists a
surjective morphism $A\to A^{\prime}$ of abelian schemes over $S$. The proof
of Poincaré’s reducibility theorem together with (6.1) shows the following:
The abelian scheme $A$ is simple if and only if $\textup{End}^{0}(A)$ is a
division algebra, and $A^{\prime}$ is a factor of $A$ if and only if there
exists an abelian scheme $B$ over $S$ such that $A^{\prime}\times_{S}B$ is
isogenous to $A$. We say that $A$ is isotypic if $A$ is isogenous to a power
of a simple abelian scheme over $S$.

##### Isogenies.

Let $\varphi\colon A\to A^{\prime}$ be a morphism of abelian schemes over $S$.
Then $\varphi$ is an isogeny if and only if $\varphi_{k}:A_{k}\to
A^{\prime}_{k}$ is an isogeny of abelian varieties over $k$. Suppose now that
$\varphi$ is an isogeny. Then $\varphi$ is flat and the rank $d=\deg(\varphi)$
of its kernel $\ker(\varphi)$ is constant on the connected scheme $S$.
Consider the ‘inverse’ isogeny ([vKK19, (5.1)])

$\varphi^{\prime}:A^{\prime}\to A,\quad\textnormal{ and
}\quad\varphi^{*}\colon\textup{End}^{0}(A^{\prime})\overset{\sim}{\to}\textup{End}^{0}(A),\quad
f\mapsto\varphi^{\prime}/d\circ f\circ\varphi.$ (6.2)

Here $\varphi^{*}$ is an isomorphism of $\mathbb{Q}$-algebras and [vKK19, Lem
5.1] gives that $\varphi^{\prime}\varphi=[d]$ and
$\varphi\varphi^{\prime}=[d]$ where $[n]$ denotes the multiplication by
$n\in\mathbb{Z}$ morphism. In particular any isogeny $A\to A^{\prime}$ becomes
invertible inside $\textup{Hom}^{0}(A,A^{\prime})$. If $A^{\prime}$ is simple
and if $A$ and $A^{\prime}$ are of the same relative dimension over $S$, then
any nonzero element $\varphi$ of $\textup{Hom}(A,A^{\prime})$ is an isogeny.
Indeed $\varphi$ corresponds via (6.1) to a nonzero $\varphi_{k}:A_{k}\to
A_{k}^{\prime}$ which is an isogeny since $A_{k}^{\prime}$ is simple and
$\dim(A_{k})=\dim(A^{\prime}_{k})$.

##### Conjugate abelian variety.

Let $A$ be an abelian variety defined over an arbitrary field $k$, let
$\sigma$ be an automorphism of $k$ and write $\sigma^{*}A$ for the base change
of $A$ via $\sigma:k\to k$. The abelian variety $\sigma^{*}A$ is called the
conjugate of $A$ by $\sigma$. We denote by

$\sigma^{*}:\textup{Hom}(A,A^{\prime})\overset{\sim}{\to}\textup{Hom}(\sigma^{*}A,\sigma^{*}A^{\prime})$

the isomorphism of abelian groups induced by base change via $\sigma$, where
$A^{\prime}$ is an abelian variety over $k$. We also denote by $\sigma^{*}$
the isomorphism of $\mathbb{Q}$-vector spaces given by
$\sigma^{*}\otimes_{\mathbb{Z}}\mathbb{Q}$. Base change properties of (group)
schemes imply the following: The map $\sigma^{*}$ is a morphism of rings if
$A=A^{\prime}$, and $\varphi:A\to A^{\prime}$ is an isogeny if and only if
$\sigma^{*}(\varphi)$ is an isogeny.

##  7 Effective Shafarevich conjecture

Let $K$ be a number field. In this section we prove new cases of the effective
Shafarevich conjecture for abelian varieties over $K$. In particular, we
establish this conjecture for abelian varieties of product
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies and for CM abelian
varieties.

Let $g\geq 1$ be a rational integer, let $S$ be a nonempty open subscheme of
$\textnormal{Spec}({\mathcal{O}_{K}})$, and let $A$ be an abelian scheme over
$S$ of relative dimension $g$. We recall that $A$ is of product
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies if $A$ is isogenous
to a product $\prod A_{i}$ of abelian schemes $A_{i}$ over $S$ of
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies as in (1.5). A
discussion of the notion of an abelian scheme over $S$ of
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies can be found in
$\mathsection$7.1.1 below.

##### Height bound.

As before, we use the quantities $d=[K:\mathbb{Q}]$, $D_{K}$ and
$N_{S}=\prod_{v\in S^{c}}N_{v}$. We define $l=(d3^{4g^{2}})!$ and we denote by
$h_{F}$ the stable Faltings height ($\mathsection$3.4) of an abelian scheme
over $S$. The main goal of this section is to prove the following result.

######  Theorem 7.1.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$.

  * (i)

If $A$ is of product $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies then

$h_{F}(A)\leq(4gl)^{144gl}\textnormal{rad}(N_{S}D_{K})^{24g}.$

  * (ii)

If $A_{\bar{K}}$ has CM then
$h_{F}(A)\leq(3g)^{(5g)^{2}}\textnormal{rad}(N_{S}D_{K})^{5g}$.

The proof shows that one can replace in (i) the exponent $24g$ by
$\max(24,5g)$, while Proposition 7.13 and (7.19) provide sharper versions of
(i) in relevant special situations. Further Proposition 7.14 and Lemma 7.10
contain more precise versions of (ii) in certain cases of interest, see also
the discussions in $\mathsection$7.2.1 for known asymptotic versions of (ii).

##### Number of classes.

Another goal of this section is to prove Theorem B (ii) on the number of
isomorphism classes of abelian schemes over $S$ of relative dimension $g$
which are of product $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies. In course of the proof given in Section 7.4, we
shall obtain various intermediate results which improve Theorem B (ii) in
certain cases of interest; see for example Proposition 7.15 and Lemma 7.17.

##### Outline of the section.

In Section 7.1 we study abelian schemes of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies. After motivating the definition
($\mathsection$7.1.1), we discuss and prove Proposition 7.2 on the factors of
such abelian schemes. This proposition involves a technical condition $(*)$
which is further studied in $\mathsection$7.1.4. We also collect in this
section some basic results which are used throughout this work; in particular
we consider abelian schemes of $\textnormal{GL}_{2}$-type (Lemma 7.5) and
isogenies between conjugate abelian varieties ($\mathsection$7.1.5).

In Section 7.2 we begin to bound the height of abelian varieties with CM. We
discuss the averaged Colmez conjecture in $\mathsection$7.2.1 and we work out
in $\mathsection$7.2.2 explicit analytic estimates for certain L-values
attached to CM fields. Then we prove Lemma 7.12 on the ramification of
endomorphism algebras of simple abelian varieties with CM.

In Sections 7.3 and 7.4 we put everything together. After reviewing known
results for the Faltings height and discussing isogeny estimates
($\mathsection$7.3.1), we prove Theorem B (ii) and Theorem 7.1 by using the
strategy and the ideas explained in $\mathsection$1.2.1.

###  7.1 Abelian schemes of $\textnormal{GL}_{2}$-type with $G$-isogenies

Let $k\subseteq K$ be fields which are algebraic over $\mathbb{Q}$ and write
$G=\textup{Aut}(\bar{k}/k)$. In this section we discuss a class of abelian
schemes which we call of $\textnormal{GL}_{2}$-type with $G$-isogenies. After
motivating our definition, we show in Proposition 7.2 that the study of this
class of abelian schemes over certain base schemes $S$ with $k(S)=K$ can be
reduced to the study of simple abelian schemes of $\textnormal{GL}_{2}$-type
over certain base schemes $T$ with $k(T)=k$.

####  7.1.1 The notion of abelian schemes of $\textnormal{GL}_{2}$-type with
$G$-isogenies

Let $k\subseteq K$ and $G$ be as above. Choose $\bar{k}$ with
$K\subseteq\bar{k}$ and let $S$ be a connected Dedekind scheme with function
field $K$. We say that an abelian scheme $A$ over $S$ of relative dimension
$g$ is of $\textnormal{GL}_{2}$-type with $G$-isogenies if the
$\mathbb{Q}$-algebra $\textup{End}^{0}(A)$ contains a number field $F$ of
degree $g$ and if for each $\sigma\in G$ there is an isogeny
$\mu_{\sigma}:\sigma^{*}A_{\bar{k}}\to A_{\bar{k}}$ such that

$\mu_{\sigma}\circ\sigma^{*}(f)=f\circ\mu_{\sigma},\quad f\in F.$ (7.1)

Here we identified $F$ with its image in $\textup{End}^{0}(A_{\bar{k}})$. More
generally, we say that an abelian scheme $A$ over $S$ is of product
$\textnormal{GL}_{2}$-type with $G$-isogenies if $A$ is isogenous to a product
$\prod A_{i}$ such that each $A_{i}$ is an abelian scheme over $S$ of
$\textnormal{GL}_{2}$-type with $G$-isogenies. If $A$ is an abelian scheme
over $S$ of $\textnormal{GL}_{2}$-type with $G$-isogenies then any abelian
scheme $A^{\prime}$ over $S$ which is isogenous to $A$ is again of
$\textnormal{GL}_{2}$-type with $G$-isogenies. Indeed one can take here for
example $F^{\prime}=\varphi\circ F\circ\varphi^{\prime}/d$ and the isogenies
$\mu^{\prime}_{\sigma}=\varphi_{\bar{k}}\mu_{\sigma}\sigma^{*}(\varphi^{\prime}_{\bar{k}})$
where $\varphi:A\to A^{\prime}$ is an isogeny of degree $d$ and
$\varphi^{\prime}:A^{\prime}\to A$ is an inverse as in (6.2).

##### Related notions.

The notion of abelian schemes over $S$ of $\textnormal{GL}_{2}$-type with
$G$-isogenies generalizes various definitions in the literature. To explain
this, we take $K=\bar{k}$ and we assume that $A$ is an abelian variety over
$K$ of $\textnormal{GL}_{2}$-type. In the case $\dim(A)=1$, we observe that
$A$ is of $\textnormal{GL}_{2}$-type with $G$-isogenies if and only if $A$ is
a $k$-curve111A $k$-curve is an elliptic curve $E$ over $K$ which is isogenous
to $\sigma^{*}E$ for all $\sigma\in G$. in the sense of Gross [Gro80], Ribet
[Rib92] and Elkies [Elk04]. Further, Ribet [Rib94] introduced the notion of
‘$k$-HBAV’ which was generalized by Wu [Wu20, Def 1.9]. She defined $A$ to be
$k$-virtual if for each $\sigma\in G$ there is an isogeny
$\mu_{\sigma}:\sigma^{*}A\to A$ such that

$\mu_{\sigma}\circ\sigma^{*}(f)=f\circ\mu_{\sigma},\quad
f\in\textup{End}^{0}(A).$ (7.2)

On comparing (7.2) with (7.1), we see that the notion of
$\textnormal{GL}_{2}$-type with $G$-isogenies generalizes the definition of
$k$-virtual abelian varieties $A$ over $K$ and thus the notion of ‘$k$-HBAV’.
The additional generality is useful in certain situations of interest in
arithmetic such as for example in $\mathsection$10.1 where we can directly
verify our weaker condition (7.1). However, if $A$ has no CM then our
condition (7.1) is in fact equivalent to the stronger $k$-virtual condition
(7.2). To prove this claim, we suppose that $A$ has no CM and that there exist
$F\subseteq\textup{End}^{0}(A)$ and isogenies $\mu_{\sigma}$ as in (7.1). Then
Lemma 7.5 gives that $A$ is isotypic and that $F$ contains the center of
$\textup{End}^{0}(A)$. Thus the isogenies $\mu_{\sigma}$ are center
compatible, and hence $A$ is $k$-virtual by Lemma 7.9. This proves the claim.

####  7.1.2 Factors of abelian schemes of $\textnormal{GL}_{2}$-type with
$G$-isogenies

We continue our notation. Further, we now assume that $k\subseteq K$ are
number fields and we suppose that
$S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ is open with complement
$S^{c}$. Then we define

$U=S\setminus p^{-1}(T^{c})\quad\textnormal{and}\quad T=p(U)$ (7.3)

where
$p:\textnormal{Spec}({\mathcal{O}_{K}})\to\textnormal{Spec}(\mathcal{O}_{k})$
is the projection and $T^{c}$ is the union of $p(S^{c})$ with the branch locus
of $p$. Weil restriction reduces the study of some abelian schemes over $S$ of
$\textnormal{GL}_{2}$-type with $G$-isogenies to the study of abelian schemes
over $T$ of $\textnormal{GL}_{2}$-type.

######  Proposition 7.2.

Let $A$ be an abelian scheme over $S$ with no CM. Suppose that $A$ is of
$\textnormal{GL}_{2}$-type with $G$-isogenies and assume $(*)$. Then there is
a simple abelian scheme $C$ over $T$ of $\textnormal{GL}_{2}$-type such that
$A_{U}$ is isogenous to a power of a simple factor of $C_{U}$.

Here condition $(*)$ is as follows: The field extension $K/k$ is normal, and
the endomorphisms of $A_{\bar{k}}$ and the isogenies $\mu_{\sigma}$ in (7.1)
are all defined over $K\subset\bar{k}$. In fact $(*)$ can always be satisfied
($\mathsection$7.1.4) when passing to a controlled field extension of $K$. The
proof of Proposition 7.2 shows in addition that any simple factor $B$ of $A$
satisfies

$\dim(C)\leq\dim(B)[K:k]$ (7.4)

for $\dim$ the relative dimension. Further we observe that $T$ is open in
$\textnormal{Spec}(\mathcal{O}_{k})$ with complement $T^{c}$, and Dedekind’s
discriminant theorem shows that $T^{c}$ is the set of finite places $v$ of $k$
such that $v$ lies under $S^{c}$ or $v$ divides the discriminant ideal of
$K/k$.

####  7.1.3 Proof of Proposition 7.2

Let $k\subseteq K\subseteq\bar{k}$, $S$ and $G$ be as in Section 7.1.1, and
let $A$ be an abelian scheme over $S$. Throughout this subsection we assume
that $A$ is of $\textnormal{GL}_{2}$-type with $G$-isogenies, that $A$ has no
CM and that $(*)$ holds. In a first step we show the following result.

######  Lemma 7.3.

There is an isogeny from $A$ to a power of an abelian scheme $B$ over $S$ of
$\textnormal{GL}_{2}$-type such that $B_{\bar{k}}$ is a simple $k$-virtual
abelian variety of $\textnormal{GL}_{2}$-type with no CM.

In particular $B_{\bar{k}}$ in Lemma 7.3 satisfies all assumptions in Wu
[Wu20, Prop 1.12]. Hence there is a simple abelian variety $C_{k}$ over $k$ of
$\textnormal{GL}_{2}$-type such that $B_{\bar{k}}$ is a factor of
$C_{\bar{k}}$. Moreover [vK13a, $\mathsection$9.4], which crucially relies on
the arguments of Ribet [Rib92, $\mathsection$6] and Wu [Wu20, Prop 1.12],
leads to the following result.

######  Lemma 7.4.

Suppose that $K$ is a number field with
$S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ open, and let $B$ be the
abelian scheme of Lemma 7.3. Then there is a simple abelian scheme $C$ over
$T$ of $\textnormal{GL}_{2}$-type such that $B_{U}$ is a simple factor of
$C_{U}$, where $U$ and $T$ are as in (7.3).

Now, on combining this result with Lemma 7.3 we deduce Proposition 7.2. The
remaining of this section is devoted to the proofs of the above two lemmas.

###### Proof of Lemma 7.3.

By assumption $S$ is a connected Dedekind scheme and $A$ is an abelian scheme
over $S$ of $\textnormal{GL}_{2}$-type with no CM. Hence (7.5) and Lemma 7.5
give a simple abelian scheme $B$ over $S$ of $\textnormal{GL}_{2}$-type and
$n\in\mathbb{Z}_{\geq 1}$ such that $A$ is isogenous to $B^{n}$.

We next prove that $B_{\bar{k}}$ is a simple abelian variety over $\bar{k}$ of
$\textnormal{GL}_{2}$-type with no CM. Firstly $B_{\bar{k}}$ is of
$\textnormal{GL}_{2}$-type, since it is a base change of $B$ which is of
$\textnormal{GL}_{2}$-type. Further, all endomorphisms of $A_{\bar{k}}$ are
defined over $K$ by $(*)$. Therefore (6.1) gives

$\textup{End}(A_{\bar{k}})\cong\textup{End}(A).$ (7.5)

This shows that $A_{\bar{k}}$ has no CM since $A$ has no CM. Then on using
that $A_{\bar{k}}$ is isogenous to $B_{\bar{k}}^{n}$, we deduce that the
abelian varieties $B_{\bar{k}}^{n}$ and $B_{\bar{k}}$ both have no CM.
Poincaré reducibility shows that any factor of $A$ also has property (7.5).
Thus $\textup{End}^{0}(B_{\bar{k}})\cong\textup{End}^{0}(B)$ is a division
algebra since $B$ is a simple factor of $A$. This implies that $B_{\bar{k}}$
is simple.

To show that $B_{\bar{k}}$ is $k$-virtual, we now use our assumption that $A$
is of $\textnormal{GL}_{2}$-type with $G$-isogenies. Hence there is a number
field $F$ inside $\textup{End}^{0}(A_{\bar{k}})$ of degree $\dim(A_{\bar{k}})$
and for each $\sigma\in G$ there is an isogeny
$\mu_{\sigma}:\sigma^{*}A_{\bar{k}}\to A_{\bar{k}}$ satisfying (7.1). As
$A_{\bar{k}}$ is an abelian variety over $\bar{k}$ of
$\textnormal{GL}_{2}$-type with no CM, an application of Lemma 7.5 with
$A_{\bar{k}}$ gives that the center $Z^{0}$ of $\textup{End}^{0}(A_{\bar{k}})$
is contained in $F$. Then, on using that
$\textup{Hom}(\sigma^{*}A_{\bar{k}},A_{\bar{k}})$ is torsion-free and that
$Z^{0}=Z\otimes_{\mathbb{Z}}\mathbb{Q}$ for $Z$ the center of
$\textup{End}(A_{\bar{k}})$, we see that (7.1) implies

$\mu_{\sigma}\sigma^{*}(z)=z\mu_{\sigma},\quad z\in Z.$

In other words the isogeny $\mu_{\sigma}:\sigma^{*}A_{\bar{k}}\to A_{\bar{k}}$
is center compatible in the sense of $\mathsection$7.1.5. Now, an application
of Lemma 7.7 with the isotypic abelian variety $A_{\bar{k}}$ and the
automorphism $\sigma$ of $\bar{k}$ gives a center compatible isogeny
$\sigma^{*}B_{\bar{k}}\to B_{\bar{k}}$. Thus an application of Lemma 7.8 with
the simple abelian variety $B_{\bar{k}}$ provides an isogeny
$\sigma^{*}B_{\bar{k}}\to B_{\bar{k}}$ which (is fully compatible and hence)
satisfies (7.2). Therefore we conclude that $B_{\bar{k}}$ is a simple
$k$-virtual abelian variety of $\textnormal{GL}_{2}$-type with no CM. This
shows that $B$ has the desired properties. ∎

In the following proof, we assume in addition that $k\subseteq K$ are number
fields and that $S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ is open.
Further, we let $p:U\to T$ be as in Section 7.1.2.

###### Proof of Lemma 7.4.

Let $B$ be the abelian scheme over $S$ of Lemma 7.3. Then $B_{\bar{k}}$ is a
simple $k$-virtual abelian variety of $\textnormal{GL}_{2}$-type with no CM.
In particular, the $k$-virtual condition assures that for each $\sigma\in G$
there is an isogeny $\mu_{\sigma}:\sigma^{*}B_{\bar{k}}\to B_{\bar{k}}$.

To show that each $\mu_{\sigma}$ is defined over $K$, we use condition $(*)$.
It assures that $K/k$ is normal and the restriction of $\sigma\in G$ to the
subfield $K\subset\bar{k}$ defines an automorphism of $K/k$ which we also
denote by $\sigma$. Then $\sigma^{*}B_{K}$ is an abelian variety over $K$. It
is simple, since it is geometrically isogenous via $\mu_{\sigma}$ to the
simple $B_{\bar{k}}$. Moreover, on using that $A_{K}$ is isogenous to
$\sigma^{*}A_{K}$ by $(*)$ and that $A$ is isogenous to $B^{n}$, we see that
$\sigma^{*}B_{K}$ is a simple factor of $A_{K}$. Hence $\sigma^{*}B_{K}$ has
to be isogenous to the simple factor $B_{K}$, and then
$\textup{Hom}^{0}(\sigma^{*}B_{K},B_{K})\cong\textup{End}^{0}(B_{K})$
identifies with
$\textup{Hom}^{0}(\sigma^{*}B_{\bar{k}},B_{\bar{k}})\cong\textup{End}^{0}(B_{\bar{k}})$
since $B_{K}$ has property (7.5). We conclude that each isogeny $\mu_{\sigma}$
is defined over $K$.

We now go into Wu’s proof of [Wu20, Prop 1.12]. The above statements show that
$B_{\bar{k}}$ satisfies all assumptions of [Wu20, Prop 1.12] and that our $K$
has all the properties of the field $F$ appearing in Wu’s proof ([Wu17,
$\mathsection$1.2]). Then Wu’s proof, which uses and generalizes Ribet’s
arguments in [Rib92, $\mathsection$6] for $\mathbb{Q}$-curves, shows in
addition that the Weil restriction $D_{k}=\textnormal{Res}_{K/k}(B_{K})$ of
$B_{K}$ has a simple factor $C_{k}$ of $\textnormal{GL}_{2}$-type.

To prove that $C_{k}$ extends to an abelian scheme $C$ over $T$, we use our
assumption that $K$ is a number field. Hence Milne [Mil72, Prop 1] gives that
$D_{k}$ has good reduction at a finite place $v$ of $k$ if $v$ does not divide
the discriminant ideal of $K/k$ and if $A_{K}$ has good reduction at all
points in $p^{-1}(v)$. Further the generic fiber $A_{K}$ of the abelian scheme
$A$ over $S$ has good reduction at each point in $S$, and Dedekind’s
discriminant theorem shows that $v$ divides the discriminant ideal of $K/k$ if
and only if $v$ lies in the branch locus $B_{p}$ of $p$. We deduce that
$D_{k}$ extends to an abelian scheme over $T=(p(S^{c})\cup B_{p})^{c}$, and
then also the factor $C_{k}$ of $D_{k}$ extends to an abelian scheme $C$ over
$T$. Furthermore (6.1) gives that $C$ is simple and of
$\textnormal{GL}_{2}$-type, since $C_{k}$ has these properties.

We next show that $B_{U}$ is a factor of $C_{U}$. The generic fiber $C_{K}$ of
$C_{U}$ identifies with the abelian variety $C_{k}\times_{k}K$ over $K$, since
the composition of $\textnormal{Spec}(K)\to U$ with $p:U\to T$ factors as
$\textnormal{Spec}(K)\to\textnormal{Spec}(k)\to T$. Further, as $C_{k}$ is a
factor of $D_{k}$, we obtain a surjective morphism $D_{k}\to C_{k}$ of abelian
varieties over $k$ which induces a surjective morphism

$\prod\sigma^{*}B_{K}\overset{\sim}{\to}D_{k}\times_{k}K\to
C_{k}\times_{k}K\overset{\sim}{\to}C_{K}$

of abelian varieties over $K$ with the product taken over all
$\sigma\in\textnormal{Aut}(K/k)$. Here we used that the field extension $K/k$
is Galois by $(*)$ which assures ([Wei82, p.5]) that the abelian variety
$D_{k}\times_{k}K$ over $K$ is isomorphic to $\prod\sigma^{*}B_{K}$. We showed
above that $B_{K}$ is isogenous to each $\sigma^{*}B_{K}$. Therefore, on
exploiting that the displayed morphism is surjective and hence nonzero, we see
that $B_{K}$ has to be a simple factor of $C_{K}$. This implies that $B_{U}$
is a simple factor of $C_{U}$. We conclude that $C$ has the desired
properties.∎

We conclude this subsection by collecting basic results for abelian schemes of
$\textnormal{GL}_{2}$-type which are essentially due to Ribet [Rib75, Rib92]
and Wu [Wu20]. In the above proofs we used (i) and (ii) of the following
lemma, while (i) - (iii) will be used in later sections.

######  Lemma 7.5.

Let $S$ be a connected Dedekind scheme whose function field $k$ embeds into
$\mathbb{C}$, and let $A$ be an abelian scheme over $S$ of
$\textnormal{GL}_{2}$-type such that $A$ has no CM.

  * (i)

The abelian scheme $A$ is isogenous to a power $B^{n}$ of a simple abelian
scheme $B$ over $S$ of $\textnormal{GL}_{2}$-type for some
$n\in\mathbb{Z}_{\geq 1}$. In particular the $\mathbb{Q}$-algebra
$\textup{End}^{0}(A)$ is isomorphic to $\textnormal{M}_{n}(D)$ where
$D=\textup{End}^{0}(B)$ is a division algebra.

  * (ii)

Let $F$ be a subfield of the $\mathbb{Q}$-algebra $\textup{End}^{0}(A)$ such
that $[F:\mathbb{Q}]$ equals the relative dimension of $A$ over $S$. Then $F$
is the centralizer of $F$ in $\textup{End}^{0}(A)$.

  * (iii)

Either $D$ identifies with a subfield $Z$ of $F$ with $[F:Z]=n$ or $D$ is a
quaternion algebra whose center identifies with a subfield $Z$ of $F$ with
$[F:Z]=2n$.

###### Proof.

To prove (i) we reduce to the case $S=\textnormal{Spec}(k)$ which was treated
in Wu [Wu20, Prop 1.5 (i)] when $k$ is an arbitrary field of characteristic
zero. Notice that Wu stated the result under the assumption that $A_{\bar{k}}$
has no CM. However, it turns out that her proof can be copied word by word
when making the weaker assumption that the abelian variety over $k$ has no CM.
Now, the reduction goes as follows: As
$\textup{End}^{0}(A)\overset{\sim}{\to}\textup{End}^{0}(A_{k})$ by (6.1) we
obtain that $A_{k}$ is of $\textnormal{GL}_{2}$-type with no CM. Then an
application of [Wu20, Prop 1.5] with $A_{k}$ gives an isogeny $A_{k}\to
B_{k}^{n}$ where $B_{k}$ is a simple abelian variety of
$\textnormal{GL}_{2}$-type and $n\in\mathbb{Z}_{\geq 1}$. In particular
$B_{k}$ is a factor of $A_{k}$ which extends to the abelian scheme $A$ over
$S$. Thus $B_{k}$ extends to an abelian scheme $B$ over $S$, and then the
isogeny $A_{k}\to B^{n}_{k}$ extends to a morphism $A\to B^{n}$ of abelian
schemes over $S$ which is again an isogeny; see for example [vKK19,
$\mathsection$5.1]. This together with
$\textup{End}^{0}(B)\cong\textup{End}^{0}(B_{k})$ implies (i).

To show (ii) and (iii) we may and do assume that $S=\textnormal{Spec}(k)$
since $\textup{End}^{0}(A)\cong\textup{End}^{0}(A_{k})$ and
$D\cong\textup{End}^{0}(B_{k})$. Our $A$ is isotypic by (i) and $A$ has no CM.
Thus no factor of $A$ has CM. Then we deduce (ii) and (iii) by using precisely
the same arguments as Ribet in his proof of [Rib92, Prop 5.2]; these arguments
exploit a trick (attributed by Ribet to Tunnel) which exploits that the field
$k$ embeds into $\mathbb{C}$ and hence $[D:\mathbb{Q}]$ divides $2\dim(B)$
since the division algebra $D$ acts $\mathbb{Q}$-linearly on the
$2\dim(B)$-dimensional homology $H_{1}(B(\mathbb{C}),\mathbb{Q})$. This
completes the proof of Lemma 7.5. ∎

####  7.1.4 Condition $(*)$ and controlled field extensions

In this section we investigate the technical condition $(*)$ appearing in
Proposition 7.2. For this purpose we let $k\subseteq K\subset\bar{k}$, $G$,
$S$ and $T$ be as in Section 7.1.2.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$ such that
$A_{\bar{k}}$ is of $\textnormal{GL}_{2}$-type with $G$-isogenies. We say that
condition $(*)$ holds over a number field $L\supseteq K$ if $L/k$ is normal
and if the endomorphisms of $A_{\bar{k}}$ and the isogenies $\mu_{\sigma}$ in
(7.1) are all defined over $L\subset\bar{k}$. On using Silverberg’s work
[Sil92] and the criterion of Néron–Ogg–Shafarevich [ST68], we obtain that
condition $(*)$ holds over a controlled field $L$.

######  Lemma 7.6.

Condition $(*)$ holds over a field $L\supseteq K$ such that $L$ is contained
in the normal closure of $K(A_{n})/k$ for each $n\in\mathbb{Z}_{\geq 3}$ and
such that $L/k$ is unramified over $T$.

Here and in what follows, for any $n\in\mathbb{Z}_{\geq 1}$, we denote by
$K(A_{n})$ the field of definition of the $n$-torsion points of $A(\bar{k})$.
The degree of $K(A_{n})$ over $K$ is at most $n^{4g^{2}}$, and Lemma 7.6
combined with Dedekind’s discriminant theorem allows to explicitly control the
discriminant of $L/k$ in terms of $g$, the discriminant of $K/k$, and
$N_{S}=\prod_{v\in S^{c}}N_{v}$.

###### Proof of Lemma 7.6.

Let $n\in\mathbb{Z}_{\geq 1}$ and denote by $K^{\textnormal{cl}}(A_{n})$ the
normal closure of $K(A_{n})$ over $k$. To show that $L=\cap_{n\geq
3}K^{\textnormal{cl}}(A_{n})$ has the desired properties, we take $\sigma\in
G$. The field $L$ is normal over $k$. Thus pulling back $A_{L}$ via the
restriction of $\sigma$ to $L$ gives an abelian variety $\sigma^{*}A_{L}$ over
$L$ which identifies over $\bar{k}$ with $\sigma^{*}A_{\bar{k}}$. We denote by
$L(\sigma^{*}A_{n})$ and $L(A_{n})$ the field of definition of the $n$-torsion
points of $\sigma^{*}A_{L}(\bar{k})$ and $A_{L}(\bar{k})$ respectively. On
using that $\sigma$ is invertible and that $L(\sigma^{*}A_{n})$ and $L(A_{n})$
are normal over $L$, we deduce

$L(\sigma^{*}A_{n})=L(A_{n}).$

Then an application of Silverberg [Sil92, Thm 4.2] with $\sigma^{*}A_{L}$ and
$A_{L}$ implies that any morphism in
$\textup{Hom}(\sigma^{*}A_{\bar{k}},A_{\bar{k}})$ is defined over the subfield
$\cap_{n\geq 3}L(A_{n})$ of $\bar{k}$. Furthermore we obtain that
$L=\cap_{n\geq 3}L(A_{n})$, since it holds that $L(A_{n})=K(A_{n})L$ and that
$K(A_{n})L\subseteq K^{\textnormal{cl}}(A_{n})$ for each $n\geq 3$. Thus we
deduce that condition $(*)$ holds over $L$.

To prove that $L/k$ is unramified over $T$, we take a rational prime $p\geq 3$
and for any scheme $S^{\prime}$ we write
$S^{\prime}[\tfrac{1}{p}]=S^{\prime}\times_{\mathbb{Z}}\mathbb{Z}[\tfrac{1}{p}]$.
The abelian variety $A_{K}$ has good reduction at each closed point of $S$,
since $A_{K}$ extends to the abelian scheme $A$ over $S$. Thus $K(A_{p})/K$ is
unramified over $S[\tfrac{1}{p}]$ by the criterion of Néron–Ogg–Shafarevich
[ST68, Thm 1]. Then the construction of $T$ in (7.3) shows that $K(A_{p})/k$
is unramified over $T[\tfrac{1}{p}]$ and hence its normal closure
$K^{\textnormal{cl}}(A_{p})/k$ is also unramified over $T[\tfrac{1}{p}]$.
Therefore, for any rational prime $q\geq 3$ with $q\neq p$, we deduce that the
subfield $L$ of $K^{\textnormal{cl}}(A_{p})\cap K^{\textnormal{cl}}(A_{q})$ is
unramified over $T=T[\tfrac{1}{p}]\cup T[\tfrac{1}{q}]$. Finally, as the field
$L=\cap_{n\geq 3}K^{\textnormal{cl}}(A_{n})$ is contained in
$K^{\textnormal{cl}}(A_{n})$ for each $n\geq 3$, we conclude that $L/k$ has
all the desired properties. ∎

####  7.1.5 Center compatible isogenies between conjugate abelian varieties

In this section we collect basic lemmas which we used in $\mathsection$7.1.1
and $\mathsection$7.1.3 above. These lemmas allow in situations where
isogenies of conjugate abelian varieties are center compatible to reduce to
the case where the isogenies are compatible with the whole endomorphism ring.
The results should be all well-known. However, we are not aware of suitable
references and for the sake of completeness we will thus give the proofs which
use and generalize arguments of Ribet [Rib94] and Pyle [Pyl04].

Let $A$ be an abelian variety defined over an arbitrary field $k$, let
$\sigma$ be an automorphism of the field $k$ and write $\sigma^{*}A$ for the
base change of $A$ via $\sigma:k\to k$.

##### Center compatible isogenies.

We say that an isogeny $\mu:\sigma^{*}A\to A$ is center compatible if any $z$
in the center of $\textup{End}(A)$ satisfies $\mu\sigma^{*}(z)=z\mu$. The
existence of such an isogeny does not necessarily guarantee a center
compatible isogeny $\sigma^{*}B\to B$ for the simple factors $B$ of $A$.
However, in the isotypic case we obtain the following result.

######  Lemma 7.7.

Suppose that $A$ is an isotypic abelian variety over $k$. If there exists a
center compatible isogeny $\sigma^{*}A\to A$, then there exists a center
compatible isogeny $\sigma^{*}B\to B$.

It is crucial for our applications in $\mathsection$7.1.3 that the statement
of Lemma 7.7 does not depend on how $\sigma$ acts on an isogeny $A\to B^{n}$
or on $\textup{End}(A)$. Luckily, these dependences ‘cancel out’ as we shall
see in the proof given at the end of this section.

##### Fully compatible isogenies.

We say that an isogeny $\mu:\sigma^{*}A\to A$ is fully compatible if any
$f\in\textup{End}(A)$ satisfies $\mu\sigma^{*}(f)=f\mu.$ In the isotypic case,
one can always modify center compatible isogenies by an inner automorphism
(obtained from the Skolem–Noether theorem) to get fully compatible isogenies.
We first consider the simple case.

######  Lemma 7.8.

Let $B$ be a simple abelian variety over $k$. If there exists a center
compatible isogeny $\sigma^{*}B\to B$, then there exists a fully compatible
isogeny $\sigma^{*}B\to B$.

###### Proof.

We may and do assume that $B\neq 0$. To modifiy a center compatible isogeny
$\tau:\sigma^{*}B\to B$, we use a trick which we learned from Ribet [Rib92]
and Pyle [Pyl04]. In fact their trick works exactly the same in our setting.
The details are as follows:

Our $B$ is simple. Thus $R=\textup{End}^{0}(B)$ is a central simple algebra
over its center $F$. Further $r\mapsto(\sigma^{-1})^{*}(\tau^{-1}\circ
r\circ\tau)$ defines an $F$-algebra morphism $\phi:R\to R$, since $\tau$ is
center compatible, $F=Z\otimes_{\mathbb{Z}}\mathbb{Q}$ for $Z$ the center of
$\textup{End}(B)$, $(\sigma^{-1})^{*}$ is a ring morphism and $\circ$ is a
pairing. Hence Skolem–Noether gives $s\in R$ such that $\phi(r)=s\circ r\circ
s^{-1}$, and as $B$ is simple we can take any $\mu:\sigma^{*}B\to B$ with
$\mu\otimes q=\tau\circ\sigma^{*}(s)$ for some $q\in\mathbb{Q}$. ∎

We now use this result to strengthen the statement of Lemma 7.7 in the sense
that we can moreover construct an isogeny $\sigma^{*}A\to A$ which is fully
compatible.

######  Lemma 7.9.

Let $A$ be an isotypic abelian variety over $k$. If there exists a center
compatible isogeny $\sigma^{*}A\to A$, then there exists a fully compatible
isogeny $\sigma^{*}A\to A$.

###### Proof.

We may and do assume that $A\neq 0$ is isotypic. Thus there is an isogeny
$\varphi:A\to B^{n}$ with $n\in\mathbb{Z}_{\geq 1}$ and $B$ a simple abelian
variety over $k$. Let $\varphi^{\prime}:B^{n}\to A$ be its inverse isogeny as
in (6.2). Further there is a center compatible isogeny $\sigma^{*}A\to A$ by
assumption. Hence Lemma 7.7 provides a center compatible isogeny
$\sigma^{*}B\to B$, and then Lemma 7.8 gives a fully compatible isogeny
$\mu:\sigma^{*}B\to B$ since $B$ is simple. After identifying
$\sigma^{*}B^{n}$ with $(\sigma^{*}B)^{n}$, we see that $\mu$ defines a fully
compatible isogeny $\mu^{n}:\sigma^{*}B^{n}\to B^{n}$. Now we consider

$\tau:\sigma^{*}A\to^{\sigma^{*}(\varphi)}\sigma^{*}B^{n}\to^{\mu^{n}}B^{n}\to^{\varphi^{\prime}}A.$

Then $\tau$ is an isogeny, since it is a composition of isogenies. Further
$\varphi^{\prime}\varphi$ and $\varphi\varphi^{\prime}$ are both
multiplication with $d=\deg(\varphi)$, and $[d]$ commutes with any morphism of
group schemes. Thus we deduce that $\tau\sigma^{*}(f)[d]=[d]f\tau=f\tau[d]$
for all $f\in\textup{End}(A)$, since $\sigma^{*}$ is a functor and since
$\mu^{n}$ is fully compatible. Then, on using that
$\textup{Hom}(\sigma^{*}A,A)$ is torsion-free, we conclude that the isogeny
$\tau:\sigma^{*}A\to A$ is fully compatible as desired. ∎

It remains to verify the statement of Lemma 7.7. The following proof consists
of checking various compatibilities (using standard arguments) in order to
assure that a certain explicitly constructed isogeny $\tau:\sigma^{*}B\to B$
is indeed center compatible.

###### Proof of Lemma 7.7.

We may and do assume that $A$ is nonzero. By assumption there exists a center
compatible isogeny $\mu:\sigma^{*}A\to A$ and there is an isogeny
$\varphi:A\to B^{n}$ where $B$ is a simple abelian variety over $k$ and
$n\in\mathbb{Z}_{\geq 1}$. Let $\varphi^{\prime}:B^{n}\to A$ be the ‘inverse
isogeny’ of $\varphi$ as in (6.2) and let $\iota:B\to B^{n}$ be the diagonal
embedding. Now, we define

$t:\sigma^{*}B\to^{\sigma^{*}(\iota)}\sigma^{*}B^{n}\to^{\sigma^{*}(\varphi^{\prime})}\sigma^{*}A\to^{\mu}A\to^{\varphi}B^{n}.$

The base change $\sigma^{*}(\iota)$ of $\iota$ is injective, since $\iota$ is
universally injective. Then, on using that $\sigma^{*}(\varphi^{\prime})$,
$\mu$ and $\varphi$ are all isogenies, we deduce that $t$ has finite kernel
and thus is nonzero since $\sigma^{*}B\neq 0$; here we used our assumption
$A\neq 0$ to assure that $B\neq 0$. Hence there is a projection $p:B^{n}\to B$
such that $t$ composed with $p$ is a nonzero morphism

$\tau:\sigma^{*}B\to B$

of abelian varieties over $k$. Moreover $\tau\neq 0$ is surjective since $B$
is simple. Then, on using that $\sigma^{*}B$ and $B$ have the same dimension,
we conclude that $\tau$ is an isogeny.

To verify that $\tau$ is center compatible, we take $z$ in the center $Z$ of
$\textup{End}(B)$. We view $\textup{End}(B)$ as a subring of
$\textup{End}(B^{n})$ via the diagonal embedding $\iota$ and we observe that
this identifies $Z$ with the center of $\textup{End}(B^{n})$. A direct
computation gives the identities

$\varphi\circ\varphi^{*}(z)=z\circ\varphi\quad\textnormal{ and
}\quad\varphi^{\prime}\circ z=\varphi^{*}(z)\circ\varphi^{\prime}$ (7.6)

inside $\textup{Hom}^{0}(A,B^{n})$ and inside $\textup{Hom}^{0}(B^{n},A)$
respectively. Here we used the definition of the $\mathbb{Q}$-algebra
isomorphism
$\varphi^{*}:\textup{End}^{0}(B^{n})\overset{\sim}{\to}\textup{End}^{0}(A)$ in
(6.2). Pulling back by $\sigma$ is a functor. Hence (7.6) shows that
$\sigma^{*}(\varphi^{\prime})\circ\sigma^{*}(z)$ equals
$\sigma^{*}(\varphi^{*}(z))\circ\sigma^{*}(\varphi^{\prime})$, and
$\sigma^{*}(\iota)$ commutes with $\sigma^{*}(z)$ since $\iota z=z\iota$.
Therefore, on using that $\mu$ is center compatible and that $\varphi^{*}(z)$
lies in the center of $\textup{End}^{0}(A)$, we deduce from (7.6) the identity

$t\circ\sigma^{*}(z)=z\circ t$

inside $\textup{Hom}^{0}(\sigma^{*}B,B^{n})$. This identity then also holds
inside the torsion-free abelian group $\textup{Hom}(\sigma^{*}B,B^{n})$, and
we observe that $pz=zp$. It follows that $\tau\sigma^{*}(z)=z\tau$ for all
$z\in Z$, which means that $\tau$ is center compatible. This completes the
proof. ∎

###  7.2 Abelian varieties with CM

In this section we prove explicit upper bounds for the Faltings height of
certain abelian varieties with CM. For this purpose we first recall the
averaged Colmez conjecture in $\mathsection$7.2.1. Then we work out in
$\mathsection$7.2.2 explicit analytic estimates for certain L-values and we
control in $\mathsection$7.2.3 the ramification of CM endomorphism algebras.

##### CM types and abelian varieties.

We now introduce some terminology which will be used throughout this section.
A CM field is a number field which has no real embeddings but which is
quadratic over a totally real subfield. Let $E$ be a CM field and let
$F\subset E$ be the maximal totally real subfield. A CM type $\Phi$ of $E$ is
a subset $\Phi\subset\textup{Hom}(E,\mathbb{C})$ such that
$\Phi\sqcup\Phi\tau=\textup{Hom}(E,\mathbb{C})$ where
$\tau\in\textup{Aut}(E/F)$ is not trivial. We say that a CM type of $E$ is
simple if it is not the full preimage under restriction of a CM type of a
proper CM subfield of $E$. For any $\varphi:E\to\mathbb{C}$, we write
$\mathbb{C}_{\varphi}$ for $\mathbb{C}$ viewed as an $E$-algebra via
$\varphi$.

Let $A$ be an abelian variety defined over a subfield $k$ of $\mathbb{C}$.
Suppose that $2\dim(A)=[E:\mathbb{Q}]$ and assume that there exists a ring
morphism $\iota:E\to\textup{End}^{0}(A)$. Then $\iota$ induces an action of
$E\otimes_{\mathbb{Q}}\mathbb{C}\cong\prod_{\varphi}\mathbb{C}_{\varphi}$ on
the Lie algebra $\textnormal{Lie}(A_{\mathbb{C}})$ of $A_{\mathbb{C}}$. This
action decomposes $\textnormal{Lie}(A_{\mathbb{C}})$ into a corresponding
product of eigenspaces $\textnormal{Lie}(A_{\mathbb{C}})_{\varphi}$ over
$\mathbb{C}$. Let $\Phi_{A}$ be the set of
$\varphi\in\textup{Hom}(E,\mathbb{C})$ such that the eigenspace
$\textnormal{Lie}(A_{\mathbb{C}})_{\varphi}$ has dimension one. Then
$\Phi_{A}$ is a CM type of $E$ which is simple if and only if $A_{\mathbb{C}}$
is a simple abelian variety; see for example [CCO14, $\mathsection$1.5]. We
call $\Phi_{A}$ the CM type of $E$ associated to $(A,\iota)$.

For any CM type $\Phi$ of $E$, we denote by $A_{\Phi}$ the usual complex
abelian variety associated to $\Phi$ which is defined for example in [CCO14,
1.5.3]. The complex abelian variety $A_{\Phi}$ has the following properties:
There exists a ring morphism
$m:\mathcal{O}_{E}\hookrightarrow\textup{End}(A_{\Phi})$ such that $\Phi$ is
the CM type of $E$ associated to $(A_{\Phi},\iota)$ for
$\iota=m\otimes\mathbb{Q}$ and
$A_{\Phi}(\mathbb{C})\cong\mathbb{C}^{g}/\Phi(\mathcal{O}_{E}).$ Here
$g=\dim(A_{\Phi})$ and $\Phi(\mathcal{O}_{E})$ denotes the lattice in
$\mathbb{C}^{g}$ given by the image of $\mathcal{O}_{E}$ under the embedding
$E\hookrightarrow\mathbb{C}^{g}$ defined by the $g$ distinct embeddings
$E\hookrightarrow\mathbb{C}$ in $\Phi$.

####  7.2.1 Averaged Colmez conjecture

To state the averaged Colmez conjecture, let $E$ be a CM field with maximal
totally real subfield $F\subset E$ and let $\Phi$ be a CM type of $E$. We
write $g=[F:\mathbb{Q}]$.

##### The height $h(\Phi)$.

Let $A_{\Phi}$ be the complex abelian variety associated to $\Phi$. Any
complex abelian variety with CM is defined over a number field and then we
define $h(\Phi)=h_{F}(A_{\Phi}).$ Here the invariance of the stable Faltings
height $h_{F}$ under geometric isomorphisms assures that $h(\Phi)$ does not
depend on the choice of a model of $A_{\Phi}$ over a number field. Moreover,
this invariance of $h_{F}$ together with Colmez’ [Col93, Thm 0.3 (ii)] implies
that

$h(\Phi)=h_{F}(A)$ (7.7)

for any abelian variety $A$ of dimension $g$ which is defined over a number
field $K\hookrightarrow\mathbb{C}$ and which has the following property: There
is a ring morphism $j:\mathcal{O}_{E}\to\textup{End}(A_{\mathbb{C}})$ such
that $\Phi$ is the CM type of $E$ associated to
$(A_{\mathbb{C}},j\otimes\mathbb{Q})$. Yuan–Zhang obtained in [YZ18,
$\mathsection$2] an alternative proof of (7.7) as a by-product of their height
decomposition.

##### The formula.

Define $\mathfrak{c}=D_{E}/D_{F}$ and let $L^{\prime}(s)$ be the derivative of
the L-function $L(s)$ of the quadratic character of $\mathbb{A}_{F}^{\times}$
corresponding to $E/F$; here $L(s)$ is without the infinite part. Now we can
state the averaged Colmez conjecture ([Col93]) which in the case $g=1$ is a
reformulation of the classical Lerch–Chowla–Selberg formula:

$\tfrac{1}{2^{g}}\sum_{\Phi}h(\Phi)=-\tfrac{1}{2}\tfrac{L^{\prime}(0)}{L(0)}-\tfrac{1}{4}\log\mathfrak{c}-\tfrac{g}{2}\log\pi$
(7.8)

with the sum taken over all distinct CM types $\Phi$ of $E$. Here the term
$-\tfrac{g}{2}\log\pi$ takes into account that we use Faltings’ normalization
of the height $h_{F}$. Colmez [Col93], Obus [Obu13] and Yang [Yan10, Yan13]
proved important special cases of the exact version [Col93, Conj 0.4] of
(7.8). In general (7.8) was established by Yuan–Zhang [YZ18] using the method
of Yuan–Zhang–Zhang [YZZ13] and independently by
Andreatta–Goren–Howard–Madapusi-Pera [AGHMP18] using the method of Yang
[Yan10, Yan13].

##### An explicit upper bound.

Colmez’ conjectures give useful bounds for the height $h(\Phi)$ of any CM type
$\Phi$ of $E$. For example Colmez [Col98, $\mathsection$3] bounded $h(\Phi)$
conditional on the exact version of (7.8) and other conjectures. Further
Colmez [Col98, p.365] for $g=1$ and Tsimerman [Tsi18, Cor 3.3] in general
deduced from (7.8) the asymptotic bound

$h(\Phi)\leq D_{E}^{o_{g}(1)}.$

The proof uses analytic estimates for $L^{\prime}(0)/L(0)$ via Brauer–Siegel
and the lower bound $h(\Phi)\geq c_{g}$ which is contained in Faltings’ work
([Fal83, p.356], [Del85, 1.22]). In fact one can take here
$c_{g}=-\tfrac{g}{2}\log(2\pi^{2})$ by Bost’s lower bound (7.10), and the
analytic estimates (except Brauer–Siegel) applied in [Tsi18, Cor 3.3] can be
worked out explicitly by using Rademacher’s arguments in [Rad60]. This leads
to the following result.

######  Lemma 7.10.

Let $h_{E}$ and $h_{F}$ be the class number of $E$ and $F$ respectively. If
$\epsilon>0$ is a real number with $\epsilon\leq 1/2$, then any CM type $\Phi$
of $E$ satisfies

$\tfrac{1}{2^{g}}h(\Phi)\leq
a\tfrac{h_{F}}{h_{E}}\mathfrak{c}^{1/2}\max\bigl{(}(\tfrac{2\pi}{2+\epsilon})^{g},\mathfrak{c}\bigl{)}^{\epsilon}+\tfrac{1}{4}\log\mathfrak{c}-\tfrac{\gamma}{2}g.$

Here
$a=\tfrac{wQ}{2\epsilon}\zeta(1+\epsilon)\tfrac{(2+\epsilon)^{g\epsilon}}{(2\pi)^{g(\epsilon+1)}}$
where $\zeta(s)$ is the Riemann zeta function, $\gamma$ is the
Euler–Mascheroni constant, $w$ is the cardinality of the group $\mu$ of roots
of unity in $E$ and
$Q=[\mathcal{O}_{E}^{\times}:\mu\mathcal{O}_{F}^{\times}]\in\\{1,2\\}$ is the
Hasse unit index of $E/F$.

###### Proof.

On combining $h(\Phi)\geq-\tfrac{g}{2}\log(2\pi^{2})$ in (7.10) with the
explicit analytic estimates worked out in (the proof of) Lemma 7.11, we see
that (7.8) implies Lemma 7.10. ∎

Class field theory gives that $h_{F}$ divides $h_{E}$ and then Lemma 7.10
provides an explicit upper bound for $h(\Phi)$ in terms of $\epsilon$, $D_{E}$
and $g$. This bound is sufficient for our purpose, but it is asymptotically
substantially weaker than Tsimerman’s bound $h(\Phi)\leq D_{E}^{o_{g}(1)}$.

####  7.2.2 Explicit analytic estimates

We now work out the explicit analytic estimate which we used in Lemma 7.10
above. As before, $\zeta(s)$ is the Riemann zeta function and $\gamma$ is the
Euler–Mascheroni constant.

######  Lemma 7.11.

Let $\epsilon>0$ be a real number with $\epsilon\leq 1/2$, let $E$ be a CM
field with maximal totally real subfield $F$, and let $L^{\prime}(s)$ be the
derivative of the L-function $L(s)$ of the character of
$\mathbb{A}_{F}^{\times}$ corresponding to $E/F$. Put $g=[F:\mathbb{Q}]$ and
$\mathfrak{c}=D_{E}/D_{F}$.

  * (i)

If the real part $\sigma$ of $s\in\mathbb{C}$ satisfies
$-\epsilon\leq\sigma\leq 1+\epsilon$ then

$|L(s)|\leq\zeta(1+\epsilon)\left(\tfrac{\mathfrak{c}}{(2\pi)^{g}}|1+s|^{g}\right)^{\tfrac{1+\epsilon-\sigma}{2}}.$

  * (ii)

Let $h_{E}$ and $h_{F}$ be the class number of $E$ and $F$ respectively, let
$w$ be the number of roots of unity in $E$ and let $Q$ be the Hasse unit index
of $E/F$. It holds

$\left|\tfrac{L^{\prime}(0)}{L(0)}\right|\leq
2a\tfrac{h_{F}}{h_{E}}\mathfrak{c}^{1/2}\max\bigl{(}(\tfrac{2\pi}{2+\epsilon})^{g},\mathfrak{c}\bigl{)}^{\epsilon}+\log\mathfrak{c}+b,$

where
$a=\tfrac{wQ}{2\epsilon}\zeta(1+\epsilon)\tfrac{(2+\epsilon)^{g\epsilon}}{(2\pi)^{g(\epsilon+1)}}$
and $b=g\log(2\pi e^{\gamma})$.

Here (ii) is a standard consequence of (i) and the analytic class number
formula, while the bound in (i) is up to a factor $\zeta(1+\epsilon)^{g-1}$
due to Rademacher [Rad60, Thm 5]. In fact we shall obtain (i) by applying his
arguments with our entire Artin L-function $L(s)$.

###### Proof of Lemma 7.11.

Let $\epsilon>0$ be a real number with $\epsilon\leq 1/2$. We first introduce
some notation. For each $s\in\mathbb{C}$ we write $s=\sigma+it$ with
$\sigma,t\in\mathbb{R}$ and we denote by $S(a,b)$ the strip $a\leq\sigma\leq
b$ for $a,b\in\mathbb{R}$ with $a<b$. Let $\chi:G\to\mathbb{C}$ be the non-
trivial irreducible character of
$G=\textup{Aut}(E/F)\cong\mathbb{Z}/2\mathbb{Z}$. In particular $\chi$ has
degree one with image $\chi(G)=\\{-1,1\\}$, and $L(s)$ coincides with the
Artin L-function of $\chi$.

We now prove (i). In the case $\sigma\geq 1+\epsilon$, it holds $\log
L(s)=\sum_{p}\sum_{n\geq 1}\tfrac{1}{n}\chi(p^{n})p^{-ns}$ with
$\chi(p^{n})\in\mathbb{Z}$ satisfying $|\chi(p^{n})|\leq 1$ and hence $|\log
L(s)|\leq\log\zeta(\sigma)$ which implies

$|L(s)|\leq\zeta(1+\epsilon).$ (7.9)

Let $\Gamma(s)$ be the Gamma function. The field $E$ is CM with maximal
totally real subfield $F$. Thus the conductor $\mathfrak{c}$ of the non-
trivial character $\chi$ of $G$ is given by $\mathfrak{c}=D_{E}/D_{F}$ and the
completed Artin L-function $\Lambda(s)$ takes the following form

$\Lambda(s)=\mathfrak{c}^{s/2}L_{\infty}(s)L(s),\quad
L_{\infty}(s)=\pi^{-g(s+1)/2}\Gamma(\tfrac{s+1}{2})^{g}.$

As the character $\chi$ is fixed by complex conjugation, the functional
equation becomes $\Lambda(s)=W(\chi)\Lambda(1-s)$ for some
$W(\chi)\in\mathbb{C}$ with absolute value 1. Therefore we obtain

$L(s)=W(\chi)\left(\frac{\mathfrak{c}}{\pi^{g}}\right)^{1/2-s}\left(\frac{\Gamma(1-\tfrac{s}{2})}{\Gamma(\tfrac{s+1}{2})}\right)^{g}L(1-s).$

Next, an application of Rademacher’s [Rad60, Lem 2] with $Q=1$ gives for all
$s$ in the strip $S(-\tfrac{1}{2},\tfrac{1}{2})$ that
$|\Gamma(1-\tfrac{s}{2})/\Gamma(\tfrac{s+1}{2})|$ is at most
$\left(\tfrac{1}{2}|1+s|\right)^{1/2-\sigma}$. Then, as $0<\epsilon\leq 1/2$
by assumption, the above displayed expression for $L(s)$ together with (7.9)
shows

$|L(-\epsilon+it)|\leq A|1-\epsilon+it|^{(1/2+\epsilon)g},\quad
A=\bigl{(}\tfrac{\mathfrak{c}}{(2\pi)^{g}}\bigl{)}^{1/2+\epsilon}\zeta(1+\epsilon).$

Further (7.9) provides $|L(1+\epsilon+it)|\leq\zeta(1+\epsilon)$, and the
Artin $L$-function $L(s)$ of our non-trivial $\chi$ is entire since $G$ is
abelian. Then an application of Rademacher’s version [Rad60, Thm 2] of the
Phragmén–Lindelöf theorem with $Q=1$, $\alpha=(1/2+\epsilon)g$ and $\beta=0$
gives for all $s$ in the strip $S(-\epsilon,1+\epsilon)$ an upper bound for
$|L(s)|$ as claimed in (i).

We now prove (ii). For this purpose we take the radius $r=\epsilon$ and the
entire function $f(z)=L(z)$ in the basic inequality
$|f^{\prime}(1)|r\leq\sup_{|z-1|=r}|f(z)|$ which follows from Cauchy’s
integral formula. This together with (i) leads to

$|L^{\prime}(1)|\leq\tfrac{1}{\epsilon}\zeta(1+\epsilon)\max\bigl{(}1,\tfrac{\mathfrak{c}}{(2\pi)^{g}}(2+\epsilon)^{g}\bigl{)}^{\epsilon}.$

We next express $L(1)$ in terms of class numbers. As
$G\cong\mathbb{Z}/2\mathbb{Z}$, the Artin L-function $L(s)$ of the non-trivial
$\chi$ is the quotient $L(s)=\zeta_{E}(s)/\zeta_{F}(s)$ of the Dedekind zeta
functions of $E$ and $F$. Then the analytic class number formula and [Was82,
Prop 4.16] imply

$L(1)=\tfrac{(2\pi)^{g}}{wQ}\tfrac{h_{E}}{h_{F}}\mathfrak{c}^{-1/2}.$

To relate the values $L^{\prime}(0)/L(0)$ and $L^{\prime}(1)/L(1)$, we
differentiate the functional equation $\Lambda(s)=W(\chi)\Lambda(1-s)$ and we
use that the Digamma function $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ has
special values $\psi(1/2)=-2\log 2-\gamma$ and $\psi(1)=-\gamma$ respectively.
Then we obtain

$-\tfrac{L^{\prime}(0)}{L(0)}=\tfrac{L^{\prime}(1)}{L(1)}+\log\mathfrak{c}-g\log(2\pi
e^{\gamma}).$

Finally, on combining the above displayed results we deduce an upper bound for
$|L^{\prime}(0)/L(0)|$ as claimed in (ii). This completes the proof of Lemma
7.11. ∎

####  7.2.3 Ramification of CM endomorphism algebras

The endomorphism algebra of a simple complex abelian variety with CM is a CM
field whose ramification over $\mathbb{Q}$ can be controlled by CM theory. The
goal of this subsection is to prove the following result in which
$N_{S}=\prod_{v\in S^{c}}N_{v}$ is as above.

######  Lemma 7.12.

Let $K$ be a number field, let $A$ be an abelian scheme over a nonempty open
subscheme $S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ and let $B$ be a
simple factor of $A_{\bar{K}}$ such that $B$ has CM.

  * (i)

The CM field $E=\textup{End}^{0}(B)$ is unramified at each rational prime
$p\nmid D_{K}N_{S}$.

  * (ii)

If $B$ and all its endomorphisms are defined over $K\subset\bar{K}$, then the
CM field $E$ is unramified at each rational prime $p\nmid D_{K}$.

###### Proof.

Let $p$ be a rational prime. To show that (ii) implies (i), we assume that
(ii) holds. Let $L$ be the field of definition of the endomorphisms of
$A_{\bar{K}}$. Poincaré’s reducibility theorem implies that $B$ is isogenous
to an abelian subvariety $B^{\prime}$ of $A_{\bar{K}}$ and that $B^{\prime}$
and all its endomorphisms are defined over $L$. Further $B^{\prime}$ is a
simple factor of $A_{\bar{K}}$ and $B^{\prime}$ has CM. Then an application of
(ii) with $B^{\prime}$ gives that $E\cong\textup{End}^{0}(B^{\prime})$ is
unramified at $p$ if $p\nmid D_{L}$. Further [Sil92, Thm 4.1] implies that
$L/K$ is unramified over $S$, since the generic fiber of the abelian scheme
$A$ over $S$ is an abelian variety over $K$ with good reduction over $S$. Thus
$\textnormal{rad}(D_{L})$ divides $D_{K}N_{S}$ by Dedekind’s discriminant
theorem and therefore we conclude that $E$ is unramified at $p$ if $p\nmid
N_{S}D_{K}$ as claimed in (i).

We now prove (ii). By assumption $B$ is a simple abelian variety with CM.
Hence the CM type $\Phi$ of $E$ associated to $(B,\textnormal{id})$ is simple.
The reflex field $E^{*}$ of $(E,\Phi)$ is defined by
$E^{*}=\mathbb{Q}(\sum_{\varphi\in\Phi}\varphi(e);e\in E)$, and [Lan83, Thm
3.1.1] gives $E^{*}\hookrightarrow K$ since $B$ and all its endomorphisms are
defined over $K$ by assumption. Thus the Galois closure
$(E^{*})^{\textnormal{cl}}$ of $E^{*}$ is unramified at $p$ if $p\nmid D_{K}$.
Further, an application of [Lan83, Thm 1.5.2] with our simple $\Phi$
identifies $E$ with the reflex field of $(E^{*},\Phi^{*})$ for some CM type
$\Phi^{*}$ of $E^{*}$ and thus $E\hookrightarrow(E^{*})^{\textnormal{cl}}$.
Hence $E$ is unramified at $p$ if $p\nmid D_{K}$ as claimed in (ii).∎

###  7.3 Proof of the effective Shafarevich conjecture

Let $K$ be a number field, let $S$ be a nonempty open subscheme of
$\textnormal{Spec}({\mathcal{O}_{K}})$ and let $g\geq 1$ be a rational
integer. Following the strategy of proof outlined in the introduction, we
prove in this section the effective Shafarevich conjecture (see Theorem 7.1)
for abelian varieties over $K$ which are of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies or which have CM.

##### No CM case.

In the first part of the proof of Theorem 7.1, we consider abelian varieties
which have no CM. In Section 7.3.2 we prove the following result by reducing
the problem via Proposition 7.2 to the known case $K=\mathbb{Q}$ stated in
(7.16).

######  Proposition 7.13.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$ with no CM.
Suppose that $A$ is of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies and assume $(*)$. Then

$h_{F}(A)\leq(4gd)^{144gd}\textnormal{rad}(D_{K}N_{S})^{24}.$

Here $h_{F}$ is the stable Faltings height ($\mathsection$3.4), while $D_{K}$,
$d=[K:\mathbb{Q}]$ and $N_{S}=\prod_{v\in S^{c}}N_{v}$ are as before. Further,
we recall that condition $(*)$ means that $K/\mathbb{Q}$ is normal and that
all endomorphisms of $A_{\bar{K}}$ and all isogenies $\mu_{\sigma}$ in (7.1)
are defined over $K\subset\bar{K}=\bar{\mathbb{Q}}$.

##### CM case.

In the second part of the proof of Theorem 7.1, we deal with the general CM
case by reducing to the situation $A=A_{\Phi}$ which was treated in Lemma
7.10. In course of the proof ($\mathsection$7.3.3) we shall obtain the
following bound which only depends on $K$ and $g$.

######  Proposition 7.14.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$ such that
$A_{\bar{K}}$ has CM. If all endomorphisms of $A_{\bar{K}}$ are defined over
$K\subset\bar{K}$, then

$h_{F}(A)\leq(3g)^{(5g)^{2}}\textnormal{rad}(D_{K})^{5g}.$

In the remaining of this section, we first review some known results and then
we give the proofs of Proposition 7.13, Proposition 7.14 and Theorem 7.1.

####  7.3.1 Properties of the Faltings height and isogeny estimates

Let $K$ be a number field and let $A$ be an abelian variety over $K$ of
dimension $g$. In this section we review properties of the stable Faltings
height $h_{F}$ of $A$ used in our proofs. In particular, we collect effective
bounds for the variation of $h_{F}$ under isogenies.

We start with the following basic property: Any abelian variety $A^{\prime}$
over $K$ satisfies $h_{F}(A\times_{K}A^{\prime})=h_{F}(A)+h_{F}(A^{\prime})$.
Further, Faltings’ work [Fal83] gives that $h_{F}(A)\geq c_{g}$ for some
constant $c_{g}$ depending only $g$; see [Del85, 1.22]. In fact Bost [Bos96a]
obtained

$h_{F}(A)\geq-\tfrac{1}{2}\log(2\pi^{2})g.$ (7.10)

See for example [GR14b, Cor 8.4] and notice that
$h_{F}(A)=h_{B}(A)-\frac{g}{2}\log\pi$ where $h_{B}$ denotes the height which
appears in the statement of [GR14b, Cor 8.4].

##### Variation under isogenies.

We now suppose that $A^{\prime}$ is an abelian variety over $K$ which is
isogenous to $A$. Faltings’ formula [Fal83, Lem 5] implies that

$|h_{F}(A)-h_{F}(A^{\prime})|\leq\tfrac{1}{2}\log\deg(\varphi)$ (7.11)

for any isogeny $\varphi:A\to A^{\prime}$. Moreover, Faltings [Fal83] bounded
$|h_{F}(A)-h_{F}(A^{\prime})|$ in terms of $A/K$ alone. On using Faltings’
arguments and some refinements of Paršin–Zarhin, Raynaud [Ray85, Thm 4.4.9]
worked out an effective bound for the variation under the assumption that $A$
has everywhere semi-stable reduction, that is

$|h_{F}(A)-h_{F}(A^{\prime})|\leq c_{A}$ (7.12)

with $c_{A}$ an effective constant (simplified in [vK21, Prop 6.10]) depending
only on $g$, $D_{K}$, $[K:\mathbb{Q}]$ and the bad reduction places of $A$. In
fact the semi-stable assumption is not required here ([vK21, Lem 3.1]). The
above bounds for the variation all do not use Diophantine approximation or
transcendence techniques. Masser–Wüstholz [MW93b, MW93a, MW95] obtained bounds
via a completely different approach using their isogeny estimates based on
transcendence theory. These isogeny estimates are fully effective ([Bos96b,
p.121]), and completely explicit constants were obtained by Gaudron–Rémond
[GR14a]: Their most recent version [GR23, Thm 1.9] gives isogenies $A\to
A^{\prime}$ and $A^{\prime}\to A$ of degree at most $\kappa(A)$ and this
combined with (7.11) proves

$\lvert
h_{F}(A)-h_{F}(A^{\prime})\rvert\leq\tfrac{1}{2}\log\kappa(A),\quad\kappa(A)=\bigl{(}(7g)^{8g^{2}}d\max(h_{F}(A),\log
d,1)\bigl{)}^{4g^{2}}$ (7.13)

for $d=[K:\mathbb{Q}]$; one can replace here $\kappa(A)$ by $\kappa(A)^{1/2}$
if $\textup{End}(A)$ identifies with a maximal order of $\textup{End}^{0}(A)$.
We remark that (7.12) improves (7.13) in some cases, and vice versa in other
cases. However, in the situations considered in this paper, it turns out that
(7.13) leads to exponentially better bounds for the same reasons as in [vK21,
Rem 5.2].

####  7.3.2 Proof of Proposition 7.13

We continue our notation. Let $K$ be a number field and let
$S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ be a nonempty open subscheme.
As before we write $d=[K:\mathbb{Q}]$ and $N_{S}=\prod_{v\in S^{c}}N_{v}$. Let
$A$ be an abelian scheme over $S$ of relative dimension $g$ which has no CM.
We suppose that $A$ is of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies and we assume that condition $(*)$ is satisfied.

###### Proof of Proposition 7.13.

Let $U\subseteq S$ and $T\subseteq\textnormal{Spec}(\mathbb{Z})$ be the open
subschemes defined in (7.3) with $k=\mathbb{Q}$. Our $A$ satisfies all the
assumptions in Proposition 7.2. Thus this proposition gives a simple abelian
scheme $C$ over $T$ of $\textnormal{GL}_{2}$-type such that $A_{U}$ is
isogenous to a power $B^{m}$ of a simple factor $B$ of $C_{U}$. As $B$ is a
factor of $C_{U}$ and $U$ is a connected Dedekind scheme, Poincaré’s
reducibility theorem and (6.1) provide an abelian scheme $D$ over $U$ such
that $B\times_{U}D$ is isogenous to $C_{U}$. Now, on using isogenies

$A_{U}\to B^{m}\quad\textnormal{and}\quad B\times_{U}D\to C_{U},$

we can estimate the heights as in [vK21, $\mathsection$5.2.2]. For any abelian
scheme $X$ over an open $S^{\prime}\subseteq S$, we denote by $v_{X}$ the
maximal variation of $h_{F}$ in the isogeny class of $X$; that is
$v_{X}=\sup\lvert h_{F}(X)-h_{F}(X^{\prime})\rvert$ with the supremum taken
over all abelian schemes $X^{\prime}$ over $S^{\prime}$ which are isogenous to
$X$. The abelian scheme $A^{\prime}=B^{m}$ over $U$ satisfies

$h_{F}(A^{\prime})=mh_{F}(B)\ \ \textnormal{ and }\ \ h_{F}(A)\leq
v_{A^{\prime}}+h_{F}(A^{\prime})$ (7.14)

since $A_{U}$ is isogenous to $A^{\prime}$. The stable height $h_{F}$ is
invariant under the base change $U\to T$, and $B\times_{U}D$ is isogenous to
$C_{U}$. Thus we obtain that $h_{F}(B)$ is at most
$v_{C_{U}}+h_{F}(C)-h_{F}(D)$ and then we see that the lower bound for
$h_{F}(D)$ in (7.10) implies

$h_{F}(B)\leq v_{C_{U}}+h_{F}(C)+\tfrac{g}{2}\log(2\pi^{2}).$ (7.15)

Here we used that the relative dimension of $D$ over $U$ is at most $g$. The
relative dimension of $B$ over $U$ is $g/m$. Thus the relative dimension $n$
of $C$ over $T$ satisfies $n\leq dg/m$ by (7.4), and applying [vK21, Thm A]
with the abelian scheme $C$ over $T$ of $\textnormal{GL}_{2}$-type gives

$h_{F}(C)\leq(3n)^{144n}N_{T}^{24}.$ (7.16)

As $N_{T}=\textnormal{rad}(D_{K}N_{S})$ we then see that (7.13), (7.14) and
(7.15) lead to an upper bound for $h_{F}(A)$ as claimed in Proposition 7.13.
We now include some details of our computations described above. For example,
one can use the inequalities

$v_{C_{U}}\leq\tfrac{1}{4}c,\quad h_{F}(B)\leq\tfrac{3}{2}c,\quad
h_{F}(A^{\prime})\leq\tfrac{3}{2}gc\ \textnormal{ and }\
v_{A^{\prime}}\leq\tfrac{1}{2}c$

as intermediate steps to deduce that $h_{F}(A)\leq 2gc$ where
$c=(3gd)^{144gd}N_{T}^{24}$.∎

####  7.3.3 Proof of Proposition 7.14 and Theorem 7.1

We continue our notation. Let $K$ be a number field of degree $d$ over
$\mathbb{Q}$, let $S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ be a
nonempty open subscheme with geometric generic point
$\bar{s}=\textnormal{Spec}(\bar{K})$ and write again $N_{S}=\prod_{v\in
S^{c}}N_{v}$. Let $A$ be an abelian scheme over $S$ of relative dimension $g$.

###### Proof of Proposition 7.14.

As in the statement of Proposition 7.14, we assume that $A_{\bar{s}}$ has CM
and that all endomorphisms of $A_{\bar{s}}$ are defined over
$K\subset\bar{K}$. Then (6.1) gives that
$\textup{End}(A)\overset{\sim}{\to}\textup{End}(A_{\bar{s}})$. We claim that
there exist simple factors $A_{i}$ of $A$ of relative dimension $g_{i}$, CM
fields $E_{i}$ of degree $2g_{i}$ with
$\mathcal{O}_{E_{i}}\cong\textup{End}(A_{i})$ and an isogeny

$A^{\prime}=\prod A_{i}^{e_{i}}\to A,\quad e_{i}\in\mathbb{Z}_{\geq 1}.$
(7.17)

To prove this claim, let $A_{i}$ be a simple factor of $A$ and write $g_{i}$
for the relative dimension of $A_{i}$ over $S$. On using that
$\textup{End}(A)\overset{\sim}{\to}\textup{End}(A_{\bar{s}})$, we see that
$\textup{End}(A_{i})\overset{\sim}{\to}\textup{End}(A_{i,\bar{s}})$. Then CM
theory gives that $A_{i}$ has CM since $A_{\bar{s}}$ has CM by assumption. As
$K$ has characteristic zero, this implies that the $\mathbb{Q}$-algebra
$\textup{End}^{0}(A_{i})$ is a CM field $E_{i}$ of degree $2g_{i}$. The order
$\mathcal{O}=\textup{End}(A_{i})$ of $E$ is contained in the maximal order of
$E$ which is $\mathcal{O}^{\prime}=\mathcal{O}_{E_{i}}$. Hence an application
of [vKK19, Lem 5.3], which is based on [CCO14, $\mathsection$1.7.4], shows
that after possibly replacing $A_{i}$ by an isogenous abelian scheme over $S$
we may and do assume that $\mathcal{O}_{E_{i}}\cong\textup{End}(A_{i})$. Then
the claim follows from Poincaré’s reducibility theorem and (6.1).

To estimate the heights, we choose an embedding $K\hookrightarrow\mathbb{C}$
and we let $\Phi_{i}$ be the CM type of $E_{i}$ associated to
$(A_{i,\mathbb{C}},j\otimes\mathbb{Q})$ where $j$ is the composition of the
ring morphisms
$\mathcal{O}_{E_{i}}\overset{\sim}{\to}\textup{End}(A_{i})\overset{\sim}{\to}\textup{End}(A_{i,\mathbb{C}})$.
An application of (7.7) with $A_{i,K}$ gives that
$h_{F}(A_{i})=h_{F}(\Phi_{i})$. This implies that $h_{F}(A^{\prime})=\sum
e_{i}h(\Phi_{i})$ and then (7.17) shows

$h_{F}(A)\leq v_{A^{\prime}}+\sum e_{i}h(\Phi_{i}),\quad g=\sum e_{i}g_{i}.$

To bound $h(\Phi_{i})$ in terms of $D_{K}$ and $g$, we first control
$D_{E_{i}}$. As $\textup{End}(A_{i})\cong\textup{End}(A_{i,\bar{s}})$ we see
that $A_{i,\bar{s}}$ has the following properties: It has CM, it is a simple
factor of $A_{\bar{s}}$, it is defined over $K\subset\bar{K}$ and all its
endomorphisms are defined over $K\subset\bar{K}$. Then an application of Lemma
7.12 (ii) with $B=A_{i,\bar{s}}$ gives that the CM field
$E_{i}\cong\textup{End}^{0}(A_{i,\bar{s}})$ of degree $2g_{i}$ is unramified
at each rational prime $p\nmid D_{K}$. Hence [vK13b, Lem 6.2], which is a
direct consequence of Dedekind’s discriminant theorem, implies

$D_{E_{i}}\leq\textnormal{rad}(D_{K})^{5g}(2g)^{24g^{2}}.$

Here we used the effective version of the prime number theorem in [RS62,
(3.12) and (3.16)] which gives that the number $n$ of rational prime divisors
of $D_{K}$ satisfies $n^{n}<\textnormal{rad}(D_{K})^{3/2}$ in the case when
$n>12$. Next, we recall that Lemma 7.10 and the subsequent discussion give an
explicit upper bound for $h(\Phi_{i})$ in terms of $\epsilon$, $D_{E_{i}}$ and
$g_{i}$. We take $\epsilon=1/2$ in this bound and then the above displayed
results together with the bound for $v_{A^{\prime}}$ in (7.13) and $d\leq
3\max(1,\log D_{K})$ lead to an estimate for $h_{F}(A)$ as claimed in
Proposition 7.14.

We now include some details of the explicit computations. For example, to
compute the final bound one can use the following inequalities as intermediate
steps:

$\zeta(\tfrac{3}{2})\leq 3,\quad a_{i}\leq\tfrac{w_{i}}{2}\leq
4g_{i}^{2},\quad
h_{F}(A_{i})=h(\Phi_{i})\leq(3g)^{24g^{2}}\textnormal{rad}(D_{K})^{5g}.$

Here $\zeta(s)$ is the Riemann zeta function and $a_{i},\omega_{i}$ are as in
Lemma 7.10 with $E=E_{i}$; a proof of the bound $w_{i}\leq 8g_{i}^{2}$ can be
found in the discussion right after (8.4). ∎

###### Proof of Theorem 7.1 (ii).

Assume that $A_{\bar{s}}$ has CM and let $L$ be the field of definition of the
endomorphisms of $A_{\bar{s}}$. The height $h_{F}$ is invariant under the base
change from $S$ to $S^{\prime}=S\times_{\mathcal{O}_{K}}\mathcal{O}_{L}$.
Then, on replacing in the proof of Proposition 7.14 the application of part
(ii) of Lemma 7.12 by part (i), we obtain the claimed bound for
$h_{F}(A)=h_{F}(A_{S^{\prime}})$.∎

###### Proof of Theorem 7.1 (i).

Assume first that $A$ is of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies such that $A_{\bar{s}}$ has no CM. Then let
$L\supseteq K$ be the number field constructed in the proof of Lemma 7.6 and
denote by $S^{\prime}$ the preimage of $S$ under the projection
$\textnormal{Spec}(\mathcal{O}_{L})\to\textnormal{Spec}({\mathcal{O}_{K}})$.

We now verify that $A_{S^{\prime}}$ satisfies all assumptions in Proposition
7.13. Lemma 7.6 provides that $(*)$ holds. In particular all endomorphisms of
$A_{\bar{s}}$ are defined over $L$ and then base change induces an isomorphism
$\textup{End}(A_{S^{\prime}})\overset{\sim}{\to}\textup{End}(A_{\bar{s}})$ by
(6.1). This shows that $A_{S^{\prime}}$ has no CM and that $A_{S^{\prime}}$ is
of $\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies, since
$A_{\bar{s}}$ has these two properties by assumption. Hence all assumptions in
Proposition 7.13 are satisfied.

We next control $L$ and $S^{\prime}$. Lemma 7.6 gives that the extension
$L/\mathbb{Q}$ is unramified over $T=\textnormal{Spec}(\mathbb{Z}[1/n])$ for
$n=N_{S}D_{K}$ and that the field $L$ is contained in the Galois closure of
$K(A_{3})/\mathbb{Q}$. Thus Dedekind’s discriminant theorem and basic Galois
theory lead to

$\textnormal{rad}(N_{S^{\prime}}D_{L})\mid\textnormal{rad}(N_{S}D_{K})\quad\textnormal{and}\quad[L:\mathbb{Q}]\leq
l=(d3^{4g^{2}})!.$ (7.18)

It holds that $h_{F}(A)=h_{F}(A_{S^{\prime}})$ since the generic fiber of
$A_{S^{\prime}}$ identifies with $A_{K}\times_{K}L$. Hence (the computation
used in the proof) of Proposition 7.13 provides a bound for
$h_{F}(A)=h_{F}(A_{S^{\prime}})$ in terms of $L$ and $S^{\prime}$ which
together with (7.18) implies

$h_{F}(A)\leq 2g(3gl)^{144gl}\textnormal{rad}(N_{S}D_{K})^{24}.$ (7.19)

This completes the proof of Theorem 7.1 (i) in the case when the abelian
scheme $A$ over $S$ is of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies such that $A_{\bar{s}}$ has no CM.

More generally, we now assume that $A$ is of product
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies. Then $A$ is
isogenous to a product $A^{\prime}=\prod A_{i}$ of abelian schemes $A_{i}$
over $S$ of $\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies. It
holds that $h_{F}(A)\leq v_{A^{\prime}}+\sum h_{F}(A_{i})$ and the relative
dimensions $g_{i}$ of $A_{i}$ over $S$ satisfy $g=\sum g_{i}$. Hence, on
combining the above established case and Theorem 7.1 (ii) with the bound for
$v_{A^{\prime}}$ provided by (7.13), we deduce Theorem 7.1 (i) in general. ∎

###  7.4 Bounding the number of isomorphism classes

Let $K$ be a number field, let
$S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ be nonempty open and let
$g\in\mathbb{Z}_{\geq 1}$. In this section we use the strategy outlined in the
introduction to bound the number of isomorphism classes of abelian schemes
over $S$ of product $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies.

Before we give in $\mathsection$7.4.1 a proof of Theorem B (ii) in full
generality, we consider a relevant special situation where we obtain a better
bound which is used in Theorem 4.1.

######  Proposition 7.15.

Suppose that $K/\mathbb{Q}$ is normal and let $n\in\mathbb{Z}_{\geq 3}$. Then
the number of isomorphism classes of abelian schemes $A$ over $S$ of relative
dimension $g$ such that $A$ is of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies and $K=K(A_{n})$ is at most
$(9gd)^{(4g)^{5}d}\textnormal{rad}(D_{K}N_{S})^{(4g)^{4}}.$

Here $d=[K:\mathbb{Q}]$, $D_{K}$ and $N_{S}=\prod_{v\in S^{c}}N_{v}$ are as
before. We now prove some lemmas which we shall use in the proof of
Proposition 7.15 given below. To control the size of each isogeny class, we
apply minimal isogeny degree estimates for abelian schemes $A$ over $S$ which
are based on transcendence ($\mathsection$7.3.1) and we let
$\kappa=\kappa(A_{K})$ be as in (7.13).

######  Lemma 7.16.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$. Up to
isomorphisms, there exist at most $\kappa^{2g}$ abelian schemes over $S$ which
are isogenous to $A$.

###### Proof.

The statement follows by using the same arguments as in step 3. of the proof
of [vK21, Thm 6.6] where the exponent $2g$ comes from Masser–Wüstholz [MW93a,
Lem 6.1]. The details are as follows: If $A^{\prime}$ is an abelian scheme
over $S$ which is isogenous to $A$, then there exists an isogeny $\varphi:A\to
A^{\prime}$ of degree at most $\kappa$ by Gaudron–Rémond [GR23, Thm 1.9] and
(6.1). As $A^{\prime}$ represents the fppf quotient $A/\ker(\varphi)$, the
number of such $A^{\prime}$ up to isomorphisms is thus at most the number of
subgroups of $A(\bar{K})^{\textnormal{tor}}$ of cardinality at most $\kappa$.
Then [MW93a, Lem 6.1] and
$A(\bar{K})^{\textnormal{tor}}\cong(\mathbb{Q}/\mathbb{Z})^{2g}$ imply Lemma
7.16. ∎

Next, we bound the number of isogeny classes. If $A$ is an abelian scheme over
$S$ of $\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies satisfying
condition $(*)$ as in $\mathsection$7.1.2, then any abelian scheme over $S$
which is isogenous to $A$ is again of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies satisfying $(*)$. Proposition 7.2 and the proof of
[vK21, Thm 6.6] lead to the following result.

######  Lemma 7.17.

Let $\mathcal{C}$ be the set of isogeny classes of abelian schemes $A$ over
$S$ of relative dimension $g$ such that $A$ is of $\textnormal{GL}_{2}$-type
with $G_{\mathbb{Q}}$-isogenies satisfying $(*)$ and such that $A$ has no CM.
It holds $|\mathcal{C}|\leq 9^{9dg}\textnormal{rad}(D_{K}N_{S})^{2}$.

###### Proof.

Let $U\subseteq S$ and $T\subseteq\textnormal{Spec}(\mathbb{Z})$ be the open
subschemes defined in (7.3). For each isogeny class $[A]$ in $\mathcal{C}$,
Proposition 7.2 provides a simple abelian scheme $C$ over $T$ of
$\textnormal{GL}_{2}$-type such that $A_{U}$ is isogenous to a power of a
simple factor $B$ of $C_{U}$. Moreover inequality (7.4) assures that the
abelian scheme $C$ over $T$ has relative dimension at most $dg$. Therefore we
see that $[A]\mapsto[B]$ defines an injective map

$\mathcal{C}\hookrightarrow\mathcal{D}$

into the set $\mathcal{D}$ of isogeny classes of the simple factors of $C_{U}$
where $C$ is a simple abelian scheme over $T$ of relative dimension at most
$dg$ which is of $\textnormal{GL}_{2}$-type. For any $n\in\mathbb{Z}_{\geq 1}$
the arguments in steps 2. and 4. of the proof of [vK21, Thm 6.6] show the
following: There are at most $\tfrac{1}{24}\nu$ distinct isogeny classes of
simple abelian schemes over $T$ of relative dimension $n$ which are of
$\textnormal{GL}_{2}$-type, where $\nu\in\mathbb{Z}$ satisfies $\nu\leq
N_{T}^{2}(2n+1)^{6\rho}$ for $\rho$ the number of rational primes at most
$2n+1$. This implies that $|\mathcal{D}|\leq 9^{9dg}N_{T}^{2}$ since the
explicit prime number theorem in [RS62, (3.6)] gives $\sum_{n\leq
dg}n(2n+1)^{6\rho}\leq 9^{9dg}$. Then the inclusion
$\mathcal{C}\hookrightarrow\mathcal{D}$ together with
$N_{T}=\textnormal{rad}(D_{K}N_{S})$ proves Lemma 7.17. ∎

Next we deal with abelian varieties which have CM. To explicitly bound their
isogeny classes, we reduce to the geometric situation over $\bar{K}$ and then
we apply CM theory which reduces the problem to count CM fields with bounded
degree and discriminant.

######  Lemma 7.18.

If $n\in\mathbb{Z}_{\geq 3}$ then up to isogenies there exist at most
$(2g)^{(4g)^{3}}\textnormal{rad}(D_{K})^{10g^{2}}$ distinct abelian varieties
$A$ over $K$ of dimension $g$ such that $A_{\bar{K}}$ has CM and $K=K(A_{n})$.

###### Proof.

We take $n\in\mathbb{Z}_{\geq 3}$. Let $A$ be an abelian variety over $K$ of
dimension $g$ such that $K=K(A_{n})$ and $A_{\bar{s}}$ has CM where
$\bar{s}=\textnormal{Spec}(\bar{K})$. Silverberg [Sil92, Thm 2.4] gives that
all endomorphisms of $A_{\bar{s}}$ are defined over $K\subset\bar{K}$ since
$K=K(A_{n})$. Hence we can apply the arguments of the proof of Proposition
7.14. They show that $A_{\bar{s}}$ is isogenous to a product $\prod
A_{i}^{e_{i}}$ with $e_{i}\in\mathbb{Z}_{\geq 1}$ and $A_{i}$ simple pairwise
non-isogenous abelian varieties over $\bar{K}$ of dimension $g_{i}$ such that
$E_{i}=\textup{End}^{0}(A_{i})$ is a CM field of degree $2g_{i}$ with
$D_{E_{i}}\leq D$ for $D=\textnormal{rad}(D_{K})^{5g}(2g)^{24g^{2}}$. The
number of isomorphism classes of CM fields $E_{i}$ of degree $2g_{i}$ and
discriminant $D_{E_{i}}\leq D$ is bounded by the explicit version (7.20) of
Hermite’s theorem, and each CM field $E_{i}$ has precisely $2^{g_{i}}$
distinct CM types; in this proof the CM types are $\bar{K}$-valued in the
sense of [CCO14, p.66]. Further CM theory ([CCO14, 1.5.4.1]) gives that the
abelian variety $A_{i}$ over $\bar{K}$ is determined up to an
$E_{i}$-compatible isogeny by the isomorphism class of the CM type associated
in [CCO14, p.66] to $A_{i}$, where two CM types $\Phi$ and $\Phi^{\prime}$, of
CM fields $E$ and $E^{\prime}$ respectively, are isomorphic if there exists an
isomorphism $\sigma:E\overset{\sim}{\to}E^{\prime}$ of fields with
$\Phi=\Phi^{\prime}\sigma$. On combining these observations, we find that up
to isogenies there exist at most $m=g2^{(2g+1)^{2}}(2D)^{2g}$ distinct abelian
varieties $A_{i}$ over $\bar{K}$ as above. As $A_{\bar{s}}$ is isogenous to
$\prod A_{i}^{e_{i}}$ and $g=\sum e_{i}g_{i}$, a combinatorial argument
([vK21, p.21]) shows that the number of isogeny classes of $A_{\bar{s}}$ with
$A$ as above is bounded by $m^{g}$. Then the number of isogeny classes of $A$
as above is at most $m^{g}$ since $K=K(A_{n})$. Indeed, if $A$ and $B$ are
abelian varieties over $K$ with $K=K(A_{n})=K(B_{n})$ then Silverberg [Sil92,
Thm 2.4] gives that
$\textup{Hom}(A,B)\overset{\sim}{\to}\textup{Hom}(A_{\bar{s}},B_{\bar{s}})$
and hence $A$ is isogenous to $B$ if and only if $A_{\bar{s}}$ is isogenous to
$B_{\bar{s}}$. This completes the proof of the lemma. ∎

We point out that Orr–Skorobogatov [OS18, Thm A] obtained a uniform finiteness
result for $\bar{\mathbb{Q}}$-isomorphism classes of abelian varieties of CM
type of given dimension which can be defined over a number field of given
degree; their arguments use results of Tsimerman [Tsi18] and Masser–Wüstholz
[MW93a] and also Zarhin’s quaternion trick. As a (polynomial) bound in terms
of $\textnormal{rad}(D_{K})$ is sufficient for our purpose, we shall use Lemma
7.18 which has the advantage of providing a dependence on $g$ and $K$.

We are now ready to prove Proposition 7.15 and we combine the above lemmas
with the explicit height bounds obtained in Propositions 7.13 and 7.14.

###### Proof of Proposition 7.15.

Let $A$ be an abelian scheme over $S$ of relative dimension $g$ which is of
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies. Suppose that
$K/\mathbb{Q}$ is normal and that $K=K(A_{n})$ for some $n\in\mathbb{Z}_{\geq
3}$. Then Lemma 7.6 assures that condition $(*)$ in $\mathsection$7.1.2 is
satisfied. In particular all endomorphisms of $A_{\bar{K}}$ are defined over
$K\subset\bar{K}$. Now, we combine Lemma 7.16 with the upper bounds for
$h_{F}(A)$ in Propositions 7.13 and 7.14. This shows that the number of
isomorphism classes of abelian schemes over $S$ which are isogenous to our
given $A$ is at most
$(7gd)^{608g^{5}d}\textnormal{rad}(D_{K}N_{S})^{96g^{4}}.$ On combining this
with the estimates for the number of isogeny classes obtained in Lemmas 7.17
and 7.18, we deduce an upper bound for the number of isomorphism classes as
claimed in Proposition 7.15. ∎

####  7.4.1 Proof of Theorem B (ii)

In this section we prove Theorem B (ii) by reducing to the situation treated
in Proposition 7.15. The reduction uses two results: An explicit bound for the
number of twists of an abelian variety over a number field (Lemma 7.19), and
an explicit estimate (7.20) for the number of number fields of given degree
and bounded discriminant.

##### Twists.

Let $A$ be an abelian variety of dimension $g\geq 1$ defined over number field
$K$. To bound the number of twists of the unpolarized $A$ over a finite $L/K$,
it seems difficult to extend the standard approach via Galois cohomology which
works in the polarized case; see Remark 7.20. Instead we exploit that each
twist embeds into the Weil restriction $R_{L/K}(A_{L})$ and then we apply the
isogeny estimates (7.13) based on transcendence. This leads to the following
result which holds moreover for twists up to isogenies.

######  Lemma 7.19.

Let $K\subseteq L$ be a finite field extension of relative degree $n$. There
exist at most $(ng)^{g}$ isogeny classes, and at most $(ng)^{g}\kappa(A)^{2g}$
isomorphism classes, of abelian varieties $B$ over $K$ such that $B_{L}$ is
isogenous to $A_{L}$ where $\kappa(A)$ is as in (7.13).

###### Proof.

We define $C=R(A_{L})$ where $R=R_{L/K}$ denotes the Weil restriction from $L$
to $K$. Let $B$ be an abelian variety over $K$ with an isogeny $B_{L}\to
A_{L}$. Its image $R(B_{L})\to C$ under the functor $R$ is a morphism of
abelian varieties over $K$. If $S$ is a $K$-scheme then $B(S\times_{K}L)\cong
R(B_{L})(S)$. On using that $\bar{K}\otimes_{K}L\cong\bar{K}^{n}$, we see that
$R(B_{L})\to C$ is quasi-finite and hence an isogeny. Further, the projection
$S\times_{K}L\to S$ induces a morphism

$B\hookrightarrow R(B_{L})\to C$

of abelian varieties over $K$. Here $B\hookrightarrow R(B_{L})$ is a closed
immersion since the identity of $B(\bar{K})$ factors through $B(\bar{K})\to
B(\bar{K}\otimes_{K}L)$ via the decomposition
$\bar{K}\to\bar{K}\otimes_{K}L\to\prod_{\varphi}\bar{K}\to\bar{K}$ of the
identity of $\bar{K}$, where $\prod_{\varphi}$ ranges over all $K$-morphisms
$\varphi:L\to\bar{K}$. Now, we can apply the arguments of steps 2 and 3 of the
proof of [vK21, Thm 6.6] to bound the number of isomorphism classes of $B$ as
above. We first control the number of isogeny classes. Let
$C_{1},\dotsc,C_{m}$ be pairwise non-isogenous simple abelian varieties over
$K$ such that $C$ is isogenous to $\prod C_{i}^{n_{i}}$ with
$n_{i}\in\mathbb{Z}_{\geq 1}$. As $B\hookrightarrow R(B_{L})\to C$ has finite
kernel, Poincaré’s reducibility theorem gives $e_{i}\in\mathbb{Z}_{\geq 0}$
such that $B$ is isogenous to $\prod C_{i}^{e_{i}}$. Here $\prod
C_{i}^{e_{i}}$ has dimension $\sum e_{i}g_{i}=\dim(B)$ where
$g_{i}=\dim(C_{i})$, and $\dim(B)=g$ since $B$ is geometrically isomorphic to
$A$. A combinatorial argument ([vK21, p.21]) shows that the number of
$e\in\mathbb{Z}_{\geq 0}^{m}$ with $\sum e_{i}g_{i}=g$ is at most
$\tbinom{a}{g}$ for $a=g+m-1$. Thus the number of distinct $\prod
C_{i}^{e_{i}}$, and hence the number of isogeny classes of $B$ as above, is at
most

$\tbinom{a}{g}\leq m^{g}.$

Let $[B_{0}]$ be such an isogeny class. We now bound the number of isomorphism
classes of abelian schemes $B$ as above which are inside $[B_{0}]$. There
exists an isogeny $B_{0}\to B$ of degree at most $\kappa(B)$ by (7.13), and
$\kappa(B)$ equals $\kappa(A)$ since the geometrically isomorphic $A$ and $B$
have the same stable Faltings height $h_{F}$. It follows that $[B_{0}]$
contains at most $\kappa(A)^{2g}$ distinct isomorphism classes of $B$ as
above. Here we used the arguments of step 3 in [vK21, p.21] which we applied
already in the proof of Lemma 7.16 and which rely on the counting result
[MW93a, Lem 6.1] of Masser–Wüstholz. Then we deduce that the number of
isomorphism classes of abelian schemes $B$ as above is at most
$\tbinom{a}{g}\kappa(A)^{2g}$ where $\tbinom{a}{g}\leq m^{g}$. This proves the
lemma since $m\leq\dim(C)$ and $\dim(C)=ng$. ∎

######  Remark 7.20.

(i) The above proof shows that one can refine Lemma 7.19 as follows. One can
replace $(ng)^{g}$ by the smaller quantity $\tbinom{a}{g}$, where $a=g+m-1$
and $m$ is the number of isogeny classes of simple factors of the Weil
restriction $R(A_{L})$. Also, if the minimal isogeny degree of abelian
varieties of dimension $g$ over $K$ would be bounded by a constant depending
only on $g$ and $K$, then one could replace $\kappa(A)$ by this constant. For
example, one can replace $\kappa(A)^{2g}$ by the number 8 if $K=\mathbb{Q}$
and $g=1$. Indeed this follows by going into the step where we bound the
number of isomorphism classes in $[B_{0}]$ and by replacing therein (7.13) by
Kenku’s refinement [Ken82, Thm 2] of Mazur’s [Maz78, Thm 5].

(ii) Let $(A,\lambda)$ be a polarized abelian variety over $K$, let $L$ be a
finite Galois extension of $K$ and write $G=\textup{Aut}(L/K)$. The set of
isomorphism classes of polarized abelian varieties
$(A^{\prime},\lambda^{\prime})$ over $K$ such that
$(A^{\prime}_{L},\lambda^{\prime}_{L})$ is isomorphic to $(A_{L},\lambda_{L})$
identifies with the first Galois cohomology group
$H^{1}\bigl{(}G,\textup{Aut}(A_{L},\lambda_{L})\bigl{)}$. It should be
possible to explicitly bound the cardinality of the finite group
$\textup{Aut}(A_{L},\lambda_{L})$ and hence of
$H^{1}\bigl{(}G,\textup{Aut}(A_{L},\lambda_{L})\bigl{)}$. However, the
automorphism group of an unpolarized abelian variety over $K$ can be infinite
and quite complicated. Thus we used a different approach in Lemma 7.19.

##### Number fields.

Let $d$ and $D$ be in $\mathbb{Z}_{\geq 1}$. Hermite proved that the number
$n$ of isomorphism classes of number fields $K$ of degree $[K:\mathbb{Q}]=d$
and discriminant $D_{K}\leq D$ is finite. On making Minkowski’s proof of this
result explicit, one obtains for example

$n\leq 2^{(d+1)^{2}}D^{d}.$ (7.20)

Stronger bounds in terms of $D$ are known, see for instance Schmidt [Sch95],
Ellenberg–Venkatesh [EV06] and the recent results of Lemke-Oliver–Thorne
[LOT22], Bhargava–Shankar–Wang [BSW22] and Anderson et al. [AGH+22]. As a
polynomial bound in terms of $D$ is sufficient for our purpose, we shall use
(7.20) which has the advantage of providing also a dependence on $d$. To
compute the bound (7.20), we go into the proof of [Neu99, Thm 3.2.13]. We find
that $\textnormal{vol}(X)=\tfrac{4}{\pi}(2\pi)^{s}cD^{1/2}$ and
$c>\tfrac{\pi}{4}(\tfrac{2}{\pi})^{s}$. Hence, if $K$ is totally complex, then
it has an integral primitive element $\alpha$ such that each coefficient $a$
of its minimal polynomial $\prod_{\sigma}(x-\sigma(\alpha))$ satisfies
$|a|=|\sum\prod\sigma(\alpha)|\leq 2^{d}D$. If $K$ has a real embedding
$\tau_{0}$, then we take $X=\\{(z_{\tau})\in
K_{\mathbb{R}};\,|z_{\tau_{0}}|<cD^{1/2},|z_{\tau}|<1\textnormal{ for
}\tau\neq\tau_{0}\\}$ with $\textnormal{vol}(X)=(2\pi)^{s}2^{r}cD^{1/2}$ and
$c>(\tfrac{2}{\pi})^{s}$ which leads again to $|a|\leq 2^{d}D$. This implies
(7.20).

We are now ready to prove Theorem B (ii). Recall that therein $K$ is a number
field of degree $d$, $S\subseteq\textnormal{Spec}({\mathcal{O}_{K}})$ is a
nonempty open subscheme, $N_{S}=\prod_{v\in S^{c}}N_{v}$ and
$g\in\mathbb{Z}_{\geq 1}$.

###### Proof of Theorem B (ii).

Let $A$ be an abelian scheme over $S$ of relative dimension $g$. To reduce to
the situation treated in Proposition 7.15, we make a base change to a
controlled scheme $T$ which is defined as follows: Choose $\bar{\mathbb{Q}}$
with $K\subset\bar{\mathbb{Q}}$, let $L\subset\bar{\mathbb{Q}}$ be the normal
closure of $K(A_{3})/\mathbb{Q}$ and let
$p:\textnormal{Spec}(\mathcal{O}_{L})\to\textnormal{Spec}({\mathcal{O}_{K}})$
be the projection. Now we define

$T=p^{-1}(S).$

It is a connected Dedekind scheme with function field $L$ of degree
$[L:\mathbb{Q}]\leq l$ where $l=(d3^{4g^{2}})!$. As in the proof of Lemma 7.6,
we see that the criterion of Néron–Ogg–Shafarevich [ST68, Thm 1] implies that
$K(A_{3})$, and hence its normal closure $L$, is unramified over each rational
prime $p\nmid 3N_{S}D_{K}$. Thus $\textnormal{rad}(N_{T}D_{L})$ divides
$\textnormal{rad}(3N_{S}D_{K})$ and [vK13b, Lem 6.2] leads to $D_{L}\leq D$
where $D=\textnormal{rad}(3N_{S}D_{K})^{3l}l^{6l^{2}}$. Here we used the
effective version of the prime number theorem in [RS62, (3.12) and (3.16)]
which gives that the number $s$ of rational prime divisors of $D_{L}$
satisfies $s^{s}<\textnormal{rad}(D_{L})^{2}$ if $s>12$.

Let $F(S)$ be the set of isomorphism classes of abelian schemes over $S$ of
relative dimension $g$ which are of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies, and let $F^{\prime}(S)$ be the set of all $[A]\in
F(S)$ with $K(A_{3})=K$. The notion of $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies is stable under the base change $T\to S$ for any
$T$ as above. Hence $[A]\mapsto[A_{T}]$ defines a map

$F(S)\to\cup_{T\in\mathcal{T}}F^{\prime}(T).$

Here $\mathcal{T}$ is the set of all schemes $T=p^{-1}(S)$ where
$p:\textnormal{Spec}(\mathcal{O}_{L})\to\textnormal{Spec}({\mathcal{O}_{K}})$
is the projection for some number field $K\subseteq L\subset\bar{\mathbb{Q}}$
with the following properties: $L/\mathbb{Q}$ is normal, $[L:\mathbb{Q}]\leq
l$, $D_{L}\leq D$, and $\textnormal{rad}(N_{T}D_{L})$ divides
$\textnormal{rad}(3N_{S}D_{K})$.

To bound $|F(S)|$ we now control $|\mathcal{T}|$ and the degree of the map
$F(S)\to\cup_{T\in\mathcal{T}}F^{\prime}(T)$. It follows from (6.1) that any
two abelian schemes over a connected Dedekind scheme are isomorphic if and
only if their generic fibers are isomorphic. Thus Lemma 7.19 gives that the
fibers of the map $F(S)\to\cup_{T\in\mathcal{T}}F^{\prime}(T)$ have
cardinality at most

$(lg)^{g}\sup\kappa([A])^{2g}$

with the supremum taken over all $[A]\in F(S)$. Here
$\kappa([A])=\kappa(A_{K})$ and $\kappa(A_{K})$ is defined in (7.13). Further,
the construction of $\mathcal{T}$ shows that $|\mathcal{T}|$ is bounded by the
number of subfields $L\subset\bar{\mathbb{Q}}$ with $[L:\mathbb{Q}]\leq l$ and
$D_{L}\leq D$. Therefore (7.20) implies

$|\mathcal{T}|\leq l^{2}2^{(l+1)^{2}}D^{l}.$

If $T\in\mathcal{T}$ then its function field $L=k(T)$ is normal over
$\mathbb{Q}$ and the definition of $F^{\prime}(T)$ assures that each $[A]\in
F^{\prime}(T)$ satisfies $L=L(A_{3})$. Therefore we can apply Proposition 7.15
which bounds $|F^{\prime}(T)|$ for all $T\in\mathcal{T}$, while Theorem 7.1
bounds $\sup\kappa([A])$. On combining these bounds with the above displayed
results, we deduce

$|F(S)|\leq(3gl)^{(6g)^{5}l^{3}}\textnormal{rad}(N_{S}D_{K})^{4l^{2}}.$ (7.21)

We now deal with the general case and we consider $A$ with the following
properties $(\dagger)$: $A$ is an abelian scheme over $S$ of relative
dimension $g$ and $A$ is of product $\textnormal{GL}_{2}$-type with
$G_{\mathbb{Q}}$-isogenies. Let $m$ be the number of isogeny classes of
abelian schemes over $S$ of relative dimension at most $g$ which are of
$\textnormal{GL}_{2}$-type with $G_{\mathbb{Q}}$-isogenies. Suppose that
$A_{1},\dotsc,A_{m}$ generate these isogeny classes, and write $g_{i}$ for the
relative dimension of $A_{i}$ over $S$. Then for any $A$ with $(\dagger)$
there exists $e\in\mathbb{Z}^{m}_{\geq 0}$ with $g=\sum e_{i}g_{i}$ such that
$A$ is isogenous to $\prod A_{i}^{e_{i}}$. Thus a combinatorial argument,
which we used already in the proof of Lemma 7.19, shows that the number of
isogeny classes of abelian schemes $A$ with $(\dagger)$ is at most $m^{g}$.
Then Lemma 7.16 gives that the number of isomorphism classes of abelian
schemes $A$ with $(\dagger)$ is at most $\sup\kappa(A)^{2g}m^{g}$ with the
supremum taken over all $A$ with $(\dagger)$ for $\kappa(A)=\kappa(A_{K})$ as
in (7.13). Theorem 7.1 bounds $\sup\kappa(A)$, and (7.21) leads to

$m\leq\sum c_{k}\leq\tfrac{g}{2}(g+1)c_{g}.$

Here the sum is taken over all $k\in\mathbb{Z}_{\geq 1}$ with $k\leq g$, and
$c_{k}$ denotes the expression obtained by replacing $g$ by $k$ in the upper
bound of (7.21). We combine the bounds and then we deduce the estimate for the
number of isomorphism classes claimed in Theorem B (ii). ∎

##  8 Polarizations

Let $L$ be a totally real number field of degree $g$ over $\mathbb{Q}$ with
ring of integers $\mathcal{O}$, and let $I$ be a nonzero finitely generated
$\mathcal{O}$-submodule of $L$ with a positivity notion $I_{+}$. In this
section we continue our study in [vKK19, $\mathsection$9] of equivalence
classes of $I$-polarizations on $\mathcal{O}$-abelian schemes of relative
dimension $g$. In particular, we explicitly bound the number of such
equivalence classes by using and generalizing the arguments of [vKK19,
$\mathsection$9].

Let $S$ be a connected Dedekind scheme with function field of characteristic
zero, let $\mathcal{C}$ be the category of $\mathcal{O}$-abelian schemes over
$S$ and let $(A,\iota)$ be an $\mathcal{O}$-abelian scheme over $S$ of
relative dimension $g$. We denote by $\Phi$ the set of $I$-polarizations on
$A$.

##### Equivalence relation.

Recall from Section 3.1 that an $I$-polarization on $A$ is an
$\mathcal{O}$-module morphism from $I$ to the $\mathcal{O}$-module
$\textup{Hom}_{\mathcal{O}}(A,A^{\vee})^{\textnormal{sym}}$, which maps
$I_{+}$ to polarizations and which induces an isomorphism
$I\otimes_{\mathcal{O}}A\overset{\sim}{\to}A^{\vee}$. We say that two
$I$-polarizations $\varphi,\varphi^{\prime}\in\Phi$ are equivalent, and we
write $\varphi\sim\varphi^{\prime}$, if there exists an automorphism $\sigma$
of the $\mathcal{O}$-abelian scheme $A$ such that
$\varphi(\lambda)=\sigma^{\vee}\varphi^{\prime}(\lambda)\sigma$ for all
$\lambda\in I$.

##### Equivalence classes.

To state our bound for the equivalence classes, we denote by $w$ and
$w^{\prime}$ the number of roots of unity in
$\textup{End}^{0}_{\mathcal{C}}(A)$ and $\textup{End}_{\mathcal{C}}(A)$
respectively.

######  Proposition 8.1.

The number of equivalence classes satisfies

$|\Phi/{\sim}|\leq\tfrac{w^{2}}{w^{\prime}}\cdot 2^{g-1}.$

The number $w$ is uniformly bounded: For example (8.4) gives that $w\leq
4g^{2}$ and $\omega\ll_{\epsilon}g^{1+\epsilon}$ for all real $\epsilon>0$,
while Lemma 8.3 shows that $w=w^{\prime}=2$ if the abelian scheme $A$ over $S$
has no CM. In particular we obtain that $w=w^{\prime}=2$ if the function field
of $S$ embeds into $\mathbb{R}$, and hence [vKK19, Prop 9.1] is a special case
of Proposition 8.1. An important feature of Proposition 8.1 is that the bound
$\tfrac{w^{2}}{w^{\prime}}\cdot 2^{g-1}$ can be controlled only in terms of
the relative dimension $g$ of $A$. This is crucial for deducing the next
result which explicitly controls the degree over $S$ of the natural forgetful
morphism $\phi_{\varphi}\colon M\to\underline{M}$ defined in [vKK19, (6.2)]
with respect to the Hilbert moduli stack $\mathcal{M}^{I}$ associated to
$(L,I,I_{+})$.

######  Corollary 8.2.

The degree of $\phi_{\varphi}(S)$ is at most $c\cdot 2^{g}$ for some constant
$c\leq 4g^{4}$.

###### Proof.

We go into the proof of [vKK19, Cor 9.2] and we replace therein the
application of [vKK19, Prop 9.1] by Proposition 8.1. This proves the
corollary. ∎

The proof of Proposition 8.1 will be given in the remaining of this section.
The strategy is as follows: After applying the formal result [vKK19, Lem 9.3]
which computes the set $\Phi$ in terms of certain isomorphisms
$A\overset{\sim}{\to}A^{\vee}\otimes_{\mathcal{O}}I^{-1}$, we use and
generalize the arguments of [vKK19, Lem 9.4] to construct in Section 8.1 an
embedding

$\Phi/{\sim}\hookrightarrow\Gamma^{\times}/U.$

Here $\Gamma$ is an order of a number field $F\supseteq L$ which comes from
Lemma 8.3 and $U$ denotes a certain subgroup of the units $\Gamma^{\times}$ of
$\Gamma$. Then in Section 8.2 we use algebraic number theory to relate the
cardinality of $\Gamma^{\times}/U$ to a ratio $R_{F}/(R_{F/L}R_{L})$ of
regulators which can be computed by using classical results from CM theory.

###  8.1 Equivalence classes of polarizations and quotients of unit groups

We continue our notation introduced at the beginning of Section 8. The goal of
the current section is to view equivalence classes of $I$-polarizations on
$\mathcal{O}$-abelian schemes inside finite quotients of certain unit groups
of orders of number fields containing $L$.

To give a precise statement, we recall that $\Phi$ denotes the set of
$I$-polarizations on an arbitrary $\mathcal{O}$-abelian scheme $(A,\iota)$
over $S$ of relative dimension $g=[L:\mathbb{Q}]$ where $S$ is a connected
Dedekind scheme with function field of characteristic zero. Notice that
$D=\textup{End}^{0}_{\mathcal{C}}(A)$ becomes an $L$-algebra via $\iota$,
since $\iota(\mathcal{O})\subseteq\textup{End}_{\mathcal{C}}(A)$ and
$L=\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$.

######  Lemma 8.3.

The following statements hold.

  * (i)

Suppose that $\dim_{L}(D)=1$. Then the ring $\textup{End}_{\mathcal{C}}(A)$
identifies with an order $\Gamma$ of $L$ and there exists an embedding
$\Phi/{\sim}\hookrightarrow\Gamma^{\times}/(\Gamma^{\times})^{2}.$

  * (ii)

Suppose that $\dim_{L}(D)\neq 1$. Then $\textup{End}_{\mathcal{C}}(A)$
identifies with an order $\Gamma\supset\mathcal{O}$ of a CM field $F$ with
maximal totally real subfield $L$ and there exists an embedding

$\Phi/{\sim}\hookrightarrow\Gamma^{\times}/N(\Gamma^{\times}).$

Here the image $N(\Gamma^{\times})$ of $\Gamma^{\times}$ under the field norm
$N:F\to L$ is a subgroup of $\Gamma^{\times}$.

The assumption that the function field of $S$ has characteristic zero is
crucial for (ii) which relies on the computation of the $L$-algebra $D$ in
Lemma 8.4 below.

In what follows in this section we write $\otimes=\otimes_{\mathcal{O}}$ for
simplicity. Our proof of Lemma 8.3 uses the formal result in [vKK19, Lem 9.3]
which computes $\Phi/{\sim}$ in terms of equivalence classes of certain
isomorphisms $A\overset{\sim}{\to}A^{\vee}\otimes I^{-1}$. To state [vKK19,
Lem 9.3], we need to introduce some notation. Firstly, we observe that
multiplication defines an isomorphism $\mu:I^{-1}\otimes
I\overset{\sim}{\to}\mathcal{O}$ of $\mathcal{O}$-modules since $I$ is
invertible. Secondly, in the category of $\mathcal{O}$-modules it holds: If
$X,Y,Z$ are $\mathcal{O}$-modules, then mapping any morphism
$\varphi:X\to\textup{Hom}(Y,Z)$ to the morphism $X\otimes Y\to Z$ defined by
$x\otimes y\mapsto\varphi(x)(y)$ gives a bijection
$\textup{Hom}(X,\textup{Hom}(Y,Z))\overset{\sim}{\to}\textup{Hom}(X\otimes
Y,Z)$ which is functorial. Therefore, on using the isomorphism $I^{-1}\otimes
I\otimes A\overset{\sim}{\to}A$ induced by $\mu$ and tensoring with $I^{-1}$,
we obtain a bijection

$\tau:\textup{Hom}(I,\textup{Hom}_{\mathcal{C}}(A,A^{\vee}))\overset{\sim}{\to}\textup{Hom}_{\mathcal{C}}(A,I^{-1}\otimes
A^{\vee}).$

Further, any $\lambda\in I$ defines a morphism $m_{\lambda}:I^{-1}\otimes
A^{\vee}\to A^{\vee}$ of $\mathcal{O}$-abelian schemes over $S$ as follows: If
$T$ is an $S$-scheme, then $m_{\lambda}(T)$ sends $j\otimes x$ in
$I^{-1}\otimes A^{\vee}(T)$ to the element $j\lambda\otimes x$ in
$\mathcal{O}\otimes A^{\vee}(T)\cong A^{\vee}(T)$. We denote by $\Psi$ the set
of isomorphisms $\psi:A\overset{\sim}{\to}I^{-1}\otimes A^{\vee}$ of
$\mathcal{O}$-abelian schemes over $S$ such that for each $\lambda\in I$ the
morphism $m_{\lambda}\psi:A\to A^{\vee}$ is symmetric and moreover a
polarization if $\lambda\in I_{+}$. Now [vKK19, Lem 9.3] says

$\tau(\Phi)=\Psi.$ (8.1)

On using the canonical bijection $\tau:\Phi\overset{\sim}{\to}\Psi$ and the
equivalence relation on $\Phi$, we obtain an equivalence relation on $\Psi$ by
transport of structure. We write again $\psi\sim\psi^{\prime}$ if two
isomorphisms $\psi,\psi^{\prime}\in\Psi$ are equivalent. We are now ready to
prove Lemma 8.3.

###### Proof of Lemma 8.3.

To show (i) we suppose that $\dim_{L}(D)=1$. Then we are in the case when
$\textup{End}^{0}_{\mathcal{C}}(A)\cong L$, and this case was treated in our
proof of [vKK19, Lem 9.4]. Therefore we deduce (i) by using exactly the same
arguments as in the proof of [vKK19, (9.2)].

We next show (ii). Assume that $\dim_{L}(D)\neq 1$ and notice that our abelian
scheme $A$ over $S$ satisfies all the assumptions made in Lemma 8.4. Therefore
Lemma 8.4 gives an $L$-algebra isomorphism $D\overset{\sim}{\to}F$ from $D$ to
a CM field $F$ with maximal totally real subfield $L$. Further,
$\textup{End}_{\mathcal{C}}(A)$ is a subring of $\textup{End}(A)$ which is a
finitely generated torsion-free abelian goup. Thus
$\textup{End}_{\mathcal{C}}(A)$ identifies with an order of $D$. Let
$\Gamma\subset F$ be the image of this order under the $L$-algebra isomorphism
$D\overset{\sim}{\to}F$. Then $\Gamma$ is an order of $F$, and $\Gamma$
contains $\mathcal{O}$ since
$\iota(\mathcal{O})\subseteq\textup{End}_{\mathcal{C}}(A)$. This proves the
first part of (ii).

It remains to construct the embedding of $\Phi/{\sim}$ appearing in the second
part of (ii). As the equivalence relation on $\Psi$ is defined by transport of
structure via (8.1), we see that $\tau:\Phi\overset{\sim}{\to}\Psi$ induces a
bijection from $\Phi/{\sim}$ to the set $\Psi/{\sim}$. To embed $\Psi$ into
$\Gamma^{\times}$, we may and do assume that there exists an isomorphism
$\psi_{0}:A\overset{\sim}{\to}I^{-1}\otimes A^{\vee}$ which lies in $\Psi$. We
observe that $\psi\mapsto\psi_{0}^{-1}\psi$ defines an embedding
$\Psi\hookrightarrow\textup{Aut}_{\mathcal{C}}(A)$, and the isomorphism
$D\overset{\sim}{\to}F$ identifies the image of
$\textup{Aut}_{\mathcal{C}}(A)\hookrightarrow D$ with $\Gamma^{\times}$.
Therefore the composition of the maps
$\Psi\hookrightarrow\textup{Aut}_{\mathcal{C}}(A)\hookrightarrow
D\overset{\sim}{\to}F$ gives an embedding

$j\colon\Psi\hookrightarrow\Gamma^{\times}.$

Further, the image $N(\Gamma^{\times})$ of $\Gamma^{\times}$ under the field
norm $N:F^{\times}\to L^{\times}$ is a subgroup of $\Gamma^{\times}$, since
the order $\Gamma$ is contained in the maximal order $\mathcal{O}_{F}$ of $F$
and since $N(\mathcal{O}_{F}^{\times})$ is contained in
$\mathcal{O}^{\times}\subseteq\Gamma^{\times}$. We now claim that
$j:\Psi\hookrightarrow\Gamma^{\times}$ induces an embedding

$\Psi/{\sim}\hookrightarrow\Gamma^{\times}/N(\Gamma^{\times}).$ (8.2)

To prove this claim, we suppose that $\psi,\psi^{\prime}\in\Psi$ and we denote
by $\varphi,\varphi^{\prime}\in\Phi$ their preimages under $\tau$
respectively. Then $\psi\sim\psi^{\prime}$ if and only if there exists
$\sigma\in\textup{Aut}_{\mathcal{C}}(A)$ with
$\varphi^{\prime}=\sigma^{\vee}\varphi\sigma$. Moreover, on applying the
formal arguments of the proof of [vKK19, Lem 9.3], we compute that
$\varphi^{\prime}=\sigma^{\vee}\varphi\sigma$ if and only if
$\psi^{\prime}=(I^{-1}\otimes\sigma^{\vee})\psi\sigma$. The latter equality is
equivalent to
$\psi_{0}^{-1}\psi^{\prime}=\psi_{0}^{-1}(I^{-1}\otimes\sigma^{\vee})\psi_{0}\psi_{0}^{-1}\psi\sigma,$
which in turn is equivalent to
$(\psi_{0}^{-1}\psi^{\prime})=\sigma^{\star}(\psi_{0}^{-1}\psi)\sigma$. Here
for any $f\in\textup{End}_{\mathcal{C}}(A)$ the involution ⋆ is defined by
$f^{\star}=\psi_{0}^{-1}(I^{-1}\otimes f^{\vee})\psi_{0}.$ Thus we obtain that
$\psi\sim\psi^{\prime}$ if and only if there exists
$\sigma\in\textup{Aut}_{\mathcal{C}}(A)$ with

$(\psi_{0}^{-1}\psi^{\prime})=\sigma^{\star}(\psi_{0}^{-1}\psi)\sigma.$

To further compute this in terms of $\Gamma^{\times}$, let $\lambda\in I_{+}$
and denote by ∗ the usual Rosati involution on $\textup{End}^{0}(A)$
associated to the polarization $\varphi(\lambda)$ of $A$, where
$\varphi=\tau^{-1}(\psi_{0})$ lies in $\Phi$ by (8.1). Then our ⋆ coincides on
$\Gamma$ with the restriction of the Rosati involution ∗ to
$D=\textup{End}_{\mathcal{C}}^{0}(A)$. Indeed, on using that any $f$ in
$\textup{End}_{\mathcal{C}}(A)$ is a morphism of $\mathcal{O}$-abelian
schemes, we compute $f^{\vee}m_{\lambda}=m_{\lambda}(I^{-1}\otimes f^{\vee})$
and then we deduce that
$f^{*}=\varphi(\lambda)^{-1}f^{\vee}\varphi(\lambda)=f^{\star}$ inside
$\textup{End}_{\mathcal{C}}^{0}(A)$ since
$m_{\lambda}\psi_{0}=\varphi(\lambda)$ by [vKK19, (9.1)]. Moreover, as
$\textup{End}^{0}(A)\overset{\sim}{\to}\textup{End}^{0}(A_{k})$ for $k$ the
function field of the connected Dedekind scheme $S$, it follows for example
from [Lan83, p.5] that the positive involution ∗ on
$D\subseteq\textup{End}^{0}(A)$ identifies via the $L$-algebra isomorphism
$D\overset{\sim}{\to}F$ with the non-trivial automorphism $c$ of $F/L$. Then
the above computations and ${}^{*}=\,^{\star}$ on $D$ show that
$\psi\sim\psi^{\prime}$ if and only if there exists $\gamma\in\Gamma^{\times}$
with

$j(\psi^{\prime})=c(\gamma)j(\psi)\gamma=N(\gamma)j(\psi),$

where the last equality exploits that $\Gamma$ is commutative. Hence our claim
in (8.2) follows and thus all statements of (ii) are proven. This completes
the proof of Lemma 8.3.∎

The above proof relies on description of the $L$-algebra
$D=\textup{End}_{\mathcal{C}}^{0}(A)$ in the following result which further
refines Lemma 5.4 (ii) in the case when the base is a connected scheme which
has a point whose residue field is of characteristic zero.

######  Lemma 8.4.

Let $L$ be a totally real number field with ring of integers $\mathcal{O}$,
let $S$ be a connected Dedekind scheme with $k(S)$ of characteristic zero, and
let $A$ be an $\mathcal{O}$-abelian scheme over $S$ of relative dimension
$g=[L:\mathbb{Q}]$. The $L$-algebra $D=\textup{End}_{\mathcal{C}}^{0}(A)$
either has dimension one or is isomorphic to a CM field with maximal totally
real subfield $L$.

###### Proof.

We first treat the case when the function field $k=k(S)$ embeds into
$\mathbb{C}$. Let $A_{\mathbb{C}}$ be the base change of $A_{k}$ to
$\mathbb{C}$ via an embedding $k\hookrightarrow\mathbb{C}$ and consider the
$\mathbb{Q}$-vector space $V$ given by its first homology
$H_{1}(A_{\mathbb{C}}(\mathbb{C}),\mathbb{Q})$. We obtain a ring morphism
$\rho:D\to\textup{End}(V)$ by composing
$D\subseteq\textup{End}^{0}(A)\to\textup{End}^{0}(A_{\mathbb{C}})$ with the
morphism $\textup{End}^{0}(A_{\mathbb{C}})\to\textup{End}(V)$ induced by the
functor $H_{1}(\,\ ,\mathbb{Q})$. By assumption $A=(A,\iota)$ is an
$\mathcal{O}$-abelian scheme over $S$. Composing $\rho$ with
$\iota\otimes_{\mathbb{Z}}\mathbb{Q}:L\to D$ gives $V$ an $L$-vector space
structure. The $L$-dimension of $V$ is two, since its $\mathbb{Q}$-dimension
is $2g$ and $[L:\mathbb{Q}]=g$. As the image of $L\to D$ lies in the center of
$D$, we then see that $\rho$ induces a morphism

$D\to\textup{M}_{2}(L)$ (8.3)

of $L$-algebras. This morphism is injective since $D$ is a division algebra by
Lemma 5.4. This implies that $\dim_{L}D\leq 4$. We claim that $\dim_{L}D$ can
not be $3$ or $4$. Indeed $\dim_{L}D=4$ would imply that
$D\overset{\sim}{\to}\textup{M}_{2}(L)$ which is not possible since
$\textup{M}_{2}(L)$ is not a division algebra. Further, the division algebra
$D$ is a central simple algebra over its center $Z$. Thus $\dim_{Z}D$ is a
square, while (8.3) and $\dim_{L}D\leq 4$ assure that $\dim_{L}Z$ lies in
$\\{1,2,4\\}$. Hence we deduce that $\dim_{L}D=3$ is not possible, proving our
claim. Suppose now that $\dim_{L}D=2$. Then the division $L$-algebra $D$ is
commutative. In particular $D$ is a number field of degree
$2\dim(A_{\mathbb{C}})$ which embeds into $\textup{End}^{0}(A_{\mathbb{C}})$.
Thus $D$ is totally imaginary. As $[L:\mathbb{Q}]=g$ and $\dim_{L}D=2$, we
then conclude that $D$ is $L$-isomorphic to a CM field with maximal totally
real subfield $L$. This proves Lemma 8.4 in the case when $k$ embeds into
$\mathbb{C}$.

The general situation can be reduced to the above case as follows. As $k$ is
of characteristic zero, we may and do choose a subfield $k^{\prime}\subseteq
k$ with the following properties: It is finitely generated over $\mathbb{Q}$,
$A_{k}$ descends to an abelian variety $A_{k^{\prime}}$ over $k^{\prime}$ and
there is a ring morphism
$\textup{End}^{0}(A_{k})\to\textup{End}^{0}(A_{\mathbb{C}})$ where now
$A_{\mathbb{C}}$ is the base change of $A_{k^{\prime}}$ to $\mathbb{C}$ via an
embedding $k^{\prime}\hookrightarrow\mathbb{C}$. Thus the arguments above
(8.3) provide again a morphism $D\to\textup{M}_{2}(L)$ of $L$-algebras and
then the arguments below (8.3) imply Lemma 8.4 in general. ∎

In the next sections we bound the cardinality of
$\Gamma^{\times}/(\Gamma^{\times})^{2}$ and
$\Gamma^{\times}/N(\Gamma^{\times})$.

###  8.2 Quotients of unit groups

In this section we prove two lemmas controlling the cardinality of quotient
groups of the form $\Gamma^{\times}/N(\Gamma^{\times})$. For this purpose we
use algebraic number theory in order to relate and compute various quotients
of unit groups in extensions of number fields.

We introduce some notation. Let $F$ be a number field and let $L\subseteq F$
be a subfield. We denote by $N:F^{\times}\to L^{\times}$ the group morphism
given by the field norm, we write $R_{F/L}$ for the relative regulator ([CF93,
$\mathsection$3]) and we put $R_{F}=R_{F/\mathbb{Q}}$. Further for any ring
$R$ we denote by $\mu(R)$ the group of roots of unity in $R$ and we write
$w_{R}=|\mu(R)|$. Our first goal is to prove the following lemma on quotients
of unit groups in general number fields.

######  Lemma 8.5.

The following statements hold.

  * (i)

If $\Gamma$ is an order of $F$ with $\mathcal{O}_{L}\subseteq\Gamma$, then
$N(\Gamma^{\times})$ is a subgroup of $\Gamma^{\times}$ and

$|\ker(N)\cap\Gamma^{\times}|\cdot|\Gamma^{\times}/N(\Gamma^{\times})|\leq|\ker(N)\cap\mathcal{O}_{F}^{\times}|\cdot|\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})|.$

  * (ii)

The cardinality of the quotient
$\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})$ satisfies

$|\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})|\leq
w_{F}\cdot|\mathcal{O}_{F}^{\times}/\mu(F)\mathcal{O}_{L}^{\times}|\cdot\tfrac{R_{F}}{R_{F/L}R_{L}}.$

###### Proof.

To show (i) we assume that the order $\Gamma$ contains $\mathcal{O}_{L}$. Any
order of $F$ is contained in the maximal order $\mathcal{O}_{F}$ of $F$, and
$N$ sends $\mathcal{O}_{F}^{\times}$ to $\mathcal{O}_{L}^{\times}$ since the
norm takes the form $x\mapsto\prod\sigma(x)$ with the product taken over all
$\sigma\in\textup{Hom}_{L}(F,\bar{F})$. Thus our assumption that $\Gamma$
contains $\mathcal{O}_{L}$ implies that the restriction of $N$ to
$\Gamma^{\times}$ is a group morphism $N:\Gamma^{\times}\to\Gamma^{\times}$.
In particular $N(\Gamma^{\times})$ is a subgroup of $\Gamma^{\times}$ and we
obtain a diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(N)\cap\Gamma^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N}$$\textstyle{\Gamma^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{\times}/N(\Gamma^{\times})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(N)\cap\mathcal{O}_{F}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{F}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N}$$\textstyle{\mathcal{O}_{F}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $\Gamma^{\times}\to\mathcal{O}_{F}^{\times}$ is the inclusion. This is a
commutative diagram of abelian groups, with exact rows and with the first
three vertical morphisms injective. Therefore an application of the snake
lemma shows that the kernel of
$\Gamma^{\times}/N(\Gamma^{\times})\to\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})$
has cardinality at most
$|\ker(N)\cap\mathcal{O}_{F}^{\times}|/|\ker(N)\cap\Gamma^{\times}|$ when
$\ker(N)\cap\mathcal{O}_{F}^{\times}$ is finite. This implies the inequality
in (i), which also holds when $\ker(N)\cap\mathcal{O}_{F}^{\times}$ is
infinite since $\infty\leq\infty$.

We next prove (ii). Costa–Friedman observed that
$|\mathcal{O}_{L}^{\times}/\mu(L)N(\mathcal{O}_{F}^{\times})|$ can be written
as the quotient of regulators $R_{F}/(R_{F/L}R_{L})$, see [CF93, (3.1)]. Then,
on using that the index of subgroups is multiplicative and that
$N(\mathcal{O}_{F}^{\times})$ is a subgroup of $\mathcal{O}_{L}^{\times}$, we
compute

$|\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})|=|\mathcal{O}_{F}^{\times}/\mathcal{O}_{L}^{\times}|\cdot|\mathcal{O}_{L}^{\times}/N(\mathcal{O}_{F}^{\times})|=|\mathcal{O}_{F}^{\times}/\mu(F)\mathcal{O}_{L}^{\times}|\cdot
n\cdot\tfrac{R_{F}}{R_{F/L}R_{L}}\cdot m.$

Here $n=|\mu(F)\mathcal{O}_{L}^{\times}/\mathcal{O}_{L}^{\times}|$ and
$m=|\mu(L)N(\mathcal{O}_{F}^{\times})/N(\mathcal{O}_{F}^{\times})|$. The
indices $n$ and $m$ are both finite, and they satisfy $n\leq w_{F}/w_{L}$ and
$m\leq w_{L}$. Therefore the displayed formula implies the upper bound for
$|\mathcal{O}_{F}^{\times}/N(\mathcal{O}_{F}^{\times})|$ stated in (ii). This
completes the proof. ∎

We now apply the above lemma in the case of CM fields. In this case CM theory
allows to compute the quantities appearing in Lemma 8.5 leading to the
following result.

######  Lemma 8.6.

Suppose that $F$ is a CM field with maximal totally real subfield $L$ and
write $g=[L:\mathbb{Q}]$. Then any order $\Gamma$ of $F$ with
$\mathcal{O}_{L}\subseteq\Gamma$ satisfies

$|\Gamma^{\times}/N(\Gamma^{\times})|\leq\tfrac{w_{F}^{2}}{w_{\Gamma}}\cdot
2^{g-1}.$

###### Proof.

We assume that $F$, $L$ and $\Gamma$ are as in the statement. Then the claimed
upper bound for $|\Gamma^{\times}/N(\Gamma^{\times})|$ is in fact contained in
the statements of Lemma 8.5. To see this we now compute the quantities
appearing in Lemma 8.5. We first show that

$\ker(N)\cap\mathcal{O}_{F}^{\times}=\mu(F)\quad\textnormal{and}\quad\mu(\Gamma)=\ker(N)\cap\Gamma^{\times}.$

For this purpose we may and do assume that the CM field $F\subset\mathbb{C}$.
Then the non-trivial automorphism of the quadratic field extension $F/L$ is
given by complex conjugation $\bar{}\,$ and we obtain that
$\ker(N)\cap\mathcal{O}_{F}^{\times}=\\{|x|=1\\}$ where we write $\\{|x|=1\\}$
for $\\{x\in\mathcal{O}_{F}^{\times};\ x\bar{x}=1\\}.$ Further $\bar{}\,$
commutes with any embedding $F\to\mathbb{C}$ since $F$ is a CM field. Thus, if
$x\in\mathcal{O}_{F}^{\times}$ satisfies $x\bar{x}=1$, then for each embedding
$\sigma:F\to\mathbb{C}$ the complex absolute value of $\sigma(x)$ is 1 and
hence $x$ has to be a root of unity. This shows that $\\{|x|=1\\}$ is
contained in $\mu(F)$ which in turn is contained in $\\{|x|=1\\}$. Thus we
obtain that $\ker(N)\cap\mathcal{O}_{F}^{\times}=\\{|x|=1\\}=\mu(F)$, which
implies that $\ker(N)\cap\Gamma^{\times}=\mu(\Gamma)$ since
$\Gamma\subseteq\mathcal{O}_{F}$. Next, we compute that

$|\mathcal{O}_{F}^{\times}/\mu(F)\mathcal{O}_{L}^{\times}|\cdot\tfrac{R_{F}}{R_{F/L}R_{L}}=2^{g-1}.$

Indeed this is a consequence of $R_{F/L}=1$ and the standard formula for the
regulator ratio $R_{F}/R_{L}$ which can be found for example in [Was82, Prop
4.16]. Here $R_{F/L}=1$ follows directly from the definition of the relative
regulator ([CF93, $\mathsection$3]), since $\\{|x|=1\\}=\mu(F)$ assures that
the relative units of $F/L$ have (free) rank zero. Now, we insert the
displayed computations into the inequalities of Lemma 8.5 and then we deduce
Lemma 8.6.∎

###  8.3 Proof of Proposition 8.1

In this section we prove Proposition 8.1 by combining the results collected in
previous sections. We continue the notation introduced at the beginning of
Section 8.

###### Proof of Proposition 8.1.

Recall that the set $\Phi$ in Proposition 8.1 denotes the set of
$I$-polarizations on an arbitrary $\mathcal{O}$-abelian scheme $(A,\iota)$
over $S$ of relative dimension $g=[L:\mathbb{Q}]$, where $S$ is a connected
Dedekind scheme with function field of characteristic zero. Hence we can apply
Lemma 8.3 and we consider again the $L$-algebra
$D=\textup{End}_{\mathcal{C}}^{0}(A)$.

We first assume that $\dim_{L}(D)=1$. Then Lemma 8.3 (i) gives an order
$\Gamma$ of $L$ with $\textup{End}_{\mathcal{C}}(A)\cong\Gamma$ and an
embedding $\Phi/{\sim}\hookrightarrow\Gamma^{\times}/(\Gamma^{\times})^{2}$.
In particular it holds

$|\Phi/{\sim}|\leq|\Gamma^{\times}/(\Gamma^{\times})^{2}|.$

Further, Dirichlet’s unit theorem [Neu99, p.81] provides that the free part of
$\Gamma^{\times}$ has rank $g-1$, since $L$ is totally real. Thus
$|\Gamma^{\times}/(\Gamma^{\times})^{2}|$ is at most $w_{L}\cdot 2^{g-1}$,
where we recall that $w_{R}$ denotes the cardinality of the group $\mu(R)$ of
roots of unity in a ring $R$. Notice that $\mu(L)=\\{1,-1\\}$ are the only
roots of unity in the totally real field $L$, the ring $D$ is isomorphic to
$L$ since $\dim_{L}(D)=1$, and the ring
$\Gamma\cong\textup{End}_{\mathcal{C}}(A)$ contains $\\{1,-1\\}$. Hence we
obtain that $w=w_{L}=w_{\Gamma}=w^{\prime}$ and it follows that
$|\Gamma^{\times}/(\Gamma^{\times})^{2}|\leq\tfrac{w^{2}}{w^{\prime}}\cdot
2^{g-1}$. Therefore the displayed inequality proves Proposition 8.1 in the
case when $\dim_{L}(D)=1$.

We now assume that $\dim_{L}(D)\neq 1$. Then an application of Lemma 8.3 (ii)
gives the following: The ring $\textup{End}_{\mathcal{C}}(A)$ identifies with
an order $\Gamma\supset\mathcal{O}$ of a CM field $F$ with maximal totally
real subfield $L$ and there exists an embedding
$\Phi/{\sim}\hookrightarrow\Gamma^{\times}/N(\Gamma^{\times})$ where we recall
that $N:F^{\times}\to L^{\times}$ denotes the field norm. It follows that

$|\Phi/{\sim}|\leq|\Gamma^{\times}/N(\Gamma^{\times})|.$

Next, we observe that the field extension $F/L$ and the order
$\Gamma\supset\mathcal{O}$ of $F$ satisfy all assumptions of Lemma 8.6. Hence
this lemma provides that
$|\Gamma^{\times}/N(\Gamma^{\times})|\leq\tfrac{w_{F}^{2}}{w_{\Gamma}}\cdot
2^{g-1}$. Moreover, in this bound we can replace $w_{\Gamma}$ by $w^{\prime}$
and $w_{F}$ by $w$, since $\Gamma\cong\textup{End}_{\mathcal{C}}(A)$ and thus
$F=\Gamma\otimes_{\mathbb{Z}}\mathbb{Q}\cong D$. Therefore the displayed
inequality proves Proposition 8.1 in the case when $\dim_{L}(D)\neq 1$. This
completes the proof of Proposition 8.1. ∎

The above proof shows that the number $w$ appearing in Proposition 8.1 can be
bounded only in terms of the relative dimension $g$ of $A$ over $S$. For
example, one obtains an explicit bound and an inexplicit but asymptotically