# Polarized products of elliptic curves with complex multiplication and field
of moduli $\mathds{Q}$

Fabien Narbonne Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
Email address<EMAIL_ADDRESS>
(March 2022)

###### Abstract

Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an
equivalence of categories between the category of polarized abelian varieties
isomorphic to a product of elliptic curves over $\mathds{C}$ with complex
multiplication (CM) by $R$ and the category of integral hermitian
$R$-lattices. Then we apply this equivalence to enumerate all the genus $2$
and $3$ curves with field of moduli $\mathds{Q}$ and with Jacobian isomorphic
to a product of elliptic curves with CM by $R$.

## Introduction

Let $E$ be an elliptic curve over $\mathds{C}$ with complex multiplication by
a maximal order $R$ in a quadratic imaginary field $K$. Then $E$ admits a
model over $\overline{\mathds{Q}}$, it even admits one over
$\mathds{Q}(j(E))$, which has degree $\\#\operatorname{Cl}(R)$ over
$\mathds{Q}$, and not over any sub-extension. It may then be surprising that
powers of CM elliptic curves may be defined over smaller fields than expected,
sometimes even over $\mathds{Q}$. In [FG18, Theorem 1.4] the authors show for
instance that there are abelian surfaces defined over $\mathds{Q}$,
$\overline{\mathds{Q}}$-isogenous to $E^{2}$, for exactly $18$ discriminants.
This study is motivated by the conjecture on the possible finiteness of the
set of endomorphism rings of abelian varieties of a given dimension over a
fixed degree extension field of $\mathds{Q}$.

A weaker requirement is to ask for the field of moduli of a (polarized)
abelian variety to be $\mathds{Q}$. In [GHR19] the authors give the finite
list of indecomposable principally polarized abelian surfaces (also known as
Jacobian of genus $2$ curves) with field of moduli $\mathds{Q}$ which are
isomorphic to $E^{2}$.

In the present article, we address the case of abelian varieties isomorphic to
products $E_{1}\times\cdots\times E_{g}$, $g>1$, of elliptic curves with CM by
a maximal order $R$ and we give an exhaustive list of these principally
polarized abelian varieties for the case $g=2$ in Table 1 and a partial list
for $g=3$ in Table 2. Before explaining the structure of the paper and our
strategy, we want to point out that we strongly believe that the restriction
on $R$ being maximal can be dropped and we hope to work on it soon. Once this
will be done, one would have an even stronger extension of loc. cit. since
every abelian varieties _isogenous_ over $\overline{\mathds{Q}}$ to $E^{g}$ is
actually isomorphic to a product of elliptic curves with CM by (possibly
distinct) orders in $K=\operatorname{Frac}{R}$ ([Kan11, Th.2]). We also hope
to address the finer question of the descend of the principally polarized
abelian variety over its field of moduli.

In Section 1 we recall some properties of the classical equivalence of
categories between the complex abelian varieties and polarizable tori and how
it behaves with polarizations. Polarizations on abelian varieties give rise to
positive definite hermitian forms with a compatibility condition on the
lattice.

In Section 2 we restrict the equivalence of categories to the abelian
varieties isomorphic to the product of elliptic curves with complex
multiplication by $R.$ The complex tori $V/\Gamma$ associated to those abelian
varieties have additional structure ; we can endow their lattice $\Gamma$ with
a structure of $R$-module. The corresponding hermitian form endows on the
$R$-module $\Gamma$ a structure of integral hermitian lattice (up to rescaling
the hermitian form which only depends on $R$). This restriction leads to an
equivalence of categories between polarized abelian varieties isomorphic to a
product of elliptic curves with CM by $R$ and integral hermitian $R$-lattices
(Theorem 2). Moreover, hermitian forms corresponding to principal polarization
give unimodular lattices through the equivalence. We may note that a similar
functor is developed in [JKP+18] in a much wider frame since it is defined for
any field, not only $\mathds{C}.$ Under some conditions it is also an
equivalence of categories. However, we chose to focus our attention on a much
simpler functor since we will need to refine it in the next section.

The goal of Section 3 is to translate the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ on abelian varieties
$\overline{\mathds{Q}}$-isomorphic to $E_{1}\times\cdots\times E_{g}$ into the
category of integral hermitian lattices through the previous functor. The
action of $\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ can be
decomposed into two steps: the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and the action of
$\operatorname{Gal}(K/\mathds{Q})$, the complex conjugation. The latter is a
continuous action and it is therefore easier to handle. For the first one, we
use the existence of a surjective morphism
$F\colon\operatorname{Gal}(\overline{\mathds{Q}}/K)\rightarrow\operatorname{Cl}(R)$
such that the action by conjugation of
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ on abelian varieties
corresponds to the tensor product of the associated lattice by a
representative of $F(\sigma)$ (and a rescaling of the hermitian form).

Still, for both actions, we need to rigidify the choice of the target objects
under the previous equivalences of categories, which are defined only up to
$\mathds{C}$-isomorphisms. Indeed, unlike [GHR19] where the compatibilities
between the various abelian varieties and their conjugate were obvious, we
could not find a simple way to impose them from abstract nonsense. We
therefore use the explicit algebraization by the Weiestrass function for
elliptic curves to be able to work out the explicit Galois action on the
associated hermitian $R$-lattices (see Proposition 2) and we then extend it,
component by component to products of elliptic curves in order to obtain our
main results (Theorems 3 and 4). Notice that the main difficulty is to be able
to keep the abelian varieties, their isogenies _and_ their analytic
representation defined over $\overline{\mathds{Q}}$ to be able to translate
the action of $\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$.

Finally, in Section 4, we look when $\mathds{Q}$ is the field of moduli of the
indecomposable principally polarized abelian varieties $A\simeq
E_{1}\times\cdots\times E_{g}$ where $E_{i}$ has with CM by $R$. We first
extend the result of [GHR19] showing that if $\mathds{Q}$ is the field of
moduli then $\operatorname{Cl}(R)$ has exponent dividing $g$. We also show
that the Steinitz class of the $R$-lattice associated to $A$ is of order at
most $2$ (see Proposition 9). From this, we deduce the surprising Corollary 1
that odd-dimensional $A$ with field of moduli $\mathds{Q}$ must be isomorphic
to the power of an elliptic curve.

For a given dimension $g$, under the Extended Riemann Hypothesis there are
only finitely many $R$ which are of exponent $g$. Their list is known up to
$g=8$, see [EKN20]. However, we do explicit computation with our Algorithm 1
only for $g=2$ and $3$ as the computations become quickly time-consuming when
$g$ and the discriminant are growing. For $g=2$, we find $137$ indecomposable
principally polarized abelian varieties with field of moduli $\mathds{Q}$
completing the $46$ found in [GHR19]. For $g=3$, we find $33$ up to one
missing discriminant.

### Acknowledgments

I would like to acknowledge Francesc Fité and Xavier Guitart who motivated
this project and helped until its fulfilment.

In addition, I want to thank Marco Streng for its attention and advises along
the way of this work.

I would also like to address a special thank to Markus Kirschmer for his
advises and patience answering all of my questions about hermitian lattices.
He also gave me an extension of the Magma library of [KNRR21] to handle more
efficiently the classification of free hermitian unimodular $R$-lattices which
we used for the computation of the dimension $g=3$ case in Section 4.3.

Finally, I want to thank Harun Kir for the rich discussions we had on this
subject and for its precious support for this work.

##  1 Complex abelian varieties and complex tori

###  1.1 Abelian varieties and polarizable tori

Let us first recall that there is an equivalence of categories between abelian
varieties over $\mathds{C}$ and the category of polarizable tori given by
$\mathbf{T}\colon A\mapsto A(\mathds{C})$, see [Mil08, Theorem 2.9]. A complex
torus is a quotient of groups $X=V/\Gamma$ with $V$ a complex vector space of
finite dimension and $\Gamma$ a $\mathds{Z}$-lattice of $V$, i.e., a subgroup
of $V$ generated by a real basis. A torus $X=V/\Gamma$, or a lattice
$\Gamma\subseteq V$, is said to be polarizable if there exists a positive
definite hermitian form $h\colon V\times V\rightarrow\mathds{C}$ such that

$\operatorname{im}h(\Gamma,\Gamma)\subseteq\mathds{Z}$

where $\operatorname{im}$ is the imaginary part. Morphisms of complex tori are
morphisms of groups $\varphi\colon X=V/\Gamma\rightarrow
X^{\prime}=V^{\prime}/\Gamma^{\prime}$. Such maps can be lifted to a linear
maps $\varphi_{\text{an}}\colon V\rightarrow V^{\prime}$ that sends $\Gamma$
on $\Gamma^{\prime}$ called the analytic representation of $\varphi$. The
group morphism
$\varphi_{\text{rat}}={\varphi_{\text{an}}}_{|\Gamma}\colon\Gamma\rightarrow\Gamma^{\prime}$
is called the rational representation of $\varphi.$ We denote the set of the
analytic representations of morphisms between $X$ and $X^{\prime}$ by
$\operatorname{Hom}_{\mathds{C}}(\Gamma,\Gamma^{\prime})$. We will often
consider analytic representations directly as morphisms of polarizable tori.

A surjective morphism $f\colon A\rightarrow B$ between abelian varieties $A$
and $B$ over $\mathds{C}$ of equal dimension is called an isogeny, its kernel
is finite and its cardinal $\\#\ker f$ is called the degree of $f$, denoted
$\deg f.$ We will also call isogeny the corresponding morphism of tori
$\varphi\colon V/\Gamma\rightarrow V^{\prime}/\Gamma^{\prime}$ by the functor
$\mathbf{T}.$ The degree of $\varphi$ is defined in the same way
$\deg\varphi=\\#\ker\varphi.$ Moreover, it satisfies

$\deg\varphi=(\Gamma^{\prime}\colon\varphi_{\text{an}}(\Gamma))=\deg f$

with
$(\Gamma^{\prime}\colon\varphi_{\text{an}}(\Gamma))=\\#(\Gamma^{\prime}/\varphi_{\text{an}}(\Gamma))$,
the index of $\varphi_{\text{an}}(\Gamma)$ in $\Gamma^{\prime}.$

###  1.2 Duality

Let $V$ be a dimension $g$ complex vector space and $\Gamma\subseteq V$ a
$\mathds{Z}$-lattice. The dual lattice associated to $\Gamma$ is defined by

$\widehat{\Gamma}=\\{\ell\in
V^{*},\operatorname{im}\ell(\Gamma)\subseteq\mathds{Z}\\}$

where $V^{*}$ denotes the set of antilinear forms $\ell\colon
V\rightarrow\mathds{C}$. This defines the dual complex torus of $V/\Gamma$,
denoted $\widehat{V/\Gamma}:=V^{*}/\widehat{\Gamma}$.
If $A$ is a complex abelian variety such that $A(\mathds{C})\simeq V/\Gamma$
then there exists an isomorphism
$\widehat{A}(\mathds{C})\simeq\widehat{V/\Gamma}$, where $\widehat{A}$ is the
dual abelian variety of $A$ (see [Mum70]). Moreover, for $f\colon A\rightarrow
B$ a morphism and $\varphi=\mathbf{T}(f)\colon V/\Gamma\rightarrow
V^{\prime}/\Gamma^{\prime}$ we have
$\mathbf{T}(\widehat{f})_{\text{an}}=\varphi_{\text{an}}^{*}$ with

$\begin{array}[]{rrl}\varphi_{\text{an}}^{*}\colon&V^{\prime*}&\longrightarrow
V^{*}\\\ &\ell&\longmapsto\ell\circ\varphi_{\text{an}}\end{array}$

where $\widehat{f}\colon\widehat{B}\rightarrow\widehat{A}$ is the dual
morphism of $f$ (see [BL04, Section 2.4]).

###  1.3 Polarizations and hermitian forms

Let $A$ be an abelian variety and let $\mathscr{L}$ be a line bundle. We
consider the map

$\begin{array}[]{rrl}a_{\mathscr{L}}\colon&A&\longrightarrow\widehat{A}\\\
&x&\longmapsto t_{x}^{*}\mathscr{L}\otimes\mathscr{L}^{-1}.\end{array}$

with $t_{x}$ the translation by $x$ map. Consider isomorphisms
$A(\mathds{C})\simeq V/\Gamma$ and $\widehat{A}(\mathds{C})\simeq
V^{*}/\widehat{\Gamma}$. Let $h=c_{1}(\mathscr{L})$ be the first Chern class
of $\mathscr{L}$ which we identify with a hermitian form on $V$ (see [BL04,
Lemma 2.4.5]). Then the map
$\rho_{h}\in\operatorname{Hom}_{\mathds{C}}(\Gamma,\widehat{\Gamma})$ defined
by

$\begin{array}[]{rrl}\rho_{h}\colon&V&\longrightarrow V^{*}\\\ &v&\longmapsto
h(v,\\_).\end{array}$

is the analytic representation of $a_{\mathscr{L}}$. A polarization on an
abelian variety $A$ is an isogeny $a_{\mathscr{L}}$ with $\mathscr{L}$ an
ample line bundle. In this case, its first Chern class $h$ is a positive
definite hermitian form (see [BL04, Proposition 4.5.2]).

A couple $(A,a)$ with $A$ an abelian variety and $a$ a polarization is called
a polarized abelian variety. Morphisms of polarized abelian varieties are
defined by maps $f\colon(A,a)\rightarrow(B,b)$ such that $\widehat{f}bf=a$ and
we call them polarized isogenies. A polarized torus is a couple
$(V/\Gamma,\rho_{h})$ (also denoted by $(\Gamma,h)$) with $V/\Gamma$ a complex
torus and
$\rho_{h}\in\operatorname{Hom}_{\mathds{C}}(\Gamma,\widehat{\Gamma})$ induced
by a positive definite hermitian form $h$, i.e., $\rho_{h}(v)=h(v,\\_)$. We
define in the same way morphisms,
$\varphi\colon(V/\Gamma,\rho_{h})\rightarrow(V^{\prime}/\Gamma^{\prime},\rho_{h^{\prime}})$
between polarized tori. Their analytic representation must satisfy
$\varphi_{\text{an}}^{*}\rho_{h^{\prime}}\varphi_{\text{an}}=\rho_{h}$ which
means that for $v,w\in V$,

$\rho_{h}(v)(w)=h(v,w)=h^{\prime}(\varphi_{\text{an}}(v),\varphi_{\text{an}}(w)).$

In particular, analytic representations of polarized isogenies define
isometries on the associated hermitian spaces. Since, the functor $\mathbf{T}$
is an equivalence of categories, it also defines an equivalence of categories
$\mathbf{T}^{p}$ between polarized abelian varieties and polarized tori
$(A,a)\mapsto(X=\mathbf{T}(A),\rho_{h}=\mathbf{T}(a))$.

For any group schemes $A$ and $B$ over a field $k$ there is an isomorphism of
groups

$A(k)\times B(k)\simeq(A\times_{k}B)(k).$

We can apply it to abelian varieties over $\mathds{C}$. For
$A(\mathds{C})\simeq V_{A}/\Gamma_{A}$, $B(\mathds{C})\simeq V_{B}/\Gamma_{B}$
and $(A\times B)(\mathds{C})\simeq V/\Gamma$ we have

$V/\Gamma\simeq(V_{A}/\Gamma_{A})\times(V_{B}/\Gamma_{B}).$

Hence, $\Gamma\simeq\Gamma_{A}\times\Gamma_{B}$ and in terms of categories,
the functor $\mathbf{T}$ commutes with products. In the same way the functor
$\mathbf{T}^{p}$ also commutes with products.

##  2 Totally split CM abelian varieties and $R$-module structure

Let $R$ be the maximal order of a quadratic imaginary field
$K=\mathds{Q}(\sqrt{-d})$ with $d$ a positive square-free integer. A
$R$-lattice is a finitely presented, torsion free $R$-module. We denote by
$\mathbfcal{L}_{R}$, the category of $R$-lattices. In this section we want to
show that there exists equivalences of categories

  * •

between $\mathbfcal{A}_{R}$, the category of complex abelian varieties
isomorphic to a product of elliptic curves with CM by $R$, and
$\mathbfcal{L}_{R}$.

  * •

between $\mathbfcal{A}_{R}^{p}$, the category of complex polarized abelian
varieties isomorphic to a product of elliptic curves with CM by $R$, and
$\mathbfcal{L}_{R}^{h,int}$, the category of integral hermitian $R$-lattices,
i.e., $R$-lattices equipped with a positive definite hermitian form $H$ on
$KL$, such that $H(L,L)\subseteq R$.

For the rest of the article we fix field extensions

$\mathds{Q}\longrightarrow
K\longrightarrow\overline{\mathds{Q}}\longrightarrow\mathds{C}.$

###  2.1 Elliptic curve with complex multiplication over $\mathds{C}$

Let $R=\mathds{Z}[\omega]$ be a maximal order of an imaginary quadratic field
$K=\mathds{Q}(\sqrt{\Delta})$ with $\Delta<0$ the discriminant of
$K/\mathds{Q}$. We chose $\omega$ a generator of $R$ with
$\alpha_{R}=\operatorname{im}\omega>0$. Notice that $\alpha_{R}$ does not
depend on the chosen generator $\omega$ with positive imaginary part.

Let $E$ be an elliptic curve over $\mathds{C}$ with complex multiplication by
$R$, i.e., there exists a ring isomorphism $\text{End}(E)\simeq R$. Let
$\Lambda\subseteq\mathds{C}$ be a lattice and consider any isomorphism
$\eta\colon\mathds{C}/\Lambda\xrightarrow[]{\sim}E(\mathds{C})$, in this case
we will denote $E$ by $E_{\Lambda}$. By [Sil94, Proposition II.1.1.] there is
a unique isomorphism

$[.]_{E}\colon R\xrightarrow[]{\sim}\text{End}(E)$

characterized by the commutativity of the diagram of Figure 1. For any
$\alpha\in R$,

$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z\mapsto\alpha
z}$$\scriptstyle{\eta}$$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{E_{\Lambda}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\alpha]}$$\textstyle{E_{\Lambda}(\mathds{C}).}$
Figure 1: The bracket isomorphism

We will always use this isomorphism when we identify $R$ with $\text{End}(E).$

According to [Sil94, Proposition 2.1] all CM elliptic curves over $\mathds{C}$
admit a model over $\overline{\mathds{Q}}.$ We denote by
$\operatorname{Ell}(R)$ the set of isomorphism classes of elliptic curves over
$\overline{\mathds{Q}}$ with CM by $R.$ We recall from [Sil94, Proposition
1.2] that $\operatorname{Cl}(R)$, the class group of $R,$ acts simply
transitively on $\operatorname{Ell}(R)$ and the action is given by

$\mathfrak{a}\star E_{\Lambda}=E_{\mathfrak{a}^{-1}\Lambda}.$

The action is well defined on the isomorphism classes because another
representative $\mathfrak{b}$ of the class of $\mathfrak{a}$ in
$\operatorname{Cl}(R)$ differs from $\mathfrak{a}$ by a scalar and then
$\mathfrak{a}^{-1}\Lambda$ and $\mathfrak{b}^{-1}\Lambda$ are homothetic
lattices.

###  2.2 Totally split complex tori and $R$-module structure

Let $L$ be a $R$-lattice, i.e., a finitely presented, torsion free $R$-module.
Since $R$ is maximal, $L$ is a module over a Dedekind domains and by [O’M00,
Theorem 81.3], we can always write $L$ as a sum

$L=\bigoplus_{i=1}^{g}\mathfrak{a}_{i}x_{i}$

with a basis $x_{1},\dots,x_{g}$ of the $K$-vector space $L\otimes_{R}K$ and
fractional ideals $\mathfrak{a}_{1},\dots,\mathfrak{a}_{g}$ of $R$. The couple
$(\mathfrak{a}_{i},x_{i})$ is called a pseudo-basis of $L$ and the Steinitz
class $\operatorname{st}(L)$ of $L$ is defined by the class of the product
$\mathfrak{a}_{1}\cdots\mathfrak{a}_{g}$ in $\operatorname{Cl}(R)$. The
Steinitz class of a lattice togeither with its rank determines its
$R$-isomorphism class [Con16, Theorem 13].
For a $R$-ideal $\mathfrak{a}$ of $R$ the norm of $\mathfrak{a}$ is defined by
$N(\mathfrak{a})=\\#\left(R/\mathfrak{a}\right)$. We can extend the definition
of the norm to any fractional ideal by the relation
$N(\lambda\mathfrak{a})=|\lambda|^{2}N(\mathfrak{a})$ for any $\lambda\in K.$

######  Theorem 1.

Let $R$ be an order in a quadratic imaginary field $K.$ There is an
equivalence of categories between $\mathbfcal{A}_{R}$, abelian varieties
isomorphic to a product of elliptic curves with CM by $R$, and
$\mathbfcal{L}_{R}$, $R$-lattices given by

$\begin{array}[]{rcl}\mathbf{F}\colon&\mathbfcal{A}_{R}&\rightarrow\mathbfcal{L}_{R}\\\
&A\text{ s.t. }A(\mathds{C})\simeq V/\Gamma&\mapsto\Gamma\\\
&(\mathds{C}\otimes L)/L&\mapsfrom L.\end{array}$

on objects and for $f\colon A(\mathds{C})\simeq V/\Gamma\rightarrow
A^{\prime}(\mathds{C})\simeq
V^{\prime}/\Gamma^{\prime},\mathbf{F}(f)=f_{\text{rat}}.$

###### Proof.

Let $A\in\mathbfcal{A}_{R}$ such that $\mathbf{T}(A)=A(\mathds{C})=V/\Gamma.$
There is an isomorphism $\varphi\colon
V/\Gamma\rightarrow\mathds{C}^{g}/\bigoplus\Lambda_{i}.$ The
$\mathds{Z}$-lattices $\Lambda_{i}$ have a natural structure of
$R=\text{End}(\Lambda_{i})$-module. The lattice $\Gamma$ is stable by
multiplication by $R$, indeed,

$\varphi_{\text{an}}(R\Gamma)=R\varphi_{\text{an}}(\Gamma)=R\bigoplus_{i}\Lambda_{i}=\bigoplus_{i}\Lambda_{i}=\varphi_{\text{an}}(\Gamma)$

so, $R\Gamma=\Gamma.$ Moreover, the isomorphism $\varphi$ endows $\Gamma$ with
a structure of $R$-module in a natural way

$r.\gamma=\varphi_{\text{an}}^{-1}(r\varphi_{\text{an}}(\gamma))=r\gamma,\text{
for }r\in R,\gamma\in\Gamma.$ (1)

Hence, $\Gamma$ has a structure of $R$-module and we can see in (1) that this
structure does not depend on the isomorphism $\varphi$ we chose.

Reciprocally, let $L$ be a $R$-lattice and $(\mathfrak{a}_{i},x_{i})$ a
pseudo-basis of it. Since, the $\mathfrak{a}_{i}$ are fractional ideals there
exists an integer $n$ such that $n\mathfrak{a}_{i}\subseteq R$, for all $i,$
so there exists an isogeny $\operatorname{Hom}_{\mathds{C}}(L,R^{g}).$ If we
endow $R^{g}$ with the canonical hermitian form
$h_{0}(x,y)={}^{t}\\!x\overline{y}$, we have $h_{0}(R^{g},R^{g})=R$, hence,

$\operatorname{im}h_{0}(R^{g},R^{g})=(\operatorname{im}\omega)\mathds{Z}=\alpha_{R}\mathds{Z}.$

Thus, $\left(R^{g},\frac{1}{\alpha_{R}}h_{0}\right)$ defines a polarized torus
and so does $\left(\Gamma,\frac{n^{2}}{\alpha_{R}}h_{0}\right)$. This proves
that the underlying $\mathds{Z}$-lattice of a $R$-lattice $\Gamma$ is
polarizable.

Finally, given a morphism $f\colon V/\Gamma\rightarrow
V^{\prime}/\Gamma^{\prime},\mathbf{F}(f)=f_{\text{rat}}\colon\Gamma\rightarrow\Gamma^{\prime}$
and since $f_{\text{rat}}={f_{\text{an}}}_{|\Gamma}$ and $f_{\text{an}}$ is
$\mathds{C}$-linear, and then $R$-linear, it is clear that $\mathbf{F}$ maps
arrows in a full and faithful way.

Hence, there is an equivalence of categories between $R$-lattices and
polarizable tori $X$ such that there exists an isomorphism
$X\rightarrow\mathds{C}/\bigoplus\Lambda_{i}.$ The later being in equivalence
with the category of abelian varieties isomorphic to a power of elliptic
curves with CM by $R.$ This proves the first equivalence of categories we want
to show between $\mathbfcal{A}_{R}$ and $\mathbfcal{L}_{R}$. ∎

It may be interesting to enhance why the CM case is so specific. If we
consider the functor $\mathbf{F}$ from abelian varieties over $\mathds{C}$
without restriction to the category of sub $\mathds{Z}$-lattices of a finite
dimensional $\mathds{C}$-vector space $\mathbf{F}$ is not essentially
surjective. Indeed, some $\mathds{Z}$-lattices $\Gamma$ of $V$ of dimension
greater than $2$ are not polarizable. Even if we restrict $\mathbf{F}$ on its
essential image it is not full. Indeed, even in dimension $1$ there are
morphisms of polarizable $\mathds{Z}$-lattices (i.e., morphisms of groups)
which are not the restriction of a $\mathds{C}$-linear map.

Hence, the structure of $\mathds{Z}$-module is not enough. Fortunately,
considering the $R$-module structure given by the complex multiplication makes
$\mathbf{F}$ an equivalence.

###  2.3 Polarizable tori and integral lattices

A hermitian $R$-lattice is defined as couple $(L,H)$ with $L$ a $R$-lattice
and $H$ a positive definite hermitian form on the ambient space $KL$. The
scale of a hermitian lattice $(L,H)$ is defined as the fractional ideal
$\mathfrak{s}(L)=H(L,L)\subseteq K$. The _dual lattice_ $L^{\\#}$ of a
hermitian lattice is the lattice defined by $L^{\\#}=\\{v\in V,H(v,L)\subseteq
R\\}$. We say that $(L,H)$ is $\mathfrak{a}$-modular if
$\mathfrak{a}L^{\\#}=L$. If $(L,H)$ is $\mathfrak{a}$-modular then its scale
satisfies $\mathfrak{s}(L)=\mathfrak{a}$. A hermitian $R$-lattice $(L,H)$ is
said to be integral if

$H(L,L)\subseteq R,$

i.e., it is integral if its scale is an integral ideal and it is also
equivalent to $L\subseteq L^{\\#}.$ A hermitian lattice $(L,H)$ which is
$R$-modular is called unimodular, it is equivalent to the conditions $(L,H)$
is integral and its scale is $\mathfrak{s}(L)=(1)=R.$ One also defines the
volume of a hermitian lattice $(L,H)$ as the fractional ideal

$\mathfrak{v}(L)=\left(\bigoplus_{i=1}^{g}N(\mathfrak{a}_{i})\right)\det(G(x_{1},\dots,x_{g}))R$

where $G(b)$ denotes the Gram matrix of a family $b$, namely,
$G(b)=(H(x,y))_{x,y\in b}.$ A hermitian lattice $(L,H)$ is
$\mathfrak{a}$-modular if, and only if,

$\mathfrak{v}(L)=\mathfrak{a}^{g}\text{ and }\mathfrak{s}(L)=\mathfrak{a}.$

(see [Hof91, Section 2]).

In this section we want to explain the link between polarized tori and
integral lattices.

Let $(V,H)$ be a hermitian $\mathds{C}$-vector space and let
$\alpha\in\mathds{R}_{>0}$. We denote by $V^{\alpha}$ the vector space $V$
provided with the hermitian form $H^{\alpha}(x,y)=\alpha H(x,y)$. For a
lattice $L$ in $(V,H)$ we denote $L^{\alpha}$ for the hermitian lattice $L$
regarded in $(V^{\alpha},H^{\alpha})$ like in [O’M00, Section 82J.].

######  Lemma 1.

Let $\mathfrak{a}$ be a sub $R=\mathds{Z}[\omega]$-module of $\mathds{C}$.
Then $\mathfrak{a}$ is an integral ideal of $R$ if, and only if,
$\operatorname{im}\mathfrak{a}\subseteq\alpha_{R}\mathds{Z}$, with
$\alpha_{R}=\operatorname{im}\omega.$

###### Proof.

The direct sense is straightforward.

Let $a=x+y\omega\in\mathfrak{a}$ with $x,y\in\mathds{R}$. Since,
$\operatorname{im}a=y\operatorname{im}(\omega)\in(\operatorname{im}\omega)\mathds{Z}$
we have $y\in\mathds{Z}$. Moreover, $\overline{\omega}\in R$ so
$a\overline{\omega}\in\mathfrak{a}$ and
$\operatorname{im}a\overline{\omega}=-x\operatorname{im}\omega$ so
$x\in\mathds{Z}.$ Hence, $a\in\mathds{Z}[\omega]=R$ and then
$\mathfrak{a}\subseteq R.$ ∎

######  Proposition 1.

Let $(V/\Gamma,\rho_{h})$ be a polarized torus such that
$\Gamma\simeq\bigoplus_{i=1}^{g}\Lambda_{i}$ with
$\text{End}_{\mathds{C}}(\Lambda_{i})\simeq R$ with $R=\mathds{Z}[\omega]$.
Then $\mathfrak{s}(\Gamma^{\alpha_{R}})$ is an integral ideal of $R$.
Moreover, we have the relation

$\deg\rho_{h}=((\Gamma^{\alpha_{R}})^{\\#}\colon\Gamma^{\alpha_{R}}).$

###### Proof.

We know that $h(\Gamma,\Gamma)$ is a sub $R$-module of $\mathds{C}$ and since
$(\Gamma,h)$ is a polarizable torus we must have
$\operatorname{im}h(\Gamma,\Gamma)\subseteq\mathds{Z}$. Hence, by Lemma 1,
$\mathfrak{s}(\Gamma^{\alpha_{R}})=\alpha_{R}h(\Gamma,\Gamma)\subseteq R.$
Moreover,

$\displaystyle\deg\rho_{h}$
$\displaystyle=(\widehat{\Gamma}\colon\rho_{h}(\Gamma))$
$\displaystyle=(\rho_{h}^{-1}(\widehat{\Gamma})\colon\Gamma)$
$\displaystyle=\\#\\{v\in
V,\operatorname{im}h(v,\Gamma)\subseteq\mathds{Z}\\}/\Gamma$
$\displaystyle=\\#\\{v\in V,(\operatorname{im}\omega)h(v,\Gamma)\subseteq
R\\}/\Gamma\text{\leavevmode\nobreak\ (by Lemma \ref{integ})}$
$\displaystyle=\\#((\Gamma^{\alpha_{R}})^{\\#}/\Gamma)$
$\displaystyle=((\Gamma^{\alpha_{R}})^{\\#}\colon\Gamma^{\alpha_{R}}).$

∎

######  Remark 1.

If moreover the lattice $(\Gamma^{\alpha_{R}},h^{\alpha_{R}})$ is
$\mathfrak{s}(\Gamma^{\alpha_{R}})$-modular then

$\deg\rho_{h}=((\Gamma^{\alpha_{R}})^{\\#}\colon\mathfrak{s}(\Gamma^{\alpha_{R}}).(\Gamma^{\alpha_{R}})^{\\#})=N(\mathfrak{s}(\Gamma^{\alpha_{R}}))^{g}.$

Hence, classes of principally polarized tori $(\Gamma,h)$, i.e., with
$\deg\rho_{h}=1$ correspond to isometry classes of integral lattices which
scale has norm $1$ and it is an integral ideal by Proposition 1. So it
corresponds to unimodular lattices.

Let $\Lambda\subseteq\mathds{C}$ be a lattice such that
$\text{End}(\Lambda)=R.$ We define $\mathbfcal{T}_{R}^{p}$ the subcategory of
polarized torus $(X=V/\Gamma,\rho_{h})$, with
$\Gamma\simeq\bigoplus\Lambda_{i}$ with $\text{End}(\Lambda_{i})\simeq R$. We
can conclude with the following theorem.

######  Theorem 2.

With the notations above there is an equivalence of categories given on
objects by

$\begin{array}[]{rrl}&\mathbfcal{T}_{R}^{p}&\rightarrow\mathbfcal{L}_{R}^{h,int}\\\
&(X=V/\Gamma,\rho_{h})&\mapsto(\Gamma^{\alpha_{R}},h^{\alpha_{R}})\\\
&((L\otimes\mathds{C})/L,H^{1/\alpha_{R}})&\mapsfrom(L,H).\end{array}$

Since, $\mathbfcal{T}_{R}^{p}$ is equivalent to the category
$\mathbfcal{A}_{R}^{p}$ by the functor $\mathbf{T}$, we have the second
equivalence we wanted between $\mathbfcal{A}_{R}^{p}$ and
$\mathbfcal{L}_{R}^{h,int}.$ We call this functor $\mathbf{F}_{h}$ and it
satisfies

$\begin{array}[]{rrl}\mathbf{F}_{h}\colon&\mathbfcal{A}_{R}^{p}&\rightarrow\mathbfcal{L}_{R}^{h,int}\\\
&(A,a)&\mapsto(\mathbf{F}(A)^{\alpha_{R}},\mathbf{F}(a)^{\alpha_{R}}).\end{array}$

Since every elliptic curve over $\mathds{C}$ with CM by $R$ has a model over
$\overline{\mathds{Q}}$, the category of polarized abelian varieties over
$\overline{\mathds{Q}}$ isomorphic to a product of $E_{i}$ with CM by $R$ is
equivalent to $\mathbfcal{A}_{R}^{p}$ by the base change functor

$A\mapsto A_{\mathds{C}}.$

Note that morphisms over $\mathds{C}$ of abelian varieties over
$\overline{\mathds{Q}}$ isomorphic to a product of CM elliptic curves are
actually defined over $\overline{\mathds{Q}}$ by [Sil94, Theorem 2.2.(c)].

##  3 Action of $\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$

The field of moduli of a polarized abelian variety $(A,a)$ over
$\overline{\mathds{Q}}$ is the fixed field by the subgroup

$\\{\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q}),(A^{\sigma},a^{\sigma})\simeq(A,a)\\}.$

Given a maximal order in an imaginary quadratic field $K$ we would like to
elaborate an algorithm to enumerate all the isomorphism classes of
$\mathbfcal{A}_{R}^{p}$ which have field of moduli $\mathds{Q}.$ In order to
do this we want to understand the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ on the hermitian
lattices through the functor $\mathbf{F}_{h}$ we developed in Section 2. In
other words given $(A,a)\in\mathbfcal{A}_{R}^{p}$ and
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ we want to
understand the isometry class of $\mathbf{F}_{h}(A^{\sigma},a^{\sigma})$ in
terms of $\mathbf{F}_{h}(A,a)$ and $\sigma.$

In order to do so we need to understand the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and the one of the complex
conjugation $\operatorname{Gal}(K/\mathds{Q})$ separately to recover the
action of
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})\simeq\operatorname{Gal}(\overline{\mathds{Q}}/K)\rtimes\operatorname{Gal}(K/\mathds{Q}).$
We describe the action of $\operatorname{Gal}(K/\mathds{Q})$ and
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$ in Theorem 3 and Theorem 4
respectively. The end of this section will be devoted to their proof.

Let us introduce the necessary notations to state the results we aim to show
in this section.

Let $(L,H)$ be a hermitian lattice. Let
$\iota\colon(KL,H)\rightarrow(K^{g},H^{\prime})$ be an isometry. We define
$(\overline{L},\overline{H})$ by $\overline{L}=\iota^{-1}\overline{\iota(L)}$
and

$\overline{H}(x,y)=\overline{H(\iota^{-1}\overline{\iota(x)},\iota^{-1}\overline{\iota(y)})}=\overline{H^{\prime}(\overline{\iota(x)},\overline{\iota(y)})}$

where $\overline{.}$ refers to the complex conjugation which is the unique
non-trivial automorphism of $\operatorname{Gal}(K/\mathds{Q}).$ The isometry
class of $(\overline{L},\overline{H})$ is independent of the choice of
$\iota$. We can now state the description of the action of the complex
conjugation.

######  Theorem 3 (Description of the action of
$\operatorname{Gal}(K/\mathds{Q})$).

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ over $\overline{\mathds{Q}}.$ Let
$\mathbf{F}_{h}(A,a)=(L,H)$ then there is an isometry

$\mathbf{F}_{h}(\overline{A},\overline{a})\simeq(\overline{L},\overline{H}).$

Let $E$ be an elliptic curve defined over $\overline{\mathds{Q}}$ with CM by
$R.$ Let $\Lambda\subseteq\mathds{C}$ be a lattice such that
$E_{\mathds{C}}=E_{\Lambda}$. In the future we will often confuse $E$ and its
base change by $E_{\mathds{C}}$ to avoid heavy notations such as
$E_{\mathds{C}}(\mathds{C})$ and we will write $E(\mathds{C})$ instead. Recall
that for $\dot{\mathfrak{a}}\in\operatorname{Cl}(R),$ the elliptic curve
$\dot{\mathfrak{a}}\star E_{\Lambda}$ is defined by
$E_{\mathfrak{a}^{-1}\Lambda}.$ By [Sil94, Proposition 2.4], there is a
surjective group morphism

$F\colon\operatorname{Gal}(\overline{\mathds{Q}}/K)\rightarrow\operatorname{Cl}(R)$

such that the elliptic curves $E^{\sigma}$ and $F(\sigma)\star E$ are
isomorphic. We can now state the description of the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$.

######  Theorem 4 (Description of the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$).

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ over $\overline{\mathds{Q}}$. Let
$\mathbf{F}_{h}(A,a)=(L,H),\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$
and $F(\sigma)=\dot{\mathfrak{a}}^{-1}\in\operatorname{Cl}(R)$. Then there is
an isometry

$\mathbf{F}_{h}(A^{\sigma},a^{\sigma})\simeq\left(\mathfrak{a}L,\frac{1}{N(\mathfrak{a})}H\right)$

where $N(\mathfrak{a})$ is the norm of $\mathfrak{a}.$

The proof of Theorem 3 is made easier by the Lemma 2 which uses the fact that
the complex conjugation in $\operatorname{Aut}(\mathds{C})$ is a continuous
map of $\mathds{C}$. With the identity they are the only automorphisms in
$\operatorname{Aut}(\mathds{C})$ with this property.

###  3.1 Positioning of the problem

Let us recall from Figure 1 that for every elliptic curve with CM by $R$ and
any isomorphism $\eta\colon\mathds{C}/\Lambda\rightarrow E(\mathds{C})$ there
is a unique isomorphism $[\cdot]_{E}\colon R\rightarrow\text{End}(E)$ such
that the following diagram commutes

$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\eta}$$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{E_{\Lambda}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\alpha]}$$\textstyle{E_{\Lambda}(\mathds{C}).}$

By, [Sil94, Theorem 2.2] the bracket isomorphism satisfies

$\text{for all }\sigma\in\operatorname{Aut}(\mathds{C}),\alpha\in
R,\left([\alpha]_{E}\right)^{\sigma}=[\alpha^{\sigma}]_{E^{\sigma}}.$

This is true with whatever $\Lambda$ and
$\eta\colon\mathds{C}/\Lambda\xrightarrow{\sim}E(\mathds{C})$ we chose and
whatever lattice identification
$\eta_{\sigma}\colon\mathds{C}/\Lambda_{\sigma}\xrightarrow{\sim}E^{\sigma}(\mathds{C})$
of $E^{\sigma}$ as soon as we again chose the same on left and right hand
sides of the diagram. Of course if we don’t take the same isomorphisms on the
right and on the left hand sides, for instance we chose $\eta$ and $-\eta$,
those properties do not hold anymore. The main difficulty we will encounter is
that we want to deal with isogenies $f\colon E\rightarrow E^{\prime}$ between
possibly non-isomorphic elliptic curves. If we expect diagrams like

$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\eta}$$\textstyle{\mathds{C}/\Lambda^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$$\textstyle{E(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E^{\prime}(\mathds{C}).}$

to have nice behaviour with $\operatorname{Aut}(\mathds{C})$ or just
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ we need a kind of
canonical way to describe the action of $\operatorname{Aut}(\mathds{C})$ (or
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$) on lattices, and a
canonical way to chose the isomorphisms $\eta$ (to avoid the $-\eta$ issue for
instance). We will show that the $\wp$-functions of Weierstrass will do the
job.

First we need to specify the isomorphisms we refer to when we consider the
analytic representation of a morphism between abelian varieties. Let $A$ and
$A^{\prime}$ be abelian varieties over $\mathds{C}$. Consider isomorphisms
$\eta\colon V/\Gamma\xrightarrow{\sim}A(\mathds{C})$ and $\eta^{\prime}\colon
V^{\prime}/\Lambda^{\prime}\xrightarrow{\sim}A^{\prime}(\mathds{C})$. Every
morphism $f\colon A\rightarrow A^{\prime}$ induces a commutative diagram as in
Figure 2.

$\textstyle{V/\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\eta}$$\textstyle{V^{\prime}/\Gamma^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$$\textstyle{A(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A^{\prime}(\mathds{C}).}$
Figure 2: Analytic representation of an isogeny

The morphism of tori $\varphi$ can be lifted to a linear map $\alpha\colon
V\rightarrow V^{\prime}$ such that $\alpha(\Gamma)\subseteq\Gamma^{\prime}$.
We call $\alpha$ the analytic representation of $f$ associated to the
isomorphisms $\eta$ and $\eta^{\prime}$ or simply the analytic representation
of $(f,\eta,\eta^{\prime}).$

We will use the Weierstrass $\wp$-functions and Eisenstein series as in
[Sil09] as a canonical way to identify an elliptic curve over $\mathds{C}$
with a complex torus and we will study the rationality of the induced analytic
representation for CM elliptic curves in Section 3.2. Then we will extend the
results for elliptic curves on product of elliptic curves component by
component in Section 3.3 to prove Theorem 3 and 4.

###  3.2 Action of $\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ on
CM elliptic curves

####  3.2.1 Fixing isomorphisms with Weierstrass $\wp$-functions

By [Sil09, Therorem 5.1], for any $A,B\in\mathds{C}$ such that
$4A^{3}-27B^{2}\neq 0$ there exists a unique lattice
$\Lambda\subset\mathds{C}$ such that

$A=g_{2}(\Lambda)=60\sum_{\omega\in\Lambda\setminus\\{0\\}}\frac{1}{\omega^{4}}\text{
and
}B=g_{3}(\Lambda)=140\sum_{\omega\in\Lambda\setminus\\{0\\}}\frac{1}{\omega^{6}}$

where $g_{2}(\Lambda)$ and $g_{3}(\Lambda)$ are the Eisenstein series of index
$4$ and $6$ of $\Lambda.$ Moreover, denoting Weierstrass $\wp$-functions by
$\wp$, by [Sil09, Proposition 3.6] the map

$\begin{array}[]{rrl}\phi_{\Lambda}\colon&\mathds{C}/\Lambda\xrightarrow{\sim}&\mathbb{P}^{2}(\mathds{C})\\\
&z\mapsto&[\wp(z,\Lambda)\colon\wp^{\prime}(z,\Lambda)\colon 1]\end{array}$

defines an isomorphism of Lie groups on its image $E(\mathds{C})$ with
$E/\mathds{C}$ with Weierstrass model

$E\colon y^{2}=4x^{3}-Ax-B.$

We will also call $\phi_{\Lambda}$ the induced isomorphism
$\mathds{C}/\Lambda\xrightarrow{\sim}E(\mathds{C}).$

_From now on, the notation $E_{\Lambda}$ means that
$E(\mathds{C})\simeq\mathds{C}/\Lambda$ with the isomorphism given by
$\phi_{\Lambda}.$ _

Let $\kappa\rightarrow\mathds{C}$ be a field extension and $E\colon
y^{2}=4x^{3}-Ax-B$ with $A,B\in\kappa$ and let $\Lambda$ be a lattice such
that $g_{2}(\Lambda)=A$ and $g_{3}(\Lambda)=B.$ For any
$\sigma\in\operatorname{Aut}(\kappa)$ we define $\Lambda_{\sigma}$ the unique
lattice such that $g_{2}(\Lambda_{\sigma})=A^{\sigma}$ and
$g_{3}(\Lambda_{\sigma})=B^{\sigma}$.

By definition of $\Lambda_{\sigma}$ we have an isomorphism
$\phi_{\Lambda_{\sigma}}\colon\mathds{C}/\Lambda_{\sigma}\rightarrow
E^{\sigma}(\mathds{C})$ with $E^{\sigma}\colon
y^{2}=4x^{3}-A^{\sigma}x-B^{\sigma}.$

######  Lemma 2.

For any lattice $\Lambda\subset\mathds{C},$

$\Lambda_{\hskip 1.42271pt\overline{.}}=\overline{\Lambda}.$

###### Proof.

The Eisenstein series $g_{2}(\Lambda)$ and $g_{3}(\Lambda)$ are absolutely
convergent and the complex conjugation is a continuous map so

$g_{k}(\overline{\Lambda})=\overline{g_{k}(\Lambda)}.$

∎

####  3.2.2 Rationality of analytic representations for CM elliptic curves

By [Sil94, Theorem 2.2.(c)] an isogeny $f\colon E\rightarrow E^{\prime}$
between two elliptic curves defined over $\overline{\mathds{Q}}$ is also
defined over $\overline{\mathds{Q}}.$ In this section we want to study the
analytic representation $\alpha$ of
$(f,\phi_{\Lambda},\phi_{\Lambda^{\prime}})$ when $E$ and $E^{\prime}$ have
Weierstrass model over $\overline{\mathds{Q}}$, with
$\mathds{C}/\Lambda\underset{\phi_{\Lambda}}{\simeq}E(\mathds{C})$ and
$\mathds{C}/\Lambda^{\prime}\underset{\phi_{\Lambda^{\prime}}}{\simeq}E^{\prime}(\mathds{C})$.

  * •

Is $\alpha$ in $\overline{\mathds{Q}}$ ?

  * •

If $\alpha\in\overline{\mathds{Q}}$, does $\alpha$ behaves well with
$\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$, i.e., is
$\alpha^{\sigma}$ the analytic representation of
$(f^{\sigma},\phi_{\Lambda_{\sigma}},\phi_{\Lambda^{\prime}_{\sigma}})$ ?

We will positively answer these questions in Proposition 2.

######  Lemma 3.

Let $\Lambda\subseteq\mathds{C}$ be a lattice and $E\colon
y^{2}=4x^{3}-g_{2}(\Lambda)x-g_{3}(\Lambda)$ be the elliptic curve over
$\mathds{C}$ associated to $\Lambda$ by $\phi_{\Lambda}.$ Let
$r\in\mathds{C}$, let $E^{\prime}$ be the elliptic curve associated to
$r\Lambda$ and let $\nu_{r}$ be the map defined by.

$\begin{array}[]{rrl}\nu_{r}&\mathds{P}^{2}(\mathds{C})\longrightarrow&\mathds{P}^{2}(\mathds{C})\\\
&[x\colon y\colon 1]\longmapsto&[\frac{1}{r^{2}}x\colon\frac{1}{r^{3}}y\colon
1]\end{array}$

Then the restriction of $\nu_{r}$ to
$E(\mathds{C})\subseteq\mathds{P}^{2}(\mathds{C})$ defines an isomorphism on
its image $E^{\prime}(\mathds{C}).$ Moreover, $r$ is the analytic
representation of
$({\nu_{r}}_{|E(\mathds{C})},\phi_{\Lambda},\phi_{r\Lambda}).$

###### Proof.

For all $z\in\mathds{C}\setminus
r\Lambda,r\in\mathds{C}\setminus\\{0\\},\wp(z,r\Lambda)=\frac{1}{r^{2}}\wp(\frac{z}{r},\Lambda)$
and
$\wp^{\prime}(z,r\Lambda)=\frac{1}{r^{3}}\wp^{\prime}(\frac{z}{r},\Lambda)$.
Hence,

$\displaystyle\phi_{r\Lambda}(z)$
$\displaystyle=[\wp(z,r\Lambda)\colon\wp^{\prime}(z,r\Lambda)\colon 1]$
$\displaystyle=[\frac{1}{r^{2}}\wp\left(\frac{z}{r},\Lambda\right)\colon\frac{1}{r^{3}}\wp^{\prime}\left(\frac{z}{r},\Lambda\right)\colon
1]$
$\displaystyle=\nu_{r}\left(\phi_{\Lambda}\left(\frac{z}{r}\right)\right).$

This proves that the following diagram commutes

$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\phi_{\Lambda}}$$\textstyle{\mathds{C}/r\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{r\Lambda}}$$\textstyle{E(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\nu_{r}}_{|E(\mathds{C})}}$$\textstyle{E^{\prime}(\mathds{C}).}$

which shows that $({\nu_{r}}_{|E(\mathds{C})},\phi_{\Lambda},\phi_{r\Lambda})$
has analytic representation $r$ ∎

Let $\mathfrak{a}$ be any fractional ideal of $R$, $\Lambda$ a lattice and
$E=E_{\Lambda}\in\mathbfcal{A}_{R}$. We defined $\mathfrak{a}\star E$ earlier
as the isomorphism class of $E_{\mathfrak{a}^{-1}\Lambda}$.

_From now on we will refer to $\mathfrak{a}\star E$ as the elliptic curve
defined by the Weierstrass equation_

$\mathfrak{a}\star E\colon
y^{2}=4x^{3}-g_{2}(\mathfrak{a}^{-1}\Lambda)x-g_{3}(\mathfrak{a}^{-1}\Lambda).$

Let $f\colon E\rightarrow E^{\prime}$ be an isogeny between elliptic curves
with $E^{\prime}=E_{\Lambda^{\prime}}$. Let $\mathfrak{a}$ be an integral
ideal of $R$ such that $\ker
f=E[\mathfrak{a}]=\cap_{a\in\mathfrak{a}}\ker[a]_{E}.$ According to [Sil94,
Proposition 1.4], $\mathfrak{a}\star E\simeq E/E[\mathfrak{a}]$ and we can
factor $f\colon E\rightarrow E^{\prime}$ and
$\alpha\colon\mathds{C}/\Lambda\rightarrow\mathds{C}/\Lambda^{\prime}$ as in
Figure 3 where
$\pi\colon\mathds{C}/\Lambda\rightarrow\mathds{C}/\mathfrak{a}^{-1}\Lambda$ is
the natural projections and $p\colon E\rightarrow\mathfrak{a}\star E$ induced
by $\pi$.

${\mathds{C}/\Lambda}$${\mathds{C}/\Lambda^{\prime}}$${\mathds{C}/\mathfrak{a}^{-1}\Lambda}$${E(\mathds{C})}$${E^{\prime}(\mathds{C})}$${(\mathfrak{a}\star
E)(\mathds{C}).}$$\scriptstyle{\alpha}$$\scriptstyle{\phi_{\Lambda}}$$\scriptstyle{\pi}$$\scriptstyle{\phi_{\Lambda^{\prime}}}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\nu_{\alpha}}$$\scriptstyle{\phi_{\mathfrak{a}^{-1}\Lambda}}$
Figure 3: Factorization of isogenies

######  Lemma 4.

Let $\Lambda$ be a lattice with complex multiplication such that
$g_{k}(\Lambda)\in\overline{\mathds{Q}}$ for $k=2,3.$ Then for any fractional
ideal $\mathfrak{a}\subseteq
K,g_{k}(\mathfrak{a}^{-1}\Lambda)\in\overline{\mathds{Q}}.$

###### Proof.

Let $E\colon y^{2}=4x^{3}-g_{2}(\Lambda)x-g_{3}(\Lambda)$ over
$\overline{\mathds{Q}}.$ Since all CM elliptic curve have a model over
$\overline{\mathds{Q}}$ we consider $E^{\prime}\colon
y^{2}=4x^{3}-A^{\prime}x-B^{\prime}$ a model of $\mathfrak{a}\star E$ over
$\overline{\mathds{Q}}$ and $\Lambda^{\prime}$ such that
$g_{2}(\Lambda^{\prime})=A^{\prime}\in\overline{\mathds{Q}}$ and
$g_{3}(\Lambda^{\prime})=B^{\prime}\in\overline{\mathds{Q}}$, i.e.,
$E^{\prime}=E_{\Lambda^{\prime}}$. Consider any isogeny $f\colon E\rightarrow
E^{\prime}$ over $\overline{\mathds{Q}}$ and $\alpha\in\mathds{C}$ the
analytic representation of $(f,\phi_{\Lambda},\phi_{\Lambda^{\prime}})$. Let
$r\in\overline{\mathds{Q}}$ be the coefficient of the induced map on
differentials $f^{*}\frac{dx^{\prime}}{y^{\prime}}=r\frac{dx}{y}$. By the
proof of [Sil09, Proposition 3.6.(b)] we have
$\phi_{\Lambda}^{*}\left(\frac{dx}{y}\right)=dz$ for any $E=E_{\Lambda}.$
Thus,

$(f\circ\phi_{\Lambda})^{*}\left(\frac{dx^{\prime}}{y^{\prime}}\right)=\phi_{\Lambda}^{*}\circ
f^{*}\left(\frac{dx^{\prime}}{y^{\prime}}\right)=\phi_{\Lambda}^{*}\left(r\frac{dx}{y}\right)=rdz$

and

$(\phi_{\Lambda^{\prime}}\circ\alpha)^{*}\left(\frac{dx^{\prime}}{y^{\prime}}\right)=\alpha
dz^{\prime}=\alpha dz$

Since the following diagram commutes

$\textstyle{\mathds{C}/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\phi_{\Lambda}}$$\textstyle{\mathds{C}/\Lambda^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Lambda^{\prime}}}$$\textstyle{E(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E^{\prime}(\mathds{C})}$

and we have equality of the differentials $rdz=\alpha dz$. Hence,
$r=\alpha\in\overline{\mathds{Q}}.$ Let $\mathfrak{b}$ an integral ideal such
that $\ker f=E[\mathfrak{b}].$ By Figure 3,
$\Lambda^{\prime}=\alpha\mathfrak{b}^{-1}\Lambda$ so
$g_{k}(\mathfrak{b}^{-1}\Lambda)=\alpha^{2k}g_{k}(\Lambda^{\prime})\in\overline{\mathds{Q}}$.
Finally, $\mathfrak{b}^{-1}\Lambda$ and $\mathfrak{a}^{-1}\Lambda$ give
isomorphic elliptic curves so they must be homothetic by some
$\lambda\in\operatorname{Hom}_{\mathds{C}}(\mathfrak{b}^{-1}\Lambda,\mathfrak{a}^{-1}\Lambda)=\\{\mu\in\mathds{C},\mu\mathfrak{b}^{-1}\Lambda\subseteq\mathfrak{a}^{-1}\Lambda\\}\subseteq
K$. Hence,

$g_{k}(\mathfrak{a}^{-1}\Lambda)=\lambda^{-2k}g_{k}(\mathfrak{b}^{-1}\Lambda)\in\overline{\mathds{Q}}.$

∎

######  Proposition 2.

Let $E\colon y^{2}=4x^{3}-Ax-B$ over $\overline{\mathds{Q}}$ and
$E^{\prime}\colon y^{2}=4x^{3}-A^{\prime}x-B^{\prime}$ over $\mathds{C}$ be
elliptic curves both with CM by $R$. Let $\Lambda$ and $\Lambda^{\prime}$ be
lattices with
$g_{2}(\Lambda)=A,g_{3}(\Lambda)=B,g_{2}(\Lambda^{\prime})=A^{\prime}$ and
$g_{3}(\Lambda^{\prime})=B^{\prime}$. Let $f\colon E\rightarrow E^{\prime}$ be
an isogeny over $\mathds{C}$. Consider $\alpha\in\mathds{C}$ the analytic
representation of $(f,\phi_{\Lambda},\phi_{\Lambda^{\prime}})$. Then

  1. 1.

$\alpha\in\overline{\mathds{Q}}$ if, and only if,
$g_{i}(\Lambda^{\prime})\in\overline{\mathds{Q}}$.

  2. 2.

If 1. is satisfied then for every
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q}),\alpha^{\sigma}$
is the analytic representation of
$(f^{\sigma},\phi_{\Lambda_{\sigma}},\phi_{\Lambda^{\prime}_{\sigma}})$ with
$f$ identified with the induced map over $\overline{\mathds{Q}}$.

  3. 3.

For every fractional ideal $\mathfrak{a}$ we have
$\left(\mathfrak{a}\Lambda\right)_{\sigma}=\mathfrak{a}^{\sigma}\Lambda_{\sigma}.$

###### Proof.

  1. 1.

Since $\Lambda^{\prime}=\alpha\mathfrak{a}^{-1}\Lambda$ we have
$g_{k}(\Lambda^{\prime})=\alpha^{-2k}g_{k}(\mathfrak{a}^{-1}\Lambda).$ By
Lemma 4, $g_{k}(\mathfrak{a}^{-1}\Lambda)\in\overline{\mathds{Q}}$ so
$\alpha\in\overline{\mathds{Q}}$ if, and only if,
$g_{k}(\Lambda^{\prime})\in\overline{\mathds{Q}}.$

  2. 2.

We define $\alpha_{\sigma}$ the analytic representation of
$(f^{\sigma},\phi_{\Lambda_{\sigma}},\phi_{\Lambda^{\prime}_{\sigma}}).$ We
want to show that $\alpha_{\sigma}=\alpha^{\sigma}.$ Since $\Lambda_{\sigma}$
and $\Lambda^{\prime}_{\sigma}$ are such that
$g_{i}(\Lambda_{\sigma})\in\overline{\mathds{Q}}$ and
$g_{i}(\Lambda^{\prime}_{\sigma})\in\overline{\mathds{Q}}$, by $1.$
$\alpha_{\sigma}\in\overline{\mathds{Q}}$. According to [Sil94, Theorem 2.2],

$\ker(f^{\sigma})=(\ker
f)^{\sigma}=\cap_{a\in\mathfrak{a}}(\ker[a]_{E})^{\sigma}=\cap_{a\in\mathfrak{a}}\ker[a^{\sigma}]_{E^{\sigma}}=E^{\sigma}[\mathfrak{a}^{\sigma}].$
(2)

Since, $f^{\sigma}=\left({\nu_{\alpha}}_{|(\mathfrak{a}\star
E)(\overline{\mathds{Q}})}\right)^{\sigma}\circ p^{\sigma}$ and
$\left({\nu_{\alpha}}_{|(\mathfrak{a}\star
E)(\overline{\mathds{Q}})}\right)^{\sigma}$ is an isomorphism, $\ker
f^{\sigma}=\ker p^{\sigma}$. By definition of $E[\mathfrak{a}]$ we have

$[\wp(z,\Lambda)\colon\wp^{\prime}(z,\Lambda)\colon 1]\in
E[\mathfrak{a}]\text{ if, and only if, }z\in\mathfrak{a}^{-1}\Lambda.$ (3)

Moreover, $[\wp(z,\Lambda_{\sigma})\colon\wp(z,\Lambda_{\sigma})\colon
1]\in\ker p^{\sigma}$ if, and only if,
$z\in(\mathfrak{a}^{-1}\Lambda)_{\sigma}$ by definition of $p^{\sigma}$ but,
$z\in{(\mathfrak{a}^{\sigma})}^{-1}\Lambda_{\sigma}$ because $\ker
p^{\sigma}=E^{\sigma}[\mathfrak{a}^{\sigma}]$. Thus, by the relations (2) and
(3), we have
$(\mathfrak{a}^{-1}\Lambda)_{\sigma}={(\mathfrak{a}^{\sigma})}^{-1}\Lambda_{\sigma}$
(this proves the point 3 of the proposition).

This proves that $(\mathfrak{a}\star E)^{\sigma}=\mathfrak{a}^{\sigma}\star
E^{\sigma}.$ We also have the factorization

$f^{\sigma}=\left({\nu_{\alpha}}_{|(\mathfrak{a}\star
E)(\overline{\mathds{Q}})}\right)^{\sigma}\circ
p^{\sigma}={\nu_{\alpha_{\sigma}}}_{|(\mathfrak{a}^{\sigma}\star
E^{\sigma})(\overline{\mathds{Q}})}\circ p^{\sigma}.$

Since $p^{\sigma}$ is an isogeny it is surjective. So, the maps
$({\nu_{\alpha}}_{|\mathds{P}^{2}(\overline{\mathds{Q}})})^{\sigma}$ and
${\nu_{\alpha_{\sigma}}}_{|\mathds{P}^{2}(\overline{\mathds{Q}})}$ coincide
and then

$\frac{1}{\alpha_{\sigma}^{2}}=\frac{1}{\left({\alpha^{\sigma}}\right)^{2}}\text{
so }\alpha_{\sigma}=\pm\alpha^{\sigma}\text{ and
}\frac{1}{\alpha_{\sigma}^{3}}=\frac{1}{\left({\alpha^{\sigma}}\right)^{3}}\text{
so }\alpha_{\sigma}=j\alpha^{\sigma}$

for some $j^{3}=1.$ Hence, $\alpha_{\sigma}=\alpha^{\sigma}.$

∎

A nice consequence is that if we take $E_{\Lambda}$ with
$g_{i}(\Lambda)\in\overline{\mathds{Q}}$ then $E_{\mathfrak{a}^{-1}\Lambda}$
has also its Weierstrass model over $\overline{\mathds{Q}}$ and
$\operatorname{Hom}_{\mathds{C}}(E_{\Lambda},E_{\mathfrak{a}^{-1}\Lambda})=\operatorname{Hom}_{\overline{\mathds{Q}}}(E_{\Lambda},E_{\mathfrak{a}^{-1}\Lambda})$
so the analytic representation of this isogenies associated to the
isomorphisms $\phi_{\Lambda}$ and $\phi_{\mathfrak{a}^{-1}\Lambda}$ are in
$\operatorname{Hom}_{\mathds{C}}(\Lambda,\mathfrak{a}^{-1}\Lambda)\subseteq
K.$ We recall from [Sil94, Proposition 1.2] that $\operatorname{Cl}(R)$ act
simply transitively on the elliptic curves with CM by $R$. Thus, given two
elliptic curves $E$ and $E^{\prime}$ over $\overline{\mathds{Q}}$ we can
always find a model $E^{\prime\prime}$ of $E^{\prime}$ such that the analytic
representations of isogenies from $E$ to $E^{\prime\prime}$ are in $K.$

Given an automorphism $\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$
for $F(\sigma)=\dot{\mathfrak{a}},$ by [Sil94, Proposition 2.4] we know that
$E^{\sigma}$ and $\mathfrak{a}\star E$ are isomorphic. This means that their
corresponding lattices $\Lambda_{\sigma}$ and $\mathfrak{a}^{-1}\Lambda$ (via
the $\wp$-functions) are homothetic by some constant $r_{\sigma}\in\mathds{C}$
which depends, a priori, on $\Lambda,\sigma$ and the choice of the
representative $\mathfrak{a}$ of $F(\sigma)$ we chose. An immediate
consequence of Proposition 2 is that $r_{\sigma}$ is in
$\overline{\mathds{Q}}.$ The issue is that chosing another lattice
$\Lambda^{\prime}$ would lead to another
$r_{\sigma}^{\prime}\in\overline{\mathds{Q}}.$ The next proposition shows
that, under some condition, we can chose the same $r_{\sigma}$ for different
lattices.

######  Proposition 3 (Action of $\operatorname{Gal}(\overline{\mathds{Q}}/K)$
on elliptic curves).

Let $\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ be an automorphism
and $\mathfrak{a}$ be a fractional ideal with
$F(\sigma)=\dot{\mathfrak{a}}\in\operatorname{Cl}(R)$. Consider an elliptic
curve $E_{\Lambda}$ over $\overline{\mathds{Q}}$. Then there exists
$r_{\sigma}\in\overline{\mathds{Q}}$ and a commutative diagram

$\textstyle{E^{\sigma}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\sigma}}$$\textstyle{E^{\prime\sigma}(\mathds{C})}$$\textstyle{\mathds{C}/\Lambda_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Lambda_{\sigma}}}$$\scriptstyle{\alpha}$$\scriptstyle{r_{\sigma}}$$\textstyle{\mathds{C}/\Lambda^{\prime}_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Lambda^{\prime}_{\sigma}}}$$\scriptstyle{r_{\sigma}}$$\textstyle{\mathds{C}/\mathfrak{a}^{-1}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathds{C}/\mathfrak{a}^{-1}\Lambda^{\prime}.}$

for all
$E^{\prime}(\mathds{C})\underset{\phi_{\Lambda^{\prime}}}{\simeq}\mathds{C}/\Lambda^{\prime}$
such that $\operatorname{Hom}_{\mathds{C}}(\Lambda,\Lambda^{\prime})\subseteq
K$ and all isogenies $f\colon E_{\Lambda}\rightarrow E_{\Lambda^{\prime}}$
over $\overline{\mathds{Q}}$ where $\alpha\in K$ is the analytic
representation of $(f,\phi_{\Lambda},\phi_{\Lambda^{\prime}})$.

###### Proof.

As we have already seen, fractional ideals act transitively on CM elliptic
curves over $\overline{\mathds{Q}}.$ Hence, there exists an isomorphism
$E^{\sigma}\rightarrow\mathfrak{a}\star E$ which induces an isomorphism on
associated tori
$r_{\sigma}\colon\mathds{C}/\Lambda_{\sigma}\rightarrow\mathds{C}/\mathfrak{a}^{-1}\Lambda.$
Let $\alpha\in K$ be the analytic representation of
$(f,\phi_{\Lambda},\phi_{\Lambda^{\prime}})$. There exists an ideal
$\mathfrak{b}\subseteq R$ such that
$\alpha\Lambda=\mathfrak{b}^{-1}\Lambda^{\prime}$ and Proposition 2.3 implies
that
$\alpha^{\sigma}\Lambda_{\sigma}=(\mathfrak{b}^{-1})^{\sigma}\Lambda^{\prime}_{\sigma}$.
Since $\alpha\in K$ and $\mathfrak{b}\subseteq K$ they are invariant by
$\sigma.$ To prove the proposition, we need to show that
$r_{\sigma}\Lambda_{\sigma}^{\prime}=\mathfrak{a}^{-1}\Lambda^{\prime}.$
On one hand we have

$\displaystyle\alpha\Lambda_{\sigma}$
$\displaystyle=\mathfrak{b}^{-1}\Lambda^{\prime}_{\sigma},$ (4) on the other
hand $\displaystyle\alpha\mathfrak{a}^{-1}\Lambda$
$\displaystyle=\mathfrak{a}^{-1}\mathfrak{b}^{-1}\Lambda^{\prime}.$ (5)
Combined with $r_{\sigma}\Lambda_{\sigma}=\mathfrak{a}^{-1}\Lambda$, (4) and
(5), it gives $\displaystyle r_{\sigma}\Lambda^{\prime}_{\sigma}$
$\displaystyle=\mathfrak{b}\alpha\mathfrak{a}^{-1}\Lambda$
$\displaystyle=\mathfrak{b}\mathfrak{a}^{-1}\mathfrak{b}^{-1}\Lambda^{\prime}$
$\displaystyle=\mathfrak{a}^{-1}\Lambda^{\prime}$

which concludes the proof. ∎

####  3.2.3 Galois action on polarizations of CM elliptic curves

The canonical polarization on an elliptic curve $E$ is defined by

$\begin{array}[]{rrl}a_{0,E}\colon&E&\longrightarrow\widehat{E}\\\
&P&\longmapsto[P]-[O]\end{array}$

with $O$ the neutral element of the group $E$. We use this isomorphism to
identify any polarization $a$ of $E$ with an element of $\text{End}(E)$ by
$a_{0,E}^{-1}\circ a$.

######  Lemma 5.

Let $E(\mathds{C})\underset{\phi_{\Lambda}}{\simeq}\mathds{C}/\Lambda$ be an
elliptic curve with $\Lambda=\mathfrak{a}x$ with $\mathfrak{a}\subseteq K$ a
fractional ideal. The principal polarization $a_{0,E}$ induces the hermitian
form

$\begin{array}[]{rrl}h_{0,\Lambda}\colon&\mathds{C}\times\mathds{C}&\longrightarrow\mathds{C}\\\
&(z,w)&\longmapsto\frac{z\overline{w}}{\alpha_{R}N(\mathfrak{a})N(x)}\end{array}$

with $\alpha_{R}=\operatorname{im}\omega>0$ and $R=\mathds{Z}[\omega].$

###### Proof.

The induced hermitian form is necessarily of the form $h_{0,\Lambda}=\mu
h_{0}$ with $\mu\in\mathds{R}_{>0}$ and $h_{0}(z,w)=z\overline{w}$ because the
conjugacy classes of hermitian forms on $\mathds{C}$-vector spaces are
determined by their rank and signature. Moreover, since $h_{0,\Lambda}$ is a
principal polarization $\operatorname{im}h_{0,\Lambda}(\Lambda,\Lambda)=\deg
a_{0,E}\mathds{Z}=\mathds{Z}$. Hence,

$\displaystyle\operatorname{im}h_{0,\Lambda}(\Lambda,\Lambda)$
$\displaystyle=\operatorname{im}h_{0,\Lambda}(\mathfrak{a}x,\mathfrak{a}x)$
$\displaystyle=\operatorname{im}\mu N(\mathfrak{a})N(x)R$ $\displaystyle=\mu
N(\mathfrak{a})N(x)\operatorname{im}R$ $\displaystyle=\mu
N(\mathfrak{a})N(x)\alpha_{R}\mathds{Z}=\mathds{Z}.$

Hence, $\mu=\frac{1}{N(\mathfrak{a})N(x)\alpha_{R}}.$ ∎

We want to investigate first how does
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$ acts on polarizations.

Let $\phi_{\Lambda}\colon\mathds{C}/\Lambda\rightarrow E(\mathds{C})$ with
$\Lambda=\mathfrak{b}x.$ We write
$\mathbf{F}_{h}(E_{\Lambda},a_{0,E})=(\Lambda,H_{0,\Lambda})$ with
$H_{0,\Lambda}=h^{\alpha_{R}}_{0,\Lambda}$ and, by Lemma 5, we can write the
Gram matrix of $H_{0,\Lambda}$ in the $K$ basis $x$ of $K\Lambda$, by
$G_{\Lambda}(x)=\frac{1}{N(\mathfrak{b})}$. It is then clear that for all
fractional ideal $\mathfrak{a}$, we have
$\mathfrak{a}\Lambda=\mathfrak{a}\mathfrak{b}x$ and thus, the Gram matrix of
the analytic representation of the canonical polarization of
$E_{\mathfrak{a}\Lambda}$ in the basis $x$ is
$G_{\mathfrak{a}\Lambda}(x)=\frac{1}{\mathfrak{a}\mathfrak{b}}=\frac{1}{N(\mathfrak{a})}G_{\Lambda}(x).$

Moreover, $a_{0,E}^{\sigma}$ is the canonical polarization $a_{0,E^{\sigma}}$
of $E^{\sigma}$ for any $E$ and $\sigma\in\operatorname{Aut}(\mathds{C})$ and
every polarization on elliptic curves is of the form $na_{0,E}$ for some
$n\in\mathds{N}^{*}$. What we did can easily be generalized for any
polarization of elliptic curves.

We can conclude with this proposition.

######  Proposition 4.

Let $E=E_{\Lambda}\in\mathbfcal{A}_{R},a_{0,E}$ the canonical polarization on
$E$ and $\mathbf{F}_{h}(E,a_{0,E})=(\Lambda,H_{0,\Lambda})$. Then for any
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and
$\dot{\mathfrak{a}}^{-1}=F(\sigma)$ there is an isometry

$\mathbf{F}_{h}(E^{\sigma},a_{0,E}){\simeq}\left(\mathfrak{a}\Lambda,\frac{1}{N(\mathfrak{a})}H_{0,\Lambda}\right).$

###### Proof.

Let
$\mathbf{F}_{h}(E^{\sigma},a_{0,E^{\sigma}})=(\Lambda_{\sigma},H_{0,\Lambda_{\sigma}}).$
By Proposition 3 and Lemma 3 we have an isomorphism

$\nu_{r_{\sigma}}\colon E^{\sigma}\longrightarrow\mathfrak{a}^{-1}\star E$

and $(\mathfrak{a}^{-1}\star
E)(\mathds{C})\simeq\mathds{C}/\mathfrak{a}\Lambda$. We have
$\mathbf{F}_{h}(\mathfrak{a}^{-1}\star
E,a_{0,\mathfrak{a}^{-1}E})=\left(\mathfrak{a}\Lambda,\frac{1}{N(\mathfrak{a})}H_{0,\Lambda}\right).$
Since every isogeny between elliptic curves is a polarized isogeny for the
canonical polarizations, the isomorphism $\nu_{r_{\sigma}}$ is a polarized
isogeny and then its analytic representation
$r_{\sigma}\colon(\Lambda_{\sigma},H_{0,\Lambda_{\sigma}})\longrightarrow\left(\mathfrak{a}\Lambda,\frac{1}{N(\mathfrak{a})}H_{0,\Lambda}\right)$
defines an isometry on the induced hermitian lattices. ∎

###  3.3 Product of CM elliptic curves

Let $A=\bigoplus_{i=1}^{g}E_{i}$ be the product of $g$ elliptic curves with CM
by $R$ over $\overline{\mathds{Q}}.$ Let $\Lambda_{i}\subseteq\mathds{C}$ be
lattices such that $\phi_{\Lambda_{i}}\colon\mathds{C}/\Lambda_{i}\rightarrow
E_{i}(\mathds{C})$ is the canonical isomorphism with
$g_{k}(\Lambda_{i})\in\overline{\mathds{Q}}.$ Then there is a canonical
isomorphism $\phi_{\Gamma}\colon\mathds{C}^{g}/\Gamma\rightarrow
A(\mathds{C})$ with $\Gamma=\bigoplus\Lambda_{i}$ and
$\phi_{\Gamma}=(\phi_{\Lambda_{i}})_{i=1\dots g}.$ In this section we show how
the results of Section 3.2 apply to products of elliptic curves and we
conclude with the proofs of Theorem 3 and 4.

####  3.3.1 Isogenies between products of elliptic curves

######  Proposition 5.

Let $A,A^{\prime}\in\mathbfcal{A}_{R}$ over $\overline{\mathds{Q}}$ with
$A=E_{\Lambda_{1}}\times\dots\times E_{\Lambda_{g}}$ and
$A^{\prime}=E_{\Lambda^{\prime}_{1}}\times\dots\times
E_{\Lambda_{g^{\prime}}}$. Let $\Gamma=\bigoplus_{i}\Lambda_{i}$ and
$\Gamma^{\prime}=\bigoplus_{j}\Lambda^{\prime}_{j}$. Then the matrix $M_{f}$
of the analytic representation in the canonical basis of $\mathds{C}\Gamma$
and $\mathds{C}\Gamma^{\prime}$ of any map $(f\colon A\rightarrow
A^{\prime},\phi_{\Gamma},\phi_{\Gamma^{\prime}})$ has coefficients in
$\overline{\mathds{Q}}$ and for any
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q}),M_{f^{\sigma}}=M_{f}^{\sigma}.$

###### Proof.

Consider the following diagram for all $k,l$

$\textstyle{\mathds{C}/\Lambda_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{k}}$$\scriptstyle{\phi_{\Lambda_{k}}}$$\textstyle{\mathds{C}^{g}/\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M_{f}}$$\scriptstyle{\phi_{\Gamma}}$$\textstyle{\mathds{C}^{g}/\Gamma^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{l}}$$\scriptstyle{\phi_{\Gamma^{\prime}}}$$\textstyle{\mathds{C}/\Lambda^{\prime}_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Lambda_{l}}}$$\textstyle{E_{\Lambda_{k}}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{k}}$$\textstyle{A(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A^{\prime}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}_{l}}$$\textstyle{E_{\Lambda^{\prime}_{l}}(\mathds{C})}$

with $j_{k},\iota_{k}$ and $p^{\prime}_{l},\pi_{l}$ the $k$-th component
inclusion and $l$-th component projection respectively.

Through those morphisms, we can see the $k,l$ coefficient $m_{l,k}$ of $M_{f}$
as a map

$m_{l,k}\colon\mathds{C}/\Lambda_{k}\rightarrow\mathds{C}/\Lambda^{\prime}_{l}$

which is in $\overline{\mathds{Q}}$ and, by Proposition 2, $m_{l,k}^{\sigma}$
is the analytic representation of $((p_{l}^{\prime}\circ
f\circ\iota_{k})^{\sigma},\phi_{\Lambda_{k}},\phi_{\Lambda_{l}})$. ∎

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ with $A\simeq
E_{\Lambda_{1}}\times\dots\times E_{\Lambda_{g}}$. Let
$\Gamma=\bigoplus_{i}\Lambda_{i}$. We denote by $h$ the hermitian form on
$\mathds{C}^{g}$ and
$\rho_{h}\in\operatorname{Hom}_{\mathds{C}}(\Gamma,\widehat{\Gamma})$ the map
induced by the polarization $a.$ Consider $a_{0}$ the product polarization of
the canonical polarizations on each elliptic curve $a_{0,E_{i}}\colon
E_{i}\rightarrow\widehat{E_{i}}.$ It is an isomorphism

$a_{0}\colon\bigoplus_{i=1}^{g}E_{i}\rightarrow\bigoplus_{i=1}^{g}\widehat{E_{i}}$

which induces

$\rho_{h_{0}}=(\rho_{h_{0,\Lambda_{i}}})\colon\mathds{C}^{g}/\bigoplus_{i}\Lambda_{i}\rightarrow(\mathds{C}^{g})^{*}/\bigoplus_{i}\widehat{\Lambda_{i}}$

with each $\rho_{h_{0,\Lambda_{i}}}$ induced by the canonical polarization on
$E_{\Lambda_{i}}$. We can now consider the analytic representation $\rho$ of
$(a_{0}^{-1}\circ a,\phi_{\Gamma},\phi_{\Gamma})$. By definition, the diagram
of Figure 4 commutes.

$\mathds{C}^{g}/\Gamma$${\mathds{C}^{g}/\Gamma}$${(\mathds{C}^{g})^{*}/\widehat{\Gamma}}$$A(\mathds{C})$${A(\mathds{C})}$${\widehat{A}(\mathds{C})}$$\scriptstyle{\rho_{h_{0}}}$$\scriptstyle\phi_{\Gamma}$$\scriptstyle{\rho_{h}}$$\scriptstyle{\phi_{\Gamma}}$$\scriptstyle{\rho}$$\scriptstyle{a_{0}}$$\scriptstyle{a_{0}^{-1}\circ
a}$$\scriptstyle{a}$ Figure 4: Analytic representation of polarizations

Recall that we denoted $\alpha_{R}=\operatorname{im}\omega>0$ with
$R=\mathds{Z}[\omega].$ Let $\Lambda_{i}=\mathfrak{a}_{i}x_{i}$ then
$\Gamma=\bigoplus\mathfrak{a}_{i}x_{i}$ and, with Lemma 1 we can show that

$\widehat{\Gamma}=\\{\ell\in(\mathds{C}^{g})^{*},\operatorname{im}\ell(\Gamma)\subseteq\mathds{Z}\\}=\\{\ell\in(\mathds{C}^{g})^{*},\alpha_{R}\ell(\Gamma)\subseteq
R\\}=\bigoplus\frac{1}{\alpha_{R}}\overline{\mathfrak{a}_{j}}^{-1}x_{j}^{*}$

with $x_{j}^{*}\in(\mathds{C}^{g})^{*}$ defined by
$x_{j}^{*}(x_{i})=\delta_{i,j}$ with $\delta_{i,j}=1$ for $i=j$ and $0$
otherwise.

######  Proposition 6.

With the notation above let $b=(x_{i})_{i=1,\dots,g}$. Let $M_{b,b}(\rho)$ be
the matrix of $\rho$ in the basis $b$ and
$G_{\Gamma}(b)=(h^{\alpha_{R}}(x_{i},x_{j}))_{i,j}$ the Gram matrix of the
hermitian form $h^{\alpha_{R}}=\alpha_{R}h$ in the basis $b$. Then

$M_{b,b}(\rho)=G_{\Gamma}(b)\cdot D$

with $D=\operatorname{diag}(N(\mathfrak{a}_{1}),\dots,N(\mathfrak{a}_{g}))$
the diagonal matrix with coefficients $d_{i,i}=N(\mathfrak{a}_{i}).$

###### Proof.

By definition of
$\rho_{h_{0,\Lambda_{i}}}\in\operatorname{Hom}_{\mathds{C}}(\Lambda_{i},\widehat{\Lambda_{i}})$
and Lemma 5,

$\begin{array}[]{rrl}\rho_{h_{0,\Lambda_{i}}}\colon&\mathds{C}/\Lambda_{i}&\longrightarrow\mathds{C}^{*}/\widehat{\Lambda_{i}}\\\
&z&\longmapsto\left(\ell_{z}\colon
w\mapsto\frac{z\overline{w}}{\alpha_{R}N(\mathfrak{a})N(x_{i})}\right).\end{array}$

Hence,
$\rho_{h_{0}}(x_{i})=\ell_{x_{i}}=\frac{1}{N(\mathfrak{a}_{i})\alpha_{R}}x_{i}^{*}$
so $\rho_{h_{0}}^{-1}(x_{i}^{*})=N(\mathfrak{a}_{i})\alpha_{R}x_{i}$. Thus,
for any $\ell=\sum_{i}u_{i}x_{i}^{*}\in(\mathds{C}^{g})^{*},$

$\rho_{h_{0}}^{-1}(\ell)=\sum_{i}u_{i}N(\mathfrak{a}_{i})\alpha_{R}x_{i}$ (6)

and then its $j$-th component is

$(\rho_{h_{0}}^{-1}(\ell))_{j}=u_{j}N(\mathfrak{a}_{j})\alpha_{R}=\alpha_{R}\ell(x_{j})$
(7)

Now, since $\rho=\rho_{h_{0}}^{-1}\circ\rho_{h}$ we have

$\displaystyle\rho_{ij}$ $\displaystyle=(\rho(x_{i}))_{j}$
$\displaystyle=(\rho_{h_{0}}^{-1}\circ\rho_{h}(x_{i}))_{j}$
$\displaystyle=N(\mathfrak{a}_{j})\alpha_{R}\rho_{h}(x_{i})(x_{j})\text{ by
(\ref{eq2}) and (\ref{eq3}) }$
$\displaystyle=N(\mathfrak{a}_{j})\alpha_{R}h(x_{i},x_{j})$
$\displaystyle=N(\mathfrak{a}_{j})h^{\alpha_{R}}(x_{i},x_{j})$
$\displaystyle=(G_{\Gamma}(b)\cdot D)_{i,j}.$

∎

######  Proposition 7.

Let $A,A^{\prime}\in\mathbfcal{A}_{R}$ with
$A=E_{\Lambda_{1}}\times\dots\times E_{\Lambda_{g}}$ and
$A^{\prime}=E_{\Lambda^{\prime}_{1}}\times\dots\times
E_{\Lambda_{g^{\prime}}}$. Assume that for all
$i,j,\operatorname{Hom}_{\mathds{C}}(\Lambda_{i},\Lambda^{\prime}_{j})\subseteq
K.$ Then there exists $r_{\sigma}\in\overline{\mathds{Q}}$ such that for any
$f\colon A\rightarrow A^{\prime}$, any
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and $\mathfrak{a}$ with
$F(\sigma)=\dot{\mathfrak{a}}\in\operatorname{Cl}(R)$ there is a commutative
diagram

$\textstyle{A^{\sigma}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\sigma}}$$\textstyle{A^{\prime\sigma}(\mathds{C})}$$\textstyle{\mathds{C}^{g}/\Gamma_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Gamma}}$$\scriptstyle{M}$$\scriptstyle{r_{\sigma}I_{g}}$$\textstyle{\mathds{C}^{g^{\prime}}/\Gamma^{\prime}_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\Gamma^{\prime}_{\sigma}}}$$\scriptstyle{r_{\sigma}I_{g^{\prime}}}$$\textstyle{\mathds{C}^{g}/\mathfrak{a}^{-1}\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M}$$\textstyle{\mathds{C}/\mathfrak{a}^{-1}\Gamma^{\prime}.}$

with $\Gamma=\bigoplus_{i}\Lambda_{i}$ and
$\Gamma^{\prime}=\bigoplus_{j}\Lambda^{\prime}_{j}$.

###### Proof.

Apply Lemma 3 and following the same steps as in the proof of Proposition 5. ∎

######  Proposition 8.

Let $E_{1},\dots,E_{g}$ be elliptic curves over $\overline{\mathds{Q}}$ with
CM by $R$ and the isomorphisms
$\phi_{\Lambda_{i}}\colon\mathds{C}/\Lambda_{i}\rightarrow E_{i}(\mathds{C})$
with $\operatorname{Hom}_{\mathds{C}}(\Lambda_{i},\Lambda_{j})\subseteq K$.
Let $a_{0,E_{i}}$ be the canonical polarization on $E_{i}$, $a_{0}$ be the
product polarization on $E_{1}\times\dots\times E_{g}$ and
$\mathbf{F}_{h}(\bigoplus_{i}E_{i},a_{0})=(\bigoplus_{i}\Lambda_{i},H_{0})$.
Then for any $\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and
$\dot{\mathfrak{a}}^{-1}=F(\sigma)$ there is an isometry

$\mathbf{F}_{h}\left(\bigoplus_{i}E_{i}^{\sigma},a_{0}^{\sigma}\right)\underset{r_{\sigma}}{\simeq}\left(\mathfrak{a}\bigoplus_{i}\Lambda_{i},\frac{1}{N(\mathfrak{a})}H_{0}\right).$

###### Proof.

This is just Lemma 4 applied component by component. ∎

####  3.3.2 Proof of Theorem 3 and 4

We have now the necessary tools to prove both Theorem 3 and 4. As both
theorems aim to describe the isometry class of
$\mathbf{F}_{h}(A^{\sigma},a^{\sigma})$, for $(A,a)\in\mathbfcal{A}_{R}^{p}$
and $\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$ or
$\operatorname{Gal}(K/\mathds{Q})$, it is enough to show there is such an
isometry for a particular object in the isomorphism class of
$(A^{\sigma},a^{\sigma}).$ By definition of $\mathbfcal{A}_{R}^{p}$, each
isomorphism class of polarized abelian variety contains an element of the form
$(E_{1}\times\dots\times E_{g},a)$ so we will show the isometry for this
particular case.

Let $(L,H)$ be an integral rank $g$ hermitian $R$-lattice. Fixing a basis $b$
of $KL$ gives an isomorphism $KL\simeq K^{g}$ and pushing forward $H$ on
$K^{g}$ gives an isometry $(KL,H)\rightarrow(K^{g},H)$. We identify $(L,H)$
with its image in $(K^{g},H).$ The hermitian form $H$ is determined by the
Gram matrix of $b$ given by $G=G(b)=(H(v,w))_{v,w\in b}\in M_{g,g}(K).$
Indeed, with $x,y\in K^{g}$, colon vectors in the basis $b$ of $KL,$ we have

$H(x,y)={}^{t}\\!xG\overline{y}$

and then, by definition,

$\overline{H}(x,y)=\overline{H(\overline{x},\overline{y})}=\overline{{}^{t}\\!\overline{x}G\overline{\overline{y}}}={}^{t}\\!x\overline{G}\overline{y}.$

Hence, $(\overline{L},\overline{H})$ has Gram matrix $\overline{G}$.

###### Proof of Theorem 3.

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ over $\overline{\mathds{Q}}$ with
$A=E_{\Lambda_{1}}\times\dots\times E_{\Lambda_{g}}$ with
$\operatorname{Hom}_{\mathds{C}}(\Lambda_{i},\Lambda_{j})\subseteq K$. Let
$\mathbf{F}_{h}(A,a)=(L,H)$ and let
$\mathbf{F}_{h}(\overline{A},\overline{a})=(L^{\prime},H^{\prime})$. We write
$\Lambda_{i}=\mathfrak{a}_{i}x_{i}$ and $b=(x_{1},\dots,x_{g})$ such that
$(\mathfrak{a}_{i},x_{i})$ is a pseudo-basis of $L=\bigoplus_{i}\Lambda_{i}$.
By Lemma 2, $L^{\prime}=\overline{L}$. Moreover, by Proposition 6,
$M_{b,b}(\rho)=G_{L}(b)\cdot D$ with
$D=\operatorname{diag}(N(\mathfrak{a}_{1}),\dots,N(\mathfrak{a}_{g}))$. By
Proposition 5 and by applying the complex conjugation to the diagram

$\textstyle{\mathds{C}^{g}/L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{L}}$$\scriptstyle{\rho}$$\textstyle{\mathds{C}^{g}/L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{L}}$$\textstyle{A(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{0}^{-1}\circ
a}$$\textstyle{A(\mathds{C})}$

the analytic representation of
$(\overline{a_{0}}^{-1}\circ\overline{a},\phi_{\overline{L}},\phi_{\overline{L}})$
is $\overline{M_{b,b}(\rho)}=\overline{G_{L}(b)}$ where $\rho$ is the analytic
representation of $(a_{0}^{-1}\circ a,\phi_{L},\phi_{L}).$ Hence,
$(\overline{L},H^{\prime})$ has Gram matrix $\overline{G_{L}(b)}$ so
$H^{\prime}=\overline{H}.$ ∎

###### Proof of Theorem 4.

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ such that
$A=E_{\Lambda_{1}}\times\dots\times E_{\Lambda_{g}}$ such that
$\text{End}(\Lambda_{i},\Lambda_{j})\subseteq K$ for all $i,j$. We will write
$E_{i}=E_{\Lambda_{i}}$ for simplicity. Let $\Gamma=\bigoplus\Lambda_{i}$ and
$\mathbf{F}_{h}(A,a)=(L,H),\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/K)$
with $L=\Gamma$ and $\dot{\mathfrak{a}}^{-1}=F(\sigma)$.

First step : find a polarized isogeny
$(E^{n},D\cdot\lambda_{0})\rightarrow(A,a)$ with $D$ diagonal with integer
entries.

Let $\Lambda=\Lambda_{1}$ and $E=E_{1}$ and $\lambda_{0}$ the product
polarization on $E^{g}.$ There exist $\alpha_{i}\in K$ such that for all
$i,\alpha_{i}\Lambda\subseteq\Lambda_{i}$. This gives us an isogeny $q\colon
E^{g}\rightarrow A$. We denote by
$Q=\operatorname{diag}(\alpha_{1},\dots,\alpha_{g})$ the analytic
representation of $(q,\phi_{\Lambda^{g}},\phi_{\Gamma})$. We consider
$\lambda\colon E^{g}\rightarrow\widehat{E}^{g}$ the pullback polarization of
$a$ on $E^{g}$, i.e., the unique polarization on $E^{g}$ that makes $q$ a
polarized isogeny. We consider
$M=\lambda_{0}^{-1}\circ\lambda\in\text{End}(E^{g})\simeq M_{g}(R)$, and since
$\lambda=\widehat{\lambda}$ the matrix $M$ is a hermitian matrix, i.e.,
${}^{t}\\!\overline{M}=M.$ Now consider a matrix $P\in M_{g}(R)$ such that
${}^{t}\\!\overline{P}MP=D$ with $D$ a diagonal matrix. Since $M$ is positive
definite and hermitian, so does $D$ and then it has positive integers entries.
So, we have a polarized isogeny $f=q\circ
P\colon(E^{g},D\cdot\lambda_{0})\rightarrow(A,a)$.

$\textstyle{E^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\scriptstyle{D}$$\textstyle{E^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{M}$$\textstyle{\bigoplus
E_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{E^{g}}$$\textstyle{E^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{t}\\!\overline{P}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widehat{\bigoplus
E_{i}}\simeq\bigoplus E_{i}}$$\scriptstyle{\widehat{q}}$

We will denote by $S=QP$ the analytic representation of
$(f,\phi_{\Lambda^{g}},\phi_{\Gamma}).$ Since $f$ is a polarized isogeny

$S\colon(K\Lambda^{g},D)\rightarrow(KL,H)$ (8)

is an isometry.

Second step : conclude.

By Proposition 8,
$\mathbf{F}_{h}\left((E^{g})^{\sigma},D^{\sigma}\right)\underset{r_{\sigma}}{\simeq}\left(\mathfrak{a}\Lambda^{g},\frac{1}{N(\mathfrak{a})}D\right).$
We now consider
$\mathbf{F}_{h}(A^{\sigma},a^{\sigma})=(L_{\sigma},H_{\sigma})$ and
$\iota_{\sigma}=\mathbf{F}_{h}(f^{\sigma})$. By Proposition 3, we have
$r_{\sigma}L_{\sigma}=\mathfrak{a}L$ and there is a commutative diagram

$\textstyle{(E^{g})^{\sigma}(\mathds{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\sigma}}$$\textstyle{A^{\sigma}(\mathds{C})}$$\textstyle{\mathds{C}^{g}/\Lambda_{\sigma}^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{P_{\sigma}}}$$\scriptstyle{S}$$\scriptstyle{r_{\sigma}I_{g}}$$\textstyle{\mathds{C}^{g}/L_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{L_{\sigma}}}$$\scriptstyle{r_{\sigma}I_{g}}$$\textstyle{\mathds{C}/\mathfrak{a}\Lambda^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{\mathds{C}^{g}/\mathfrak{a}L.}$

Moreover, we have the isometries of hermitian spaces

$r_{\sigma}I_{g}\colon(KL_{\sigma},H_{\sigma})\rightarrow\left(K\mathfrak{a}L,H^{\prime}=\frac{1}{N(r_{\sigma})}H_{\sigma}\right)$
(9)
$S\colon\left(K\mathfrak{a}\Lambda^{g},\frac{1}{N(\mathfrak{a})}D\right)\rightarrow\left(K\mathfrak{a}L,H^{\prime}\right).$
(10)

By the isometries (8) and (10) we have
$H^{\prime}=\frac{1}{N(\mathfrak{a})}{}^{t}\\!\overline{S^{-1}}DS^{-1}=\frac{1}{N(\mathfrak{a})}H$.
Hence,

$\mathbf{F}_{h}(A^{\sigma},a^{\sigma})=(L_{\sigma},H_{\sigma})\simeq\left(\mathfrak{a}L,\frac{1}{N(\mathfrak{a})}H\right).$

This concludes the proof of Theorem 4.

∎

##  4 Application to the field of moduli of varieties in
$\mathbfcal{A}_{R}^{p}$

###  4.1 General results on the field of moduli of principally polarized
abelian varieties in $\mathbfcal{A}_{R}^{p}$

We recall that the field of moduli of a polarized abelian variety $(A,a)$ over
$\overline{\mathds{Q}}$ is the fixed field by the subgroup

$\\{\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q}),(A^{\sigma},a^{\sigma})\simeq(A,a)\\}.$

In the same way we define the field of moduli of a curve $C$ over
$\overline{\mathds{Q}}$ by the fixed field of
$\\{\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q}),C^{\sigma}\simeq
C\\}.$

The Abel-Jacobi map $C\mapsto(\operatorname{Jac}(C),j)$ which associates a
curves to its polarized Jacobian variety induces a morphism

$\begin{array}[]{rrl}[\operatorname{Jac}]\colon&M_{g}\longrightarrow&A_{g}\\\
&[C]\longmapsto&[\operatorname{Jac}(C),j]\end{array}$

between the moduli space of smooth absolutely irreducible projective genus $g$
curves and the moduli space of principally polarized abelian varieties of
dimension $g$. By [CS86, Chapter VII,Corollary 12.2], $[\operatorname{Jac}]$
is injective so $C$ and $(\operatorname{Jac}(C),j)$ have the same field of
moduli.

For $g=2$ and $g=3$ the moduli spaces $M_{g}$ and $A_{g}$ have the same
dimension and are irreducible. Thus, the Jacobian map is dominant if the field
is algebraically closed. More specifically, every indecomposable principally
polarized abelian variety is the Jacobian of a curve (see [Hoy63]).

We give a necessary condition on the class group of $R$ and on
$\mathbf{F}_{h}(A,a)$ for an abelian variety $(A,a)\in\mathbfcal{A}_{R}^{p}$
to have field of moduli $\mathds{Q}.$

######  Proposition 9.

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ and $\mathbf{F}_{h}(A,a)=(L,H)$ be a
hermitian lattice of rank $g$. If $(A,a)$ has field of moduli $\mathds{Q}$
then $R$ has exponent dividing $g$ and $\operatorname{st}(L)$ has order at
most $2.$

###### Proof.

If $(L,H)$ corresponds to a field of moduli $\mathds{Q}$ polarized abelian
variety then
$(L,H)\simeq\left(\mathfrak{a}L,\frac{1}{N(\mathfrak{a})}H\right)$ for all
fractional ideal $\mathfrak{a}.$ In particular, $L\simeq\mathfrak{a}L$ for all
$\mathfrak{a}$ and then, their Steinitz class are the same

$\operatorname{st}(L)=\operatorname{st}(\mathfrak{a}L)=\mathfrak{a}^{g}\operatorname{st}(L)\in\operatorname{Cl}(R).$

Hence, $\dot{\mathfrak{a}}^{g}=1\in\operatorname{Cl}(R)$ so
$\operatorname{Cl}(R)$ has exponent dividing $g.$

The hermitian lattice isometry class must also be invariant by the action of
the complex conjugation so

$\operatorname{st}(L)=\operatorname{st}(\overline{L})=\overline{\operatorname{st}(L)}.$

By the formula $\overline{\mathfrak{a}}\mathfrak{a}=N(\mathfrak{a})R$ it means
that $\operatorname{st}(L)$ must have order at most $2$. ∎

######  Corollary 1.

Let $(A,a)\in\mathbfcal{A}_{R}^{p}$ of dimension $g$ odd. Suppose $(A,a)$ has
field of moduli $\mathds{Q}$ then there exists an elliptic curve $E$ with CM
by $R$ such that $A\simeq E^{g}.$

###### Proof.

Let $(L,H)=\mathbf{F}_{h}(A,a)$ be the corresponding unimodular hermitian
lattice. By Proposition 9, its Steinitz class $\operatorname{st}(L)$ has order
dividing $g$ and $2$ hence it must be $1.$ In other words $L$ is free over
$R$, i.e., $L\simeq R^{g}$ and then $A\simeq E^{g}$ with $\mathbf{F}(E)\simeq
R.$ ∎

###  4.2 Enumeration of the indecomposable principally polarized abelian
varieties in $\mathbfcal{A}_{R}^{p}$ with field of moduli $\mathds{Q}$

Since we are able to understand the action of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$ and
$\operatorname{Gal}(K/\mathds{Q})$ throught the equivalence of categories
$\mathbf{F}_{h}$ developed in Section 2 we are able to check when
$(A,a)\simeq(A^{\sigma},a^{\sigma})$ for all
$\sigma\in\operatorname{Gal}(\overline{\mathds{Q}}/\mathds{Q})$ when
$(A,a)\in\mathbfcal{A}_{R}^{p}$ by looking for isometries between hermitian
$R$-lattices. This is what Algorithm 1 does.

_We denote by $\mathscr{A}_{R}(g)$ the set of all classes of dimension $g$
indecomposable principally polarized abelian varieties
$(A,a)\in\mathbfcal{A}_{R}^{p}$ and by $\mathscr{A}_{R,\mathds{Q}}(g)$ the
subset of $\mathscr{A}_{R}(g)$ corresponding to elements with field of moduli
$\mathds{Q}.$_

0: A maximal order $R$ with exponent dividing $g$.

0: The list of unimodular indecomposable hermitian lattices $(L,H)$
corresponding to the elements of $\mathscr{A}_{R,\mathds{Q}}(g)$.

$\operatorname{LList}\leftarrow\\{\text{Unimodular indecomposable hermitian
lattices of rank }g\\}/\simeq$

$\operatorname{LList}_{\text{FM}-\mathds{Q}}\leftarrow\\{\leavevmode\nobreak\
\\}$ { List of abelian varieties with field of moduli $\mathds{Q}$.}

for $(L,H)\in\operatorname{LList}$ do

$\operatorname{bool}\leftarrow\textbf{true}{}$

for $\mathfrak{a}\in\\{$generators of $\operatorname{Cl}(R)\\}$ do

$\operatorname{bool}\leftarrow\operatorname{bool}$ and
$(L,H)\simeq\left(\mathfrak{a}L,\frac{1}{N(\mathfrak{a})}H\right)$.

end for

if $\operatorname{bool}$ and
$(L,H){\simeq}\left(\overline{L},\overline{H}\right)$ then

$\operatorname{LList}_{\text{FM}-\mathds{Q}}\leftarrow\operatorname{LList}_{\text{FM}-\mathds{Q}}\cup\\{(L,H)\\}$

end if

end for

return $\operatorname{LList}_{\text{FM}-\mathds{Q}}$

Algorithm 1 Enumeration algorithm

To compute the list of elements of $\mathscr{A}_{R,\mathds{Q}}(g)$ for a given
maximal order $R$ we need the list of all unimodular indecomposable hermitian
$R$-lattices of rank $g$. This can be done the classification of these
lattices developed by authors in [KNRR21].

We want to run the algorithm over all maximal orders of a given exponent
dividing $g$. By [EKN20], the complete list of the corresponding discriminants
is finite for all $g$ and known for $g$ up to $8$ under the Extended Riemann
Hypothesis.

###  4.3 Enumeration of dimension $2$ and $3$ principally polarized abelian
varieties of $\mathbfcal{A}_{R}^{p}$ with field of moduli $\mathds{Q}$

As the computations become quickly time-consuming as the dimension $g$ and the
discriminant of the order grows we restricted to $g=2$ and $3$ to be able to
have complete tables (up to one discriminant for $g=3$). We used the Magma
library developed in [KNRR21].

We use Algorithm 1 to compute the cardinal of $\mathscr{A}_{R,\mathds{Q}}(2)$
and copy the results in the Table 1. In the column $\mathscr{P}$ we copy the
number of $(A,a)\in\mathbfcal{A}_{R}^{p}$ such that $A$ is the square of an
elliptic curve with field of moduli $\mathds{Q}$ to confirm we find the same
values as in [GHR19, Table 2].

$h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$ |  $h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$ |  $h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1  | -3 | 0  | 0  | 0 | 2  | -15 | 0  | 1  | 1 | 2  | -115 | 0  | 3  | 11
| -4 | 0  | 0  | 0 |  | -20 | 1  | 1  | 3 |  | -123 | 0  | 4  | 12
| -7 | 0  | 0  | 0 |  | -24 | 1  | 3  | 3 |  | -148 | 3  | 5  | 13
| -8 | 1  | 1  | 1 |  | -35 | 0  | 1  | 5 |  | -187 | 0  | 3  | 17
| -11 | 1  | 1  | 1 |  | -40 | 2  | 4  | 4 |  | -232 | 5  | 10  | 20
| -19 | 1  | 1  | 1 |  | -51 | 0  | 2  | 6 |  | -235 | 0  | 5  | 21
| -43 | 2  | 2  | 2 |  | -52 | 2  | 3  | 5 |  | -267 | 0  | 6  | 24
| -67 | 3  | 3  | 3 |  | -88 | 2  | 6  | 8 |  | -403 | 0  | 3  | 35
| -163 | 7  | 7  | 7 |  | -91 | 0  | 1  | 9 |  | -427 | 0  | 3  | 37
$h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$ |  $h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$ |  $h_{R}$ |  $\Delta$ |  $\mathscr{P}$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}$
4  | -84 | 0  | 2  | 18 | 4  | -340 | 0  | 2  | 60 | 4  | -595 | 2  | 2  | 106
| -120 | 3  | 4  | 24 |  | -372 | 0  | 2  | 66 |  | -627 | 0  | 0  | 112
| -132 | 1  | 2  | 26 |  | -408 | 0  | 4  | 72 |  | -708 | 1  | 2  | 122
| -168 | 0  | 4  | 32 |  | -435 | 0  | 2  | 80 |  | -715 | 0  | 2  | 126
| -195 | 0  | 2  | 40 |  | -483 | 0  | 0  | 88 |  | -760 | 1  | 4  | 130
| -228 | 1  | 2  | 42 |  | -520 | 3  | 4  | 90 |  | -795 | 2  | 2  | 140
| -280 | 0  | 4  | 50 |  | -532 | 0  | 2  | 92 |  | -1012 | 0  | 2  | 172
| -312 | 1  | 4  | 56 |  | -555 | 0  | 2  | 100 |  | -1435 | 0  | 2  | 246

  * •

$h_{R}\colon$ Class number of the maximal order $R$ of
$\mathds{Q}(\sqrt{\Delta})$.

  * •

$\\#\mathscr{A}_{R}\colon$ Number of elements of $\mathscr{A}_{R}(2)$ defined
in Section 4.2.

  * •

$\\#\mathscr{A}_{R,\mathds{Q}}\colon$ Number of elements of
$\mathscr{A}_{R,\mathds{Q}}(2)$.

  * •

$\mathscr{P}\colon$ Number of elements $(A,a)$ of
$\mathscr{A}_{R,\mathds{Q}}(2)$ such that $A\simeq E^{2}$ for some $E$.

  * •

Table 1: Computations for $g=2$

In fact, for $g=2$, the following proposition shows that it is not necessary
to check if $(L,H)\simeq(\overline{L},\overline{H})$.

######  Proposition 10.

Let $(A,a)$ be a dimension $2$ principally polarised abelian variety over
$\mathds{C}$ with $A$ isomorphic to the power of elliptic curves $E_{i}$ with
CM by $R$ maximal. Let $\mathbf{F}_{h}(A,a)=(L,H)$ be a hermitian integral
lattice and let $\mathfrak{a}$ be the Steinitz class of $L$. Then

$\mathbf{F}_{h}\left(\overline{A},\overline{a}\right)=\left(\overline{L},\overline{H}\right)\simeq\left(\overline{\mathfrak{a}}L,\frac{1}{N(\mathfrak{a})}H\right).$

Hence, in this particular case, the action of the complex conjugation
corresponds to the action of an automorphism of
$\operatorname{Gal}(\overline{\mathds{Q}}/K)$.

###### Proof.

Let us write $L=Rx\oplus\mathfrak{a}y.$ Let $G=\begin{pmatrix}\alpha&\beta\\\
\overline{\beta}&\delta\end{pmatrix}$ be the Gram matrix of $H$ in the basis
$(x,y)$ of $KL$. Since $(A,a)$ is principally polarized $(L,H)$ must be
unimodular and then its volume $\mathfrak{v}(L)=N(\mathfrak{a})\det(G)R=R$ so
$N(\mathfrak{a})\det(G)$ is invertible and real in $R$ so
$N(\mathfrak{a})\det(G)=1.$

Now consider $P=N(\mathfrak{a})\begin{pmatrix}\overline{\beta}&\delta\\\
-\alpha&-\beta\end{pmatrix}$, matrix of a linear map
$K\overline{x}+K\overline{y}\rightarrow Kx+Ky.$ It satisfies the relation

${}^{t}\\!P\frac{1}{N(\mathfrak{a})}G\overline{P}=N(\mathfrak{a})\det(G)\overline{G}=\overline{G}.$

So, $P$ defines an isometry between hermitian spaces.

Moreover, $\alpha=H(x,x)=N(x)\in R\cap\mathds{R}=\mathds{Z},\beta=H(x,y)$, so
$\overline{\mathfrak{a}}\beta=H(x,\mathfrak{a}y)\subseteq R$ so
$\beta\in\overline{\mathfrak{a}}^{-1}=\frac{\mathfrak{a}}{N(\mathfrak{a})}$
and, in the same way, $\delta\in\frac{1}{N(\mathfrak{a})}\mathds{Z}.$ Hence,

$\displaystyle P\overline{x}$
$\displaystyle=N(\mathfrak{a})(\overline{\beta}x-ay)\in\overline{\mathfrak{a}}x+N(\mathfrak{a})y=\overline{\mathfrak{a}}L$
$\displaystyle P\overline{\mathfrak{a}y}$
$\displaystyle=N(\mathfrak{a})\overline{\mathfrak{a}}(\delta
x-by)\in\overline{\mathfrak{a}}x+N(\mathfrak{a})y=\overline{\mathfrak{a}}L.$

Thus, $P$ defines an isometry
$(\overline{L},\overline{H})\rightarrow\left(\overline{\mathfrak{a}}L,\frac{1}{N(\mathfrak{a})}H\right)$
∎

In dimension $g=3$ computations are more time consuming. Fortunately, by
Corollary 1, in odd dimension all isomorphism class of $(A,a)$ in
$\mathscr{A}_{R,\mathds{Q}}(3)$ are actually isomorphic to some a power of an
elliptic curve. Hence, we can run the algorithm only on free unimodular
hermitian lattices and the latter are easier to enumerate and to work with.

We copy the results of the computations in Table 2. Pay attention that the
table is complete up to the discriminant $\Delta=-4027$ with class group
isomorphic to $\mathds{Z}/3\mathds{Z}\times\mathds{Z}/3\mathds{Z}$ for which
the algorithm is still running.

$h_{R}$ |  $\Delta$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}^{\text{free}}$ |  $h_{R}$ |  $\Delta$ |  $\\#\mathscr{A}_{R,\mathds{Q}}$ | $\\#\mathscr{A}_{R}^{\text{free}}$
---|---|---|---|---|---|---|---
1  | -3 | 0  | 0 | 3  | -107 | 2  | 44
| -4 | 0  | 0 |  | -139 | 1  | 79
| -7 | 0  | 0 |  | -211 | 0  | 209
| -8 | 0  | 0 |  | -283 | 1  | 417
| -11 | 0  | 0 |  | -307 | 0  | 507
| -19 | 1  | 1 |  | -331 | 2  | 613
| -43 | 3  | 5 |  | -379 | 0  | 851
| -67 | 5  | 13 |  | -499 | 1  | 1665
| -163 | 13  | 103 |  | -547 | 1  | 2059
|  |  |  |  | -643 | 1  | 3075
3  | -23 | 0  | 3 |  | -883 | 0  | 6703
| -31 | 0  | 6 |  | -907 | 1  | 7163
| -59 | 1  | 10 |  |  |  |
| -83 | 0  | 24 | 9  | -4027 | ?  | ?

  * •

$h_{R}\colon$ Class number of the maximal order $R$ of
$\mathds{Q}(\sqrt{\Delta})$.

  * •

$\\#\mathscr{A}_{R}^{\text{free}}\colon$ Number of classes $(A,a)$ in
$\mathscr{A}_{R}(3)$ with $A\simeq E^{3}$ for some $E$.

  * •

$\\#\mathscr{A}_{R,\mathds{Q}}\colon$ Number of elements of
$\mathscr{A}_{R,\mathds{Q}}(3)$.

  * •

Table 2: Computations for $g=3$

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