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\begin{document}

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%\mbox{{\bfseries UNIVERSITAT DE BARCELONA}}
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%\mbox{Departament d'\`{A}lgebra i Geometria}
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%\mbox{{\bfseries \LARGE{ABELIAN VARIETIES WITH}}}
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%\mbox{{\bfseries \LARGE{QUATERNIONIC MULTIPLICATION}}}
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%\mbox{{\bfseries \LARGE{AND THEIR MODULI}}}
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\begin{titlepage}
%\title{ABELIAN VARIETIES WITH QUATERNIONIC MULTIPLICATION \\ AND THEIR MODULI}

%\author{V\'{\i}ctor Rotger}






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\mbox{{\bfseries ABELIAN VARIETIES WITH QUATERNIONIC}}
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\mbox{{\bfseries MULTIPLICATION AND THEIR MODULI}}
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\mbox{Mem\`{o}ria presentada per optar al grau de Doctor en
Matem\`{a}tiques per}
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\mbox{\bfseries V\'{\i}ctor Rotger Cerd\`{a}}
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\mbox{\bfseries Universitat de Barcelona, 2002}
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\newpage


Departament d'\`{A}lgebra i Geometria

Programa de Doctorat d'\`{A}lgebra i Geometria, Bienni 1998-2000

Doctorand: V\'{\i}ctor Rotger Cerd\`{a}

Tutora i Directora de Tesi: Pilar Bayer Isant




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Pilar Bayer Isant, Catedr\`{a}tica d'\`{A}lgebra

de la Facultat de Matem\`{a}tiques de la Universitat de Barcelona,

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FAIG CONSTAR

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que el senyor V\'{\i}ctor Rotger Cerd\`{a} ha realitzat aquesta mem\`{o}ria

per optar al grau de Doctor en Matem\`{a}tiques sota la meva direcci\'{o}.

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Barcelona, novembre de 2002.


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Signat: Pilar Bayer Isant


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Aquesta tesi ha estat parcialment finan\c{c}ada per

DGCYT PB97-0893 i BFM2000-0627.

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1991 Mathematics Subject Classification: 11G18, 14G35.
\newpage





\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}

\setcounter{page}{1}
In this monograph, we explore several arithmetical, geometrical and diophantine
questions concerning {\em quaternion algebras}, {\em abelian varieties} and
{\em Shimura varieties} and the rich relationships existing between
them. Specifically, these notes are focused on

\begin{itemize}

\item
Abelian varieties whose ring of endomorphisms is a maximal order in a
totally indefinite quaternion algebra $B$ over a totally real
number field $F$, and on the

\item
Shimura varieties $X_B/\Q $ that naturally occur as their moduli spaces.
\end{itemize}

As we aim to make apparent, many of the arithmetical and geometrical
properties of these abelian varieties are encoded either in the quaternion algebra
$B$ or in the Shimura variety $X_B$. In turn, the nature of these
Shimura varieties cannot be handled
without an understanding of the objects parametrized by them.

Due to Albert's classification of involuting division
algebras (cf.\,\cite{Mu}) and the work of Shimura \cite{Sh63}, there
is a limited number of rings that can occur as the
endomorphism ring of an abelian variety. Namely, if $A$ is a simple abelian
variety over an algebraically closed field,
$\End (A)$ is an order in either a
totally real number field, a quaternion algebra over a
totally real number field or a division algebra over a complex multiplication field.
We stress that abelian varieties with distinct endomorphism rings
tend to exhibit very different behaviour in many aspects.

There is a considerable body of work on abelian varieties
with complex multiplication due to Shimura and
Taniyama \cite{ShTa}, Mumford \cite{Mu}, Lang \cite{Lan} and others. The impact of
these contributions on class field theory and the Birch and
Swinnerton-Dyer conjecture is enormous.

There is also abundant literature on abelian varieties
with real multiplication due to Humbert \cite{Hu}, Shimura
\cite{Sh63}, Ribet \cite{Ri80}, \cite{Ri94},
Lange \cite{La}, Wilson \cite{Wi} and others. These works are
highly relevant to the generalized
Shimura-Taniyama-Weil conjectures due to the fact that modular abelian varieties
$A_f/\Q $ attached
to cusp forms $f\in S_2(\Gamma _0(N))$ have totally
real multiplication over $\Q $. We refer the reader to \cite{HaHaMo}
and \cite{Ri90} for more details.

However, few authors have studied quaternionic
multiplication (QM) on abelian varieties. In this case, the
arithmetic of their endomorphism ring is considerably more
involved than in the commutative cases, forcing their
N\'{e}ron-Severi groups to be less accessible. We refer the reader to
\cite{No}, \cite{JoMo}, \cite{HaMu}, \cite{HaHaMo} and \cite{Oh}
for some recent contributions.

Shimura \cite{Sh63}, \cite{Sh67} was the first to consider the coarse
moduli spaces of abelian varieties with quaternion
multiplication, which admit a canonical model $X_B/\Q $
over the field $\Q $ of
rational numbers. As complex manifolds, the varieties $X_B(\C )$
can be described as compact quotients of certain bounded symmetric domains
by arithmetic groups acting on them. Shimura
explored their arithmetic, showing that the coordinates of
the so-called {\em Heegner points} on $X_B$ generate class
fields whose Galois action on them can be described by
explicit reciprocity laws.

Interest in Shimura
varieties has increased in recent years. They have been crucial in several
studies of major questions in number theory. Let us quote
some of them.

With regard to modular conjectures, Shimura curves play a
fundamental role in
Ribet's proof of the Epsilon conjecture which, in turn, implies that
Fermat's Last Theorem follows from the Shimura-Taniyama-Weil
conjecture (cf.\,\cite{Ri89}, \cite{Ri90} and \cite{Pr}).

In connection to the Birch and Swinnerton-Dyer conjecture, Vatsal
\cite{Va} and Cornut \cite{Cor} have recently proved,
independently, several conjectures of Mazur on higher Heegner
points on elliptic curves by means of modular curves, Gross and Shimura
curves and Ratner's ergodic theory. Moreover,
Bertolini and Darmon \cite{BeDa96}, \cite{BeDa98}, \cite{BeDa99}
have exploited the \v{C}erednik-Drinfeld theory on the bad
reduction of Shimura curves to prove anti-cyclotomic versions of
conjectures of Mazur, Tate and Teitelbaum on p-adic variants of
the Birch and Swinnerton-Dyer conjecture.

Concerning the finiteness and squareness conjectures of the
Shafarevich-Tate group of an abelian variety over a number field,
Poonen and Stoll \cite{PoSt} have recently made a careful study of
the Cassels-Tate pairing and have given explicit criteria for the
squareness of the torsion part of the Shafarevich-Tate group of
the Jacobian variety of a curve. On the basis of these results, Jordan
and Livn\'{e} \cite{JoLi99} have exhibited Atkin-Lehner quotients of
Shimura curves such that the cardinality of the finite part of the
Shafarevich-Tate group of their Jacobian varieties is not a
perfect square but twice a perfect square (cf.\,also \cite{Bab}).
In a recent work, Stein \cite{St} provides explicit examples of
abelian varieties $A/\Q $ such that $\sharp \mathrm{Sha }(A/\Q ) =
p\cdot n^2$, $n\in \Z $, for every odd prime $p<10000$, $p\not =
37$.

In addition to the many applications of Shimura curves and varieties, many
authors have also been interested in their
geometrical and diophantine properties themselves. Indeed, integral models of
Shimura curves and their special fibres have been considered by
Morita \cite{Mo}, Boutot and Carayol \cite{BoCa}, Buzzard
\cite{Bu}, \v{C}erednik \cite{Ce}, Drinfeld \cite{Dr} and Zink
\cite{Zi} among many others. Also of great interest are the series
of papers by Kudla and Kudla-Rapoport on height pairings on
Shimura curves, intersection numbers of special $0$-cycles and the
values at the centre of their symmetry of the derivatives of
certain Eisenstein series, along the pattern initiated by the
classical papers of Hirzebruch-Zagier and Gross-Zagier. See
\cite{Kud} and \cite{KudRa}, for instance.

Effective results and computations on Shimura curves
have also been worked out by Kurihara \cite{Ku}, Elkies
\cite{El}, Alsina \cite{Al} and Bayer \cite{Ba} among
others. These are particularly valuable
since the absence of cusps on these curves make these approaches more
difficult than in the classical modular case.

A notable and very re\-cent re\-sult has been obtained by
Edix\-ho\-ven and Ya\-fa\-ev in \cite{EdYa} and its sequels,
pro\-ving part of the An\-dr\'{e}-Oort Con\-jec\-tu\-re on the
dis\-tri\-bu\-tion of spe\-cial points on Shi\-mu\-ra
va\-rie\-ties.

In a diffe\-rent di\-rec\-tion,
I\-ha\-ra \cite{Ih}, Jor\-dan and Liv\-n\'{e}
\cite{JoLi85}, \cite{Jo86}, \cite{JoLi86},
Ogg \cite{Ogg83}, \cite{Ogg84},
Mil\-ne \cite{Mi} and Ka\-mienny \cite{Ka} have
stu\-died the sets of ra\-tio\-nal points on Shi\-mu\-ra cur\-ves, their
At\-kin-Leh\-ner quo\-tients and their Ja\-co\-bian va\-rie\-ties over
glo\-bal, lo\-cal and fi\-ni\-te fields. Fi\-nally, we re\-fer the
rea\-der to \cite{Gr} for an approach to Shi\-mura sur\-faces.

%\vspace{1cm}
\newpage

This report is organized as follows. In Chapter \ref{back}, we fix
some notation and review some facts on number fields, quaternion
algebras and abelian varieties that we will need
throughout the monograph.

In Chapter \ref{AQA}, we look into several questions on the arithmetic
of quaternion algebras and orders which arise naturally in our
subsequent study of abelian varieties and Shimura varieties. Firstly,
in Section \ref{distance}, we consider a problem on the existence of
suitable integral bases of quaternion algebras which
is related to several papers by Chinburg and
Friedman in relation to arithmetic 3-orbifolds (cf.\,\cite{ChFr86}, \cite{ChFr99},
\cite{ChFr00}). Secondly,
in Section \ref{poll}, we study the set of conjugation classes of pure
quaternions of given reduced norm in a quaternion algebra $B$ under an
action introduced by O'Connor and Pall \cite{CoPa} in the 1930s and further
studied by Pollack \cite{Po} in the 1960s. We express the number
of conjugation classes in terms of class numbers of quadratic
extensions embedded in $B$ and groups of units.

In Chapter \ref{GAV} we study abelian varieties
$A$ with quaternionic multiplication and give an arithmetic
criterion for the existence of principal polarizations on them. In
particular, we prove that abelian varieties
with quaternionic multiplication over a totally real number field $F$ of
narrow class number $h_+(F)=1$ are always principally polarizable.
Moreover, we give an expression for the
number of isomorphism classes of principal
polarizations on $A$ in terms of relative class numbers of orders in CM-fields
by means of Eichler's theory of optimal embeddings. As a
consequence, we exhibit simple abelian varieties of any even
dimension admitting arbitrarily many nonisomorphic principal
polarizations. In turn, we obtain that
there exist arbitrarily large sets of pairwise non isomorphic curves of
genus $2$ sharing isomorphic unpolarized Jacobian varieties.

In Chapter \ref{GSV} we consider the moduli spaces of polarized abelian varieties
with multiplication by a maximal order in a totally indefinite
quaternion algebra $B$ over a totally real number field $F$. Shimura
constructed canonical models $X_B/\Q $ of these moduli spaces which
are proper schemes over $\Q $ of dimension $[F:\Q ]$.

We introduce several
subgroups $V_0\subseteq W_0\subseteq W^1\subseteq \Aut _{\Q }(X_B)$ of
the group of automorphisms of these Shimura varieties and we study
their modular interpretation by extending some results due to
Jordan \cite{Jo81}. We then consider several maps that occur naturally between
the Shimura varieties $X_B$, Hilbert modular varieties $\cH _S$ and the
moduli spaces of polarized abelian varieties $\cA _g$ which appear when we
forget certain endomorphism structures. We prove that, up to birational
equivalences, these forgetful maps coincide with the natural
projection by suitable groups of Atkin-Lehner involutions.

We then derive several applications of our main result in this chapter. In
Section \ref{qlocus}, we study the quaternionic locus in the moduli
space $\cA _g$ of principally polarized abelian varieties and we describe it
as a union of Atkin-Lehner quotients of Shimura varieties.
Subsequently, in Section \ref{FM}, we study the field of moduli of the
quaternionic multiplication on a principally polarized abelian variety.

In Chapter \ref{AAV}, we
explore the diophantine properties of abelian surfaces
$A/K$ with quaternionic multiplication over a number field. In Section
\ref{FD}, based on our results in Chapter \ref{GAV},
we look into the representations of the absolute Galois group $G_K$
in the ring of endomorphisms $\End _{\bar K}(A)$ and the N\'{e}ron-Severi
group $\NS _{\bar K}(A)$, respectively. This allows us to derive a
substantial amount of information on the minimal field of definition of
the endomorphisms of
$A\otimes \mathrm{Spec } \bar K$ and the intermediate endomorphism ring
$\End _K(A)$.

In Section \ref{JacSurf}, we compare the field of moduli and the field
of definition of the quaternionic multiplication on the Jacobian
variety of a curve of genus two. Finally, in Section \ref{examplesHashi}, we
illustrate our results
with several explicit examples in dimension $2$ based on explicit computations
due to Hashimoto, Murabayashi and Tsunogai \cite{HaMu}, \cite{HaTs}.

In Chapter \ref{ASV} we look into several diophantine questions on
Shimura curves. Our main tool in this part is provided by the
theory of \v{C}erednik-Drinfeld on the special fibres of the
integral models of Shimura curves at the ramified primes. Let
$X_D/\Q $ be the Shimura curve over $\Q $ attached to the
indefinite rational quaternion algebra $B$ of discriminant $D$. In
Section \ref{secaut} we investigate the group of automorphisms of
$X_D$ and prove that, in many cases, $\Aut (X_D\otimes \mathrm
{Spec }\qbar ) = \Aut (X_D) = W^1$ is the positive Atkin-Lehner group. In
Section \ref{bielliptic}, we determine the family of bielliptic
Shimura curves and we use it in Section \ref{secQu} to study the
set of points on $X_D$ rational over quadratic fields.
In this way we answer a question posed by Kamienny \cite{Ka}. Finally, we
obtain explicit equations of {\em all} elliptic quotients
of $X_D$ of degree $2$.

{\em Design of the cover.} The design of the cover is due to the
illustrator Maria Vidal. It is inspired from
a drawing that may be found in the PhD. Thesis of Jordan \cite{Jo81}
and it makes use of the
fundamental domain of the Shimura curve $X_6$ due to Alsina \cite{Al}.



{\em Acknowledgements.} I am deeply indebted to Pilar Bayer for introducing
me to this wonderful area of mathematics and for her
guidance throughout this work.

I am also grateful to the components of the Seminari de Teoria de
Nombres de Barcelona of the Universitat de Barcelona, the Universitat
Polit\`{e}cnica de Catalunya and the Universitat Aut\`{o}noma de Barcelona for
their valuable
help during these years, their enthusiasm and many enriching and
thought-provoking discussions.

I thank Ju\-liusz Brze\-zins\-ki, Jo\-s\'{e} Ig\-na\-cio Bur\-gos,
Peter Clark, Eduardo Fried\-man, Ge\-rard van der Geer,
Chan\-dra\-she\-khar Kha\-re, Ulf Kuehn,
St\'{e}\-pha\-ne Lou\-bou\-tin, Joan Car\-les Na\-ran\-jo, Di\-pen\-dra Pra\-sad
and Gerald E. Welters for help\-ful con\-versa\-tions and corres\-pon\-dence.

I thank the Tata Institute of Fundamental Research at Mumbai;
Dipendra Prasad, Maneesh Thakur and the Mehta Research Institut
at Allahabad; Norbert Schappacher and the Institut de Rech\'{e}rche
Math\'{e}ma\-ti\-que Avanc\'{e}e at Strasbourg; Jurg Kramer, Ulf
Kuehn and the Humboldt-Universit\"{a}t zu Berlin; Gerard van der Geer and
the Universiteit van Amsterdam;
and Barry Mazur and the Harvard University at Boston
for their warm hospitality in several periods from 1999 to 2002.

I express my deep gratitude to Luis Dieulefait for allowing me to include in
Chapter \ref{AAV} part of our joint studies in this monograph.

The memory of these years preparing this work will always be linked to
my friends Elisenda Feliu, Jes\'{u}s Fern\'{a}ndez, Marta Casanellas and Joaquim
Ro\'{e} from the Departament d'\`{A}lgebra i Geometria of the Universitat of
Barcelona. I am also indebted to Josep Gonz\'{a}lez and Jordi Gu\`{a}rdia
from the Universitat Polit\`{e}cnica de Catalunya for
their help and hospitality at Vilanova i la Geltr\'{u} during the fall of
2002.

Finally, this thesis could have not be completed without the
wonderful times I have shared with my non-mathematical family and friends
from Cardedeu and Barcelona.


\tableofcontents


\chapter{Background}\label{back}

\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{back}. Background}
{\ref{IntroA} Introduction}\label{IntroA}


In this chapter we establish some notations and
review some well known facts on number fields, quaternion algebras and
abelian varieties that we will use in this monograph.

\section{Basic facts on number fields}
\markright{\ref{NF} Basic facts on number fields}
\label{NF}

Let $\Q $ denote the field of rational numbers and let $\qbar $ be an
algebraic closure of $\Q $ in the field $\C $ of complex numbers.

Let $F$ be a number field, that is, a finite field extension of $\Q $. The ring of
integers $R_F$ of $F$ is a Dedekind domain: the set of fractional
ideals of $R_F$ is a free abelian group generated by the prime ideals
of $R_F$.

Let $R$ be an order in $F$ over $\Z $: a subring of $R_F$ such
that $F=R\otimes _{\Z }\Q $.

Let $\Pic (R)$ denote the group of fractional projective ideals of $R$
modulo principal ideals and let $h(R)=|\Pic (R)|$ denote the class number of $R$.
We will simply write $\Pic (F)$ instead of $\Pic (R_F)$ and $h(F)$ instead of
$h(R_F)$.

The class number of an order $R$ in $F$ is related to the
class number $h(F)$ as follows:

$$
h(R) = \frac {h(F)}{(R_F^*:R^*)}\frac {|(R_F/\mathfrak f_R\cdot
R_F)^*|}{|(R/\mathfrak f_R)^*|},
$$
where we let $\mathfrak f_R$ denote the conductor of $R$.

For any place $v$ of $F$, we will let $F_v$ denote the completion of
$F$ at $v$. We freely identify the set of non archimedean places of $F$
with the set of prime ideals of $F$ and the finite set of archimedean places
of $F$ with the set of immersions $\sigma :F\hookrightarrow \C $ up to
complex conjugation.

A totally real number field is a number field $F$ all whose
archimedean places $\sigma $ factor through the field $\R $ of
real numbers. A number field $F$ is totally imaginary if none of
its archimedean places factor through $\R $. A complex muliplication (CM)
field is a totally imaginary quadratic extension of a totally real number
field.

An element $a\in F^*$ is
called totally positive if $\sigma (a)>0$ for any real archimedean
place of $F$. We let $F^*_+$ denote the subgroup of totally positive
elements of $F^*$. For any subset $S$ of $F^*$, we let $S_+ = S\cap
F^*_+$. In particular, we let $R_{F +}^* = R_F^*\cap F^*_+$.


A principal ideal of $F$ is called totally positive if it
can be generated by a totally positive element of $F$. We let
$\Pic _+(F)$ stand for the
narrow class group of fractional ideals of $F$ up to totally positive
principal ideals.

More generally, for any subset $\infty = \{ \sigma _1, ..., \sigma
_r\}$ of real archimedean places on a number field $F$, we let $F^*_{\infty
}$ denote the subgroup of elements $a\in F^*$ such that $\sigma (a)>0$
for all $\sigma \in \infty $.
We let $\Pic _{\infty
}(F)$ denote the class group of fractional ideals of $F$ up to
principal ideals which can be generated by an element $a\in F^*_{\infty }$. We
similarly let $h_{\infty }(F) = |\Pic _{\infty }(F)|$.

For any subfield $K$ of $F$, we let
$R^{\sharp }_{F/K}=\{a\in F: \mathrm{tr}_{F/K}(a R_F)\subseteq R_K\}$
be the codifferent of $R_F$ over $R_K$. The inverse ideal of the
codifferent of $F$ over $F$ is the different $\vartheta _{F/K} =
{R^{\sharp -1}_{F/K}}$.


\section{Basic facts on quaternion algebras}
\markright{\ref{QA} Basic facts on quaternion algebras}
\label{QA}

\subsection{Quaternion algebras, orders and ideals}

Let $F$ be either a number field or the completion of a number field at
a local place. Unless $F=\R $ or $\C $,
let $R_F$ denote the ring of integers of $F$.

\begin{definition}

A quaternion algebra $B$ over $F$ is a central simple
algebra over $F$ of $\rank _F(B)=4$.

\end{definition}

Quaternion algebras can be classically described and constructed as follows.
Let $L$ be a quadratic separable algebra over
the field $F$, let $\sigma $ denote the
non trivial involution on $L$ over $F$
and let $m\in F^*$ be any non zero element. Then, the
algebra

$$
B = L + L e
$$
with
$$
e^2=m \mbox{ and } e \beta = {\beta }^{\sigma } e
\mbox{ for any } \beta \in L,
$$
is a quaternion algebra over $F$.
The classical notation for it is $B = (L, m)$. As it is shown
in \cite{Vi}, any quaternion algebra over $F$ is of this form.

A second and alternative construction of quaternion algebras is the
following. Let $a$, $b\in F^*$ be non zero elements. Then, the algebra

$$
B = \left ( \frac {a, b}{F}\right ) = F + F i + F  j + F i j,
$$
with
$$
i^2 = a, j^2 = b \mbox{ and } i j = -j i,
$$
is again a quaternion algebra over $F$
and again any quaternion algebra admits such a description. Note
that the two constructions are related since $B=(\frac {a, b}{F})=
(F(i), b)$.

Let $B$ be a quaternion algebra over $F$. The
algebra $B$ comes equipped with an anti-involuting conjugation
map $\beta \mapsto \bar \beta $ such that, when restricted to a quadratic
extension $F(\beta )$, $\beta \in B^*\setminus F^*$, is
the nontrivial automorphism of $F(\beta )/F$. More
explicitly, if $\beta = a + b i + c j  + d i j$, then
$\overline \beta = a - b i - c j -d i j$.
Elements $\beta \in B$ are roots of the quadratic polynomial
$x^2-\mathrm{tr}(\beta )x+\mathrm{n}(\beta )$, where

$$
\mathrm{tr}(\beta
)=\beta +\overline \beta
$$
and
$$\n (\beta )=\beta \overline \beta
$$
denote the {\em reduced trace} and the {\em reduced norm} of $\beta \in B$,
respectively.

For any subset $S$ of $B$, we denote by $S_0 = \{\beta
\in S: \mathrm{tr}(\beta )=0\}$ the subgroup of {\em pure} quaternions
of $S$. We let $B^*_+$ and $B^*_-$ denote the subgroup of elements of $B^*$ of
totally positive and totally negative reduced norm, respectively. For
any subset $S$ of $B^*$, we let $S_+ = B^*_+\cap S$ and $S_- = B^*_-\cap
S$. More generally, for any subset $\infty = \{ \sigma _1, ..., \sigma
_r\}$ of real archimedean places on $F$, we let $B^*_{\infty
}$ denote the subgroup of elements $\beta \in B^*$ such that $\sigma
(\n (\beta ))>0$ for all $\sigma \in \infty $.

Let $v$ be a place of $F$ and let $F_v$ denote the completion of $F$
respect to $v$. If $v$ is a complex archimedean place, necessarily
$B\otimes F_v\simeq \M _2(\C )$ since this is the only quaternion
algebra over $\C $ up to isomorphism. If $v$ is a finite or a real
archimedean place, then there are two isomorphism classes of quaternion
algebras over the local field $F_v$: the split algebra
$\mbox{M}_2(F_v)$ and a division algebra that we will denote by
$\mathbb H_v$. In the real case $F_v=\R $, $\mathbb H_v = (\frac{-1,
-1}{\R })$ is the classical skew-field of Hamilton's quaternions.

A place $v$ of $F$ {\em ramifies} in $B$ if
$B\otimes F_v\simeq \mathbb H_v$ is the nonsplit algebra over $F_v$. In
particular, a real archimedean place $v$ ramifies in $B$ if $B\otimes
F_v = (\frac{-1, -1}{\R })$. It is a classical theorem of Hasse that
there is a finite and even number of places $v$ on $F$ that ramify in
$B$. We will say that $B$ is {\em totally indefinite} over $F$ if no real
place ramifies in $B$.

Let us assume from now on that $F$ is either a number field or the
completion of a number field with respect to a finite place.

\begin{definition}

An element $\beta \in B$ is {\em integral} if $\tr (\beta )$, $\n (\beta )\in
R_F$.

\end{definition}

Unlike number fields or local fields, the set of integral elements
of $B$ is not a ring anymore.

\begin{definition}

An {\em order} $\cO \subset B$ over $R_F$ is a ring of
integral elements in $B$ which is finitely generated as $R_F$-module and
such that $\cO \otimes _{R_F}F = B$. It is a {\em maximal order} if it
is not properly contained in any other order.

\end{definition}

Let us agree to say that two orders $\cO $ and $\cO '$ of $B$ are
{\em conjugate} if $\cO = \gamma ^{-1}\cO '\gamma $ for some $\gamma \in
B^*$. The conjugation class $\{ \gamma ^{-1}\cO \gamma : \gamma \in
B^*\}$ is also known as the {\em type} of $\cO $.

Maximal orders in quaternion algebras are in general not unique, often
not even up to conjugation by elements of $B^*$. The finite number of
conjugation classes of maximal orders in $B$ is called the {\em type
number} of $B$ and it is denoted by $t(B)$. In general, we have the
following

\begin{definition}

Let $\cO $ be an order in $B$. For any finite place $\wp $ of $F$,
let $\cO _{\wp } = \cO \otimes _{R_F} R_{F_{\wp }}$. The {\em type
number} $t(\cO )$ of $\cO $ is the number of conjugation classes
of orders $\cO '$ of $B$ such that $\cO '_{\wp } \simeq \cO _{\wp
}$ for all prime ideals $\wp $ of $F$.


\end{definition}

It follows from \cite{Vi} that the type number $t(\cO )$ of an
arbitrary order $\cO $ of $B$ is always a finite number.

The following result is known as the Hasse-Schilling-Maass
Norms Theorem for
quaternion algebras (cf.\,\cite{HaSc}, \cite{Ei37}, \cite{Ei38}, \cite{Vi}).

\begin{proposition}
\label{norms}

Let $\cO $ be a maximal order in $B$. Let
$\infty = \mbox{Ram}_{\infty }(B)$ be the set of archimedean places
of $F$ that ramify in B. Then

$$
\n (B^*) = F^*_{\infty }
$$
and
$$
\n (\cO ^*) = R_{F {\infty }}^*.
$$

\end{proposition}

Note that the set $\mbox{Ram}_{\infty }(B)$ may be empty.

\begin{definition}

Let $\cO $ be an order in $B$. A {\em left ideal} (respectively {\em right ideal})
of $\cO $ is a finitely generated $R_F$-module $I$ with
$I\otimes _{R_F}F = B$ and such that $\cO I\subseteq I$ (respectively
$I \cO \subseteq I$).

\end{definition}

There are several ideals related to a left
ideal $I$ of a maximal order $\cO $. Firstly, the {\em inverse ideal} of a
$\cO $-left ideal $I$ is defined to be $I^{-1}=\{\beta \in B: I\beta
I\subseteq I\}$. It is a right $\cO $-ideal such that $I I^{-1}=\cO $.

The conjugate ideal of $I$ is the right ideal
$\overline I = \{ \overline \alpha : \alpha \in I\} $. The
product ideal $I\cdot \overline I = \{ \sum
\alpha _i \overline {\alpha _i}: \alpha _i\in I\} $ is a two-sided ideal of
$\cO $. If $\n (I) = \{ \n (\alpha ): \alpha \in I\}\subset
F$ denotes the norm ideal of $I$, we have that $I\cdot \overline I = \n
(I)\cdot \cO $ (cf.\,\cite{Sh63J}). We introduce the following

\begin{definition}

Let $I$ be a left ideal of a maximal order $\cO $ in $B$. Then, we
define

$$
\cN (I)=\n (I) \cO =I \overline I
$$
to be the two-sided ideal of $\cO $ generated by the ideal $\n (I)$ of $F$.

\end{definition}


The set of classes of left ideals of a maximal order $\cO $ is

$$ \mbox{Pic}_{\ell }(\cO )=\{I\subset B: \cO I\subseteq I\}/\{\cO
\beta : \beta \in B^*\}. $$ By its adelic description and approximation
theorems, Pic$_{\ell }(\cO )$ is a finite set (cf.\,\cite{Vi}) and the map
$I\mapsto I^{-1}$ induces a bijection between $\mbox{Pic}_{\ell }(\cO
)$ and the set $\mbox{Pic}_{r}(\cO )$ of classes of right ideals of
$\cO $. In fact, its cardinality does not depend on the choice of the
maximal order $\cO $ and it is denoted by $h(B)$. The following can be
found in \cite{Ei55}.

\begin{proposition}
\label{pic} Let $\cO $ be a ma\-xi\-mal order in $B$. If there is
an archime\-dean place $v$ on $F$ which does not ramify in $B$,
then the class number of $B$ coincides with that of the field $F$:
$h(B)=h(F)$. More precisely, there is a bijection of sets

$$ n: \Pic _{\ell }(\cO )\stackrel {\sim }{\rightarrow }\Pic (F) $$
induced by the reduced norm of $B$ over $F$.

\end{proposition}

The existence of an unramified archimedean place on $B$ is called
{\em Eichler's condition} in the literature (cf.\,\cite{Vi}). We refer the reader
to \cite{Joh}, \cite{Re} and
\cite{Vi} for more details on quaternion algebras and orders.



\subsection{Codifferent and discriminant}

As above, let $F$ be either a number field or the completion of a number
field with respect a non archimedean place, let $R_F$ denote the ring
of integers of $F$ and let $B$ be a quaternion algebra over $F$.

\begin{definition}

Let $\cO $ be an order of $B$ and let $I$ be a left ideal of $\cO $.
Let $K\subset F$ be a subfield of $F$ and let $R_K\subset R_F$ denote its ring of
integers. Then, the {\em codifferent } of $I$ over $R_K$ is defined by
$I^{\sharp }_{B/K}=\{\beta \in B: \mathrm{tr}_{B/K}(I\beta )\subseteq R_K\}$.
It is a right ideal of $\cO $ over $R_F$.

\end{definition}

The following facts are well known, but we include here a proof of
them due to the lack of a suitable reference.

\begin{proposition}
\label{different}

Let $\cO $ be a maximal order of $B$ and let $I$ be a left ideal of $\cO $.

\begin{enumerate}


\item

$I^{\sharp }_{B/K} = I^{-1}\cdot \cO ^{\sharp }_{B/K}.$

\item

$\cO ^{\sharp }_{B/K} = \cO ^{\sharp }_{B/F}\cdot R^{\sharp }_{F/K}$.


\item

$\cO ^{\sharp }_{B/F} = \prod _{\wp }D_{\wp }^{-1},$
where the product runs over the finite set of prime ideals $\wp $ of
$F$ that ramify in $B$ and $D_{\wp }$ are two-sided integral ideals of
$\cO $ such that $D_{\wp }^2 = \wp $.

\end{enumerate}

\end{proposition}

{\em Proof. } 1. For any prime $\wp $ of $F$, let $I_{\wp } = I\otimes
R_{F_{\wp }}$. Since it is a local statement, it suffices to prove that
$(I_{\wp })^{\sharp }_{B_{\wp }/K_{\wp }}=I_{\wp }^{-1}\cdot (\cO _{\wp
})^{\sharp }_{B_{\wp }/K_{\wp }}$ for any prime $\wp $, where $K_{\wp }$
denotes the completion of $K$ respect to the prime below $\wp $. As it
is shown in \cite{Vi}, $I_{\wp }$ must be a principal ideal and hence
$I_{\wp }=\cO _{\wp } \beta $ for some $\beta \in B_{\wp }$. Then
obviously $I_{\wp }^{\sharp } = \beta ^{-1}\cdot \cO _{\wp }^{\sharp }$.

2. The proof translates step by step the one given for number fields in
\cite{Se}, Chapter III, Proposition 8.

3. This fact is stated, without proof, in \cite{Sh63J}. By
Proposition \ref{pic} and Proposition \ref{norms}, for any ramifying prime
ideal $\wp $ there is an integral $\cO $-left ideal $D_{\wp }$ such
that $\n (D_{\wp }) = \wp $. At all finite prime ideals $\mathfrak
q\not = \wp $, $D_{\wp }\otimes R_{F_{\mathfrak q}} \simeq \cO \otimes
R_{F_{\mathfrak q}}$ and $D_{\wp }\otimes R_{F_{\wp }}$ is the (unique,
two-sided) maximal ideal of $\cO \otimes R_{F_{\wp }}$. Since so it is
locally, $D_{\wp }$ is a two-sided ideal.

We show now that $\cO ^{\sharp }_{B/F} = \prod _{\wp }D_{\wp }^{-1}$ locally at
any finite prime ideal of $F$. If $\wp $ is a ramified prime in $B$,
then $B_{\wp }$ is discrete valuation division algebra over $F_{\wp }$
(cf.\,\cite{Vi}). Let $\pi $ be an uniformizer of $B_{\wp }$ such that $\pi ^2
= \pi _{F_{\wp }}$ is an uniformizer of the completion of $F$ at $\wp
$. In this case, necessarily $I_{\wp }=\cO _{\wp }\pi ^n$ for some $n\in
\Z $ (cf.\,\cite{Vi}) and $I_{\wp }^{\sharp } = \pi ^{(-n-1)} \cO _{\wp } =
I^{-1}\cdot \cO _{\wp }^{\sharp }$. If
$\mathfrak q$ does not ramify, $\cO _{\mathfrak q}\simeq \mbox{M}_2
(R_{F_{\mathfrak q}})$ and $I_{\mathfrak q} = \cO _{\mathfrak q}\cdot
\beta $, $\beta \in B_{\mathfrak q}$, is a principal ideal. Then
$I_{\mathfrak q}^{\sharp } = \beta ^{-1}\cdot \cO _{\mathfrak q} = I_{\mathfrak
q}^{-1}\cdot \cO _{\mathfrak q}^{\sharp }$. $\Box $


\begin{definition}

Let $\cO $ be an order of $B$ and let $I$ be a left ideal of $\cO $.
Let $K\subset F$ be a subfield of $F$ and let $R_K$ denote its
ring of integers. The discriminant of $I$ over $R_K$ is

$$
\mathrm {disc }_{B/K}(I) := \mathrm{n}_{B/K}(I^{\sharp }_{B/K})^{-1}\cdot
\mathrm{n}_{B/K}(I).
$$


\end{definition}


In particular, the discriminant of $B$ is defined to be

$$
\mathrm{disc }(B)=\mathrm{disc}_{B/F}(\cO ),
$$
for any maximal order $\cO $ of $B$. The following
result ensures that the definition of $\disc (B)$
does not depend of the choice of the maximal order $\cO $.

\begin{proposition}
\label{discriminant}

Let $\cO $ be a maximal order of $B$ and let $I$ be a left ideal of $\cO $.

\begin{enumerate}

\item

$\mathrm{disc }_{B/K}(I) = \mathrm{n}_{B/K}(I)^2\cdot \mathrm{disc
}_{B/K}(\cO )$.

\item

$\mathrm{disc }_{B/K}(\cO ) = \mbox{N}_{F/K}(\mathrm{disc }_{B/F}(\cO
))\cdot \mathrm{disc }_{F/K}(R_F)^2$.


\item

$\mathrm{disc }(B) = \mathrm{disc }_{B/F}(\cO ) = \wp _1\cdot ... \cdot
\wp _r$, where $\wp _i$ are the prime ideals of $F$ which ramify in $B$.

\item

$\mathrm{disc }_{B/K}(I)^2 = \langle \mathrm{det}(\mathrm{tr}_{B/K}
(\alpha_i\cdot \alpha_j))\rangle _{R_F} = \langle
\mbox{det }(\tr _{B/K}(\alpha_i\cdot \overline {\alpha_j}))\rangle
_{R_F}$, where $\{\alpha_i \}$ run on all $K$-bases of $B$ that are
contained in $I$.

\end{enumerate}

\end{proposition}

{\em Proof. } The first three statements are consequence of Proposition
\ref{different}. The first equality of 4. is proved in \cite{Vi}, p.\,25. For the
last equality, we may suppose by 2. that $F = K$ and it suffices to
show that, for any prime ideal $\wp $ of $F$, $\langle
\mathrm{det}(\mathrm{tr}_{B_{\wp }/F_{\wp }} (\alpha_i\cdot
\alpha_j))\rangle _{R_{F_{\wp }}} = \langle
\mathrm{det}(\mathrm{tr}_{B_{\wp }/F_{\wp }}(\alpha_i\cdot
\overline {\alpha_j})) \rangle _{R_{F_{\wp }}}$, where $B_{\wp } = B\otimes F_{\wp
}$ and $\{\alpha_i \}$ run on all $F_{\wp }$-bases of $B_{\wp }$
that are contained in $I_{\wp } = I\otimes R_{F_{\wp }}$. Now all left
ideals of $\cO _{\wp }$ are principal and, for any $\alpha $, $\alpha
_{i j}\in B_{\wp }$, we have that $\mathrm{det } (\mathrm{tr }(\alpha
\alpha _{i j})) = \n (\alpha )^2 \mathrm{det }(\mathrm{tr }(\alpha _{i
j}))$. Hence we may suppose that $I_{\wp } = \cO _{\wp }$. If $\wp
\nmid \mathrm{disc }(B)$, then $\cO _{\wp }\simeq \mathrm{M}_2(R_{F_{\wp
}}) = \langle \begin{pmatrix}
  _{1} & _{0} \\
  _{0} & _{0}
\end{pmatrix}, \begin{pmatrix}
  _{0} & _{1} \\
  _{0} & _{0}
\end{pmatrix}, \begin{pmatrix}
  _{0} & _{0} \\
  _{1} & _{0}
\end{pmatrix}, \begin{pmatrix}
  _{0} & _{0} \\
  _{0} & _{1}
\end{pmatrix}\rangle $ and a simple computation shows the claim.

Analogously, if $\wp \vert \mathrm{disc }(B)$,

$$
\cO _{\wp } \simeq \left \{ \begin{pmatrix}
  _{a} & _{b} \\
  _{\pi b^{\sigma }} & _{a^{\sigma }}
\end{pmatrix}: a, b\in R_{F_{\wp }^2} \right \} ,
$$
where $R_{F_{\wp }^2}$ denotes the ring of integers of the quadratic
unramified extension $F_{{\wp }^2}$ of $F_{\wp }$, $\pi $ is a local
uniformizer and $\sigma $ is the nontrivial element in the Galois
group $\mathrm{Gal } (F_{\wp }^2/F_{\wp })$. As in the case $\wp \nmid
\mathrm{disc }(B)$, the consideration of an explicit basis for this order allows us
to conclude. $\Box $

\begin{definition}

An order $\cO $ of $B$ is an {\em Eichler order} if it is the intersection
of two maximal orders. It is an {\em hereditary order} if all left ideals $I$
of $\cO $ are projective over $R_F$.

\end{definition}

The proof of the following facts can be found in \cite{Re} and
\cite{Vi}.

\begin{proposition}

\begin{enumerate}


\item
Let $\cO $ be an order of $B$. Then $\disc (B)\vert \disc (\cO )$ and
$\cO $ is maximal if and only if $\disc (B) = \disc (\cO
)$.

\item
Let $\cO $ be an Eichler order of $B$. Then, there exists an ideal
$\cN$ of $F$, $(\cN , \disc (B)) = 1$, such that $\disc (\cO ) =
\disc (B)\cdot \cN $. The ideal $\cN $ is called the level of $\cO $.

\item
An order $\cO $ of $B$ is hereditary if and only $\disc (\cO )$ is a
square-free ideal and if and only if $\cO $ is an Eichler order of
square-free level.


\end{enumerate}


\end{proposition}



\subsection{Eichler theory on optimal embeddings}\label{OE}

Let $B$ be a quaternion algebra over a field $F$. The following
two statements are equivalent versions of the Skolem-Noether Theorem.

\begin{theorem}\label{Sk}

\begin{enumerate}

\item
Any automorphism $\varphi : B\stackrel{\sim }{\ra }B$ of $F$-algebras
is inner: there exists $\gamma \in B^*$ such that $\varphi (\beta ) =
\gamma ^{-1}\beta \gamma $ for all $\beta \in B$.

\item
Let $L$ be a quadratic separable algebra of $F$ and let $i:L\hookrightarrow
B$ and $j:L\hookrightarrow B$ be two embeddings of $L$ into $B$. Then
there exists $\gamma \in B^*$ such that $i (L) = \gamma ^{-1} j(L)\gamma
$.


\end{enumerate}
\end{theorem}

Let us assume that $F$ is a number field and let
$\cO $ be an order in $B$. Let us define the
groups $\cO ^*\supset \cO ^*_+\supseteq \cO ^1$ of units in $\cO
$, units in $\cO $ of totally positive reduced norm and units in
$\cO $ of reduced norm $1$ respectively.
We also let $\Nm _{B^*}(\cO ) = \{ \gamma \in B^*: \gamma ^{-1}\cO \gamma
\subseteq \cO \} $ be the normalizer of $\cO $ in $B^*$. Notice that
the Skolem-Noether Theorem \ref{Sk} can also be rephrased by saying
that $\Aut _F(B)\simeq B^*/F^*$ and that the automorphism group of $\cO
$ is $\Aut _{R_F}(\cO )\simeq \Nm _{B^*}(\cO )/F^*$.


A positive anti-involution $\varrho $ on $B$ is a map $\varrho
:B\rightarrow B$ such that $(\beta _1+\beta _2)^{\varrho }=\beta
_1^{\varrho }+\beta _2^{\varrho } $ and $ (\beta _1\cdot \beta
_2)^{\varrho }=\beta _2^{\varrho }\cdot \beta _1^{\varrho } $ for
any $\beta _1$, $\beta _2\in B$, and such that $\tr (\beta \cdot
\beta ^{\varrho })\in F^*_+ $ for any $\beta \in B^*$. By the
Skolem-Noether Theorem, if $\varrho $ is a positive
anti-involution, there exists $\mu \in B^*$ such that $\beta
^{\varrho }=\mu ^{-1}\overline \beta \mu$. Further, it is easily shown
that the positiveness of $\varrho $ implies
that $\tr (\mu )=0$ and $\n (\mu )\in F^*_+$ (cf.\,\cite{Mu}, \cite{LaBi}). The
element $\mu $ is determined by $\varrho$ up to
multiplication by elements of $F^*$ and we will sometimes use the
notation $\varrho = \varrho _{\mu }$.


\begin{definition}

Let $\cO $ be an order in $B$ and
let $S$ be an order over $R_F$ in a quadratic algebra $L$ over $F$.

\begin{enumerate}

\item
An embedding $i:S\hookrightarrow \cO $ is {\em optimal} if
$i(S)=i(L)\cap \cO $.

\item
Two optimal embeddings $i:S\hookrightarrow \cO $ and $j:S\hookrightarrow \cO
$ are Eichler conjugate over $\cO ^*$ if there exists
$\gamma \in \cO ^*$ such that $i (L) = \gamma ^{-1} j(L)\gamma
$. We will denote it by $i\sim _e j$.

\item
We let $E(S, \cO )$ denote
the set of Eichler conjugation classes of optimal embeddings of $S$
in $\cO $ and $e(S, \cO ) = |E(S, \cO )|$.
\end{enumerate}

\end{definition}

As we quote in the theorem below, the set $E(S, \cO  )$ is indeed
finite and it makes sense to consider its cardinality.

\begin{definition}

Let $L$ be a quadratic separable extension of
$F$ and let $S$ be an order in $L$ over $R_F$ of conductor $\mathfrak f _S$. Let
$\wp $ be a prime ideal of $F$. The {\em Eichler symbol} of $S$ and $\wp $ is

$$
(\frac {S}{\wp }) =
  \begin{cases}
    1 & \mbox{ if } \wp \vert \mathfrak f _S\mbox{ or } \wp
    \mbox{ decomposes in }L, \\
    -1 & \mbox{ if } \wp \nmid \mathfrak f _S \mbox{ and }\wp \mbox{ remains
    inert in }L, \\
    0 & \text{otherwise}.
  \end{cases}
$$


\end{definition}

A proof of next statement can be found e.\,g.\, in \cite[p.\,96-97]{Vi}.

\begin{theorem}\label{Eic1}
Let $B$ be a division quaternion algebra over a number field $F$ and
let $\infty $ be the set of real archimedean places of $F$ which are
ramified in $B$. Assume that there is at least one archimedan place of $F$ which does not belong to $\infty $.
Let $\cO $ be an Eichler order of square-free
level $\cN $ in $B$ and let $S$ be an order in a quadratic algebra $L$ over
$F$. Then, the number of Eichler conjugation classes of optimal
embeddings $i:S\hookrightarrow \cO $ is

$$
e(S, \cO ) = \frac {h(S)}{h_{\infty }(F)} \prod _{\wp \vert \disc (B)}
(1-(\frac {S}{\wp })) \prod _{\wp \vert \cN }(1+(\frac {S}{\wp })).
$$


\end{theorem}

Notice that, in particular, Eichler's Theorem \ref{Eic1} establishes a
criterion for the embeddability of a quadratic order $S$ into an
Eichler order $\cO $.

Since it will be of use later in Chapter \ref{GAV}, we consider in Proposition
\ref{Galois} below a stronger form of a particular case of
Eichler's Theorem. Let
$B$ be a totally indefinite division quaternion algebra over a
totally real number field $F$. Let $\cO $ be an Eichler order of
level $\cN $ in $B$ and let $S$ be an order over $R_F$ in a quadratic field
extension $L/F$.

Assume that all prime ideals $\wp \vert \disc (\cO )$ ramify in
$L$ but do not divide the conductor $\mathfrak f_S$ of $S$. This is a
strong restriction on $S$ which we will naturally encounter in Chapter
\ref{GAV}. Let $H_S$
be the {\em ring class field} of $S$ over $L$. The Galois group
$\mathrm{Gal} (H_S/L)$ is isomorphic, via the Artin reciprocity
map, to the Picard group $\Pic (S)$ of classes of locally
invertible ideals of $S$. In the particular case that $S$ is the
ring of integers of $L$, then $H_S$ is the {\em Hilbert class
field} of $L$.  It follows from our assumption on $S$ that
$L$ and $H_S$ are linearly disjoint over $F$, that
is, $F = L\cap H_S$. The norm induces a map $N_{L/F}:\Pic
(S)\rightarrow \Pic (R_F)$ that, by the reciprocity isomorphism
can be interpreted as the restriction map $\mathrm{Gal} (H_S/L)
\rightarrow \mathrm{Gal} (L\cdot H_F/L)\simeq \mathrm{Gal}
(H_F/F)$ (cf.\,\cite{Ne}, Chapter VI, Section 5). In particular,
we have an exact sequence

$$
0\rightarrow \Delta \rightarrow \Pic (S)\stackrel
{N_{L/F}}{\rightarrow } \Pic (R_F)\rightarrow 0 .
$$

Here, $\Delta = \mbox{Ker }(N_{L/F})$ can be viewed as the Galois
group of $H_S$ over the fixed field $L_{\bigtriangleup }$ of $H_S$
by $\Delta $. The group $\Delta = \mathrm{Gal}
(H_S/L_{\bigtriangleup })$ acts on $E (S, \cO )$ by a
reciprocity law as follows: let $i:S\hookrightarrow \cO $ be an
optimal embedding and let $\tau \in \mathrm{Gal}
(H_S/L_{\bigtriangleup })$. Then let $\mathfrak b= [\tau , H_S/L]$
be the locally invertible ideal in $S$ corresponding to $\tau $ by
the Artin's reciprocity map. Since the reduced norm on $B$ induces a
bijection of sets $\n :\Pic _{\ell }(\cO )\simeq \Pic (F)$ and
$N_{L/K}(\mathfrak b)$ is a
principal ideal in $F$, it follows that $i(\mathfrak b)\cO =\beta \cO $ is a
principal right ideal of $\cO $ and we can choose a
generator $\beta \in \cO $. Then $\tau $ acts on $i\in E (S, \cO )$ as

$$
i^{\tau } = \beta ^{-1}i \beta .
$$
It can be checked that this
action does not depend on the choice of the ideal $\mathfrak b$ in its
class in $\Pic (S)$ nor on the choice of the
element $\beta \in \cO $. Moreover, a local
argument shows that this action is free. Since
$|\Delta | = |E (S, \cO )|$, we obtain

\begin{proposition}
\label{Galois}

The action of $\Delta $ on the set of Eichler conjugacy classes of
optimal embeddings $E (S, \cO )$ is free and transitive.

\end{proposition}

The above action acquires a real arithmetic meaning and coincides
with Shimu\-ra's reciprocity law in the particular case that $L$
is a CM-field over $F$. In this situation, $E (S, \cO )$ can also
be interpreted as the set of {\em Heegner points} on
a Shimura variety $\mathfrak X$ on which the Galois group $\Delta
$ is acting (cf.\,\cite{Sh67}, Section 9.10).

Finally, several manuscripts deal with the computation of the numbers
$e(S, \cO )$. See \cite{HiPiSh} and \cite{Br} for Gorenstein and Bass orders.

\markright{\ref{AV} Basic facts on abelian varieties}
\section{Basic facts on abelian varieties}
\markright{\ref{AV} Basic facts on abelian varieties}
\label{AV}

Let $k$ be a subfield of the field $\C $ of complex numbers and let $\bar
k$ denote an algebraic closure of $k$ in $\C $. Let $A$ be an
abelian variety of dimension $g\geq 1$ defined over $k$.

For any field extension $K/k$, let $G_K=\Gal (\bar k/K)$. We let
$A_K = A \otimes _{\Spec k} \Spec K$ and we let $K(A)$ denote the
function field of $A_K$.

Let $\Div (A_{\bar k})$ denote the group of Weil divisors of
$A_{\bar k}$. For any field extension $K/k$ in $\bar k$, a Weil
divisor $\Theta $ is rational over $K$, that is, $\Theta \in \Div
(A_K) = H^0(G_K, \Div (A_{\bar k}))$, if it is stable by the
action of the absolute Galois group of $K$.

Let $\Pic (A_{\bar k})$ denote the group of invertible sheaves on
$A_{\bar k}$. There is a natural exact sequence

$$
0\ra \bar k(A)^*/\bar k^*\ra \Div (A_{\bar k})\ra \Pic (A_{\bar
k})\ra 0
$$
which induces a long exact sequence of Galois cohomology groups
and an isomorphism $\Div (A_{K})/K(A)^*\simeq \Pic (A_{\bar k})$
for any $K/k$, because $A(K)\not =\emptyset $ (cf.\,\cite{PoSt}).

Let $\Pic ^0_{\bar k}(A)$ denote the subgroup of $\Pic _{\bar
k}(A)$ of invertible sheaves on $A$ which are algebraically
equivalent to $0$ and let $\Pic _K^0(A) = \Pic _K(A)\cap \Pic
^0_{\bar k}(A)$ for any field extension $K/k$.

The N\'{e}ron-Severi group $\NS (A)$ of algebraic equivalence classes
of invertible sheaves on $A$ is the group $\NS (A) = \Pic (A)/\Pic
^0(A)$. We note that not all elements in $\NS (A_{\bar k})^{G_K} =
H^0(G_K, \NS _{\bar k}(A))$ are represented by an invertible sheaf
$\cL $ on $A$ defined over $K$. This translates into the fact that
$\NS (A_K)\subseteq \NS (A_{\bar k})^{G_K}$ but these two groups
do not need to be equal.

By the N\'{e}ron Basis Theorem, $\NS (A_{\bar k})^{G_K}$ is a finitely
generated and torsion-free abelian group and we will agree to
define the Picard number of $A_K$ to be

$$
\rho (A_K)= \rank _{\,\Z \,}\NS (A_{\bar k})^{G_K}.
$$

For any closed point $P\in A(\bar k)$, let $t_P:A_{\bar
k}\rightarrow A_{\bar k}$ denote the translation-by-$P$ map. As is
well known, an invertible sheaf $\cL \in \NS (A_{\bar k})^{G_k}$
induces a morphism

$$
\begin{matrix}
 \varphi _{\cL }: & A &\rightarrow &\hat A  \\
  & P & \mapsto & t_P^*(\cL )\otimes \cL ^{-1}
\end{matrix}
$$
defined over $k$.

We let $K(\cL )$ denote the kernel of $\varphi _{\cL }$. An
invertible sheaf $\cL \in \NS(A_{\bar k})$ is {\em nondegenerate}
if $K(\cL )$ is a finite group. Since the image and kernel of the
morphism $\varphi _{\cL }$ attached to a degenerate invertible
sheaf $\cL \in \NS (A_{\bar k})^{G_k}$ is a proper subabelian
variety of $A_{k}$, it follows that all non algebraically
equivalent to zero invertible sheaves $\cL \in \NS (A_{\bar
k})^{G_k}$ on a simple abelian variety $A$ over $k$ are
nondegenerate.

\begin{definition}

The degree of a nondegenerate invertible sheaf $\cL $ on $A$ is

$$
\deg (\cL ) = |K(\cL )|^{1/2}.
$$

An invertible sheaf $\cL $ is {\em principal}
if and only if $K(\cL )$ is trivial, that
is, if $\varphi _{\cL }:A\rightarrow \hat A$ is an isomorphism.

\end{definition}

Let us agree to say that two invertible sheaves $\cL $ and $\cL
'\in \NS (A_{\bar k})^{G_k}$ are isomorphic, denoted by $\cL
\simeq \cL '$, if $(A, \cL )\simeq (A, \cL ')$ as polarized
abelian varieties, that is, there exists an automorphism $\alpha
\in \Aut (A)$ such that $\cL = \alpha ^*(\cL ')$.

\begin{definition}
We let

$$
\Pi (A)=\{ \cL \in \NS (A_{\bar k})^{G_k}: \deg (\cL )=1\}
/_{\simeq }
$$
denote the set of isomorphism classes of principal invertible
sheaves  on $A$.

\end{definition}

The following theorem follows from a result of Narasimhan and Nori
\cite{NaNo}. See also \cite{GoGuRo}.

\begin{theorem}
The set $\Pi (A)$ is a finite set.
\end{theorem}

We shall let $\pi (A) = |\Pi (A)|$ denote the cardinality of $\Pi (A)$.

Let $\cL \in \NS (A_{\bar k})^{G_k}$ be a nondegenerate invertible
sheaf on an abelian variety $A$ over $k$. By Mumford's Vanishing
Theorem (cf.\,\cite{Mu}, $\S 16$), there is a unique integer
$i(\cL )$ such that $H^{i(\cL )}(A,\cL )\not =0$ and $H^j(A,\cL
)=0$ for all $j\not =i(\cL )$. The integer $i(\cL )$ is called the
{\em index of} $\cL $ and it only depends on the class of $\cL $
in $\NS (A_{\bar k})$. We have $0\leq i(\cL )\leq g=\dim (A)$. The
class of algebraic equivalence $\cL \in \NS (A_{\bar k})^{G_k}$ of
an invertible sheaf on $A$ is a {\em polarization over }$k$ if
$i(\cL ) = 0$ or, equivalently, if $\cL $ is ample. By the
Riemann-Roch Theorem, $|K(\cL )| = |\mbox{Ker }\varphi _{\cL
}:A\rightarrow \hat A| = \dim _k(H^{i(\cL )}(A,\cL ))$.

In particular, $\cL $ is {\em principal} if and only if $\dim
_{\bar k}(H^{i(\cL )}(A_{\bar k},\cL ))=1$. In consequence, $\cL
\in \NS (A_{\bar k})^{G_k}$ is a {\em principal polarization} if
and only if $H^{0}(A_{\bar k},\cL ) = \bar k\cdot \vartheta $, for
a certain nonzero automorphic form $\vartheta $.


\begin{definition}

For any nonnegative integer $0\leq i \leq g$, let

$$
\Pi _i(A)=\{ \cL \in \NS (A_{\bar k})^{G_k}: i(\cL )=i, \deg (\cL
)=1\} /_{\simeq }
$$
denote the set
of isomorphism classes of principal invertible sheaves on $A$ of index $i$.

\end{definition}

The set $\Pi (A)$ can be naturally written as the disjoint union
$\Pi (A) = \Pi _0(A)\cup ... \cup \Pi _g(A)$. If we let $\pi
_i(A)=|\Pi _i(A)|$, it then holds that $\pi (A) = \pi _0(A) + ...
+ \pi _g(A)$. Moreover, as is shown in \cite{Mu}, it holds for any
invertible sheaf $\cL $ on $A_{\bar k}$ that $i(\cL ) + i(\cL
^{-1}) = g$. Consequently, the map $\cL \mapsto \cL ^{-1}$ induces
a one-to-one correspondence between $\Pi _i(A)$ and $\Pi
_{g-i}(A)$ and therefore $\pi _i(A)=\pi _{g-i}(A)$.

Let now $A(\C )$ be the set of complex-valued points of the abelian
variety $A_{\C }$. There exists a complex vector space $V$ of dimension
$g$ and a lattice $\Lambda \subset V$ of rank $2 g$ over $\Z $ such
that $V/\Lambda \simeq A(\C )$.

A Riemann form on $\Lambda $ is an $\R $-alternating bilinear form
$E :V\times V\rightarrow \R $ such that $E(\Lambda \times \Lambda
)\subset \Z $ and $E(\sqrt {-1} u, \sqrt {-1} v)=E(u, v) $ for all
$u$, $v\in V$.

By the Appell-Humbert Theorem (cf.\,\cite{Mu}, \cite{LaBi}), the
first Chern class induces an isomorphism of additive groups
between the N\'{e}ron-Severi group $\NS (A_{\C })$ of $A_{\C }$ and
the group of Riemann forms on $\Lambda $. An invertible sheaf $\cL
$ on $A$ is a polarization if and only if the corresponding
Riemann form $E_{\cL }$ attached to $\cL $ by the Appell-Humbert
Theorem satisfies that $E(\sqrt {-1} u, u)>0$ for all $u\in
V\setminus \{ 0\}$.


Let
$$
\begin{matrix}
  H: & V\times V &\ra & \C \\
     & (u, v)    & \mapsto & E(\sqrt {-1} u, v) + \sqrt {-1} E(u, v)
\end{matrix}
$$
be the hermitian form attached to a Riemann form $E$ on $\Lambda $. If
$H$ is the hermitian form
associated to an invertible sheaf $\cL $, the index $i(\cL )$ agrees with
the number of negative eigen-values of $H$ (cf.\,\cite{Mu}, $\S 16$).


Let $\cL $ be a polarization on $A_{\C }$ and $E_{\cL }$ the
corresponding Riemann form. There exists a suitable basis of $\Lambda $
such that the matrix expression of $E_{\cL }$ is

$$
\begin{pmatrix}
      &   &  & d_1 &     & \\
      & 0 &  &     & ... & \\
      &   &  &     &     & d_g\\
 -d_1 &   &  &  &     & \\
      & ... &  &     & 0 & \\
      &   & -d_g &     &     &
   \end{pmatrix}
$$
for some positive integers $d_1\vert d_2\vert ...\vert d_g$. The
sequence $(d_1, ..., d_g)$ is called the {\em type} of the polarization
$\cL $. A polarization $\cL \in \NS (A)$ is called {\em primitive} if $\cL \not
\in d\cdot \NS (A)$ for any $d\in \Z, d\geq 2$. Its type is then
$(1, d_2, ..., d_g)$ for $d_i\vert d_{i+1}$, $i=2, ..., g-1$. The
polarization $\cL $ is principal if and only if $d_1=...=d_g=1$.


For any field extension $K/k$ in $\C $, let $\End _K(A)$ denote the
ring of endomorphisms of the abelian variety $A_K$. This is a possibly
non commutative $\Z $-algebra of finite rank.

Let $\End _K^0(A) = \Q \otimes \End _K(A)$. It is well known that this
is a semisimple algebra of finite rank over $\Q $. Moreover,
$\End _K^0(A)$ is a simple division algebra
if and only if $A_K$ is a simple abelian variety, i.\,e., $A$
contains no proper subabelian varieties defined over $K$. The following
theorem follows from Albert's classification of anti-involuting
division algebras.

\begin{theorem}\label{Albert}

Let $A/k$ be a simple abelian variety over $k$ of dimension $g$.
Let $B=\End ^0(A)$, $F$ be the centre of $B$ and $F_0$ be the
maximal totally real subfield of $F$. Let $e=[F:\Q ]$,
$e_0=[F_0:\Q ]$ and $d^2=[B:F]$. Then one of the following
possibilities must hold:

\begin{enumerate}

\item $B=F=F_0$ and $e\vert g$.

\item
$F=F_0$, $B$ is a totally indefinite quaternion algebra over $F$ and
$2 e\vert g$.

\item $F=F_0$, $B$ is a totally definite quaternion algebra over $F$ and
$2 e\vert g$.

\item $F$ is a CM-field over $F_0$, $B$ is a division algebra over $F$ and
$e_0 d^2\vert g$.

\end{enumerate}
\end{theorem}


\chapter[Arithmetic of quaternion algebras]{Arithmetic of quaternion
algebras}\label{AQA}

\section*{Introduction}\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{AQA}. Arithmetic of quaternion algebras}
{\ref{IntroB} Introduction}\label{IntroB}


\sk\noindent

In this chapter, we study two questions on the arithmetic of quaternion
algebras and their orders which arise from our work on abelian
varieties with quaternionic multiplication and the Shimura varieties
which occur as their moduli spaces.

Let $B$ be a quaternion algebra over either a number field or the completion of a
number field with respect to a finite place and let $\cO $ be an
arbitrary order in $B$. In Section \ref{distance}, we consider the problem of
finding suitable integral bases of $B$ lying in $\cO $. This question is
related to several papers by Chinburg and
Friedman in relation to arithmetic 3-orbifolds (cf.\,\cite{ChFr86}, \cite{ChFr99},
\cite{ChFr00}) and also arises naturally in Chapter \ref{GSV} in the
study of forgetful maps between Shimura varieties and the field of
moduli of the quaternionic multiplication on an abelian variety.

In Section \ref{poll}, we reconsider a quaternionic equation introduced
by Pall and O'Connor in \cite{Pa} and \cite{CoPa} and further
explored by Pollack in \cite{Po}. Motivated by their work, we introduce
a conjugation relation in the set of pure quaternions in $\cO $ of given
reduced norm and we compute the number of orbits in terms of class
numbers of quadratic orders embedded in $\cO $ by making use of
Eichler's theory of optimal embeddings. Our results in this section are
crucial to our study of the set of isomorphism classes of line bundles
on abelian varieties with quaternionic multiplication in Chapter
\ref{GAV}.

The results of this chapter are contained in \cite{Ro2} and \cite{Ro4}.

\section{Integral quaternion bases and distance $\\ $ideals}
\markright{\ref{distance} Integral
quaternion bases and distance ideals}\label{distance}

In this section we focus on the following questions on quaternion
algebras and orders which naturally arise from our results in
Chapter \ref{GSV}.

\begin{question}\label{questionbasis}

Let $F$ be either a number field or the completion of a number field with respect
to a finite place and let
$B$ be a quaternion algebra over $F$. Let $\cO $ be an arbitrary order in $B$.

\begin{enumerate}
\item
If $B\simeq (\frac {a, b}{F})$ for some $a, b\in R_F$, can one
find integral elements $\iota $, $\eta \in \cO $ such that $\iota
^2=a$, $\eta ^2=b$, $\iota \eta  = - \eta \iota $?
\item
If $B\simeq (L, m)$ for a quadratic separable algebra over $F$ and
$m\in R_F$, can one find $\chi \in \cO $ such that $\chi ^2=m$,
$\chi \beta = \overline {\beta } \chi $ for any $\beta \in L$?
\end{enumerate}
\end{question}


We note that the second question may be considered as a refinement
of the first. Indeed, let $\cO $ be an order in $B=(\frac {a,
b}{F})$ and fix an arbitrary element $j\in \cO $ such that
$i^2=a$. Then, while our first question asks whether there exist
arbitrary elements $\iota $, $\eta \in \cO $ such that $\iota
^2=a$, $\eta ^2=b$ and $\iota \eta = -\eta \iota $, the second
wonders whether such an integral basis exists with $\iota = i$.

If $B = \left ( \frac {a, b}{F}\right ) = F + F i + F  j + F i j$,
let $\cO _0 = R_F[i, j]$. Obviously, the first part of the question is answered
positively whenever $\gamma ^{-1}\cO \gamma \supseteq \cO _0$ for
some $\gamma \in B^*$. The following proposition asserts that this
is actually a necessary condition. Although it is not stated in
this form in \cite{ChFr00F}, it is due to Chinburg and Friedman, and
follows from the ideas therein. It is a consequence of Hilbert's
Satz 90.

\begin{proposition}
\label{90}

Let $B = F + F i + F  j + F i j = \left ( \frac {a, b}{F}\right ) $
with $a$, $b\in R_F$. Let $\cO _0=R_F [i, j]$.

An order $\cO $ in $B$ contains a basis $\iota $, $\eta \in \cO
$, $\iota ^2=a$, $\eta ^2=b$, $\iota \eta  = - \eta \iota $ of $B$ if
and only if the type of $\cO _0$ is contained in the type of $\cO $.


\end{proposition}

{\em Proof. } Assume that there exist $\iota $, $\eta \in \cO $
satisfying the above relations. By the Skolem-Noether Theorem
(cf.\,\cite{Vi}), $j$ and $\eta $ are conjugate (by, say, $\alpha \in
B^*$). Thus, by replacing $i$ by $\alpha ^{-1}i \alpha $ and $\cO _0$
by $\alpha ^{-1}\cO _0\alpha $, we may assume that $j=\eta \in \cO $.
We then need to show the existence of an element $\gamma \in F(j)=F(\eta
)$ such that $\gamma ^{-1}\iota \gamma = i$.

We have $i \eta = -\eta i$ and thus $\eta = -i^{-1} \eta i$. In
addition, since $\iota \eta = -\eta \iota $, $\iota i^{-1} \eta i = \eta \iota
$. Hence, $(\iota i^{-1}) \eta = \eta (\iota i^{-1})$ and we deduce
that $\iota i^{-1}\in F(\eta )$ is an element of norm
$\Norm _{F(\eta )/F}(\iota i^{-1})=1$.

By Hilbert's Satz 90, there exists $\om \in F(\eta )$ such that $\iota i^{-1} =
\om \overline {\om }^{-1}$, that is, $\iota = \om \overline {\om }^{-1} i$. Stated in this
form, we need to find an element $\gamma \in F(\eta )$ with
$\gamma ^{-1}\om \overline {\om }^{-1} i \gamma = i$. Since $\gamma i = i\overline {\gamma
}$, we can choose $\gamma =\om $. $\Box $


\begin{corollary}

Assume that $F$ is the completion of a number field with respect
to a finite place. Let $\cO $ be an Eichler order of
level $\cN $ in $B=(\frac {a, b}{F})$, $a$, $b\in R_F$. Then, there
exist $\iota $, $\eta \in \cO $, $\iota ^2=a$, $\eta ^2=b$, $\iota \eta = -\eta
\iota $ if and only if $\cN \vert 4 a b$.

\end{corollary}

{\em Proof. } By \cite{Vi}, $\S 2$, there is only one type of
Eichler orders of fixed level $\cN $ in $B$. Remark that, if $B$
is division, necessarily $\cN =1$. Let $\cO _0=R_F [i, j]$. Since
$\disc (\cO _0)= 4 a b$, as one can check, a necessary and
sufficient condition on $\cO $ to contain a conjugate order of
$\cO _0$ is that $\cN \vert 4 a b$. The corollary follows from
Proposition \ref{90}. $\Box $

In the global case, the approach to question \ref{questionbasis}
\S 1 can be made more effective under the assumption that $B$
satisfies the {\em Eichler condition}. Namely, a quaternion
algebra $B$ over $F$ is said to satisfy the Eichler condition if
some archimedean place $v$ of $F$ does not ramify in $B$, that is,
$B\otimes_F F_v\simeq \M _2(F_v)$. Here, we let $F_v\simeq \R $ or
$\C $ denote the completion of $F$ at $v$.

The following theorem of Eichler describes the set $\cT (\cN )$ of
types of Eichler orders of given level $\cN $ purely in terms of
the arithmetic of $F$. Let $\Pic _{\infty }(F)$ be the narrow class group
of $F$ of fractional ideals up to principal fractional ideals
$(a)$ generated by elements $a\in F^*$ such that $a>0 $ at any
real archimedean place $v$ that ramifies in $B$ and let
$h_{\infty }(F)=|\Pic _{\infty }(F)|$.

\begin{definition}

The group $\overline {\Pic }_{\infty }^{\cN} (F)$ is the quotient of $\Pic
_{\infty }(F)$ by the subgroup generated by the squares of fractional
ideals of $F$, the prime ideals $\wp $ that ramify in $B$ and the
prime ideals $\mathfrak q$ such that $\cN $ has odd $\mathfrak
q$-valuation.

\end{definition}

The group $\overline {\Pic }_{\infty }^{\cN} (F)$ is a $2$-torsion finite
abelian group. Therefore, if $h_{\infty }(F)$ is odd, then $\overline
{\Pic }_{\infty }^{\cN} (F)$ is trivial.

The following can be found in \cite{Ei37}, \cite{Ei38} and \cite{Vi}, p.\,89.

\begin{proposition}
\label{type}
The reduced norm $\n $ induces a bijection of sets

$$
\cT (\cN )\stackrel {\sim }{\longrightarrow }\overline {\Pic
}_{\infty }^{\cN }(F).
$$

\end{proposition}

The bijection is not canonical in the sense that it depends on the
choice of an arbitrary Eichler order $\cO $ in $B$. For $\cN =1$, the
bijection is explicitly described as follows. For any two maximal
orders $\cO $, $\cO '$ of $B$ over $R_F$, define
the {\em distance ideal} $\rho (\cO , \cO ')$ to be the order-ideal of
the finite $R_F$-module $\cO /\cO \cap \cO '$ (cf.\,\cite{Re}, p.\,49).
Alternatively, $\rho (\cO , \cO ')$ can also be defined locally in
terms of the local distances between $\cO \otimes _{R_F}R_{F_{\wp }}$
and $\cO '\otimes _{R_F}R_{F_{\wp }}$ in the Bruhat-Tits tree $\cT _{\wp
}$ for any (nonarchimedean) prime ideal $\wp $ of $F$ that does not
ramify in $B$ (cf.\,\cite{ChFr99}). Finally, $\rho (\cO , \cO ')$ is also
the {\em level} of the Eichler order $\cO \cap \cO '$. This notion of distance
proves to be suitable to classify the
set of types of maximal orders of $B$, as the assignation $\cO '\mapsto
\rho (\cO , \cO ')$ induces the bijection claimed in Proposition
\ref{type}.


\begin{corollary}
Let $B = \left ( \frac {a, b}{F}\right )$, $a$, $b\in R_F$ be a
quaternion algebra over a global field $F$. If $B$ satisfies
Eichler's condition and $h_{\infty }(F)$ is odd then, for any Eichler
order $\cO $ in $B$, there is an integral basis $\iota $, $\eta
\in \cO $, $\iota ^2=a$, $\eta ^2=b$, $\iota \eta = -\eta \iota $
of $B$.

\end{corollary}

As for question \ref{questionbasis} \S 2,
let $B = F+F i +F j +F i j= \left ( \frac
{a, b}{F}\right )= (L, b)$ with $a$, $b\in R_F$ and $L=F(\sqrt {a})$.
Choose an arbitrary order $\cO $ of $B$. For given $\eta \in \cO $, $\eta
^2=a$, we ask whether there exists $\chi \in \cO $, $\chi ^2=b$, such
that $\eta \chi = -\eta \chi $. By Proposition \ref{90}, a necessary
condition is that $\cO _0 = R_F[i, j]\subseteq \cO $ up to conjugation
by elements of $B^*$ and, without loss of generality, we assume that this
is the case. With these notations, we have

\begin{definition}


Let $\cO \supseteq \cO '$ be two arbitrary orders in $B$. The
transportator of $\cO '$ into $\cO $ over $B^*$ is $(\cO :\cO ')
:=\{ \gamma \in B^*: \gamma ^{-1}\cO '\gamma \subset \cO \} $.


\end{definition}

Note that $\Norm _{B^*}(\cO )$ is a subgroup of finite index
of $(\cO :\cO ')$.


\begin{proposition}
\label{Q2} Let $\cO \supseteq \cO _0$ be an order in $B$ and let
$\eta \in \cO $, $\eta ^2=a$. Then, there exists $\chi \in \cO $
such that $\chi ^2=b$ and $\eta \chi = -\chi \eta $ if and only if
$\eta = \gamma ^{-1} i \gamma $ for $\gamma \in (\cO :\cO _0)$.



\end{proposition}

Let $f=|(\cO :\cO _0) : \Norm _{B^*}(\cO )|$ be the index of the
normalizer group $\Norm _{B^*}(\cO )$ in $(\cO :\cO _0)$. Consider
$\eta _1$, ... $\eta _h$ representatives of elements in $\cO $
such that $\eta _i^2 = a$ up to conjugation by elements in $\Norm
_{B^*}(\cO )$. Then, it follows from the above proposition that
there exist $\eta _{i_1}$, ..., $\eta _{i_f}$ such that, for a
given element $\eta \in \cO $, there exists $\chi \in \cO $, $\chi
^2=b$, $\eta \chi = -\chi \eta $ if and only if $\eta $ lies in
one of the $f$ $\Norm _{B^*}(\cO )$-conjugation classes generated
by $\eta _{i_1}$, ..., $\eta _{i_f}$.


$\\ $
Again, the number of $\Norm _{B^*}(\cO )$-conjugation classes of
elements $\eta \in \cO $ such that $\eta ^2=a$ can be explicitly
computed in many cases in terms of class numbers
by means of the theory of Eichler of optimal embeddings. We refer the reader to
\cite{Vi} for details.


\markright{\ref{poll} Pollack conjugation}
\section{Pollack conjugation}
\markright{\ref{poll} Pollack conjugation}
\label{poll}

Let $F$ be a number field and let $B$ be a division quaternion
algebra over $F$. Let $\cO $ be an order in $B$.


\begin{definition}\label{defpoll}

Two quaternions $\mu _1$, $\mu _2\in B$ are {\em Pollack
conjugate} over $\cO $ if $\mu _2 = \overline {\alpha }\mu _1 \alpha $
for some unit $\alpha \in \cO ^*$. We will denote it by $\mu
_1\sim _p \mu _2$.

\end{definition}


The motivation for the above definition comes from Pollack's work \cite{Po}, in
which he studied the obstruction
for two pure quaternions $\mu _1$ and $\mu _2\in B$ with
the same reduced norm to be conjugate over $B^*$ in the above sense, that is, $\mu
_1=\overline {\alpha }\mu _2\alpha $ with $\alpha \in B^*$. He expressed
this obstruction in terms of the $2$-torsion subgroup $\mathrm{Br}_2(F)$ of the
Brauer group $\mathrm{Br}(F)$ of $F$. Furthermore, he
investigated the solvability of the
equation $\mu _2=\overline {\alpha }\mu _1\alpha $ over $\cO ^*$
for quaternions $\mu _1$ and $\mu _2$ in a maximal order $\cO $ of $B$.

As a refinement of his considerations, it is natural to consider
the set of orbits of pure quaternions $\mu \in \cO $ of fixed
reduced norm $\n (\mu )=d\in F^*$ under the action of the group of
units $\cO ^*$ by Pollack conjugation. We drop
the restriction on $\cO $ to be maximal in our statements.

\begin{definition}

For any integral element $d\in R_F$, we let

$$
\cP (d, \cO ) = \{ \mu \in \cO : \tr (\mu ) = 0, \n (\mu ) = d\}.
$$

Let
$$
P (d, \cO ) = \cP (d, \cO )/\sim _p
$$
and $p (d, \cO ) = |P (d, \cO )|$.

\end{definition}

In order to avoid trivialities, we assume
for the rest of this section that $-d$ is not a perfect
square in $R_F$. Then, any quaternion $\mu \in \cP (d, \cO )$ induces an embedding

$$
\begin{array}{clc}
i_{\mu }: F(\sqrt {-d})&\hookrightarrow &B \\ a+b\sqrt{-d}&\mapsto
&a+b\mu
\end{array}
$$
for which $i_{\mu }(R_F[\sqrt {-d}])\subset \cO $.

Let $S$ be an order over $R_F$ in a quadratic algebra $L$ over $F$. Recall from
Section \ref{OE} that an embedding $i:S\hookrightarrow \cO $ is {\em optimal} if
$i(S)=i(L)\cap \cO $. For any $\mu \in \cP (d, \cO )$ there is a uniquely
determined order
$S_{\mu }\supseteq R_F[\sqrt {-d}]$ such that $i_{\mu }$ is optimal at $S_{\mu }$.

Moreover, two equivalent quaternions $\mu _1$, $\mu _2\in \cP (d, \cO )$,
$\mu _1\sim _p\mu _2$, are optimal at the same order $S$. Indeed, if
$\alpha \in \cO^*$ is such that $\mu _1=\overline {\alpha }\mu _2 \alpha $,
then $\alpha $ is forced to have reduced norm $\mathrm{n}(\alpha )=\pm
1$. Hence $\overline \alpha =\pm \alpha ^{-1} \in \cO ^*$ and the
observation follows since $\alpha $ normalizes $\cO $. Conversely,
any optimal embedding $i:S\hookrightarrow \cO $,
$S\supseteq R_F[\sqrt {-d}]$ determines a quaternion $\mu
=i(\sqrt {-d})\in \cP (d, \cO )$.

Hence, a necessary condition for
$\mu _1\sim _p\mu _2$ over $\cO ^*$ is that $\mu _1$ and $\mu _2$
induce an optimal embedding at the same quadratic order
$F(\sqrt {-d})\supset S\supseteq R_F[\sqrt {-d}]$.

\begin{definition}

For any quadratic order $S$ over $R_F$, we let

$$
\cP (S, \cO ) = \{ \mbox{ Optimal embeddings }i: S\hookrightarrow \cO
\}.
$$

Let
$$
P (S, \cO ) = \cP (S, \cO )/\sim _p
$$
and $p(S, \cO )=|P (S, \cO )|$.
\end{definition}

In contrast to Pollack conjugation, we recall from Section \ref{OE} that two
optimal embeddings $i, j:S\hookrightarrow \cO $
lie on the same conjugation class in the sense of Eichler, written
$i\sim _e j$, if there exists $\alpha \in \cO ^*$ such that $i =
\alpha ^{-1} j \alpha $. We also recall that, according to our above definition,
we let $E(S, \cO )=\cP (S, \cO )/_{\sim _e}$ and $e(S, \cO ) = |E(S, \cO )|$.

\begin{proposition}\label{SS}

Let $S$ be an order in a quadratic algebra $L$ over $F$ and let $\cO $
be an order in a division quaternion algebra $B$ over $F$. Then, the number of
Pollack conjugation classes of optimal embeddings of $S$ into $\cO $ is

$$
p(S, \cO ) = |\n (\cO ^*)/N_{L/F}(S^*)|\cdot \frac {e(S, \cO )}{2}.
$$


\end{proposition}

{\em Proof. } Let us agree to say that two pure quaternions
$\mu _1$ and $\mu _2\in \cO
$ lie in the same $\pm $Eichler conjugation class if there exists
$\alpha \in \cO ^* $ such that $\mu _1 = \pm \alpha ^{-1}\mu _2 \alpha
$. We shall denote it by $\mu _1 \sim _{\pm e} \mu _2 $
and $E _{\pm } (S, \cO ) = \cP (S, \cO )/_{\sim _{\pm e}}$. The identity
map $\mu \mapsto \mu$ descends to a natural surjective map
$$
\begin{array}{clc}
\rho :P (S, \cO )&\ra &E _{\pm } (S, \cO )\\
\end{array}
$$
and the proposition now follows from the following lemma.

\begin{lemma}
\label{epsilon2}

Let $e_S = \mathrm{dim }_{\F _2}(\n (\cO ^*)/N_{L/F}(S^*))$. Let $\mu \in
\cP (S, \cO )$ and let $\varepsilon _{\mu }=1$ if $\mu \sim _e -\mu
$ and $\varepsilon _{\mu }=2$ otherwise.
Then, in the $\pm $Eichler
conjugation class $\{ \pm \alpha ^{-1}\mu \alpha : \alpha \in \cO ^*\}$ of $\mu $,
there are exactly $\varepsilon _{\mu } 2^{e_S-1}$
Pollack conjugation classes of pure quaternions.

\end{lemma}

{\em Proof of Lemma \ref{epsilon2}. } Suppose first that
$\varepsilon _{\mu }=1$. Then, the $\pm $Eichler conjugation class of $\mu \in
\cP (S, \cO )$ is $\{\alpha ^{-1}\mu \alpha :\alpha \in \cO ^*\} $. Let
$\gamma \in \cO ^*$ be such that $-\mu = \gamma \mu \gamma ^{-1} =
\gamma ^{-1}\mu \gamma $. We claim that, for any given $\alpha \in \cO
^*$, it holds that
$\mu \sim _p \alpha ^{-1}\mu \alpha $ if and only if $\n (\alpha
)\in N_{L/F}(S^*)\cup (-\n (\gamma )N_{L/F}(S^*))$. Indeed,
if $\mu \sim _p \alpha ^{-1}\mu \alpha $, let $\beta \in \cO
^*$ with $\n (\beta )=\pm 1$ be such that $\overline \beta \alpha ^{-1}\mu
\alpha \beta = \mu $. If $\n (\beta)=1$, then $\alpha \beta \mu = \mu
\alpha \beta $ and hence $\alpha \beta \in L\cap \cO ^*=S^*$. Thus $\n
(\alpha \beta )=\n (\alpha )\in N_{L/F}(S^*)$. If $\n (\beta )=-1$, a
similar argument shows that $\n (\alpha )\in -\n (\gamma
)\cdot N_{L/F}(S^*)$. Conversely, let $\n (\alpha ) = v\in N_{L/F}(S^*)\cup
(-\n (\gamma )N_{L/F}(S^*))$ and let $s\in S^*$ be such that $N_{L/F}(s) = v \mbox{
or} -v \n (\gamma )^{-1}$. Since $\mu $ induces an embedding
$S\hookrightarrow \cO $, we can regard $s$ as an element in $\cO ^*$
such that $\n (s) = v \mbox{ or} -v \n (\gamma )^{-1}$ and $s \mu
= \mu s$. Hence $\alpha ^{-1} \mu \alpha = \alpha ^{-1} s \mu s^{-1}
\alpha = (\alpha ^{-1} s)\mu \overline {(\alpha ^{-1} s)} \mbox{ or }
(\alpha ^{-1} s \gamma ^{-1})\mu \overline {(\alpha ^{-1} s \gamma
^{-1})}$. This proves the claim.

Since $B$ is division, Pollack's Theorem on Pall's Conjecture
(\cite{Po}, Theorem 4) applies to show that $-\n (\gamma ) \not
\in N_{L/F}(S^*)$. We then conclude that the distinct Pollack
conjugation orbits in $\{\alpha ^{-1}\mu \alpha :\alpha \in
\cO ^*\} $ are exactly the classes

$$
\mathcal C _u = \{ \alpha ^{-1}\mu \alpha :
\n (\alpha )\in u N(S^*)\cup (-\n (\gamma ) u N(S^*))\}
$$
as $u\in \n (\cO ^*)$ runs through a set of representatives in $\n (\cO ^*)/ \langle
-\n (\gamma ), N(S^*)\rangle $. There are $2^{e_S-1}$ of them.

Assume that $\varepsilon _{\mu }=2$, that is, $\mu \not \sim _{e} -\mu
$. Then, the $\pm $Eichler
conjugation class of $\mu \in \cP (S, \cO )$ is
$\{\alpha ^{-1}\mu \alpha :\alpha \in \cO ^*\}
\cup \{-\alpha ^{-1}\mu \alpha :\alpha \in \cO ^*\} $. As in the
previous case, it is shown that $\mu \sim _p \alpha
^{-1}\mu \alpha $ if and only if $\n (\alpha )\in N_{L/F}(S^*)$ and
$\mu \sim _p -\alpha ^{-1}\mu \alpha $ if and only if $\n (\alpha
)\in -N_{L/F}(S^*)$. We obtain that, as $u\in \n (\cO ^*)$ runs through a set of
representatives in $\n (\cO ^*)/N(S^*)$, the
$2^{e_S}$ distinct Pollack conjugation classes in the $\pm $Eichler
conjugation class of the quaternion $\mu \in \cP (S, \cO )$ are

$$
\mathcal C '_u = \{ \alpha ^{-1}\mu \alpha :\n (\alpha )\in u N(S^*)\} \cup \{
-\alpha ^{-1}\mu \alpha :\n (\alpha )\in -u N(S^*)\}. \quad \Box
$$

\begin{remark} In view of Proposition \ref{SS}, the effective computation of the
number of Pollack conjugation classes $p(S, \cO )$ for arbitrary orders lies on the
computability of the groups $N_{L/F}(S^*)$ and $\n (\cO ^*)$ and
the number $e(S, \cO )$. The study of the former depends on the
knowledge of the group of units $S^*$ and there is abundant
literature on the subject. If $\cO $ is an Eichler order, the
Hasse-Schilling-Maass Theorem in its integral version describes $\n (\cO ^*)$ in
terms of the archimedean ramified places of $B$ (cf.\,\cite{Vi}). Finally, we
refer the reader to
Section \ref{OE} for details on the computation of the number of Eichler
conjugation classes of optimal embeddings.
\end{remark}

\begin{corollary}

Let $\cO $ be an order in a division quaternion algebra $B$ over $F$ and let
$-d\in R_F$ be not a perfect square.  Then, the number of
Pollack conjugation classes of pure quaternions $\mu \in \cO _0$ of
reduced norm $\n (\mu ) = d$ is

$$
p(d, \cO ) = \sum _{S\supseteq R_F[\sqrt {-d}]}
|\n (\cO ^*)/N_{L/F}(S^*)|\cdot \frac {e(S, \cO )}{2},
$$
where $S$ runs through the finite set of quadratic orders in $F(\sqrt
{-d})$ which contain $R_F[\sqrt {-d}]$.
\end{corollary}



\chapter[Abelian varieties with quaternionic multiplication]{Abelian
varieties with quaternionic multiplication}\label{GAV}

\section*{Introduction}\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{GAV}. Abelian varieties with quaternionic
multiplication}{\ref{Intro1} Introduction}\label{Intro1}

\sk\noindent

It is well known that an elliptic curve $E$ over an arbitrary
algebraically closed field always admits a unique principal polarization
up to translations. This is in general no longer shared by
higher dimensional abelian varieties. In fact, it is a
delicate question to decide whether a given abelian variety $A$ is
principally polarizable. Even, if
this is the case, it is an interesting problem to investigate the
set $\Pi _0(A)$ of isomorphism classes of principal polarizations
on $A$. By a theorem of Narasimhan
and Nori \cite{NaNo}, $\Pi _0(A)$ is a finite set. We shall
denote its cardinality by $\pi _0(A)$.

The aim of this chapter is to study these
questions on abelian varieties with quaternionic multiplication. It
will be made apparent that the geometrical properties of these abelian
varieties are encoded in the arithmetic of their ring of
endomorphisms. As we will see in Chapter \ref{GSV}, our
results shed some light on the geometry and arithmetic of
the Shimura varieties that occur as moduli spaces of abelian
varieties with quaternionic multiplication. Also, these are
the basis of our study of the
diophantine properties of abelian varieties with quaternion
multiplication over number fields carried in Chapter \ref{AAV}. The
results of this chapter are an expanded version of \cite{Ro2}.


\section{Main results}\markright{\ref{mainres} Main results}\label{mainres}

A generic principally polarizable abelian variety admits a single
class of principal polarizations. Humbert \cite{Hu} was the first
to exhibit simple complex abelian surfaces with two nonisomorphic
principal polarizations on them. Later, Hayashida and Nishi
(cf.\,\cite{HaNi} and \cite{Ha}) computed $\pi _0(E_1\times E_2)$ for
isogenous elliptic curves $E_1/\C $ and $E_2/\C $ with complex
multiplication. In positive characteristic, Ibukiyama, Katsura and
Oort \cite{IbKaOo} related the number of principal polarizations on the power
$E^n$ of a supersingular elliptic curve to the class number of
certain hermitian forms. Lange \cite{La}
translated this problem into a number-theoretical one involving
the arithmetic of the ring $\End (A)$ and
produced examples of simple abelian varieties of high dimension
with several principal polarizations. However, he showed that for
an abelian variety with endomorphism algebra $\End (A)\otimes \Q
=F$, a totally real number field, the number $\pi _0(A)$ is
uniformly bounded in terms of the dimension of $A$: $\pi _0(A)\le
2^{\dim (A)-1}$. That is, abelian varieties with real
multiplication may admit several but not {\em arbitrarily many}
principal polarizations.

It could be expected that Lange's or some other bound for $\pi _0(A)$ held for any
simple abelian variety. Hence the
$\\ $
\begin{question} Given
$g\ge 1$, are there simple abelian varieties of dimension $g$ with
arbitrarily many nonisomorphic principal polarizations?
\end{question}

As was already observed, this is not the case in dimension $1$. In
$g=2$, only simple abelian surfaces with at most $\pi _0(A)=2$ were known.
One of our main results, stated in a particular case, is the
following.

\begin{theorem}
\label{mainGAV}

Let $F$ be a totally real number field of degree $[F:\Q ]=n$, let
$R_F$ denote its ring of integers and $\vartheta _{F/\Q }$ the
different of $F$ over $\Q $. Let $A$ be a complex abelian variety
of dimension $2 n$ whose ring of endomorphisms $\End (A)\simeq \cO$ is
a maximal order in a totally indefinite quaternion division
algebra $B$ over $F$.

Assume that the narrow class number $h_+(F)$ of $F$ is $1$ and
that $\vartheta _{F/\Q }$ and $ \disc (B)$ are coprime ideals.
Then,

\begin{enumerate}
\item
$A$ is principally polarizable.

\item
The number of isomorphism classes of principal polarizations on $A$ is

$$
\pi _0(A)= \frac {1}{2} \sum _S h(S),
$$
where $S$ runs through the set of orders in the CM-field $F(\sqrt
{-D})$ that contain $R_F[\sqrt {-D}]$, $D\in F^*_+$ is taken to be
a totally positive generator of the reduced discriminant ideal
$\cD $ of $B$ and $h(S)$ denotes its class number.

\end{enumerate}

In particular, if $A$ is an abelian surface,

$$
\pi _0(A)=
  \begin{cases}
    \dfrac {h(-4 D)+h(-D)}{2} &
    \text{ if }D\equiv 3\quad \mbox{ mod } 4, \\
    \dfrac {h(-4 D)}{2} & \text{otherwise.}
  \end{cases}
$$


\end{theorem}

We prove Theorem \ref{mainGAV} in the more general form of
Proposition \ref{polpral2} and our main Theorem \ref{pio}. In
order to accomplish it, we present an approach to the problem
which stems from Shimura's classical work \cite{Sh63} on analytic
families of abelian varieties with prescribed endomorphism ring.

Our approach is essentially different to Lange's in \cite{La} or
Ibukiyama-Katsura-Oort's in \cite{IbKaOo}. Indeed, whereas in \cite{La}
and \cite{IbKaOo} the (noncanonical) interpretation of line
bundles as symmetric endomorphisms is exploited, we translate the
questions we are concerned with to Eichler's language of optimal
embeddings. In particular, this leads us to solve a problem that has its roots in
the work of O'Connor, Pall and Pollack (see \cite{Po})
and that has its own interest: see Section \ref{poll} for
details.

In regard to the question above, the second main
result of this chapter is the following.

\begin{theorem}
\label{ass}

Let $g$ be a positive integer. Then
\begin{enumerate}
\item
If $g$ is even, there exist simple abelian varieties $A$ of
dimension $g$ such that $\pi _0(A)$ is arbitrarily large.
\item
If $g$ is odd and square-free, $\pi _0(A)\leq 2^{g-1}$ for
any simple abelian variety $A$ of dimension $g$ over $\C $.

\end{enumerate}

\end{theorem}

The boundless growth of $\pi _0(A)$ when $g$ is even stems from our main Theorem
\ref{pio} combined with analytical results on the asymptotic
behaviour and explicit bounds for relative class numbers of
CM-fields due to Horie-Horie \cite{HoHo} and Louboutin
\cite{Lo00}, \cite{Lo}. The second part of Theorem \ref{ass} follows
from the ideas of Lange in \cite{La}. The details of the proof are
completed in Section \ref{asym}.

The following corollary follows from Theorem \ref{ass} and the fact that
any simple principally polarized abelian surface is
the Jacobian of a smooth curve of genus $2$ which, by Torelli's Theorem,
is unique up to isomorphism.

\begin{corollary}
\label{jac}

There are arbitrarily large sets $C_1$, ..., $C_N$
of pair-wise noniso\-mor\-phic genus $2$ curves
with isomorphic simple unpolarized Jacobian
varieties $J(C_1)\simeq J(C_2)\simeq ... \simeq J(C_N)$.

\end{corollary}

In view of Theorem \ref{ass}, it is natural to wonder whether there
exist arbitrarily large sets of pairwise nonisomorphic curves of given
even genus $g\geq 4$ with isomorphic unpolarized Jacobian varieties. In this
direction, Ciliberto and van der Geer \cite{CivdGe} proved
the existence of two nonisomorphic curves of
genus 4 whit isomorphic Jacobian varieties. Explicit examples of
curves with isomorphic (nonsimple) Jacobians have been
constructed by Howe \cite{Ho}, while examples of pairs of
distinct modular curves of genus $2$ defined over $\Q $ with isomorphic
unpolarized absolutely simple Jacobian varieties
have been obtained in \cite{GoGuRo}.

Finally, let us note that the statement of Theorem \ref{ass} does
not cover odd non square-free dimensions.

\begin{conjecture}
Let $g$ be a non square free positive integer. Then there exist
simple abelian varieties of dimension $g$ such that $\pi _0(A)$ is
arbitrarily large.

\end{conjecture}

The conjecture is motivated
by the fact that, when $g$ is non square-free, there exist
abelian varieties whose ring of
endomorphisms is an order in a noncommutative division algebra
over a CM-field and there is a strong similitude between the
arithmetic of the N\'{e}ron-Severi groups of these abelian varieties
and those in the quaternion case.








\section{Abelian varieties with
quaternionic multiplication}\markright{\ref{AVQM} Abelian varieties with
quaternionic multiplication}\label{AVQM}

The main object of study of this chapter and the rest of the monograph will be
the following class of abelian varieties.

\begin{definition}\label{defquat}

Let $k$ be a field and let $\bar k$ be an algebraic closure of $k$.
An abelian variety $A/k$ over $k$ is an
abelian variety with quaternionic multiplication (QM) if the following
conditions are fulfilled:

\begin{itemize}
\item[(i)]
$\End _{\bar k}(A)\simeq \cO $ is a maximal order in a totally indefinite quaternion
algebra $B$ over a totally real number field $F$.

\item[(ii)]
$\dim (A) = 2 [F:\Q ]$.
\end{itemize}
\end{definition}

\begin{remark}
In this chapter we focus our attention on abelian
varieties with quaternionic multiplication over the field $\C $ of
complex numbers. However, we make the above definition in greater generality
since in the subsequent chapters we will consider abelian varieties with
quaternionic multiplication over number fields.
\end{remark}

\begin{remark}
We warn the reader that there are several and non coincident notions
of quaternionic multiplication in the literature.

According to our definition,
there do not exist abelian varieties with quaternionic multiplication over
finite fields nor over the algebraic closure of a finite field. Indeed,
this follows from the theory
of Honda-Tate (cf.\,\cite{Ta66}, \cite{Ta68}).

In particular, if we let $A/K$ be an abelian variety with
quaternionic multiplication over a number field $K$ and we let
$\wp $ be a prime ideal of $K$ of good reduction of $A$, let $k$
be the residue field of $K$ at $\wp $ and let $\widetilde A =
A\otimes \mbox{Spec }k$. Then it holds that $\End _{\bar
K}(A)\simeq \cO $ is a maximal quaternion order, whereas $\End
_{\bar k} (\widetilde A)\varsupsetneq \cO $. Indeed, $\widetilde
A$ fails to be simple and it isogenous to the product of two
abelian varieties of dimension $\dim (A)/2$.
\end{remark}

Let thus $F$ be a totally real number field of degree $[F:\Q ]=n$ and
let $R_F$ be its ring of integers. Let $B$ denote a totally indefinite
division quaternion algebra over $F$ and let $\cD = \disc (B) =
\prod _{i=1}^{2 r}\wp _{i}$, where $\wp _i$ are finite prime
ideals of $F$ and $r\geq 1$, be its reduced discriminant ideal.

We may fix an isomorphism of $F$-algebras

$$
(\eta _{\sigma }): B\otimes
_{\Q }\R \simeq \bigoplus _{\sigma }\mbox{M}_2(\R ^{\sigma }),
$$
where $\sigma $ runs in $\Gal (F/\Q )$ and
$\R ^{\sigma }$ denotes $\R $ as a $F$-vector space via the
immersion $\sigma :F\hookrightarrow \R $. For any $\beta \in B$, we
will often abbreviate $\beta ^{\sigma }=\eta _{\sigma }(\beta )\in
\mbox{M}_2(\R )$.

Let $A/\C $ be an abelian variety with quaternionic multiplication by $\cO $.
As a complex manifold, $A(\C )\simeq V/\Lambda $ for $V$ a complex vector
space of dimension $g$ and $\Lambda \subset V$ a co-compact
lattice that may be identified with the first group of integral
singular homology $H_1(A,\Z )$. The lattice $\Lambda $ is
naturally a left $\cO $-module and $\Lambda \otimes \Q $ is a left
$B$-module of the same rank over $\Q $ as $B$. Since every left $B$-module is
free (cf.\,\cite{We}, $\S 9$), there is an
element $v_0\in V$ such that $\Lambda \otimes \Q =
B\cdot v_0$ and therefore $\Lambda = \cI \cdot v_0$ for some left
$\cO $-ideal $\cI \subset B$.

Note that $\cI $ is determined by $A$ up to principal ideals and
we can choose (and fix) a representative of $\cI $ in its class in
$\Pic _{\ell }(\cO )$ such that $\mathrm{n}(\cI )\subset F$ is
coprime with the discriminant $\cD $. This is indeed possible
because $B$ is totally indefinite: it is a consequence of
Proposition \ref{norms}, Proposition \ref{pic} and the natural
epimorphism of ray class groups $\Pic ^{\cD }(F)\ra \Pic (F)$ of
ideals of $F$ (cf.\,\cite{Ne}, $\S 6$).

Let $\rho _a: B\hookrightarrow \End (V)\simeq \mbox{M}_{2n}(\C )$
and $\rho _r: \cO \hookrightarrow \End (\Lambda )\simeq
\mbox{M}_{4n}(\Z )$ denote the analytic and rational
representations of $B$ and $\cO $ on $V$ and $\Lambda $, respectively. It
is well known that $\rho _r\sim \rho _a \bigoplus \bar {\rho _a}$.


\begin{lemma}\label{porfin}

There exists a basis of the complex vector space $V$ such that

\begin{enumerate}
\item[(i)]
$\rho _a(\beta )=\mathrm{ diag }(\eta _{\sigma _i}(\beta))\in \osum
_{i=1}^n\M _2(\R )\subset \M _{2 n}(\R )$ for any
$\beta \in B$, and

\item[(ii)]
The coordinates of $v_0$ are $(\tau _1, 1, ..., \tau _n, 1)$ for certain
$\tau _i\in \C$, Im$(\tau _i)>0$.


\end{enumerate}
\end{lemma}

{\em Proof. }  The existence of an appropiate basis $\{ e_1, ..., e_{2 n}\}$
of $V$ such that the analytic representation $\rho _a$
of $B$ in $V$ is of the form $(i)$ is
shown in \cite{LaBi}, $\S 9$, 1.1. Note that the same holds if we replace the above
basis for the basis $\{ S\cdot e_1, ..., S\cdot e_{2 n}\}$, for any invertible matrix
$S\in \osum _{i=1}^n\M _2(\R )\subset \M _{2 n}(\R )$. The lemma now
follows from the fact that $S$ can be chosen such that the coordinates
of $v_0$ in the basis $\{ S\cdot e_1, ..., S\cdot e_{2 n}\}$ are of the
form $(ii)$. Indeed, this follows from linear algebra and the fact that
$\cI \cdot v_0\subset V$ is a lattice of maximal rank. $\Box $

We note that the choice of the element $v_0$ fixes an isomorphism of real
vector spaces $B\otimes \R \simeq V$.

Reciprocally, for any choice of a left $\cO $-ideal $\cI $ in $B$ and a
vector $v_0\in V$, $v_0 = (\tau _1, 1, ..., \tau _n, 1)$ with $\mathrm{Im }(\tau
_i)>0$, we may consider the complex torus $V/\Lambda $
with $\Lambda = \cI \cdot v_0$ and $B$ acting on $V$ via the fixed
diagonal analytic representation $\rho _a$. The torus $V/\Lambda $ admits
a polarization and can be
embedded in a projective space (cf.\,\cite{Sh63} and \cite{LaBi}). In consequence,
it is the set of complex points of an abelian variety $A$ such that $\End
(A)\supseteq \cO $.

Moreover, it holds that for the
choice of $v_0$ in a dense subset of $V$, we exactly have
$\End (A) = \cO $. Besides, for $v_0$ in a subset of measure zero of $V$, $A$ fails
to be simple and it is isogenous to the product of two abelian
varieties of dimension $n$ with complex multiplication (cf.\,\cite{Sh63},
\cite{LaBi}).

\markright{\ref{NSNS} The N\'{e}ron-Severi group}

\markright{\ref{NSNS} The N\'{e}ron-Severi group}
\section{The N\'{e}ron-Severi group}\markright{\ref{NSNS} The N\'{e}ron-Severi
group}\label{NSNS}

We fix the following notation for this section. We let
$A/\C $ denote an abelian variety with
quaternionic multiplication by a maximal order $\cO $ in $B$. In addition, we let
$\cI $ be a left ideal of $\cO $ in $B$ such that $H_1(A, \Z )\simeq \cI
$ as a left $\cO $-module and $(\n (\cI ), \cD ) = 1$.


Let $\NS (A)=\mathrm{Pic}(A)/\mathrm{Pic}^0(A)$ be the N\'{e}ron-Severi
group of invertible sheaves  on $A$ up to
algebraic equivalence. We recall from Section \ref{AV}
that two invertible sheaves  $\cL _1$, $\cL _2\in
\mathrm{NS}(A)$ are isomorphic, denoted by $\cL _1\simeq \cL _2$, if there is
an automorphism $\alpha \in \Aut (A)$ such that $\cL _2=\alpha
^*(\cL _1)$. Let us also recall from Section \ref{QA}
that for an arbitrary $\cO $-left ideal $I$, we
write $I ^{\sharp }=\{\beta \in B: \tr _{B/\Q
}(I \beta )\subseteq \Z \}$ and
$\cN (I)=\n (I) \cO =I \overline I$.

The following theorem stems from Shimura's work \cite{Sh63} and
describes $\mathrm{NS}(A)$ intrinsically in terms of the
arithmetic of $B$. The theorem establishes when two line
bundles on $A$ are isomorphic and translates this into a certain
conjugation relation in $B$. We keep the notations as above.


\begin{theorem}
\label{NeSe}
There is a natural isomorphism of groups

$$
\begin{matrix}
c_1: & \mathrm{NS}(A)&\stackrel {\sim }{\rightarrow } & \cN (\cI )_0^{\sharp } \\
   & \cL &\mapsto &\mu =c_1(\cL )
\end{matrix}
$$
between the N\'{e}ron-Severi group of $A$ and the group of pure
quaternions of the codifferent of the
two-sided ideal $\cN (\cI )$.

Moreover, for any two invertible sheaves  $\cL _1$, $\cL _2\in
\mathrm{NS}(A)$, we have that $\cL _1\simeq \cL _2$ if and only if
there exists $\alpha \in \cO ^*$ such that $c_1(\cL _2) =
\overline \alpha c_1(\cL _1) \alpha $.

\end{theorem}

{\em Proof. } By the Appell-Humbert Theorem (cf.\,\cite{Mu},
\cite{LaBi}), the first Chern class allows us to interpret an
invertible sheaf $\cL \in \mathrm{NS}(A)$ as a Riemann form: an
$\R $-alternate bilinear form $E_{\cL }:V\times V\rightarrow \R $
such that $E_{\cL }(\Lambda \times \Lambda )\subset \Z $ and
$E_{\cL }(\sqrt {-1} u,\sqrt {-1} v)=E(u,v) $ for all $u$, $v\in
V$.

Fix an invertible sheaf $\cL $ on $A$ and let $E_{\cL }$ be the
corresponding Riemann form. The linear map $B\rightarrow \Q $,
$\beta \mapsto E_{\cL }(\beta v_0, v_0)$ is a trace form on $B$
and hence, by the nondegeneracy of $\tr _{B/\Q }$, there is a
unique element $\mu \in B$ such that $E_{\cL }(\beta v_0, v_0)=
\mathrm{tr}_{B/\Q }(\mu \beta )$ for any $\beta \in B$. Since
$E_{\cL }$ is alternate, $E_{\cL }(a v_0,a v_0)= \mathrm{tr}_{F/\Q
}(a^2 \mathrm{tr}_{B/F}(\mu ))=0$ for any $a\in F$. It follows
again from the nondegeneracy of $\mathrm{tr}_{F/\Q }$ and the fact
that the squares $F^{*2}$ span $F$ as a $\Q $-vector space that
$\mathrm{tr}_{B/F}(\mu )=0$. Thus $\mu ^2+\delta =0$ for some
$\delta \in F$.

The invertible sheaf $\cL $ induces an anti-involution $\varrho $
on $B$ called the Rosati involution. It is characterized by the
rule $E_{\cL }(u, \beta v)=E_{\cL }(\beta ^{\varrho }u, v)$ for
any $\beta \in B$ and $u$, $v\in V$. It follows that $\beta
^{\varrho }=\mu ^{-1}\overline \beta \mu $ and we conclude that
the Riemann form $E_{\cL }$ attached to the invertible sheaf $\cL
$ on $A$ is $E_{\cL }=E_{\mu }: V\times V\rightarrow \R $,
$(u,v)\mapsto \tr _{B\otimes _{\Q }\R /\R }(\mu \beta ^{\varrho }
\gamma )= \tr _{B\otimes _{\Q }\R /\R }(\overline \beta \mu \gamma
) = \tr _{B\otimes _{\Q }\R /\R }(\mu \gamma \overline \beta )$,
where $\mu \in B$ is determined as above and $\gamma $, $\beta $
are elements in $B\otimes _{\Q }\R  \simeq \M _2(\R )^n$ such that
$u=\gamma  v_0$ and $v=\beta  v_0$. Since $E_{\cL }(\Lambda \times
\Lambda ) \subset \Z $ and $\tr (\mu )=0$, we deduce that $\mu \in
\cN (\cI )_0^{\sharp }$.

Conversely, one checks that any element $\mu \in \cN (\cI )_0^{\sharp }$
defines a Riemann form $E_{\mu }$ which is in
turn the first Chern class of an invertible sheaf $\cL $ on $A$. Indeed, since
$\mu \in \cN (\cI )^{\sharp }$, $E_{\mu }$ is integral over the
lattice $\Lambda =\cI \cdot v_0$ and $E_{\mu }$ is alternate because
$\mathrm{tr}(\mu )=0$. Moreover, let $\iota =\mathrm{diag}(\iota _1, ...,
\iota _n)\in \mathrm{GL} _{2n}(\R )$, $\iota _i\in \mathrm{GL} _2(\R )$, $\iota
_i^2+1=0$, be a matrix such that $\iota \cdot v_0=\sqrt{-1} v_0$. Then
$E_{\mu }(\sqrt{-1}u, \sqrt{-1}v) = E_{\mu }(\gamma \sqrt{-1}v_0, \beta
\sqrt{-1}v_0) = \mathrm{tr} (\mu \gamma \iota \overline \iota \overline
\beta ) = E_{\mu }(u, v)$ for all $u$, $v\in V$.
This concludes the proof of the first part of the theorem.

As for the second, we note that the first Chern class of the
pull-back $\alpha ^*\cL $ of a line bundle $\cL $ on $A$ by an
automorphism $\alpha \in \Aut (A)=\cO ^*$ is represented by the
Riemann form $\alpha ^*E_{\cL }: V\times V\rightarrow \R $, $(u,
v)\mapsto E_{\cL }(\alpha u, \alpha v)$. Hence, if $\cL _2=\alpha
^* (\cL _1)$, then $\tr (\mu _2 \gamma \overline \beta
)=\mathrm{tr}(\mu _1 \alpha \gamma \overline \beta \overline
\alpha )=\mathrm{tr}(\overline \alpha \mu _1 \alpha \gamma
\overline \beta )$ for all $\gamma $, $\beta \in B$ and this is
satisfied if and only if $\mu _2=\overline \alpha \mu _1\alpha $,
by the nondegeneracy of the trace form. Reciprocally, one checks
that if $\mu _2=\overline \alpha \mu _1\alpha $ for some $\alpha
\in \cO ^*$, then $E_{\mu _2}=\alpha ^*E_{\mu _1}$ and therefore
$\cL _2=\alpha ^*\cL _1$. $\Box $

\begin{remark}

As we claimed in Section \ref{mainres}, the isomorphism between $\NS
(A)$ and $\cN (\cI )_0^{\sharp }$ is canonical in the sense that it
does not depend on the choice of a possibly nonexisting principal
polarization on $A$. However, we note that it does depend on the choice
of an isomorphism between the quaternionic order $\cO $ and the ring of
endomorphisms $\End (A)$ of $A$.

\end{remark}

According to Definition \ref{defpoll}, the last statement of
Theorem \ref{NeSe} can be rephrased by saying that two invertible
sheaves $\cL _1$, $\cL _2\in \mathrm{NS}(A)$ are isomorphic if and
only if $c_1 (\cL _1)\sim _p c_1 (\cL _2)$.

We now address the question whether the abelian variety
$A$ admits a principal bundle. We provide an explicit criterion in
terms of the arithmetic of the order $\cO $ and the left ideal $\cI $.
Crucial in the proof of the main result in this direction is the
theory of Eichler on optimal embeddings (cf.\,Section \ref{OE}).

\begin{proposition}
\label{degree}

Let $\cL $ be an invertible sheaf on $A$ and let $c_1(\cL )=E_{\mu }$ be its first
Chern class for some element $\mu \in
B$, $\mu ^2+\delta =0$, $\delta \in F$. Then

$$
\deg (\cL ) = \mathrm{N}_{F/\Q }( \vartheta _{F/\Q }^2\cdot
\n (\cI )^2\cdot \cD \cdot \delta ).
$$

\end{proposition}

{\em Proof.} The degree $\deg (\cL )=\deg (\varphi _{\cL })^{1/2}$
can be computed in terms of the Riemann form as follows:

$$
\mathrm{deg }(\varphi _{\cL }) = \det (E_{\mu }(x_i, x_j)) = \det
(\mathrm{tr}_{B/\Q } (\mu \beta _i \overline {\beta _j})),
$$
where $x_i=\beta _i v_0$ runs through a $\Z $-basis of the lattice
$\Lambda $. Now, it holds that $\mathrm{det }(\mathrm{ tr}_{B/\Q
}(\mu \beta _i \overline {\beta _j})) = \mathrm{n}_{B/\Q }(\mu
)^2\det (\mathrm{ tr}_{B/\Q }(\beta _i \cdot \overline {\beta
_j}))= \mathrm{n}_{B/\Q }(\mu )^2 \disc _{B/\Q }(\cI )^2 =
(\mathrm{N}_{F/\Q }(\delta \cdot \mathrm{n}(\cI )^2 \cdot
\vartheta _{F/\Q }^2\cdot \cD ))^2$ by Proposition
\ref{discriminant}. $\Box $


For the sake of simplicity and unless otherwise stated, we
assume for the rest of the chapter the following

\begin{assumption}

The ideals
$\vartheta _{F/\Q }$ and $\cD $ are coprime.

\end{assumption}

The general case can be dealt by
means of the remark below.

\begin{theorem}
\label{pral}

The abelian variety $A$
admits a {\em principal} invertible sheaf if and only if
the ideals $\cD $ and
$\vartheta _{F/\Q }\cdot \mathrm{n}(\cI )$ of $F$ are {\em principal}.

\end{theorem}

{\em Proof. } Let $\cL $ be a principal invertible sheaf on $A$ and let
$E_{\mu }=c_1(\cL )$ be the associated Riemann form for some $\mu \in
\cN (\cI )_0^{\sharp }$, $\mu ^2+\delta =0$. Since $\cL $ is principal,
the induced Rosati involution $\varrho $ on $\End (A)\otimes \Q =B$
must also restrict to $\End (A)=\cO$ and we already observed that
$\beta ^{\varrho }= \mu ^{-1}\overline \beta \mu $. Therefore $\mu $ belongs
to the normalizer group $\mathrm{Norm}_{B^*}(\cO )$ of $\cO $ in $B$.
The quotient $\mathrm{Norm}_{B^*}(\cO )/\cO ^*F^* \simeq W$ is a finite
abelian 2-torsion group and representatives $w$ of $W$ in $\cO $ can be
chosen so that the reduced norms $\n (w)\in R_F$ are only divisible by
the prime ideals $\wp \vert \cD $ (cf.
\cite{Vi}, p.\,39, 99, \cite{Br}). Hence, we may express $\mu =u\cdot
t \cdot w^{-1}$ for some $u\in \cO ^*$, $t \in F^*$ and $w\in W$.

Recall that $(\n (\cI ), \cD )=1$ and
$(\vartheta _{F/\Q }, \cD )=1$. Since, from Proposition
\ref{degree}, $\mathrm{n}(\cI )^{2} \cdot \vartheta _{F/\Q }^{2}\cdot \cD =
(\delta ^{-1}) = (t ^{-2}\cdot \mathrm{n}(w))$, we conclude that
$\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }=(t ^{-1})$ and
$\cD =(\mathrm{n}(w))$ are principal ideals.

Conversely, suppose now that $\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }=(t)$ and
$\cD =(D)$ are principal ideals, generated by
some elements $t $, $D\in F^*$. Let $R_L$ be the ring of integers in $L = F(\sqrt
{-D})$. From Theorem \ref{Eic1}, since any
prime ideal $\wp \vert D$ ramifies in $L$, Eichler's theory of optimal
embeddings guarantees the existence of an embedding $\iota :R_L\hookrightarrow \cO $
of $R_L$ into the quaternion order $\cO $. Let $w=\iota (\sqrt
{-D})\in \cO $ and let $\mu = (t \cdot w)^{-1}$. As one checks locally,
$\mu\in \mathrm{Norm}_{B^*}(\cO )\cap \cN (\cI )_0^{\sharp }$ and,
by Theorem \ref{NeSe} and Proposition \ref{degree}, $\mu $
is the first Chern class of a principal invertible sheaf on A. $\Box $

Let us recall that an abelian variety $A$ is self-dual if there is an
isomorphism $\varphi :A\stackrel {\sim }{\ra }\hat A$. Acoording to
Section \ref{AV}, if $\cL $ is a principal invertible sheaf on $A$,
then $\varphi _{\cL }:A\ra \hat A$ is an isomorphism. Conversely, it
does not need to hold that any isomorphism between $A$ and $\hat A$ is
of this form. We obtain from the above the following

\begin{corollary}
\label{dual}

If $\cD $ and
$\vartheta _{F/\Q }\cdot \mathrm{n}(\cI )$ are principal
ideals, then $A$ is self-dual.

\end{corollary}


\begin{remark} The case when $\vartheta _{F/\Q }$ and $\cD $ are
non necessarily coprime is reformulated as follows: $A$ admits a
principal invertible sheaf if
and only if there is an integral ideal $\mathfrak a=\wp _1^{e_1}\cdot ...\cdot
\wp _{2r}^{e_{2r}}\vert \vartheta _{F/\Q }$ in $F$ such that both $\cD
\cdot \mathfrak a^2$ and $\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }\cdot
\mathfrak a^{-1}$ are principal ideals. In this case, $A$ is also self-dual. The
proof is {\em mutatis
mutandi} the one given above.
\end{remark}


Let us recall from Section \ref{AV} the finite set $\Pi (A)$ of isomorphism
classes of principal invertible sheaves on $A$. Let

$$
\cP (\cO ) = \{ \mu \in \cO : \tr (\mu ) = 0, \n (\mu ) R_F = \cD .\}
$$
and let
$$
P(\cO ) = \cP (\cO )/\sim _p,
$$
where, as in Section \ref{poll}, we let $\sim _p$ denote Pollack
conjugation over $\cO ^*$. We note that $\cP (\cO )$ is nonempty if and
only if $\cD $ is a principal ideal of $R_F$.

The above proof, together with Theorem \ref{NeSe}, yield the following
corollary, which shall serve us in the next section to compute the number of
isomorphism classes of principal line bundles on $A$.

$$
\\
\\
\\
\\
$$


\begin{corollary}
\label{bij}

Let $A$ be an abelian variety with quaternion multiplication by
a maximal order $\cO $. If $\cD =(D)$ and $\vartheta _{F/\Q
} \cdot \mathrm{n}(\cI )=(t)$ are principal ideals, the
assignation

$$
\cL \mapsto (t \cdot c_1(\cL ))^{-1}
$$ induces a
bijection of sets between $\Pi (A) \mbox{ and } P (\cO )$.

\end{corollary}
\markright{\ref{lineEichler} Principal invertible
sheaves and Eichler optimal embeddings}
\section{Principal invertible sheaves and Eichler optimal
embeddings}\markright{\ref{lineEichler} Principal invertible
sheaves and Eichler optimal embeddings}
\label{lineEichler}


The aim of this section is to compute
the cardinality $\pi (A)= |\Pi (A)|$ of the set of isomorphism classes
of principal invertible sheaves  on an abelian variety $A$ with
quaternionic multiplication by a maximal order $\cO $ in a
totally indefinite quaternion algebra
$B$ over a totally real number field $F$.

Theorem \ref{pi} below exhibits a close relation
between $\pi (A)$ and the class number of $F$
and of certain orders in quadratic extensions $L/F$ that embed in $B$.

\begin{theorem}
\label{pi}

Let $A/\C $ be a principally polarizable
abelian variety with quaternionic multiplication by a
maximal order $\cO $ in $B$ over $F$. Let $\cD = \disc (B) = (D)$ for
some $D\in F^*$ and assume that $(\cD, \vartheta _{F/\Q })=1$. Then the
number of isomorphism classes of principal invertible sheaves  on $A$ is

$$
\pi (A) = \frac {1}{2 h(F)}\sum _u \sum _{S_{\mu }} 2^{e_{S_{\mu }}} h(S_{\mu }),
$$
where $S_{\mu }$ runs through the finite set of orders in the
number field $F(\sqrt {-u D})$ such that $R_F[ \sqrt {-u D}]
\subseteq S_{\mu }$ and $u\in R_F^*/R_F^{*2}$ runs through a set
of representatives of units in $R_F$ up to squares. Here,
$2^{e_{S_{\mu }}} = |R_F^*/N_{F(\sqrt {-u D})/F}(S_{\mu }^*)|$.

\end{theorem}

The proof of Theorem \ref{pi} will be completed during the rest of this
section. Let us make before several remarks for the sake of its
practical application.


\begin{remark}
First of all, by Dirichlet's Unit Theorem, $R_F^*/R_F^{*2}\simeq (\Z
/2\Z )^{n}$ and thus $e_S\leq n$. The case $F=\Q $ is trivial since
$\Z ^*/\Z ^{*2} = \{ \pm 1\} $. In the case of real quadratic fields
$F$, explicit fundamental units $u\in R_F^*$ such that $R_F^*/R_F^{*2}
= \{ \pm 1, \pm u\} $ are classically well known (cf.\,\cite{Ne}).
For higher degree totally real number fields there is abundant
literature on systems of units. See for instance \cite{La} for an
account.
\end{remark}

\begin{remark}\label{remark}
Let $2 R_F = \mathfrak q_1^{e_1}\cdot ... \cdot \mathfrak q_m^{e_m}$ be
the decomposition of $2$ into prime ideals in $F$. For any $u\in
R_F^*$, the conductor $\mathfrak f$ of $R_F[\sqrt {-u D}]$ over
$R_F$ in $L$ is

$$
\mathfrak f = \prod _{\stackrel {\mathfrak q\vert 2}{\mathfrak q\not
\vert D}} \mathfrak q^{a_{\mathfrak q}}, \quad 0\leq a_{\mathfrak q}\leq
e_{\mathfrak q}.
$$

For a prime ideal $\mathfrak q\vert 2$, $\mathfrak
q\nmid D$, let $\pi $ be a local uniformizer of the completion of $F$
at $\mathfrak q$ and $k = \F _2^f$ be the residue field. Let $e =
e_{\mathfrak q}\geq 1$. Since $-u D\in R_{F_{\mathfrak q}}^*$, we have that

$$
-u D = x_0 + x_k \pi ^{k} + x_{k+1} \pi ^{k+1} + ...,
$$
for some $1\leq k\leq \infty$ and
$x_i$ running in a system of representatives of $\F _{2^f}$ in
$R_{F_{\mathfrak q}}$ such that $\overline {x_0}$, $\overline {x_k}\not = 0$.
Here, we agree to set $k=\infty $ in case that $-u D = x_0$.

It then holds that

$$
\mbox{min }([\frac {k}{2}], e)\leq a_{\mathfrak q}\leq e.
$$

More precisely, we have

$$
a_{\mathfrak q}=
  \begin{cases}
    [\frac {k}{2}] & \text{if } k\leq e+1, \\
    e & \text{if } [\frac {k}{2}]\geq e.
  \end{cases}
$$

If $[\frac{k}{2}]<e<k-1$, then
the determination of $a_{\mathfrak q}$ depends on the
choice of the system of representatives of $\F _{2^f}$ in
$R_{F_{\mathfrak q}}$ and it deserves a closer inspection. This applies in many
cases for deciding whether $R_F[\sqrt {-u D}]$ is
the ring of integers of $F(\sqrt {-u D})$, i.e., $\mathfrak f = 1$.

In order to show the above formula, we note that the discriminant of $R_F[\sqrt
{-u D}]$ over $R_F$ is $-4 u D$. Since $u\in R_F^*$ and $D$ has
square-free decomposition, $\mathfrak f\vert 2$. Further, by \cite{Se},
I, 6, $R_F[\sqrt {-u D}]$ is also maximal at the places $\wp _i\vert D$.
Let thus $\mathfrak q\vert 2$, $\mathfrak q\nmid D$. The ring of
integers of $F_{\mathfrak q}(\sqrt {-u D})$ is $R_{F_{\mathfrak q}}
[\alpha ]$ for some $\alpha =v \pi ^r+w \pi ^s \sqrt {-u D}$ with $v$,
$w\in R_{F_{\mathfrak q}}^*$ and $r$, $s\in \Z $. Then $\mbox{Tr
}(\alpha )=2 v \pi ^r= v' \pi ^{r+e}$, $v'\in R_{F_{\mathfrak q}}^*$,
and $\mbox{N} (\alpha )=v^2 \pi ^{2r}-D w^2 \pi ^{2s}$. Since $\alpha $
must be integral, we obtain that $r=s=-a_{\mathfrak q}$ with $0\leq
a_{\mathfrak q}\leq e$. Computing the local expression of $v^2 \pi
^{2r}-D w^2 \pi ^{2r}$, the rest of our claim follows.

In addition, let us note that
the set of orders $S$ in $L = F(\sqrt {-u D})$ that contain $R_F[\sqrt
{-u D}]$ can be described as follows. Any order $S\supseteq R_F[\sqrt
{-u D}]$ has conductor $\mathfrak f _S\vert \mathfrak f $ and for every
ideal $\mathfrak f '\vert \mathfrak f $ there is a unique order
$S\supseteq R_F[\sqrt {-u D}]$ of conductor $\mathfrak f '$. Further,
$\mathfrak f _S\vert \mathfrak f _T$ if and only if $S\supseteq T$.
\end{remark}

In order to prove Theorem \ref{pi}, we begin by an equivalent formulation
of it. As it was pointed out in Corollary \ref{bij}, the first Chern
class induces a bijection of sets between $\Pi (A)$ and the set of
Pollack conjugation classes $P (\cO )$. As in Section \ref{poll}, for
any $u\in R_F^*$, we write $\cP (-u D, \cO ) =\{ \mu \in \cO : \mu
^2 + u D = 0\} $. Observe that $\cP (\cO )$ is the disjoint
union of the sets $\cP (-u_k D, \cO )$ as $u_k$ run through units in
any set of representatives of $R_F^*/R_F^{*2}$.

As in Section \ref{poll}, for any quadratic order $S$
over $R_F$, let $\cP (S, \cO )$ denote the set of optimal
embeddings of $S$ in $\cO $ and $P(S, \cO )= \cP (S, \cO )/_{\sim _p}$. We
obtain a natural identification of sets

$$
P (\cO ) = \sqcup _{k} \sqcup _{S} P (S, \cO ),
$$
where $S$ runs through the
set of quadratic orders $S\supseteq R_F[\sqrt {-u_k
D}]$ for any unit $u_k$ in a set of representatives of $R_F^*/R_F^{*2}$. Hence, in
order to prove Theorem
\ref{pi}, it is enough to show that, for any quadratic order $S
\supseteq R_F[\sqrt {-u D}]$, $u\in R_F^*$, it holds that

$$
p(S, \cO ):=|P (S, \cO )|=\frac {2^{e_S-1} h(S)}{h(F)}.
$$

Since this question is interesting on its own, we have studied it separately in
Chapter \ref{AQA}, where we have proved the above statement in greater
generality. We are now in position to complete the

{\em Proof of Theorem \ref{pi}. } Firstly, under the assumptions of Theorem
\ref{pi}, Proposition \ref{norms} asserts that $\n (\cO ^*)=R_F^*$.
Secondly, we have that

$$
e (S, \cO )= \frac {h(S)}{h(F)}
$$
for any $S\supseteq R_F[\sqrt {-u D}]$, $u\in R_F^*$. This follows from
Eichler's Theorem \ref{Eic1} together with Remark \ref{remark}. The
combination of these
facts together with the preceding discussion
and Proposition \ref{SS} yield the theorem. $\quad \Box $

\markright{\ref{nondege} The
index of a nondegenerate invertible sheaf}
\section{The index of a nondegenerate invertible
sheaf}\markright{\ref{nondege} The
index of a nondegenerate invertible sheaf}\label{nondege}

Let $A/\C $ be an abelian variety with quaternionic multiplication by a maximal
order $\cO $ in $B$. We have that $A(\C )\simeq V/\Lambda $ for a
complex vector space $V$ and a lattice $\Lambda $. As in Lemma
\ref{porfin}, we fix for the rest of the section a basis of $V$ such that

\begin{enumerate}
\item[(i)]
The analytic representation of any endomorphism $\beta \in \cO $ is
$$
\rho _a(\beta ) = \mbox{diag }_{\sigma :F\hookrightarrow \R }(\beta ^{\sigma
}), \beta ^{\sigma }\in \M _2(\R )
$$
and
\item[(ii)]
$\Lambda = \cI \cdot v_0$, where $\cI $ is a left ideal of $\cO $ such
that $(\cD ,\n (\cI )) = 1$ and $v_0=(\tau _1, 1, ..., \tau _n, 1)\in
V$ with $\mbox{Im }(\tau _i)>0$.
\end{enumerate}

We now compute the
index of an invertible sheaf $\cL $ on an abelian variety $A$ with
quaternionic multiplication in terms of the quaternion $\mu =c_1(\cL )$.


\begin{lemma}
Let $\mu \in B_0$ be a pure quaternion and let $\n (\mu ) =
\delta \in F^*$. Then, for
any immersion $\sigma :F\hookrightarrow \R$, there exists

$$
\nu _{\sigma }= \nu
_{\sigma }(\mu )\in \mathrm{GL} _2(\R )
$$
such that $\nu _{\sigma } \mu
^{\sigma } \nu _{\sigma }^{-1}=\omega _{\sigma }$, where

$$
\omega _{\sigma } =
\begin{cases}
    \begin{pmatrix} _{0}& _{\sqrt {\sigma (\delta )}}\\
_{-\sqrt {\sigma (\delta )}}&{0}
\end{pmatrix} & \text{if } \sigma (\delta )>0, \\
    \begin{pmatrix} _{\sqrt {\sigma (-\delta )}}&_{0}\\_{0}&_{-\sqrt {\sigma (-\delta
)}}
\end{pmatrix} & \text{otherwise}.
  \end{cases}
$$

Moreover, although $\nu _{\sigma }$ is not
uniquely determined by $\mu $, $\sign (\mathrm{det }(\nu
_{\sigma }))$ is.

\end{lemma}

{\em Proof. } Since $\mu \in B\hookrightarrow B\otimes _{\Q }\R \simeq
\M _2(\R )\oplus ... \oplus \M _2 (\R )$ satisfies that $\mu ^2 + \delta =
0$, the lemma follows from linear algebra. $\Box $

Motivated by the following theorem, we have

\begin{definition}
\begin{enumerate}
\item[(i)]
The signature of a nonzero pure quaternion $\mu \in B_0$ such that
$\mu ^2+\delta =0$ for $\delta \in F^*$ is

$$
\mbox{sgn } (\mu )= (\mbox{sgn }(\mbox{det } (\nu _{\sigma })
))\in \{ \pm 1\} ^n.
$$

\item[(ii)]
We say that a pure quaternion $\mu \in B^*$ is {\em ample}
if $\mathrm{sgn }(\mu )= (1, ..., 1)$.

\item[(iii)]
For any real immersion $\sigma :F\hookrightarrow \R $, we
define the local archimedean index $i_{\sigma }(\mu )$ of $\mu $ by

$$
i_{\sigma }(\mu ) =
  \begin{cases}
    0& \text{if } \sigma (\delta )>0
\mbox{ and det}(\nu _{\sigma })>0,
\\
    1 & \text{if } \sigma (\delta )<0, \\
    2 & \text{if } \sigma (\delta )>0
\mbox{ and det}(\nu _{\sigma })<0.
  \end{cases}
$$
\end{enumerate}
\end{definition}


\begin{theorem}
\label{index}

Let $A/\C $ be an abelian variety with quaternionic multiplication by a
maximal order $\cO $.

Let $\cL \in \mathrm{NS}(A)$ be an invertible sheaf on $A$ and let $\mu = c_1(\cL
)$. Then, the index of $\cL $ is

$$
i(\cL ) = \sum _{\sigma
:F\hookrightarrow \R } i_{\sigma }(\mu ).
$$

\end{theorem}

{\em Proof. } The index of $i(\cL )$ coincides with the number of
negative eigenvalues of the hermitian form $H_{\mu }$ associated
to the invertible sheaf $\cL $. If we
regard $\M _2(\R )\times \stackrel {n}{...}\times \M _2(\R )$
embedded diagonally in $\M _{2n}(\R )$, there is an
isomorphism of real vector spaces between
$B\otimes _{\Q }\R $ and
$\M _2(\R )\times \stackrel {n}{...}\times \M _2(\R )$
explicitly given by the map
$\beta \mapsto \beta \cdot v_0$. The complex structure that
$\M _2(\R )^n$ inherits from that of $V$ is such that $\{ 0\} \times
...\times \M _2(\R )\times ...\times \{ 0\} $ are complex vector
subspaces of $\M _2(\R )^n$ and we may choose a $\C $-basis of $V$ of
the form $\{ \mbox{diag }(\beta _1, 0, ..., 0)\cdot v_0, $ $\mbox{diag
}(\gamma _1, 0, ..., 0)\cdot v_0, ..., \mbox{diag }(0, ..., 0,\beta
_n)\cdot v_0,$ $\mbox{diag }(0, ..., \gamma _n)\cdot v_0\} $ for $\beta
_i$, $\gamma _i\in \M _2(\R )$.

Let $\iota =\mathrm{diag} (\iota _{\sigma })\in \mathrm{GL}
_{2n}(\R )$ be such that $\iota \cdot v_0=\sqrt{-1} v_0$. For any
$\beta =\mbox{diag }_{\sigma }
(\beta _{\sigma })$, we have that $\gamma =\mbox{diag }_{\sigma }
(\gamma _{\sigma })\in \M _{2n}(\R )$ and

$$
H_{\mu }(\beta v_0, \gamma v_0)=
\sum _{\sigma }\mbox{tr }(\mu ^{\sigma }\beta
_{\sigma } \iota _{\sigma }\overline {\gamma _{\sigma }}) + \sqrt{-1}\sum
_{\sigma } \mbox{tr }(\mu ^{\sigma }\beta
_{\sigma } \overline {\gamma _{\sigma }}).
$$

Thus, if we let $H_{\sigma }\in
\mathrm {GL}_2(\C )$ denote the restriction of $H_{\mu }$ to
$V_{\sigma }=\M _2(\R )\cdot
{\begin{pmatrix} {\tau _{\sigma }}, & {1} \end{pmatrix}}^t$, the matrix of $H_{\mu
}$ respect to the chosen basis has diagonal form
$H_{\mu }=\mbox{diag }(H_{\sigma })$.

In order to prove Theorem \ref{index}, it suffices to show that the
hermitian form $H_{\sigma }$ has $i_{\sigma }(\mu )$ negative
eigenvalues. Take $\beta \in \M _2(\R )$ and let $v=\beta \cdot
{\begin{pmatrix} {\tau _{\sigma }}, & {1}\end{pmatrix}}^t\in V_{\sigma }$. Then,
$H_{\sigma }(v, v)=
\mathrm{tr }(\mu ^{\sigma }\beta _{\sigma }\iota _{\sigma }\overline
{\beta _{\sigma }})=\mathrm{tr }(\omega _{\sigma }\beta '_{\sigma }
\iota '_{\sigma }\overline {\beta '_{\sigma }})$, where $\beta '_{\sigma
}=\nu _{\sigma }\beta _{\sigma } \nu _{\sigma }^{-1}$ and $\iota
'_{\sigma }=\nu _{\sigma }\iota _{\sigma }\nu _ {\sigma }^{-1}$. Denote
$w _{\sigma }= {\begin{pmatrix} w_1, & w_2 \end{pmatrix}}^t =
\nu _{\sigma } \beta _{\sigma }\cdot {\begin{pmatrix} \tau _{\sigma }, & 1
\end{pmatrix}}^t \in \C ^2$ and $\| w_{\sigma }\| ^2 = w_1 \overline {w_1}+
w_2 \overline {w_2}$.

Some computation yields that

$$
H_{\sigma }(v, v)=\sum _{\sigma }
\dfrac{C_{\sigma }\sqrt {| \sigma (\delta )|}}{\mathrm{det}(\nu
_{\sigma }) \mathrm{Im} (\tau _{\sigma })},
$$
where $C_{\sigma }=\|
w\| ^2 $ if $\sigma (\delta )>0$ and $C_{\sigma }= 2\mathrm{
Re}(w_1 \overline {w_2})$ if $\sigma (\delta )<0$. From this, the result
follows. $\Box $

\begin{remark}
From the above formula, the well known relation $ i(\cL )+i(\cL
^{-1})=\dim A $ (\cite{Mu}, Chapter III, Section 16, p.\,150) is
reobtained.
\end{remark}



\section{Principal polarizations and self-duality}
\markright{\ref{PPSD} Principal polarizations and self-duality}\label{PPSD}


We devote this section to consider the following

\begin{question}\label{Quest}

Let $A$ be an abelian variety and assume that it
is self-dual, that is, $A\simeq \hat A$. Then, is $A$
principally polarizable?

\end{question}

Let us note that in the generic case in which $A$ is an abelian
variety whose ring of endomorphisms is $\End (A)=\Z $, it holds
that $A$ is principally polarizable if and
only if it is self-dual.

As in the previous sections, we let $\cO $ denote a maximal order in a
totally indefinite quaternion algebra $B$ over a totally real field
$F$ and let $\cD =\disc (B)$. Let $\cI $ be an $\cO $-left ideal
such that $(\n (\cI ), \cD ) = 1$.


Let $\cO ^*_+$ be the subgroup of units in $\cO $ of totally
positive reduced norm. By Proposition \ref{norms}, we have $\n (\cO
^*)=R_F^*$ and we let $\Sigma = \Sigma (R_F^*)\subseteq \{ \pm 1\}
^n$ be the $\F _2$-subspace of signatures of units in $R_F^*$. As
$\F _2$-vector spaces, $\Sigma \simeq R_F^*/R_{F +}^*$ and, by
Dirichlet's Unit Theorem, $|\Sigma | = \dfrac {2^n h(F)}{h_+(F)}$. Note
that $h_+(F) = h(F)$ if and only if $\Sigma (R_F^*)=\{ \pm 1\} ^n$.
By Proposition \ref{norms}, the group $\Sigma $ fits in the exact sequence

$$
1\rightarrow \cO ^*_+\rightarrow \cO ^*\stackrel {\mbox{sgn}\cdot \n }{\ra }\Sigma
\rightarrow 1
$$


\begin{proposition}\label{polpral}

There exist principally polarizable abelian varieties $A/\C $ with
quaternionic multiplication
by $\cO $ and $H_1(A,\Z )\simeq \cI $ if and only if $\cD $ and
$\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }$ are principal ideals and
$\cD =(D)$ is generated by a totally positive element $D\in F^*_+$.

\end{proposition}

{\em Proof. } Assume first that there exists a principally polarizable
abelian variety $A/\C $ with quaternionic multiplication
by $\cO $ and $H_1(A,\Z )\simeq \cI $. Then, by Theorem \ref{pral},
$\cD $ and $\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }$ are principal
ideals. Let $D\in F^*$ be any generator of $(\cD )$.
Let $\cL $ be a principal polarization on $A$ and let $\mu =c_1(\cL
)$. By Corollary \ref{bij}, $\n (\mu ) = u D\in F^*/F^{*2}$ for some $u\in
R_F^*$. Since $i(\cL )=0$, it follows from Theorem \ref{index} that $u D\in
F^*_+$.

Conversely, we assume that $\cD = (D)$ and $\mathrm{n}(\cI )\cdot
\vartheta _{F/\Q }=(t)$ for some $D\in F^*_+$ and $t\in F^*$. Since,
from Theorem \ref{pral}, $A$ admits a principal line bundle, it follows
from Corollary \ref{bij} that the set $\cP (\cO )$ is nonempty. Let
$\mu \in \cP (\cO )$ and let $S=\mbox{diag }_{\sigma :F\hookrightarrow \R }
(\begin{pmatrix}\mbox{sgn }(\mu ^{\sigma }) & 0 \\ 0 & 1\end{pmatrix})$.

Let us fix an immersion $B\hookrightarrow B\otimes _{\Q }
\R \simeq \M _2(\R )\times \stackrel {n}{...}\times \M _2(\R )
\hookrightarrow \M _{2 n}(\R )$, $\beta \mapsto \mbox{diag }_
{\sigma :F\hookrightarrow \R }(\beta ^{\sigma })$, by
embedding diagonally $\M _2(\R )\times
\stackrel {n}{...}\times \M _2(\R )$ in $\M _{2 n}(\R )$.
Upon conjugating the embedding $B\hookrightarrow \M _{2 n}(\R )$ by $S$, we can
assume that $\mbox{sgn}(\mu ) = (1, ..., 1)$. Let $V$ be a
complex vector space of dimension $2 n$ and let $B$ act on $V$ through
the above embedding. Let $\mathfrak H$ denote Poincar\'{e}'s upper half plane. For
any choice of $(\tau _1, ..., \tau _n)$ in a dense
subset of $\mathfrak H\times ...\mathfrak H$,
the complex torus $V/\Lambda $, $\Lambda = \cI \cdot v_0$, $v_0=(\tau _1, 1, ...,
\tau _n, 1)$, is the set of complex points of an abelian variety $A/\C
$ with quaternionic multiplication by $\cO $ such that $H_1(A, \Z )\simeq \cI
$ as left $\cO $-modules. Since conditions $(i)$ and $(ii)$ of Lemma
\ref{porfin} hold for $A$, we deduce from Theorem \ref{index} that $A$
is principally polarizable. $\Box $


\begin{remark}
As a consequence of the above proposition, we
obtain that there exist {\em self-dual but non principally
polarizable abelian varieties}. Indeed, let $F= \Q (\sqrt{3})$ and
let $\cO $ be a maximal order
in the quaternion algebra $B$ of discriminant $\cD =
2 \sqrt {3}$. Let $\cI = \cO $. Since
$\cD $  and $\n (\cI )\cdot \vartheta _{F/\Q }$ are principal ideals but the
former can be generated by no totally real elements, it follows from
Corollary \ref{dual} and
Proposition \ref{polpral} that any abelian variety with quaternionic
multiplication by $\cO $ and such that $H_1(A, \Z )\simeq \cO $ is
self-dual but non principally polarizable. These abelian varieties
indeed exist, as we have shown at the end of Section \ref{AVQM}.
\end{remark}

Let now $A/\C $ be an abelian variety
with quaternionic multiplication by
$\cO $. We choose $\cI $ to be a $\cO $-left ideal such that $H_1(A, \Z )
\simeq \cI $ as $\cO $-left modules and $(\n (\cI ), \cD ) = 1$. We
fix a complex vector space $V$ and a vector $v_0\in V$ such that
$A(\C )=V/\Lambda $ with $\Lambda =\cI \cdot v_0$ under the same
conditions of Lemma \ref{porfin}.

In regard to the Question \ref{Quest}, note that, from
Corollary \ref{dual}, a sufficient condition for $A$ to be
self-dual is that $\cD $ and $\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }$ are
principal ideals. By Corollary \ref{bij} and
Theorem \ref{index}, a necessary condition for $A$ to
be principally polarizable is that $\cD $ be generated by a
totally positive element $D$ in $F$. However, in general these conditions
are not sufficient for the existence of a principal polarization on $A$.


\begin{definition}

We denote by $\Omega \subseteq \{\pm 1\} ^n$ the set of signatures
$$
\Omega = \{ (\mbox{sgn }(\mbox{det  } \nu _{\sigma }(\mu ))\}
_{\mu \in \cP (\cO ) }.
$$

\end{definition}

The set $\Omega $ can be identified with a subset of the set of connected
components of $\R ^n\setminus \cup _{i=1}^n\{x_i = 0\}$. With the
notations as above, we obtain the following corollary of Theorems
\ref{pral} and \ref{index}.


\begin{proposition}
\label{polpral2}
A complex abelian variety $A/\C $
with quaternionic multiplication
by $\cO $ and $H_1(A,\Z )\simeq \cI $
admits a principal polarization if and only if
$\cD $ and $\mathrm{n}(\cI )\cdot \vartheta _{F/\Q }$ are principal ideals,
$\cD =(D)$ is generated by a totally positive element $D\in F^*_+$ and
$(1, ..., 1)\in \Omega $.

\end{proposition}


Signature questions on number fields are delicate. In order to have a
better understanding of Proposition \ref{polpral2}, we describe $\Omega
$ as the union (as sets) of linear varieties in the affine space
$\mathbb A _{\F _2}^n = \{ \pm 1\} ^n$ as follows. Let $\{ u_k\} $ be a
set of representatives of units in $R_F^*/R_F^{*2}$ and, for any order
$S\supseteq R_F[\sqrt {-u_k D}]$ in $L=F(\sqrt {-u_k D})$, choose $\mu
_S\in \cP (S, \cO )$. We considered in Section \ref{poll} the Galois
group $\Delta = \mbox{Ker}(N:\Pic(S)\rightarrow \Pic (F))$. Naturally
associated to it there is a sub-space of signatures $\Sigma (\Delta
)$ in the quotient space $\mathbb A_{\F _2}^n/\Sigma (R_F^*)$ as follows: if
$\mathfrak b$ is an ideal of $S$ such that $N_{L/F}(\mathfrak b)=(b)$ for
some $b\in F^*$, the signature of $b$ does not depend on the
class of $\mathfrak b$ in $\Pic (S)$ and only depends on the choice of
the generator $b$ up to signatures in $\Sigma (R_F^*)$. By an abuse of
notation, we still denote by $\Sigma (\Delta )$ the sub-space of
$\mathbb A^n_{\F _2}$ generated by $\Sigma (R_F^*)$ and the signatures
of the norms of ideals in $\Delta $. Then, from Proposition
\ref{Galois} we obtain that

$$
\Omega = \bigcup _{k, S} \Sigma (\Delta )\cdot \mathrm{sgn } (\mu _S).
$$
This allows us to compute $\Omega $ in many explicit examples and to show
that, in many cases, the set $\Omega $ coincides with the whole space of signatures
$\{ \pm 1\} ^n$. The following corollary, which remains valid even if
we remove the assumption
$(\vartheta _{F/\Q }, \cD ) = 1$, illustrates this fact.

\begin{corollary}
\label{cor}

Assume that

\begin{enumerate}

\item[(i)]
$\cD = (D)$ for some $D\in F^*_+$ and that
$\n (\cI )\cdot \vartheta _{F/\Q } = (t)$ for some $t\in F^*$.

\item[(ii)]
$h_+(F)=h(F)$.
\end{enumerate}

Then, any abelian variety $A/\C $ with quaternionic multiplication by
$\cO $ and $H_1(A, \Z )\simeq \cI $ is principally polarizable.

In particular, if $h_+(F)=1$, then the above two conditions on $\cO $ and $\cI
$ are always accomplished.

\end{corollary}

{\em Proof. } Since $\Sigma (R_F^*) = \{ \pm 1\} ^n$, we have that
$\Omega =\{ \pm 1\} ^{n}$. The result follows from
Proposition \ref{polpral2}. $\Box $

As we shall see in Chapter \ref{GSV}, this
is highly relevant in the study of certain {\em Shimura
varieties}. As was already known to the specialists
in dimension $2$, we obtain that
{\em any} abelian surface with quaternionic multiplication by a
maximal order in an indefinite quaternion algebra $B/\Q $
admits a principal polarization. It can actually be shown
that any abelian surface with multiplication by an hereditary quaternion
order admits a principal polarization (cf.\,\cite{Ro2}). This extends our
result to a wider class of quaternion orders.

\markright{\ref{Sectionpio} Isomorphism
classes of principal polarizations}
\section{The number of isomorphism classes of
principal polarizations}\markright{\ref{Sectionpio} Isomorphism
classes of principal polarizations}\label{Sectionpio}


Let $A$ be a complex abelian variety with quaternion multiplication by a
maximal order $\cO $. Let $V$ be a complex vector space of dimension $2 n$ and let
$B\hookrightarrow \osum _{i=1}^n\M _2(\R )\subset \M _{2 n}(\R )$ act on
$V$ diagonally in $2\times 2$ boxes as in Lemma \ref{porfin}
in Section \ref{AVQM}. Let
$v_0 = (\tau _1, 1, ..., \tau _n, 1)\in V$, $\mbox{Im }(\tau _i)>0$, be such that
$A(\C )=V/\Lambda $ for $\Lambda = \cI \cdot v_0$.
As they were introduced in Section \ref{AV}, for any
integer $0\leq i\leq g$, the set $\Pi _i(A)$ denotes the set
of isomorphism classes of principal invertible sheaves  $\cL \in
\mathrm{NS}(A)$ of index $i(\cL )=i$. The set $\Pi (A)$ naturally splits
as the disjoint union $\Pi (A)= \sqcup \Pi _i(A)$. Moreover, we recall that
due to the relation $i(\cL ) + i(\cL ^{-1}) = g$, the map $\cL \mapsto \cL ^{-1}$
induces a one-to-one correspondence between $\Pi _i(A)$ and $\Pi
_{g-i}(A)$ and therefore $\pi _i(A)=\pi _{g-i}(A)$.

Formulas for $\pi _i(A)$, $0\leq i\leq g$, analogous to that of
Theorem \ref{pi} can be derived. Due to its significance, we will only
concentrate on the number $\pi _0(A)$
of classes of principal polarizations. The Galois action on the
sets $E (S, \cO )$
of Eichler classes of optimal embeddings and its behaviour respect to
the index of the associated invertible sheaves  will play an important role.

Assume then that $\Pi _0(A)\not =\emptyset $. For
simplicity, recall that we also assume that $(\vartheta
_{F/\Q }, \cD )=1$. By Proposition \ref{polpral2}, we may choose $D\in
F^*_+$ and $t \in F^*$ such that $\cD =(D)$ and $\mathrm{n}(\cI )\cdot
\vartheta _{F/\Q }=(t )$. With these notations, we introduce the
following

\begin{definition}

Let $u\in R^*_{F +}$ be a totally positive unit. An order
$S\supseteq R_F[\sqrt {-u D}]$ is
{\em ample} respect to $A$ if
there exists an optimal embedding $i:S\hookrightarrow \cO $ such that
$\mu =i(\sqrt {-u D})$ is ample. We define $\cS _u$ to be the set of ample orders
$S\supseteq R_F[\sqrt {-u D}]$ in $F(\sqrt {-u D})$.

\end{definition}

The existence of a principal polarization $\cL $ on $A$ implies that
there is at least some $\cS _u$ nonempty. With these notations, we obtain the
following expression for $\pi _0(A)$ in terms of the
narrow class number of $F$ and the class numbers of certain CM-fields
that embed in $B$.

\begin{theorem}
\label{pio}

The number of isomorphism classes of principal polarizations on $A$ is

$$
\pi _0(A) = \frac {1}{2 h_+(F)}\sum _{u\in R_{F_+}/R_F^{*2}} \sum _{S\in \cS _u}
 2^{e^+_S}  h(S),
$$
where $2^{e_S^+} = |R_{F +}^*/N(S^*)|$.
\end{theorem}

{\em Proof. } By the existing duality
between $\Pi _0(A)$ and $\Pi _g(A)$, it is equivalent to show that
$\pi _0(A) + \pi _g(A) = \sum _u \sum _{S\in \cS _u} 2^{e^+_S}
h(S)/h_+(F)$.

Let us introduce the set
$\cP _{0, g}(\cO )  = \{ \mu \in \cO : \n (\mu )\in R^*_{F +}\cdot D,
\mathrm{sgn } (\mu ) = \pm (1, ..., 1)\} $.
By Theorems \ref{NeSe} and
\ref{index}, the set $P _{0, g}(\cO )=
\cP _{0, g}(\cO ) ^{\pm }/_{\sim
_p}$ is in one-to-one correspondence with $\Pi_0(A)\cup \Pi _g(A)$
and we have a natural decomposition $P _{0, g}(\cO ) =
\sqcup P _{0, g}(S, \cO )$ as $S$ runs through ample orders in $\cS _u$
and $u\in R^*_{F +}/R_F^{*2}$.

Fix $u\in R^*_{F +}$ and $S$ in $\cS _u$. In order to compute the
cardinality of $P _{0, g}(S, \cO )$, we relate it to the
set $E_{0, g}(S, \cO )=\cP _{0, g}(S, \cO )/_{\sim _{e}}$ of
$\cO ^*_{\pm }$-Eichler conjugacy
classes of optimal embeddings $i_{\mu
}:S\hookrightarrow \cO $. Here, we agree to say that
two quaternions $\mu _1$ and $\mu _2\in \cP _{0, g}(S, \cO )$
are Eichler conjugate by $\cO ^*_{\pm }$ if there is
a unit $\alpha \in
\cO ^*_{\pm }$ of either totally positive or totally negative reduced norm
such that $\mu _2 = \alpha ^{-1} \mu _1\alpha $. Note that, by
Theorem \ref{index}, the action of $\cO ^*_{\pm }$-conjugation
on the invertible sheaf $\cL $ associated to an element in $\cP (S, \cO ) $
either preserves the index $i(\cL )$ or switches it to $g-i(\cL
)$. This makes sense of the quotient $E _{0, g}(S, \cO )$.

We have the following exact diagram:

$$
\begin{matrix}

0   &\ra & \Delta _+   &  \ra   &  \Pic (S)   & \stackrel {N_{L/F}}{\ra } & \Pic
_+(F)  &  \ra  &  0  \\
      &    &    \downarrow   &  &  \parallel   &   &   \downarrow
      &   & \\


 0    & \ra  &  \Delta    &  \ra   &  \Pic (S)   & \stackrel {N_{L/F}}{\ra } &
 \Pic (F)  &  \ra  &  0.

\end{matrix}
$$

Indeed, there is a natural map $\Pic (S)\stackrel
{N_{L/F}}{\rightarrow } \Pic _+(F)$, since the norm of an element
$a+b\sqrt {-u D}\in L$ for $a$, $b\in F$ is $a^2+u b^2D\in F^*_+$.
The surjectivity of the map $\Pic (S)\rightarrow
\Pic _+(F)$ is argued as in Section \ref{poll} by replacing the Hilbert class
field $H_F$ of $F$ by the big Hilbert class field $H^+_F$, whose
Galois group over $F$ is $\mathrm{Gal} (H_F^+/F)=\Pic _+(F)$.
By Proposition \ref{Galois}, $\Delta $ acts freely and transitively
on $E (S, \cO )$. Therefore, by Theorem \ref{index},
there is also a free
action of $\Delta _+$ on $E _{0, g}(S, \cO )$. Up to sign,
the $\cO ^*_{\pm }$-Eichler conjugation class of an
element $\mu \in \cP (S, \cO )$ has a well defined signature
$\pm \mathrm{sgn } (\mu)$. Note also that two inequivalent $\cO
^*_{\pm }$-Eichler classes that fall in the same $\cO
^*$-conjugation class never have the same signature, even
not up to sign. Taken together, this shows that $\Delta _+$ also
acts transitively on $E _{0, g}(S, \cO )$.
This means that

$$
|E _{0, g}(S, \cO )| = \frac {h(S)}{h_+(F)}.
$$

Arguing as in Section \ref{lineEichler}, Theorem \ref{pio} follows. $\Box $

\markright{\ref{exxx} Examples in
low dimensions}
\section{Examples in low dimensions}\markright{\ref{exxx} Examples in
low dimensions}\label{exxx}

\subsection{Dimension $2$}

The simplest examples to be considered are abelian surfaces with
quaternionic multiplication. Let $B$ be an indefinite division
quaternion algebra over $\Q $ of discriminant $D = p_1\cdot ...\cdot
p_{2r}$, for $p_i$ prime numbers and $r\geq 1$. Let $\cO \subset B$ be
a maximal order in $B$. Since $h(\Q )=1$,
$\Pic _{\ell }(\cO )$ is trivial by Proposition \ref{pic}. Thus,
all left $\cO $-ideals are
principal. Moreover, the maximal order $\cO $ is unique up to
conjugation.

By Corollary \ref{cor}, any abelian surface $A$ with $\End (A) = \cO $
admits a principal polarization. In fact, $A$ admits principal line
bundles of each index $0$, $1$ and $2$. In order to compute $\pi _i(A)$
for $i=0, 1$ and $2$ we may use Theorems \ref{pi} and \ref{pio}.

Let $d\in \Z \setminus \Z ^2$. As is well known, the quadratic
orders in $\Q (\sqrt {d})$ containing $\Z [\sqrt {d}]$ are
$\Z [\sqrt {d}] \mbox{ and }\Z [\frac {1+\sqrt {d}}{2}] \mbox{ if
}d\equiv 1 (\mbox{ mod }4)$ or $\Z [\sqrt {d}]\mbox{ if }d\not \equiv
1 (\mbox{ mod }4)$. Furthermore, for $S = \Z [\sqrt {d}] \mbox{ or }\Z
[\frac {1+\sqrt {d}}{2}]$,

$$
e_S =
  \begin{cases}
    0 & \text{if } d>0 \text{ and there are units of negative norm in }S, \\
    1 & \text{otherwise}.
  \end{cases}
$$
and $e_S^+ = 0 \mbox{ in any case.}$ We conclude that

$$
\pi _0(A)=\pi _2(A)=
  \begin{cases}
    \frac {h(-4 D)+h(-D)}{2} & \text{ if }D\equiv 3\quad (\mbox{ mod }4), \\
    \frac {h(-4 D)}{2} & \text{otherwise}
  \end{cases}
$$
and
$$
\pi _1(A) =
  \begin{cases}
    \varepsilon _{4 D} h(4 D)+ \varepsilon _D h(D) &
    \text{ if }D\equiv 1\quad (\mbox{ mod }4), \\
    \varepsilon _{4 D} h(4 D) & \text{otherwise,}
  \end{cases}
$$
where $\varepsilon _{D}$, $\varepsilon _{4 D} = 1$ or $\frac {1}{2}$ is
computed from the above formula for $e_S$.

As a simple example, the number of isomorphism classes of principal
polarizations on an abelian surface $A$ with QM by a maximal order in a
quaternion algebra of discriminant $D = 2\cdot 3\cdot 5\cdot 7\cdot
11\cdot 13\cdot 17\cdot 19$ is  $\pi _0(A)=1040$. This
also implies the existence of $1040$ pair-wise nonisomorphic smooth
algebraic curves $\{C_1, ..., C_{1040}\}$ of genus $2$ such that their
respective Jacobian varieties are isomorphic as abelian surfaces:
$J(C_1)\simeq \cdot \cdot \cdot \simeq J(C_{1040}).$

Using a programming package like PARI (cf.\,\cite{PA}) and our results, we
obtain the following table for
the numbers of isomorphism classes of principal invertible sheaves  of index
0, 1 and 2 respectively on an abelian surface with QM by a maximal
order in a quaternion algebra of discriminant D. Recall that $\pi
(A)=\pi _0(A)+\pi _1(A)+\pi _2(A)$ and that $\pi _0(A)=\pi _2(A)$.




\begin{table}\label{praline}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline \multicolumn{9}{|c|}{\bfseries Table \ref{praline}:
Principal invertible sheaves on abelian surfaces with QM}\\

\hline $\mbox{Disc }(B)$&$\pi _0(A)$&$\pi _1(A)$&$\pi _2(A)$&
&$\mbox{Disc }(B)$&$\pi _0(A)$&$\pi _1(A)$&$\pi _2(A)$ \\ \hline \hline
$6$  & $1$ & $1$ & $1$ &      & $111$ & $8$ & $2$ & $8$           \\
$10$ & $1$ & $1$ & $1$ &      & $115$ & $4$ & $2$ & $4$        \\ $14$
& $2$ & $1$ & $2$ &      & $118$ & $3$ & $1$ & $3$              \\ $15$
& $2$ & $2$ & $2$ &      & $119$ & $10$ & $2$ & $10$
\\ $21$ & $2$ & $2$& $2$ &       & $122$ & $5$ & $1$ & $5$
\\ $22$ & $1$ & $1$ & $1$ &      & $123$ & $4$ & $2$ & $4$
\\ $26$ & $3$ & $1$ & $3$ &      & $129$ & $6$ & $2$ & $6$
\\ $33$ & $2$ & $2$ & $2$ &      & $133$ & $2$ & $2$ & $2$
\\ $34$ & $2$ & $2$ & $2$ &      & $134$ & $7$ & $1$ & $7$
\\ $35$ & $4$ & $2$ & $4$ &      & $141$ & $4$ & $4$ & $4$         \\
$38$ & $3$ & $1$ & $3$ &      & $142$ & $2$ & $3$ & $2$              \\
$39$ & $4$ & $2$ & $4$ &      & $143$ & $10$ & $2$ & $10$
\\ $46$ & $2$ & $1$ & $2$&       & $145$ & $4$ & $4$ & $4$
\\ $51$ & $4$ & $2$ & $4$&       & $146$ & $8$ & $2$ & $8$         \\
$55$ & $4$ & $2$ & $4$ &      & $155$ & $8$ & $2$ & $8$              \\
$57$ & $2$ & $2$ & $2$ &      & $158$ & $4$ & $1$ & $4$       \\ $58$ &
$1$ & $1$ & $1$&       & $159$ & $10$ & $2$ & $10$      \\ $62$ & $4$ &
$1$ & $4$ &      & $161$ & $8$ & $2$ & $8$      \\ $65$ & $4$ & $2$ &
$4$ &      & $166$ & $5$ & $1$ & $5$              \\ $69$ & $4$ & $2$ &
$4$ &      & $177$ & $2$ & $2$ & $2$      \\ $74$ & $5$ & $1$ & $5$ & &
$178$ & $4$ & $2$ & $4$           \\ $77$ & $4$ & $2$ & $4$ &      &
$183$ & $8$ & $2$ & $8$              \\ $82$ & $2$ & $2$ & $2$  &     &
$185$ & $8$ & $2$ & $8$   \\ $85$ & $2$ & $2$ & $2$ &      & $187$ &
$4$ & $2$ & $4$                \\ $86$ & $5$ & $1$ & $5$  &     & $194$
& $10$ & $2$ & $10$    \\ $87$ & $6$ & $2$ & $6$ &      & $201$ & $6$ &
$2$ & $6$         \\ $91$ & $4$ & $2$ & $4$ &      & $202$ & $3$ & $1$
& $3$               \\ $93$ & $2$ & $2$ & $2$  &     & $203$ & $8$ &
$2$ & $8$        \\ $94$ & $4$ & $1$ & $4$  &     & $205$ & $4$ & $4$ &
$4$     \\ $95$ & $8$ & $2$ & $8$  &     & $206$ & $10$ & $1$ & $10$
\\ $106$ & $3$ & $1$ & $3$ &     & $209$ & $10$ & $2$ & $10$  \\

\hline
\end{tabular}
\end{table}

\newpage

\subsection{Dimension $4$}

Let $F$ be the real quadratic field $\Q (\sqrt {2})$. Its ring of
integers is $\Z [\sqrt {2}]$ and all its ideals are principal and can
be generated by totally positive elements. That is, the narrow class
number of $F$ is $h_+(F)=1$. The group of units is $\Z [\sqrt {2}]^* =
\{ \pm (1+\sqrt {2})^n\}_{n\in \Z }$. Representatives for
$R_F^*/R_F^{*2}$ are $\{ \pm 1, \pm (1+\sqrt {2})\}$.

Let $B$ be the quaternion algebra over $F$ that ramifies exactly at the
two prime ideals $(3\pm \sqrt {2})$ above $7$. By the
second remark to Theorem \ref{pi}, the orders $S_{\pm} = \Z
[\sqrt {2} ,\sqrt {\pm 7 (1+\sqrt {2} )}]$ are the maximal rings of
integers in their respective quotient fields. With the help of the
programming package PARI we have that the class number
is $h=2$ and $|\Z [\sqrt {2}]^*/N(S_{\pm}^*)|=4$ in both cases.

A similar inspection yields that the conductor of the $\Z [\sqrt {2}
,\sqrt {7}]$ in its quotient fields is the dyadic ideal $\mathfrak f =
(\sqrt {2} )$. This means that the maximal ring of integers properly
contains this order but also that there are no intermediate orders
between them.

The ring of integers of $\Q (\sqrt {2} ,\sqrt {7})$ is $\widetilde S = \Z
[\sqrt {2} , \frac {3+\sqrt {7}}{\sqrt {2} }]$ and its class number is
$h(\widetilde S)=1$. Generators of the unit group, up to torsion, are
$1+\sqrt {2} $, $\frac {3+\sqrt {7}}{\sqrt {2}}$ and $2 \sqrt {2}+\sqrt
{7}$. Computing their norm over $\Q (\sqrt {2})$ we obtain that $|\Z
[\sqrt {2}]^*/N(\widetilde {S}^*)|=4$.

We can compute the class number of $S = \Z [\sqrt {2}, \sqrt {7}]$
in terms of that of the field $\Q (\sqrt {2} ,\sqrt {7})$
by the formula given in Section \ref{NF}. We have that
$h(S)=1$ and $|\Z [\sqrt {2}]^*/N(S^*)|=4$.

Finally, the ring of integers of $\Q (\sqrt {2}, \sqrt {-7})$ is
$\widetilde S = \Z [\sqrt {2}, \frac {1-2 \sqrt {2}-\sqrt {-7}}{2}]$ and
the conductor of $S = \Z [\sqrt {2}, \sqrt {-7}]$ is $\mathfrak f = 2$.
Hence there is exactly one order $\hat S$ that sits between $S$ and
$\widetilde S$. We have that $\widetilde {S}^* = \{ \pm (1+\sqrt {2})^n\}
_{n\in \Z }$, $h(S)=h(\hat S)=h(\widetilde S)=2$ and $e_S = e_{\hat S} =
e_{\widetilde S} = 2$.

By applying Theorems \ref{pi}, \ref{index} and \ref{pio}, we conclude
that for any abelian variety $A$ of dimension $4$ such that $\End (A)$
is a maximal order in the quaternion algebra $B/\Q (\sqrt {2})$ of
discriminant $7$, the number of isomorphism classes of principal line
bundles of index 0, 1, 2, 3 and 4 respectively are:

\newpage
\begin{table}
\begin{tabular}{|l|l|l|l|l|}
\hline


$\pi _0(A)$&$\pi _1(A)$&$\pi _2(A)$& $\pi _3(A)$&$\pi _4(A)$ \\ \hline
\hline $6$ & $4$ & $4$ & $4$ & $6$              \\
\hline
\end{tabular}
\end{table}

\newpage

\markright{\ref{asym} Asymptotic behaviour of $\pi _0(A)$}
\section{Asymptotic behaviour of $\pi _0(A)$}
\markright{\ref{asym} Asymptotic behaviour of $\pi _0(A)$}\label{asym}

In this section, we combine Theorem \ref{pio} with analytical tools to estimate
the asymptotic behaviour of $\log (\pi _0(A))$. This will yield a
stronger version of part $1$ of Theorem \ref{ass} in the introduction.

For any number field $L$, we let $D_L$ and $\mathrm{Reg }_L$ stand for
the absolute value of the discriminant and the regulator, respectively.
For any two sequences of real numbers $\{ a_n\} _{n\geq 1}$ and
$\{ b_n\} _{n\geq 1}$, we write $\{ a_n\} \sim \{ b_n\} $ if
and only if $\mathrm{lim }_{n\rightarrow \infty }\dfrac {a_n} {b_n} = 1$.


\begin{theorem}
\label{log}

Let $F$ be a totally real number field of degree $n$. Let $A$
range over a sequence of principally polarizable abelian varieties of
dimension $2n$
with $\End (A)$ a maximal order in a quaternion algebra $B/F$ of
discriminant $D\in F^*_+$ with $|\mbox{N}_{F/\Q }(D)|\rightarrow \infty
$. Then

$$
\log \pi _0(A)\sim \log \sqrt {|\mbox{N}_{F/\Q }(D)|\cdot D_F}.
$$

\end{theorem}


The proof of Theorem \ref{log} adapts an argument of Horie-Horie
(\cite{HoHo}) on estimates of relative class numbers of CM-fields. We
first show that there indeed exist families of abelian
varieties satisfying the properties quoted in the theorem.

By \v{C}ebotarev's Density Theorem, we can find
infinitely many pairwise different totally positive principal
prime ideals $\{\wp _i\}_{i\geq 1}$ in $F$. We can also choose them such
that $(\wp _i, \vartheta _{F/\Q })=1$. We then obtain principal
ideals $(D_j) = \wp _1\cdot \wp _2\cdot ...\cdot \wp _{2 j -
1}\cdot \wp _{2 j}$ with $D_j\in F^*_+$ and $(D_j, \vartheta
_{F/\Q })=1$.

According to \cite{Vi}, p.\,74, there exists a
totally indefinite quaternion algebra $B_j$ over $F$ of
discriminant $D_j$ for any $j\geq 1$. Then, Proposition \ref{polpral}
asserts that there
exists an abelian variety $A_j$ of dimension $2n$ such that $\End
(A_j)$ is a maximal order in $B_j$ and $\Pi _0(A_j)\not =
\emptyset $.

\vspace{0.5cm}
{\em Proof of Theorem \ref{log}.} Let $A$ be a principally polarizable
abelian variety with quaternion multiplication by
a maximal order in a totally indefinite division quaternion algebra $B$ over $F$ of
discriminant $D\in F^*_+$.

For any totally positive
unit $u_k\in R^*_{F +}$, let $L_k= F(\sqrt
{-u_k D})$. For any order $S\supseteq R_F[\sqrt {-u_k D_j}]$ in the CM-field
$L_k$, it holds that $h(S) = c_S h(L_k)$ for some positive constant $c_S\in \Z $
which is uniformly bounded by $2^n$.

The class number $h(F)$ turns out to divide $h(L_k)$
and the {\em relative class number} of $L_k$ is
defined to be $h^-(L_k) = h(L_k)/h(F)$ (cf.\,\cite{Lo00}). Since
$h_+(F) = 2^m h(F)$ for
$m = n - \mathrm{dim }_{\F _2}(\Sigma (R_F^*))$, Theorem
\ref{pio} can be rephrased as $\pi _0(A) = \sum 2 ^{(e^+_S
- 1 -m)} c_S h^-(L_k)$.

In order to apply the Brauer-Siegel Theorem, the key point is to relate
the several absolute discriminants $D_{L_k}$ and regulators $\mathrm{Reg } _{L_k}$
as $u_k$ vary among totally positive units in $F$.

Firstly, we have the relations
$D_{L_k} = |N_{F/\Q }(D_{{L_k}/F})\cdot D_F^2| = 2^{p_k} |N_{F/\Q }(D)| D_F^2$
for some $0\leq p_k \leq 2 n$. Secondly, by \cite{Wa}, p.\,41, it holds that
$\mathrm {Reg }_{L_k}=2^{c} \mathrm {Reg }_F$ with $c=n-1$ or $n-2$.

Let $\varepsilon $ be a sufficiently small
positive number. By the Brauer-Siegel Theorem, it holds that
$D_{L_k}^{(1-\varepsilon )/2}\leq h(L_k)\mathrm {Reg }_{L_k}\leq
D_{L_k}^{(1+\varepsilon )/2}$ for $D_{L_k}\gg 1$. Thus
$$
\frac{D_F^{(1-\varepsilon )/2}}{h(F)\mathrm {Reg }_F} (D_{L_k}/D_F)
^{(1-\varepsilon )/2}\leq h^-(L_k)\leq
\frac{D_F^{(1+\varepsilon )/2}}{h(F)\mathrm {Reg }_F} (D_{L_k}/D_F)
^{(1+\varepsilon )/2}.
$$
Fixing an arbitrary CM-field $L$ appearing in the expression for $\pi
_0(A)$, this boils down to
$$
C_-\cdot \frac{D_F^{(1-\varepsilon )/2}}{h(F)\mathrm {Reg }_F} (D_{L}/D_F)
^{(1-\varepsilon )/2}\leq \pi _0(A)\leq
C_+\cdot \frac{D_F^{(1+\varepsilon )/2}}{h(F)\mathrm {Reg }_F} (D_{L}/D_F)
^{(1+\varepsilon )/2}
$$
for some positive constants $C_-$ and $C_+$. Taking logarithms,
these inequalities yield Theorem \ref{log}. $\Box $

\begin{remark}

The argument above is not effective since it relies on the
classical Brauer-Siegel Theorem on class numbers. However,
recent work of Louboutin (\cite{Lo00}, \cite{Lo}) on lower and upper
bounds for relative class numbers of CM-fields, based upon estimates of residues at
$s=1$ of Dedekind zeta functions, could be used to obtain explicit lower and
upper bounds for $\pi _0(A)$.
\end{remark}

Finally, we conclude this section with the proof of the second main
result of this chapter quoted in Section \ref{mainres}.

$\\ $
{\em Proof of Theorem \ref{ass}. } Part $1$ is an immediate consequence
of Theorem \ref{log}. Let us explain how part $2$ follows. Assume that $A$
is a simple complex abelian variety of odd and square-free
dimension $g$. Then, by Theorem \ref{Albert},
$\End (A)\simeq S$ is an order in
either a totally real number field $F$ or a CM-field $L$ over a
totally real number field $F$. In any case, $[F:\Q ]\leq g$.

In the former case, by Theorem 3.1 of Lange in \cite{La}, $\pi
_0(A)=|S^*_+/S^{*2}|\leq 2^{g-1}$. In the latter, let $S_0\subset F$
be the subring of $S$ fixed by complex conjugation. If $\cL $
is a principal polarization on $A$,
the Rosati involution precisely induces complex
conjugation on $\End (A)\simeq S$ and we have that $\pi _0(A)=
|S_{0 +}^*/\Norm _{L/F}(S^*)|\leq |S_{0 +}^*/S_0^{*2}|\leq 2^{g-1}$, by
applying Theorem 1.5 of \cite{La}. $\Box $


\chapter{Shimura varieties and forgetful maps}
\label{GSV}


\section*{Introduction}\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{GSV}. Shimura varieties and forgetful maps}{\ref{Intro2}
Introduction}\label{Intro2}

\sk\noindent

Let $F$ be a totally real number field of degree $[F:\Q ]=n$ and
let $B$ be a totally indefinite quaternion algebra over $F$. In
this chapter we will be concerned with certain Shimura varieties $X_B$ attached
to an arithmetic datum arising from $B$ and several
maps that occur naturally between them.

As complex manifolds, these varieties can be described throughout
quotients of certain bounded symmetric domains by arithmetic
groups acting on them and, by the theory of Baily-Borel
(cf.\,\cite{BaBo}), they become quasi-projective complex algebraic
varieties. Shimura discovered a moduli interpretation of these
varieties which allowed him to construct canonical models
$X_B/\Q $ over the field $\Q $ of rational numbers. Shimura also
explored their arithmetic, showing that the coordinates of
so-called {\em Heegner points} on $X_B$ generate certain class
fields and that the Galois action on them can be described by
explicit reciprocity laws (cf.\,\cite{Sh63} and \cite{Sh67}).

The nature of the Shimura varieties $X_B$ differs notably
depending on the existence or absence of zero divisors in $B$.
When $B$ is the split algebra $\mathrm {M}_2(F)$, the varieties
$X_B$ are classically called {\em Hilbert modular
varieties}. These are not complete and suitable compactifications
of them can be constructed, though at the cost of producing new
singularities. The literature on them is abundant, especially on
the low dimensional cases. In dimension $1$, these are called
modular curves and have become crucial in many aspects of number
theory. In dimension $2$, a reference to Hilbert modular surfaces
is van der Geer's book \cite{vdGe}.

On the other hand, when $B$ is nonsplit, that is, it is a
division algebra, then the emerging Shimura varieties $X_B/\Q $
are already projective. This fact makes the study of their
arithmetic highly difficult since, in the Hilbert modular case,
much of it is encoded in the added cusps. In remarkable contrast
to Hilbert modular varieties, Shimura proved that, when $B\not
\simeq \M _2(F)$, $X_B$ do not have real points and therefore
do not have rational points over any number field that admits a
real embedding. In the last years, there has been increasing
interest on Shimura curves arising from rational indefinite
quaternion algebras, since they play a crucial role in modularity
questions (cf.\,\cite{HaHaMo} or \cite{Ri90}, for instance).

From the modular point of view, there are natural maps

$$
\pi : X_B\rightarrow \mathcal A _{g, (d_1, ..., d_g)}
$$
from $X_B$ into the moduli space of polarized abelian
varieties that involve {\em forgetting} some additional
structures. Further, for any totally real field $L$ containing $F$
and embedded in $B$, these maps factorize through a morphism $\pi
_L: X_B\rightarrow \mathcal H _L$ into a Hilbert
modular variety (cf.\,Section \ref{ShimuraVarieties}).

It is the purpose of this chapter to describe in detail the nature of
these morphisms and their image in the several moduli spaces of
abelian varieties. As we will see, $\pi $ and $\pi _L$ are finite
maps onto their image and the mere computation of their degree turns out to be
unexpectedly subtle. It is based on arithmetic questions on $B$
that were recently studied by Chinburg and Friedman in a series of
papers in connection with arithmetic hyperbolic 3-orbifolds (cf.
\cite{ChFr86}, \cite{ChFr00}).

In Section \ref{ShimuraVarieties}, we review the construction due to Shimura of the
varieties we will be dealing with and the above mentioned maps. In
Section \ref{Atkin}, we introduce several Atkin-Lehner groups of
automorphisms acting on them and describe their modular
interpretation. Particularly, in
Section \ref{twstable} we introduce what we call the {\em
stable} and {\em twisting Atkin-Lehner groups}.

We state the main result of the chapter as Theorem \ref{mainGSV} in Section
\ref{theorems} and we devote Section \ref{proofs} to present its proof.

In closing this introduction, we point out two different
applications of the results presented in this chapter. The first one is
pursued in Section \ref{qlocus} and concerns the geometry of the
quaternionic locus $\cQ _{\cO }$ of abelian varieties
admitting multiplication by a maximal order $\cO $ in the moduli
space $\cA _g$ of principally polarized abelian varieties of even
dimension $g$. By means of Theorem \ref{mainGSV} and Eichler's theory
on optimal embeddings, the number of irreducible components of
$\cQ _{\cO }$ can be related to certain class numbers and
conditions can be given for its irreducibility. We also refer
the reader to \cite{Ro5} for an exposition of these results.

Secondly, Theorem \ref{mainGSV} can also be used to explore the
arithmetic of abelian varieties with quaternionic multiplication. Indeed, in
Section \ref{FM}, we
investigate the field of moduli $k_B$ of the quaternion
multiplication on a principally polarized abelian variety $(A, \cL )/\qbar $. This
is a fundamental arithmetic invariant of the $\qbar $-isomorphism class of
$(A, \cL )$ which is in intimate relationship with the possible fields of
definition of $(A, \cL )$.

The main results developed in this chapter can be found in \cite{Ro3} and
\cite{Ro4}.

\markright{\ref{ShimuraVarieties} Shimura varieties}
\section{Shimura varieties}
\markright{\ref{ShimuraVarieties} Shimura varieties}
\label{ShimuraVarieties}


Let $F$ be a totally real number field of degree $n$ and let
$B$ be a totally
indefinite quaternion algebra over $F$. Let $\mathcal D = \mathrm{disc}(B) =
\mathfrak p_1\cdot ...\cdot \mathfrak p_{2r}$.

Fix a datum $( \cO , \cI , \varrho )$ consisting of a maximal
order $\cO $ in $B$, a left $\cO $-ideal $\cI $, or rather its
class in $\mathrm{Pic}_{\ell }(\cO )$, and a positive anti-involution
$\varrho = \varrho _{\mu }: B\ra B$, $\beta \mapsto
\beta ^{\varrho } = \mu ^{-1} \overline {\beta } \mu $,
with $\mu \in \cO $, $\mu ^2+\delta =0$, $\n (\mu )=\delta
\in F^*_+$. We will refer to $( \cO , \cI , \varrho )$ as a
{\em quaternionic datum attached to $B$}.

Attached to the datum $( \cO , \cI , \varrho )$ there
is the following moduli problem over $\Q $: classifying
isomorphism classes of triplets $(A, \iota , \cL )$ given by

\begin{itemize}
\item

An abelian variety $A$ of dimension $g=2n$.

\item

A ring homomorphism $\iota :\cO \hookrightarrow \mbox{ End }(A)$
such that $H_1(A, \Z )$, regarded as a left $\cO $-module, is
isomorphic to the left ideal $\cI $.

\item
A pri\-mitive pola\-riza\-tion $\cL $ on $A$ such that the Rosa\-ti involu\-tion
$\circ :\mathrm{End }^0(A)\rightarrow \mathrm{End }^0(A)$ with
respect to $\cL $ on $\mathrm{End }^0(A) = \mathrm{End }(A)\otimes _{\Z }\Q $
coincides with $\varrho $ when restricted to $\iota (\cO )$:

$$
\circ |_{\iota (\cO )}=\varrho \cdot \iota .
$$

\end{itemize}

A triplet $(A, \iota , \cL )$ will be referred to as a polarized abelian variety
with multiplication by $\cO $. Two triplets $(A_1, \iota _1, \cL
_1)$, $(A_2, \iota _2, \cL _2)$ are isomorphic if there exists an
isomorphism $\alpha \in \Hom (A_1, A_2)$ such that $\alpha \iota
_1 (\beta ) = \iota _2(\beta ) \alpha $ for any $\beta \in \cO $
and $\alpha ^* (\cL _2) = \cL _1\in \NS (A_1)$.
Remark also that, since a priori there is no canonical
structure of $R_F$-algebra on $\End (A)$, the immersion $\iota
:\cO \hookrightarrow \End (A)$ is just a homomorphism of rings.

As it was proved by Shimura, the
corresponding moduli functor is coarsely represented by an
irreducible and reduced quasi-projective scheme $X_B/\Q = X_{(\cO
, \cI , \varrho )}/\Q $ over $\Q $ and of dimension $n=[F: \Q ]$.
Moreover, if $B$ is division (that is, if $r>0$), the Shimura variety
$X_B$ is complete (cf.\,\cite{Sh63}, \cite{Sh67}).

Complex analytically, the manifold $X_B(\C )$ can be described as
the quotient of a symmetric space by the action of a discontinuous
group as follows. Since $B$ is totally indefinite, we may fix an
embedding $B\hookrightarrow B\otimes _{\Q }\R \stackrel{\sim }
{\rightarrow }\mbox{M}_2(\R )\oplus ...\oplus \M _2(\R )$ and
regard the group

$$
\Gamma _B = \cO
^1 = \{\gamma \in \cO ^*: \mbox{n}(\gamma )=1\}
$$
as a discrete subgroup of $\mbox{SL}_2(\R )^n$. An
element $\gamma = (\gamma _1, ..., \gamma _n)\in \Gamma _B$ acts on
the cartesian product ${\bf \mathfrak H}^n$ of $n$ copies of Poincar\'{e}'s
upper half plane ${\bf \mathfrak H}= \{ x+y i: x, y\in \R , y>0\} $ by
Moebius transformations:

$$
\gamma \cdot (\tau _1, ..., \tau _n)^t =
(\frac{a_1 \tau _1+b_1}{c_1
\tau _1+d_1}, ..., \frac{a_n \tau _n+b_n}{c_n \tau _n+d_n})^t
$$
where
$\gamma _i = \begin{pmatrix}
  a_i & b_i \\
  c_i & d_i
\end{pmatrix}\in \mathrm{SL}_2(\R )$. Then

$$
\Gamma _B\backslash \mathfrak H^n\simeq X_B(\C ).
$$

\section{Forgetful maps into $\cA _g$ and the Hilbert modular varieties
$\cH _S$ and $\cH _F$}\markright{\ref{Shimuramaps} Forgetful maps}
\label{Shimuramaps}

It is the purpose of this section to describe several natural maps that occur
between the Shimura varieties $X_B$, the moduli spaces of
polarized abelian varieties and several Hilbert
modular varieties $\cH _S$ and $\cH _F$.

The following notion will be convenient for us in
order to introduce these Hilbert modular varieties.

\begin{definition}

An Eichler pair $(S, \varphi )$ for $\cO $ is a pair consisting of
an order $S$ over $R_F$ in a quadratic extension $L$ of $F$ and
an optimal embedding $\varphi :S\hookrightarrow \cO $ of orders over $R_F$.

An Eichler pair is said to be totally real if $L$ is.

\end{definition}

We recall from Section \ref{OE} than an embedding $\varphi :S\hookrightarrow \cO
$ is optimal if $\varphi (S) = \varphi (L)\cap \cO $. Note
that not all orders $S$ in quadratic extensions $L$ of $F$
can be optimally embedded in $\cO $. Namely, there exists an embedding $\varphi
$ of the ring of integers $R_L$ of $L$ into $\cO $ if and only if any prime
ideal $\mathfrak p$ of $F$ that ramifies in $B$ either remains
inert or ramifies in $L$. Here, the fact that $B$ is division and
splits at least at one archimedean place makes the condition for
the embeddability of $R_L$ in $\cO $ particularly neat. Otherwise,
it depends heavily on the conjugation class of $\cO $.

Given an order $S$ over $R_F$ in a totally real quadratic extension $L$ of
$F$ and a sequence of
positive integers $(1, d_2, ..., d_g)$, $d_{i-1}\vert d_{i}$ for $2\leq i
\leq g=2 n$, we
may consider the Hil\-bert mo\-dular va\-rie\-ty
$\mathcal H _S = \mathcal H _{S, (1, d_2, ..., d_g)}$ that
classifies isomorphism classes of triplets $(A, i, \cL )$ given by

\begin{itemize}

\item
An abelian variety $A$ of dimension $[L:\Q ] = 2n$.

\item
A ring homomorphism $i:S\hookrightarrow \End (A)$.

\item
A primitive polarization $\cL $ of type $(1, d_2, ..., d_g)$ on $A$.

\end{itemize}

The scheme $\mathcal H _S$ is $2 n$-dimensional, noncomplete and
defined over $\Q $. Actually, this is a particular case of the
Shimura varieties introduced above: $\cH _S$ is the union of
several irreducible components, all them isomorphic to Shimura
varieties $X_{(\mathrm{M}_2(S), \cI , \varrho )}$ for several
$\mathrm{M}_2(S)$-left ideals $\cI $ and anti-involutions $\varrho $.

In addition to these, we will also be interested in the
(reduced) scheme $\cH _F/\Q $ that coarsely represents the functor
attached to the moduli problem of classifying $(1, d_2, ...,
d_g)$-polarized abelian varieties $A$ of dimension $2 n$ together
with an homomorphism $R_F\hookrightarrow \End (A)$. The variety
$\cH _F$ has dimension $3 n$ and $\cH _F(\C )$ is the quotient of
$n$ copies of the $2$-Siegel half space $\mathfrak H_2$ by a
suitable discontinuous group (cf.\,\cite{Sh63}, \cite{LaBi}).

However, the Hilbert modular variety $\cH _F$ does not
correspond to any of the Shimura varieties $X
_B$ introduced above for any quaternion algebra $B$, not even
$\mathrm{M}_2(F)$.

Notice that, when $F=\Q $, $\cH _F = \cA _{2, (1, d)}$ is Igusa's
three-fold of level $d\geq 1$, the moduli space of $(1,
d)$-polarized abelian surfaces.


Let now $(A, \iota ,\cL )$ be a (primitively) polarized abelian
variety with multiplication by $\cO $ with respect to the datum
$(\cO , \cI ,\varrho )$. Note that the type $(1, d_2, ..., d_g)$
of $\cL $ was not specified when we posed the moduli problem attached
to the Shimura variety $X_{(\cO , \cI ,\varrho )}$.
However, since $X_{(\cO , \cI , \varrho )}$ is connected, the
type of $\cL $ only depends on the datum $(\cO , \cI , \varrho )$
and not on the particular triplet $(A, \iota , \cL )$. Thus $(1,
d_2, ..., d_g)$ will often be referred to as the type of $(\cO ,
\cI , \varrho )$.

Let $\mathcal A _{g, (1, d_2, ..., d_g)}/\Q $ be
the moduli space of polarized abelian varieties of type $(1, d_2, ...,
d_g)$. It is a reduced scheme over $\Q $ of dimension $\frac {g
(g+1)}{2}$. The observation above allows us to define a natural
morphism

$$
\begin{matrix}
 \pi : & X_{(\cO , \cI , \varrho )} &\longrightarrow  & \mathcal A _{g,
(1, d_2, ..., d_g)}  \\
       &(A, \iota , \cL )& \mapsto & (A, \cL )
\end{matrix}
$$
from the Shimura variety to $\cA _{g, (1, d_2, ..., d_g)}$ that
consists of {\em
forgetting} the quaternion endomorphism structure. This morphism
is representable, proper and defined over the field $\Q$ of
rational numbers. Moreover, as we now explain, the morphism $\pi
:X_B\rightarrow \mathcal A _{g, (1, d_2, ..., d_g)}$ factorizes
in a natural way through suitable Hilbert modular varieties
$\cH _{S, (1, d_2, ..., d_g)}$ and the Hilbert modular variety $\cH _F$.

Indeed, let $(S, \varphi )$ be any totally real Eichler pair. Imitating the
construction of $\pi $ we obtain a morphism

$$
\begin{matrix}
\pi _{(S, \varphi )}: & X_{(\cO , \cI , \varrho)} &
\longrightarrow & \mathcal H _S \\
& (A, \iota , \cL ) & \mapsto & (A, \iota \cdot \varphi , \cL )
\end{matrix}
$$
where $\iota \cdot \varphi :S\hookrightarrow \cO
\hookrightarrow \End (A)$.

Note that, although the construction of the
scheme $\cH _S$ does not depend on the embeddability of the order $S$
in $\cO $ nor on the choice of a possibly existing embedding
$\varphi :S\hookrightarrow \cO $, the morphism
$\pi _{(S, \varphi )}: X_{(\cO , \cI , \varrho)}\rightarrow \mathcal H
_S$ does indeed depend on it.

Finally, we also have a morphism

$$
\pi _F:\quad X_{(\cO , \cI , \varrho )}\longrightarrow \mathcal H _F
$$
from $X_B$ into the Hilbert modular variety $\cH _F$. The
map $\pi _F:\quad X_{(\cO , \cI , \varrho
)}\longrightarrow \cH _F$ is constructed as above: by restricting the
endomorphism structure of a triplet $(A, \iota , \cL )$ to $\iota
|_{R_F}:R_F\hookrightarrow \End (A)$. The
whole picture can be summarized with the following commutative
diagram of morphisms:

$$
\begin{matrix}
   &    &    & \cH _{S_1} &    &    &   &   \\
 & &\stackrel {\pi _{(S_1, \varphi _1)}}{\nearrow }&\stackrel{:}{.}
 &\searrow & & & \\
\pi : & X_B & \stackrel {\pi _{(S_2, \varphi _2)}}
{\longrightarrow } & \cH _{S_2} & \longrightarrow & \cH _F &
\rightarrow & \mathcal A _{g, (1, d_2, ..., d_g)} \\
  &  & \stackrel {\searrow }{\pi _{(S_n, \varphi _n)}} & \stackrel{:}{.} &
\nearrow  & & & \\
  &  &  & \cH _{S_n} &  &  &  & \\
  &  &  &    &  &  &  & \\
\hline \\
\mbox{dimension }  & n  &  &  2 n   &  & 3 n  &  & n (2 n + 1)
\end{matrix}
$$

Note that, while there is a canonical forgetful map from $X_B$
to $\cH _F$, we obtain distinct maps from $X_B$ to $\cH _S$ as
$\varphi :S\hookrightarrow \cO $ varies among all possible Eichler
embeddings.


\markright{\ref{Atkin} The Atkin-Lehner group of a Shimura variety}
\section{The Atkin-Lehner group of a Shimura variety}
\markright{\ref{Atkin} The Atkin-Lehner group of a Shimura variety}
\label{Atkin}

As before, let $F$ be a totally real number field of degree $n$.
Let $B$ be a totally indefinite quaternion algebra over $F$ of
discriminant $\mathcal D = \mathfrak p_1\cdot ...\cdot \mathfrak
p_{2r}$ and let $\cO $ be a maximal order in $B$. We recall from Section
\ref{QA} the
groups $\cO ^*\supset \cO ^*_+\supseteq \cO ^1$ of units in $\cO
$, units in $\cO $ of totally positive reduced norm and units in
$\cO $ of reduced norm $1$, respectively. The two latter are related by
the exact sequence

$$
1\rightarrow \cO ^1\rightarrow \cO ^*_+\stackrel{\n }{\rightarrow }
R^*_{F +}\rightarrow 1.
$$

We also recall that we let $B^*_+$ the subgroup of elements of $B^*$ of
totally positive reduced norm and we let $\mathrm{Norm}_{B^*}(\cO )$
and $\mathrm{Norm}_{B^*_+}(\cO )$ denote the normalizer group of $\cO $
in $B^*$ and $B^*_+$, respectively.


\begin{definition}

The {\em positive Atkin-Lehner group} $W^1$ of $\cO $ is

$$
W^1 = \Nm _{B_+^*}(\cO )/(F^*\cdot \cO ^1).
$$

\end{definition}

In addition to the above, although less relevant for our purposes, we
also introduce the groups

$$
W=\Nm _{B^*}(\cO )/(F^*\cdot \cO ^*)
$$
and
$$
W_+ = \Nm _{B_+^*}(\cO )/(F^*\cdot \cO ^*_+).
$$


Let us recall that, by the Skolem-Noether Theorem \ref{Sk}, the group
of automorphisms of $\cO $ is precisely the group $\mathrm{Norm}
_{B^*}(\cO )/F^*$. The
group $W$ is identified with the group of principal
two-sided ideals of $\cO $ by the assignation $\omega \in W \mapsto \cO
\cdot \omega $ and it is a finite abelian $2$-group. More precisely,
by results of Eichler (cf.\,\cite{Ei37}, \cite{Ei38} and \cite{Vi}, Theorem 5.7), the
reduced norm $\n :B^*\rightarrow F^*$ induces an isomorphism

$$
W\simeq \Z /2 \Z \times \stackrel{2 r}{...}\times \Z /2 \Z .
$$

Moreover, it also follows from loc.\,cit.\,that any
element $[\om ]\in W$ can be represented by an
element $\om \in \cO $ whose reduced norm $\n (\om )$ is supported at
the prime ideals $\mathfrak p\vert \mathcal D$.

The group $W_+$ may and will be regarded as the subgroup $W_+=
\{ [\omega ] \in W : \n (\omega )\in F^*_+\} $ of $W$ and both coincide whenever
$\Pic _+(F)\simeq \Pic (F)$. From
the above, we obtain the exact sequence

$$
1\rightarrow R^*_{F +}/R_F^{*2}\stackrel{\alpha }{\rightarrow }W^1\ra W_+\ra
1,
$$
where the map $\alpha : R^*_{F +}/R_F^{*2}\ra W^1$ maps
a totally positive unit $u\in R^*_{F +}$ to any $\alpha _u \in \cO
^*_+$ with $\n (\alpha _u )=u$, whose existence is guaranteed
by Proposition \ref{norms}. In this
way, we obtain that $W^1$ is isomorphic to the {\em direct} product

$$
W^1\simeq R^*_{F +}/R_F^{*2}\times W_+\simeq (\Z /2\Z )^s
$$
for $s\leq (n-1) + 2 r$. The bound for $s$ follows from
Dirichlet's Unit Theorem and the inclusion
$W_+\subseteq W$. The precise value of $s$ can
be determined in terms of the behaviour of
signatures of elements of $R_F^*$.

\subsection{Modular interpretation of the positive Atkin-Leh\-ner group}
\label{ModularAtkin}

In his Ph.D thesis, Jordan described
the modular interpretation of the action of the positive
Atkin-Lehner group $W^1$ on Shimura curves (cf.\,\cite{Jo81}). As we now explain,
this interpretation can be extended to the higher dimensional cases. We keep
the same notations of above.

\begin{definition}\label{popo}

Let $A/\C $ be an abelian variety with quaternionic multiplication by $\cO
$. Let $\cL \in \NS (A)\otimes \Q $ be a fractional
invertible sheaf and let $\om \in \End (A)=\cO $ be
a non zero endomorphism of $A$.

We define $\cL _{\om }\in \NS (A)\otimes \Q $ to be the fractional
invertible sheaf on $A$ such that

$$
c_1(\cL _{\om }) = \om ^{-1} c_1(\cL ) \om  .
$$

\end{definition}

$\\ $
In other words, if we regard the
first Chern class $c_1(\cL )$ of $\cL $ as an alternate bilinear
form $E$ on $V = \mathrm{Lie }(A(\C ))$, then $c_1(\cL _{\om }):
V\times V\longrightarrow \R $, $(u, v)\mapsto
E (\frac {\om }{\n (\om )}(u), \om (v))$. Hence,
$c_1(\cL _{\om }) = \frac {1}{\n (\om )} c_1 (\om ^*(\cL ))$.


From Theorem \ref{NeSe}, it is checked
that this does correspond to the first Chern class of a fractional line
bundle on $A$. Moreover, if $\cL \in \NS (A)$ and $\om \in \cO \cap \Nm _{B^*}(\cO
)$, then it also follows from Theorem \ref{NeSe} that $\cL _{\om }\in \NS
(A)$. Finally, if $\cL \in \NS (A)$ is a polarization and $\om \in \cO \cap
\Nm _{B_+^*}(\cO )$, we obtain from Theorem \ref{index} that $\cL _{\om }\in \NS
(A)$ is also a polarization.

Let $\cI $ be a left $\cO $-ideal
and let $\varrho $ be a positive involution on $B$. The action of
$B^*_+\subset \mathrm{GL }^+_2(\R )^n$ on $\mathfrak
H^n$ by Moebius transformations descends to a free action of $W^1$
on the set $\cO ^1\backslash \mathfrak H^n$ of complex points of
the Shimura variety $X_B = X_{(\cO , \cI , \varrho )}$,


The following is due to Jordan for the case of abelian
surfaces. The proof is along the same lines the one given in \cite{Jo81}
and we consequently omit it.

\begin{proposition}\label{Jordan}

Let $X_{(\cO , \cI, \varrho )}/\Q $ be a Shimura variety and let
$P=[(A, \iota , \cL )]$ denote the isomorphism class of a
polarized abelian variety with
multiplication by $\cO $ viewed as a closed point on it.

Then, an element $\om \in W^1$ acts on $P$ by

$$
\om (P) = [(A, \iota _{\om }, \cL _{\om })]
$$
where
$$
\begin{matrix}
\iota _{\om }: & \cO & \hookrightarrow & \End (A) \\
    & \beta & \mapsto &
\om ^{-1} \iota (\beta ) \om .
\end{matrix}
$$

\end{proposition}

Therefore, according to Proposition \ref{Jordan}, an Atkin-Lehner
involution $\om \in W^1$ acts on a triplet $[(A, \iota , \cL )]$ by
leaving stable the isomorphism class of the underlying abelian variety
$A$ but conjugating the endomorphism structure $\iota :\cO
\hookrightarrow \End (A)$ and switching the polarization $\cL $. It
readily follows the compatibilty of $\cL $ with $\iota $
that the polarization $\cL _{\om }$ is compatible with
$\iota _{\om }$.

From this
interpretation and by standard moduli considerations, it follows
that $W^1\subseteq \Aut (X_B)$ acts on $X_B$ as a subgroup
of algebraic involuting automorphisms over $\Q $.

\begin{remark} If $\alpha \in \cO ^*_+$, $\n (\alpha )=u\in R^*_{F +}$, then
$\alpha $ can be simultaneously viewed as an
automorphism $\alpha \in \Aut (A)\simeq \cO ^*$
of $A$ and as a representative of an Atkin-Lehner
involution $[\alpha ]\in W^1$. For any polarized abelian variety with
quaternionic multiplication $[(A, \iota , \cL )]$, this
automorphism induces an isomorphism of triplets
$\alpha [(A, \iota , \cL )] = [(A, \iota _{\alpha },
\cL _{\alpha })] = [(A, \iota , \cL _{\alpha })]$ where $\cL _{\alpha }$ is a
polarization on $A$ such that, although $\cL \not \simeq
\cL _{\alpha }$, both induce the same Rosati involution on $\End (A)\otimes \Q $.

In the literature (cf.\,\cite{Sh63}, \cite{Mi}), $\cL $ and $\cL _{\alpha }$
are called {\em weakly isomorphic}. It is checked that $\cO ^*_+/\cO
^1$ acts freely and transitively on the set of isomorphism classes
of a given weak polarization class on $A$.
\end{remark}


\begin{remark} There is a natural moduli theory for
polarized abelian varieties with quaternionic multiplication up
to weak isomorphism which is also
considered in \cite{Sh63}. Both theories coincide in dimension 2 (because it
corresponds to $F=\Q $), but for higher dimensions the
latter is coarser and less suitable for our purposes.
\end{remark}



\begin{remark} We wonder in what circumstances $W^1$ is the full
group of automorphisms of $X_B$. The impression is that, {\em
generically}, it does hold that $\Aut (X_B) = W^1$, but of
course the term {\em generic} should be made precise in any case.
The split case of an order $\cO $ in $B=\mathrm{M}_2(\Q )$ was
classically studied by Ogg, who found that the modular curve
$X_0(37)$ is an interesting exception to the prediction that,
whenever the genus of a modular curve is at least 2, $W=W^1=\Aut
(X_{\cO })$. For Shimura curves attached to a maximal order in a
rational division quaternion algebra $B/\Q $, this question is
investigated in Section \ref{secaut} of Chapter \ref{ASV}. For higher
dimensional Shimura
varieties, it seems to hold that whenever $\cO ^*$ does not
contain torsion units (besides $\pm 1$), $W^1 = \Aut (X_{\cO ,
\cI , \varrho })$ (cf.\,Theorem \ref{ellpoints}).
\end{remark}

\subsection{The twisting and stable Atkin-Lehner subgroups}\label{twstable}


Let $B$ be a totally indefinite quaternion algebra over a totally real
field $F$. Let us assume that the reduced discriminant is a
totally positive principal ideal of $F$ and let $D\in F^*_+$ be such that
$\disc (B) = (D)$. This is an
assumption that we will naturally encounter in several situations and
which is always satisfied whenever
the narrow class number $h_+(F)=1$ of $F$ is trivial. In particular, this
is always satisfied by indefinite rational quaternion algebras $B/\Q $.

The following definition is motivated by Theorems \ref{NeSe} and
\ref{index}.

\begin{definition}

A {\em polarized maximal order} of $B$ is a pair $(\cO , \mu )$ where $\cO \subset B$
is a maximal order and $\mu \in \cO $ is a pure quaternion of totally
positive reduced norm $\n (\mu )\in F^*_+$.

If $(\n (\mu )) = (D)$, we say that $(\cO , \mu )$ is a
{\em principally polarized order} of $B$.

\end{definition}

\begin{definition}

Let $(\cO , \mu )$ be a polarized
maximal order in $B$. A {\em twist} of $(\cO ,\mu )$ is an element $\chi \in \cO \cap
\Norm _{B^*}(\cO )$ such that $\chi ^2+\n (\chi )=0$ and $\mu \chi =-\chi \mu
$.


\end{definition}

In particular, if $(\cO , \mu )$ is a principally polarized maximal
order for some $\mu \in \cO $ with $\mu ^2 + u D=0$ and
$u\in R^*_{F +}$, a twist of $(\cO , \mu )$ is a pure
quaternion $\chi \in \cO \cap \Norm _{B^*}(\cO )$ such that
$$
B = F+F\mu +F\chi +F \mu \chi = (\frac {-u D, -\n (\chi )}{F}).
$$

We say that a principally polarized maximal order $(\cO , \mu )$ in $B$ is
{\em twisting} if it admits some twist $\chi $ in $\cO $. We will say
that a maximal order $\cO $ is {\em twisting} if
there exists $\mu \in \cO
$ such that $(\cO , \mu )$ is a twisting principally polarized order.
Finally, we will say that $B$ is {\em twisting} if there exists a
twisting maximal order $\cO $ in $B$. Note that $B$ is twisting if and
only if $B\simeq (\frac {-u D, m}{F})$ for some $u\in R^*_{F +}$ and $m\in
F^*$ such that $m\vert D$.

\begin{definition}
\label{twist}

Let $(\cO , \mu )$ be a polarized maximal order of $B$. A {\em twisting
involution} $\om \in W^1$ is an Atkin-Lehner involution such that
$[\om ]=[\chi ]\in
W$ is represented in $B^*$ by a twist $\chi $ of $(\cO , \mu )$.

We let $V_0(\cO , \mu )$ denote the subgroup of $W^1$ of twisting
involutions of $(\cO , \mu )$.

\end{definition}

For any subring $S\subset \cO $, we say that $\chi $ is a twist of
$(\cO , \mu )$ in $S$ if $\chi \in S$. We say that a twisting involution $[\om ]$
is in $S\subseteq \cO $ if it can be represented by a twist $\chi \in S$.
We will let $V_0(\cO , \mu , S)$ denote the subgroup of $W^1$ generated by the
twisting involutions of $(\cO , \mu )$ in $S$, or simply $V_0(S)$ if
the polarized order $(\cO , \mu )$ is well understood. We will also
simply write $V_0$ for the set of twists of $(\cO , \mu )$ in $S=\cO $.

Let us remark that, since $B$ is totally indefinite, no $\chi \in B^*_+$
can be a twist of a polarized order
$(\cO , \mu )$ because a necessary condition for
$B\simeq (\frac {-n (\mu ), -\n (\chi )}{F})$ is that $\n (\chi )$ be
totally negative. In fact, twisting involutions $\om \in W^1$ are
always represented by twists $\chi \in B^*$ of totally negative reduced
norm.

For a polarized order $(\cO , \mu )$, let $R_{\mu }=F(\mu )\cap \cO $
and let $\Omega =
\Omega (R_{\mu })=\{ \xi \in R_{\mu }: \xi ^f=1, f\geq 1\} $ denote the
finite group of roots of unity in the CM-order $R_{\mu }$.


\begin{definition}

The {\em stable group} of a polarized order
$(\cO , \mu )$ is the subgroup

$$
W_0=U_0\cdot V_0
$$
of $W^1$ generated by

$$
U_0 = U_0(\cO , \mu ) = \Norm _{F(\mu )^*}(\cO )/(F^*\cdot \Omega (R_{\mu })),
$$
and the group $V_0$ of twisting involutions of $(\cO , \mu )$.

\end{definition}

\section{Main theorem}
\markright{\ref{theorems} Main theorem}
\label{theorems}

As remarked in the introduction, the type of the primitive
polarizations of the abelian varieties parametrized by the Shimura
variety $X_{(\cO , \cI , \varrho )}$ is determined by the datum
$(\cO , \cI , \varrho )$. The following has been proved in Chapter \ref{GAV}
and makes this observation explicit.

\begin{proposition}
\label{Ro}

The po\-lari\-zations of the abe\-lian varie\-ties with qua\-ter\-nio\-nic
mul\-tipli\-cation $(A, \iota , \cL )$ parametrized by $X_{(\cO , \cI ,
\varrho )}$ are {\em principal} if and only if:

\begin{enumerate}

\item [(i)]
$\mathrm{Disc }(B) = (D)$ is a principal ideal of $F$ generated
by a totally positive element $D\in F^*_+$.

\item [(ii)]
$\n _{B/F}(\cI )$ and $\vartheta _{F/\Q }^{-1}$
lie in the same ideal class in $\Pic (F)$.

\item [(iii)]
The positive anti-involution on $B$ is
$\varrho = \varrho _{\mu }: B\rightarrow B$, $\beta \mapsto
\mu ^{-1} \overline \beta \mu $ for some
$\mu\in \cO $ such that $\mu ^2 + u D = 0$, $u\in R^*_{F +}$.

\end{enumerate}

\end{proposition}

For the rest of this chapter, we will focus on moduli spaces of
principally polarized abelian varieties and therefore we place
ourselves under the conditions on $(\cO , \cI , \varrho _{\mu })$ of
Proposition \ref{Ro}. We thus assume in particular that
$\disc (B)$ is a totally positive principal ideal and that we can choose a
generator $D\in F^*_+$ of it such that $\mu \in \cO $ satisfies
$\mu ^2+D=0$. In this case
we say that $(\cO , \cI , \varrho _{\mu })$ is of
{\em principal type}.

Our main Theorem \ref{mainGSV} below describes how the modular maps
introduced in Section \ref{Shimuramaps} factorize through the quotient of $X
_B$ by certain subgroups of Atkin-Lehner involutions that
were introduced in Section \ref{twstable}.


Let
$$
\widetilde {X}_B = \widetilde {X}_{(\cO
, \cI , \varrho )} = \pi (X_{(\cO , \cI , \varrho )})\hookrightarrow
\mathcal A_g
$$
denote the image in the
moduli space $\mathcal A_g = \mathcal A_{g, (1, ..., 1)}$ of principally
polarized abelian varieties by $\pi $ of the
Shimura variety $X_{(\cO , \cI , \varrho )}$. Similarly, define
$\widetilde {X}_{B/F}\subset \mathcal H _F$ and
$\widetilde {X}_{B/(S, \varphi )}\subset \mathcal H _S$
respectively to be the algebraic subvarieties $\pi
_{F}(X_{(\cO, \cI , \varrho )})$ and $\pi _{(S, \varphi )}(X
_{(\cO, \cI , \varrho )})$ of the Hilbert modular varieties  $\cH _{F}$
and $\cH _{S}$.

\begin{remark}
Classical objects in the literature are
{\em Humbert varieties}, which are introduced as quotients of Hilbert
modular varieties (cf. \cite{vdGe}). We note that Humbert varieties can be
considered as a degenerate case of the above defined varieties $\widetilde {X
}_B$, since they correspond to the split algebra $B=\mathrm{M}_2(F)$.
\end{remark}

\begin{definition}
A closed point $[(A, \iota ,\cL )]$ on $X
_B$ and its image on $\widetilde {X}_{B/(S, \varphi )}$, $\widetilde {X}_{B/F}$
or $\widetilde {X}_B$ is called a {\em Heegner point} if $\End (A)\otimes \Q
\simeq \M _2(M)$ for a CM-field $M$ over $F$. Equivalently, if $\End (A)
\varsupsetneq \cO $.
\end{definition}

It follows from the work of Shimura in \cite{Sh67} that the set of
Heegner points on these varieties is discrete and
dense.

Let us say that a morphism $\pi : X\ra Y$ of schemes of non
necessarily the same dimension is {\em quasifinite} if the fibres
of $\pi $ are finite (cf.\,\cite{Har77}, p.\,91).

With the same notations as above, we have

\begin{theorem}
\label{mainGSV}

Let $(\cO , \cI , \varrho _{\mu })$ be a quaternionic datum of
principal type and let $X_B = X_{(\cO , \cI , \varrho _{\mu })}$
be the Shimura variety attached to it. For any totally real Eichler pair
$(S, \varphi )$, let

$$
\begin{matrix}
     &    &        & \cH _S &  & & & \\
     &        &\stackrel {\pi _{S, \varphi }}{\nearrow } & & \searrow
     & & & \\
\pi :& X_B  & &\stackrel {\pi _F}{\longrightarrow }  & & \cH _F & \rightarrow &\cA
_g.\\
\end{matrix}
$$
be the diagram of forgetful morphisms introduced above. Then

\begin{enumerate}

\item The morphism $\pi _F:X_B\rightarrow \cH _F$ is a quasifinite morphism that
factorizes over $\Q $ into the natural projection $X
_B\rightarrow X_B/W_0 $ from $X_B$ onto its quotient by the
stable group $W_0\subseteq \Aut (X_B)$ and a birational
morphism $b_F: X_B/W_0\dashrightarrow \widetilde {X}_{B/F}$
onto the image of $X_B$ in $\cH _F$.

The domain of definition of $b_F^{-1}$ is
$\widetilde {X}_{B/F}\setminus \mathcal T _F$,
where $\mathcal T _F$ is a finite set of Heegner points.

\item
The morphism $\pi _{(S, \varphi )}:X_B\rightarrow \cH _S$ is a
quasifinite morphism that factorizes over $\Q $ into the projection $X
_B\rightarrow X/V_0(\varphi (S))$ of $X_B$ onto its quotient
by the finite 2-group $V_0(\varphi (S))\subseteq W_0$ and a
birational morphism $b_{(S, \varphi )}: X_B/V_0(\varphi
(S))\dashrightarrow \widetilde {X}_{B/(S, \varphi )}$ into the
image of $X_B$ in $\cH _S$ by $\pi _{(S, \varphi )}$.

As before, $b_{(S, \varphi )}^{-1}$ is defined on the whole $\widetilde {X}_{B/(S,
\varphi )}$ but at a finite set $\mathcal T _{(S, \varphi )}$ of Heegner points.

\end{enumerate}

\end{theorem}

We call the sets $\mathcal T _F$ and $\mathcal T _{(S, \varphi )}$
the {\em singular Heegner loci} of $\widetilde {X}_{B/F}$ and
$\widetilde {X}_{B/(S, \varphi )}$ respectively, as they are
indeed the sets of singular points (of quotient type) of these
varieties.

We present the proof of the theorem in the next section; let us now
make the statement more explicit and precise.

Recall that for a polarized order $(\cO , \mu )$, we let $R_{\mu }= F(\mu )\cap
\cO $ be the order in the CM-field $F(\mu )\simeq F(\sqrt {-D})$
that optimally embeds in $\cO $. Note that, since $\mu \in \cO $,
$R_{\mu }\supseteq R_F[\sqrt {-D}]$. We also let $\Omega = \Omega
(R_{\mu })= \{ \xi \in R_{\mu }: \xi ^f=1, f\geq 1\}$ and $\Omega _{odd}=
\{ \xi \in R_{\mu }: \xi ^{f}=1, f\mbox{ odd }\} $. Their
cardinalities will respectively be denoted by $\omega $ and $\omega _{odd}$.
Note that $U_0$ is indeed a subgroup of $W^1$ because $\Omega =F(\mu )
\cap \cO ^1$.


\begin{theorem}\label{finitedeg}

\begin{enumerate}
\item
Let $(\cO , \mu)$ be a non twisting principally polarized maximal order. Then,

\begin{enumerate}
\item[(i)]
For any totally real Eichler pair $(S, \varphi )$, the morphism
$\pi _{(S, \varphi )}:X_B\rightarrow \cH _S$ is a birational
equivalence and

\item[(ii)]
$\deg (\pi _F:X_B\rightarrow \cH _F) = 2^{\omega _{odd}}$.

\end{enumerate}

\item
Let $(\cO , \mu)$ be a twisting principally polarized maximal order. Then,

\begin{enumerate}
\item[(i)]
For any totally real Eichler pair $(S, \varphi )$
such that $\varphi (S)$ contains no twists of $(\cO , \mu )$, the
morphism $\pi _{(S, \varphi )}$ is a birational equivalence.

\item[(ii)]
For any twist $\chi \in \cO $ of $(\cO , \mu )$ and any
totally real Eichler pair $(S, \varphi )$ such that
$\varphi (S)\supseteq R_F[\chi ]$, the morphism
$\pi _{R_F[\chi ]}: X_B\rightarrow \cH _{R_F[\chi ]}$ is $2:1$.

\item
$\deg (\pi _F:X_B\rightarrow \cH _F) = 2^{2 \omega _{odd}}$.

\end{enumerate}
\end{enumerate}

\end{theorem}

In particular, note that the forgetful map $\pi _{\Q }: X
_B\longrightarrow \cA _2$ from a Shimura curve into Igusa's moduli
space of principally polarized abelian surfaces is either of
degree $2$ or of degree $4$ and that a necessary condition for the
latter is that
$$
B\simeq (\frac{-\disc (B), m}{\Q })
$$
for some $m>0$, $m\vert \disc (B)$.

It turns out, for instance, that for any
choice of a quaternionic datum $(\cO , \cI , \varrho )$ of principal type of
discriminant $D=6$ or $10$, the corresponding forgetful map of the
Shimura curve $X_{(\cO , \cI , \varrho )}$ into $\cA _2$ has
degree $4$.

In order to prove Theorem \ref{finitedeg}, we introduce the following
lemma.


\begin{lemma}\label{nontw}
Let $(\cO , \mu)$ be a principally
polarized maximal order. Then $U_0\simeq C_2^{\omega _{odd}}$.
\end{lemma}

{\em Proof. } Let us identify $F(\mu )$ and $F(\sqrt {-D})$
through any fixed isomorphism. As $U_0$ naturally embeds in
$F(\sqrt {-D})^*/(F^*\cdot \Omega )$, we first show that the maximal
$2$-torsion subgroup $H$ of $F(\sqrt {-D})^*/(F^*\cdot \Omega )$ is
isomorphic to $C_2^{\omega _{odd}}$.

If $\om \in F(\sqrt {-D})^*$ generates a subgroup of $F(\sqrt {-D})^*/(F^*\cdot
\Omega )$ of order $2$, then $\om ^2=\lambda \xi $ for some
$\lambda \in F^*$ and some
root of unity $\xi \in \Omega $. In particular, note that if $\om \in F(\sqrt
{-D})^*$, then $\om ^2\in F^*$ if and only if $\om \in F^*\cup F^*\sqrt
{-D}$. Let us write $\overline {H}=
H/\langle \sqrt {-D}\rangle $.

We then have that, if $\xi \in \Omega $, there exists at most a
single subgroup $\langle \om \rangle \subseteq \overline {H}$ such that
$\om \in F(\sqrt {-D})^*$, $\om ^2\in F^* \xi $. Indeed, if $\om
_1$, $\om _2\in F(\sqrt {-D})$, $\om _i^2=\lambda _i \xi $ for
some $\lambda _i\in F^*$ then ${\om _1\cdot \om _2^{-1}}^2\in F^*$
and hence $\om _1\cdot \om _2^{-1}\in F^*\cup F^* \sqrt {-D}$. This
shows that $[\om _1]=[\om _2]\in \overline {H}$.

Observe further that, if $\xi _f\in \Omega $ is a root of unity of
odd order $f\geq 3$, then $\om = \xi _f^{\frac {f+1}{2}}\in
F(\sqrt {-D})^*$ generates a $2$-torsion subgroup of $F(\sqrt
{-D})^*/(F^*\cdot \Omega )$ such that $\om ^2=\xi _f$.

It thus suffices to show that $\overline {\overline {H}}=H/\langle \sqrt
{-D}, \{ \xi _f ^{\frac {f+1}{2}}\} _{f\geq 3 \mbox{ odd}}\rangle
$ is trivial. Let $\om \in F(\sqrt {-D})^*$, $\om ^2=\lambda \xi
$, $\xi $ a root of unity of primitive order $f\geq 1$. If $f$ is
$2$ or odd we already know that the class $[\om ]\in \overline {\overline
{H}}$ is trivial. Further, it cannot exist any $\xi \in F(\sqrt {-D})$
of order $f=2^N$, $N\geq 2$, because otherwise $\xi ^{2^{N-1}}$
would be a square root of $-1$ and we would have that $F(\sqrt
{-D})=F(\sqrt {-1})$. This is a contradiction since $D R_F= \wp
_1\cdot ...\cdot \wp _{2 r}$, $r>0$.

Finally, it is also impossible that there should exist $\om \in
F(\sqrt {-D})$, $\om ^2=\lambda \xi $, $\xi ^f=1$, $f=2^N f_0$
with $N\geq 1$ and $f_0\geq 3$ odd. Indeed, since in this case
$\xi '=\xi ^{2^N}\in F(\sqrt {-D})$ is a primitive root of unity
of order $f_0$, $\om '=\xi ^{\frac {f_0+1}{2}}$ satisfies ${\om
'}^2=\xi '$. Then we would have ${\om '\cdot \om ^{-1}}^2=(\xi
^{2^{N-1}}) \xi $ and this would mean that $[\frac {\om '}{\om
}]=[\om ]\in \overline {H}$, which is again a contradiction. This shows
that $\overline {\overline H}$ is trivial and therefore $H=\langle \sqrt
{-D}, \{ \xi _f ^{\frac {f+1}{2}}\} _{\xi _f\in \Omega
_{odd}}\rangle $. In order to conclude the lemma, we only need to
observe that both $\mu $ and $\xi _f ^{\frac {f+1}{2}}\in F(\mu )$
normalize the maximal order $\cO $ for any odd $f$, because their
respective reduced norms divide the discriminant $D$. $\Box $


{\em Proof of Theorem \ref{finitedeg}: } Let us firstly assume that $(\cO , \mu )$
is a non twisting polarized order. It is then clear from the
definitions that the
groups of twisting involutions $V_0(S)$ are trivial for any
subring $S$ of $\cO $. In addition, by the above lemma, $W_0=U_0\simeq C_2^{\omega
_{odd}}$. By Theorem \ref{mainGSV}, this yields the
first part of Theorem \ref{finitedeg}. The second follows from the
following lemma, which shows that, when $(\cO , \mu )$ is twisting, the
situation is more subtle and less homogenous.

\begin{lemma}
\label{lemmatw}

Let $(\cO , \mu )$ be a {\em twisting} principally polarized maximal
order in a totally
indefinite quaternion algebra $B$ over $F$ of discriminant $\disc
(B)=(D)$, $D\in F^*_+$. Then $U_0\subset V_0$ is a subgroup of
$V_0$ and $V_0/U_0\simeq U_0$. In particular, $W_0=V_0\simeq
C_2^{2 \om _{odd}}$.
\end{lemma}

{\em Proof. } Let $\om \in U_0$ be represented by an element $\om
\in \Norm _{F(\mu )^*}(\cO )\cap \cO $ and let $\nu \in V_0$ be a
twisting involution. We know that the class of $\nu $ in $\Norm
_{B^*}(\cO )/(F^*\cdot \cO ^*)$ is represented by a twist $\chi \in \Norm
_{B^*}(\cO )\cap \cO $ that satisfies $\chi ^2+\n (\chi )=0$ and
$\mu \chi =-\chi \mu $. Then we claim that $\om \nu \in V_0$ is
again a twisting involution of $(\cO , \mu )$. Indeed, first $\om
\chi \in \Norm _{B^*} (\cO )\cap \cO $, because both $\om $ and
$\chi $ do. Second, since $\om \in F(\mu )$, $\mu (\om \chi )= \mu
\om \chi = \om \mu \chi = -\om \chi \mu = -(\om \chi ) \mu $ and
finally, we have $\tr (\mu (\om \chi ))= \mu \om \chi +
\overline {\om \chi }\overline {\mu }= \mu \om \chi -\overline {\om \chi }\mu =-\tr (\om
\chi )\mu \in F$ and thus $\tr (\om \chi )=0$.

This produces a natural action of $U_0$ on the set of twisting
involutions of $(\cO , \mu )$ which is free simply because $B$ is
a division algebra. In order to show that it is transitive, let
$\chi _1$, $\chi _2$ be two twists. Then $\om =\chi _1 \chi
_2^{-1}\in F(\mu )$ because $\mu \om = \mu \chi _1 \chi
_2^{-1}=-\chi _1 \mu \chi _2^{-1}=\chi _1 \chi _2^{-1}\mu =\om \mu
$ and $F(\mu )$ is its own commutator subalgebra of $B$; further
$\om \in \Norm _{B^*}(\cO )$ because its reduced norm is supported
at the ramifying prime ideals $\wp \vert \disc (B)$. Let us remark
that, in the same way, $\chi _1 \chi _2\in \Norm _{F(\mu )^*}(\cO
)$.

We are now in a position to prove the lemma. Let $\nu \in V_0$ be
a fixed twisting involution. Then $U_0\subset V_0$: for any $\om
\in U_0$ we have already shown that $\om \nu $ is again a twisting
involution and hence $(\om \nu )\nu = \om \in V_0$ because $V_0$
is a $2$-torsion abelian group. In addition, the above discussion
shows that any element of $V_0$ either belongs to $U_0$ or is a
twisting involution and that there is a noncanonical isomorphism
$V_0/U_0\simeq U_0$. $\Box $

This concludes the proof of Theorem \ref{finitedeg}. Observe
that the above lemma can be rephrased by asserting that, in the twisting
case, $U_0$ acts
freely and transitively on the set of twisting involutions of
$W^1$ with respect to $(\cO , \mu )$.


In view of Theorem \ref{finitedeg}, the behaviour of the
forgetful maps introduced above
differs considerably depending on whether $(\cO , \mu )$
is a twisting polarized order or not. For a maximal order $\cO $ in a
totally indefinite
quaternion algebra $B$ of principal reduced discriminant $D\in F^*_+$,
it is then obvious to ask the following questions.

\begin{enumerate}
\item[(i)] Is there any $\mu \in \cO $, $\mu ^2+D=0$, such that
$(\cO , \mu )$ is twisting?

\item[(ii)]
If $(\cO , \mu )$ is twisting, which is its twisting group $V_0$?
\end{enumerate}

Both questions are particular instances of the ones that we consider in
Section \ref{distance}.


\markright{\ref{proofs} Proof of Theorem \ref{mainGSV}}
\section{Proof of Theorem \ref{mainGSV}}
\markright{\ref{proofs} Proof of Theorem \ref{mainGSV}}
\label{proofs}

Let $(\cO , \cI , \varrho )$ be a quaternion datum attached to a
totally indefinite quaternion algebra $B$ over a totally real
number field $F$ of degree $[F:\Q ]=n$. We assume that it is of principal type (cf.
Proposition \ref{Ro} and below). This means in particular that
$\mathrm {disc }(B)$ can be generated by an element $D\in F^*_+$
such that $\varrho = \varrho _{\mu }$ for some $\mu \in \cO $ with $\mu
^2+D=0$.

Let $X_B = X_{(\cO , \cI , \varrho )}$ be the Shimura
variety attached to $(\cO , \cI , \varrho )$. Let us first show that $\pi _F:X
_B\rightarrow \cH _F$ is a quasifinite map that factorizes into the
natural projection

$$
X_B\rightarrow X_B/W_0
$$
from $X_B$ onto its quotient by the stable group
$W_0 = U_0\cdot V_0 \subseteq \Aut _{\Q }(X_B)$ attached to $(\cO , \mu )$.

To this end, we firstly make the following

\begin{claim}

There is a free and transitive action of the stable group
$W_0$ on the geometric fibres of the morphism $\pi _F: X
_B\rightarrow \cH _F$ at the set of non Heegner points of
$\widetilde {X}_{B/F}$.

\end{claim}

{\em Proof of the claim. } Let $(A, j_F, \cL )$ be a complex principally
polarized abelian variety together with a homomorphism of $\Z
$-algebras $j_F:R_F\hookrightarrow \End (A)$. Without making any
further mention of it, we will regard $\End (A)$ as an
$R_F$-algebra through the given immersion $j_F$.

The isomorphism class $[(A, j_F, \cL )]$ may be interpreted as a closed point in
$\cH _F$. If it is nonempty, the elements in $X_B$ of the fibre of $\pi
_F$ at this point can then be interpreted as those isomorphism classes of triplets
$(A, \iota , \cL )$ where $(A, \cL )$ is a principally polarized
abelian variety of dimension $2 n$ and
$\iota : \cO \hookrightarrow \End (A)$ is a
homomorphism of $R_F$-algebras such that the Rosati involution
that $\cL $ induces on $\cO $, via $\iota $, coincides with $\varrho _{\mu }$.

Choose a triplet $(A, \iota , \cL )$ and assume that
$\iota : \cO \simeq \End (A)$ is actually an isomorphism, that is, $P=[(A,
\iota , \cL )]$ is a non Heegner point of $X_B$. Through the first Chern
class, Theorem
\ref{NeSe} identifies $\NS (A)$ with a lattice in $B_0$ and
it allows us to regard polarizations on $A$ as pure
quaternions of $B$. In particular, since $\circ |_{\iota (\cO )} =
\varrho _{\mu }\cdot \iota $, we have that
$c_1(\cL )=\mu $ up to multiplication by elements in $F^*$.

Recall now that $W_0 = U_0\cdot V_0$ for certain
subgroups $U_0$ and $V_0\subseteq W^1$ that were defined in Section \ref{Atkin}.

Let $\omega \in U_0$. Let $L = F(\mu )\simeq F(\sqrt {-D})$ be the
CM-field generated by $\mu \in B$ over $F$ and let $S\supseteq R_F[\mu ]$
be the order in $L$ at which $\mu $ is optimal. Then
$\omega $ is represented by an element that we still denote $\omega \in S$
and we wish to show that the closed points $[(A, \iota , \cL )]$
and $\om [(A, \iota , \cL )]$ in $X_B$ lie on the same fibre of $\pi _F$.

From Proposition \ref{Jordan},
the iso\-mor\-phism class of $\om [(A, \iota , \cL )]$ is
represented by the triplet $(A, \iota _{\om }, \cL _{\om })$ where
$\iota _{\om } = \om ^{-1}\iota \om :\cO \hookrightarrow \End (A)$ and
$c_1 (\cL _{\om }) = \om ^{-1} c_1 (\cL ) \om $.
Since $c_1(\cL ) = \mu \in B^*/F^*$ and
$\om $ belong to the same quadratic algebra $L$ embedded in $B$, they
commute and it holds that

$$
c_1(\cL _{\om }) = \om ^{-1} c_1(\cL ) \om = c_1(\cL ).
$$

We then conclude that $(A, \cL )$ and $(A, \cL _{\om })$ are isomorphic
polarized varieties. Moreover, $\iota $ and $\iota _{\om
}$ coincide when restricted to the centre $R_F$ of $\cO $ and we obtain that
$(A, \iota |_{R_F}, \cL )\simeq (A, \iota _{\om }|_{R_F}, \cL _{\om
})$. Therefore, we obtain that the group $U_0$ acts
on the geometric fibres of the morphism $\pi _F: X
_B\rightarrow \cH _F$ at non Heegner points.

Let now $\omega \in V_0$ be an element represented by $\omega \in
\cO _+$ with $\n (\om ) = m\in F^*_+$ and such that

$$
B=F+F \mu +F \chi +F \mu \chi
$$
for some $\chi \in \cO $, $\chi ^2=m$ , $\mu
\chi = -\chi \mu $. Let $[(A, \iota , \cL )]\in X_B$ be a
closed point over $[(A, j_F, \cL )]\in \cH _F$. We have that $\om
[(A, \iota , \cL )] = [(A, \iota _{\om }, \cL _{\om })]$ and we
must show that $(A, \iota |_{R_F}, \cL )\simeq (A, \iota _{\om
}|_{R_F}, \cL _{\om })$. Again, since $\iota |_{R_F}=\iota _{\om
}|_{R_F}$, we only need to see that $\cL $ and $\cL _{\om }$ are
isomorphic polarizations on $A$.

To do so, we first note that
$\alpha = \om ^{-1}\cdot \chi $ is a unit in $\cO $ of reduced
norm $\n (\alpha )=-1$. This is due to the fact that the principal
ideal $m R_F$ is supported at the prime ideals $\wp\vert
D$, as can be checked locally.

Let us identify $\alpha $ with the
automorphism $\iota (\alpha )$ of $A$. We now show that $\alpha ^*(\cL
_{\omega })=\cL \in \NS (A)$. Indeed, since $c_1(\cL )$ and $\chi $
anti-commute, we have by Theorem \ref{NeSe} that
$c_1(\alpha ^*(\cL _{\om })) = \overline {\alpha } \om
^{-1} c_1(\cL ) \om \alpha =\frac {1}{m}\overline {\chi } c_1(\cL )
\chi  = -\chi ^{-1} c_1(\cL ) \chi  = \chi
^{-1} \chi  c_1(\cL ) = c_1(\cL )$. Since $W_0=U_0\cdot V_0$,
this proves part of our claim.

Let us now show that the action of $W_0$ on the fibres of $\pi
_F$ at non Heegner points is free and transitive.
By the Skolem-Noether Theorem, if a closed point
$[(A, \iota , \cL )]\in X_B(\C )$
lies over $[(A, j_F, \cL )]$, any element on the
fibre of $\pi _F$ at this point must be represented by a triplet
$(A, \om ^{-1}\iota \om , \cL _{\om })$ for some $\om \in
\Nm _{B^*}(\cO )$. Moreover, since $\cL _{\om }$ is a
polarization on $A$, it follows from
Theorem \ref{index} that $\om \in B^*_+$.

Since $(A, \iota ,
\cL )\simeq (A, \iota _{\om }, \cL _{\om })$ for any $\om \in
F^*\cdot \cO ^1$, we deduce that there exists a subgroup of
$W^1 = \Nm _{B_+^*}(\cO )/(F^*\cdot \cO ^1)$ that acts
freely and transitively on the fibre of $\pi _F$ at any
non Heegner point. It follows from the discussion above that this
subgroup must contain $W_0$.

Let us now see that it cannot be larger than $W_0$.
Assume that $\om \in W^1$ is such
that $\cL \simeq \cL _{\om }$. Then there exists $\alpha \in \cO
^*\stackrel {\iota }{\simeq }\Aut (A)$ such that $\overline {\alpha
}\om ^{-1}\mu \om \alpha =\mu $. Taking reduced norms, this
already implies that $\n (\alpha )^2=1$.

If $\n (\alpha )=1$, then $\overline {\alpha }=\alpha ^{-1}$ and the
above yields that $\mu \om \alpha =\om \alpha \mu $. This means that
$\om \alpha \in F(\mu )\cap \cO = S$ and therefore $\om \in U_0$.
If $\n (\alpha )=-1$, then $\mu \om \alpha = -\om \alpha \mu $.
Write $\chi = \om \alpha $. We then have
that $\tr (\mu \chi )= \mu \chi + \overline {\chi }\overline {\mu }= \mu
\chi -\overline {\chi }\mu = -\tr (\chi )\mu \in F$. Thus, since $\tr
(\chi )\in F$, we deduce that actually $\tr (\chi )=0$.
Since $\n (\chi )=\n (\om ) \n(\alpha ) = -\n (\om )$, we obtain
that $\chi ^2=\n(\om )$. This says that $\omega \in V_0$. $\Box $
\sk

Since the action of $W_0\subseteq \Aut _{\Q }(X_B)$ is algebraic, it
extends to an action on the fibres of $\pi _F$ at the points
in the closure $\widetilde {X}_{B/F}$ of the dense subset of
non Heegner points of $\widetilde {X}_{B/F}$. Similarly, the
action must be free and transitive on the fibres at
points in an algebraic subset $\cU _F$ of $\widetilde {X}_{B/F}$.

We obtain as a consequence that $\pi _F:
X_B \rightarrow \cH _F$ factorizes through the natural
projection of $X_B$ onto the quotient $X_B/W_0$ and a
morphism $b_F:X_B/W_0 \rightarrow \cH _F$ that is one-to-one
outside the Heegner locus of $X_B/W_0$. Since
Heegner points are isolated and $b_F$ is algebraic,
$b_F$ must be a birational equivalence between
$X_B/W_0$ and its image in $\cH _F$ whose inverse is
defined everywhere but at a finite set $\mathcal T _F$ of Heegner
points.

Moreover, since $W_0\subseteq \Aut _{\Q }(X_B)$, the
projection $X_B\rightarrow X_B/W_0$ is defined
over $\Q $. Since $\pi _F$ is also a morphism over $\Q $ which is
the composition of the above projection and the birational
equivalence $b_F$, it follows that $b_F$ is also defined over $\Q
$. This finishes the proof of the first part of Theorem
\ref{mainGSV}.

Now let $(S, \varphi )$ be an Eichler pair for $\cO $ and identify
$S$ with its image $\varphi (S)$ in $\cO $. As we saw, it induces
a natural morphism $\pi _{S, \varphi }: X_B\rightarrow \cH _S$
from $X_B$ into the Hilbert modular variety $\cH _S$ in such a
way that we have

$$
\pi _F: X_B\stackrel {\pi _{(S, \varphi )}}{\rightarrow }\cH _S
\rightarrow \cH _F.
$$

As the situation is very
similar to the one studied above, we will limit ourselves to
showing that the subgroup $V_0(S)$ of the stable group $W_0$ acts
freely and transitively on the fibre of $\pi _{(S, \varphi )}$ at
any non Heegner point of $\pi _{(S, \varphi )}(X_B) =
\widetilde {X}_{B/(L, \varphi )}$ in $\cH _S$.

Let $[(A, \iota , \cL )]$ thus be a closed point on $X_B$
represented by a principally polarized abelian variety with
quaternionic multiplication. From the above discussion, it is
clear that any other point at the same geometric fibre of $\pi
_{(S, \varphi )}$ as $[(A, \iota , \cL )]$ is represented by $(A,
\iota _{\om }, \cL _{\om })$ for some $\om \in W_0$.

If $(A, \iota \cdot \varphi , \cL )\simeq (A, \iota _{\om }\cdot \varphi ,
\cL _{\om
})$ is an isomorphism given by $\alpha \in \cO ^*$, then, as we already
saw, $\n (\alpha )=\pm 1$. Write $\chi =\om \alpha $.

If $\n (\alpha )=1$, then $\chi \in S\subset F(\mu )$. At the same
time, we must have that, for any $\beta \in S\subset \cO $, $\chi
\beta = \beta \chi $. This means that $\chi $ commutes
element-wise with $L$ and $F(\mu )$. Since the two quadratic
extensions are distinct because the first is totally real while
the second purely imaginary, $\chi \in R_F^*$ and thus $\om $ is
trivial in $W_0$.

If $\n (\alpha )=-1$, then $\alpha ^*(\cL_{\om })=\cL $ implies
that

$$
B=F+F\mu +F\chi +F\mu \chi
$$
while $\alpha ^{-1}\iota _{\om
}|_{S} \alpha = \iota |_{S}$ says that $\chi \in S$. Thus $\om \in
V_0(S)$. The converse also holds and the theorem follows as
before. $\Box $


\markright{\ref{qlocus} The quaternionic locus}
\section{The quaternionic
locus in the moduli space of principally polarized
abelian varieties}
\markright{\ref{qlocus} The quaternionic locus}
\label{qlocus}

Let $g= 2 n$ for some positive integer $n$ and let
$\cA _g/\Q $ be the moduli space of principally polarized abelian
varieties of dimension $g$. As in the preceding sections, we
let $[(A, \cL )]$ denote
the isomorphism class of a principally polarized
abelian variety $(A, \cL )$ regarded as a closed point in $\cA
_g$.

Let $F$ be a totally real number field of degree
$[F:\Q ]=n$. Let $\cO $ be a maximal order in a totally indefinite
division quaternion algebra $B$ over $F$. As in
Chapter \ref{GAV}, assume that $\vartheta _{F/\Q }$ and
$\disc (B)$ are coprime ideals.

It is our aim to use the results of the preceding
chapters to investigate the nature of the following object.


\begin{definition}

The quaternionic locus

$$
Q_{\cO } \subset \mathcal A _{g}(\C )
$$
is the locus of complex principally polarized abelian
varieties $[(A, \cL )]$ with $\End (A)\supseteq \cO $.

\end{definition}


We naturally wonder about the geometry of $Q_{\cO }$. Let us remark
that the algebraicity of $Q_{\cO }$ is not granted from the definition.
A reformulation of
Proposition \ref{polpral} in Chapter \ref{GAV} yields

\begin{proposition}\label{empty}

The quaternionic locus $Q_{\cO }$ is non empty
if and only if $\disc (B)$ is a totally positive principal ideal of $F$.

\end{proposition}

{\em Proof. } By Propositions \ref{norms} and \ref{pic}, there exists a
unique $\cO $-left ideal
$\cI _{\vartheta }$ such that $\n (\cI _{\vartheta })=\vartheta ^{-1}$.
From Proposition \ref{polpral}, there exist principally polarized
abelian varieties $A$ with quaternionic multiplication by $\cO $ if and
only if $H_1(A, \Z )\simeq \cI _{\vartheta }$ as left $\cO $-modules and $\disc (B)$
is trivial in $\Pic _+(F)$.

\vspace{0.5cm}
In consequence, and for the sake of simplicity, we assume that
$h_+(F)=1$. This automatically implies that the quaternionic locus $Q_{\cO
}$ is non empty. We fix a totally positive generator $D\in F^*_+$ of $\disc (B)$.
Note that $Q_{\cO }$ is the disjoint union of the set of principally
polarized abelian varieties $[(A, \cL )]$ with quaternionic
multiplication by $\cO $ and the set of Heegner points. The former
are distinguished by the fact that $\End (A)\simeq \cO $ while the latter
satisfy that $\End (A)\varsupsetneq \cO $.

Let $\cI _{\vartheta }$ be the $\cO $-left ideal as in the above proof
and let $\mu \in \cO _0$ be a pure quaternions such that
$\n (\mu ) = u D$ for some unit $u\in R^*_{F +}$. We
also note that the set $\widetilde {X}_{\mu }(\C )$ of complex points of
the Shimura variety $\widetilde {X}_{(\cO , \cI _{\vartheta },
\varrho _{\mu })}/\Q $ attached in Section \ref{theorems} to the datum
$(\cO , \cI _{\vartheta }, \varrho _{\mu })$ sits inside $Q_{\cO }$.

\begin{proposition}\label{inter}

Let $\mu $, $\mu '\in \cO _0$ be two pure quaternions such that
$\n (\mu ) = u D$ and $\n (\mu ') = u' D$ for some units $u$, $u'\in
R_{F +}^*$. If $\widetilde {X}_{\mu }(\C )$ and
$\widetilde {X}_{\mu '}(\C )$ are different subvarieties of $\cA _{g}(\C
)$, then $\widetilde {X}_{\mu }(\C )\cap \widetilde {X}_{\mu
'}(\C )$ is a finite set of Heegner points.

\end{proposition}

{\em Proof. } Assume that the isomorphism class $[A, \cL ]$
of a principally polarized abelian variety falls at the intersection of
$\widetilde {X}_{\mu }$ and $\widetilde {X}_{\mu '}$ in $\cA _g$. Write
$[A, \cL ]=\pi ([A, \iota ,\cL ]) = \pi ([A', \iota ',\cL '])$ as the
image by $\pi $ of points in $X_{\mu }$ and $X_{\mu '}$,
respectively. Since $[(A, \cL )]=[(A', \cL ')]\in
Q _{\cO }$, we can identify the pair $(A, \cL ) = (A', \cL ')$
through a fixed isomorphism of polarized abelian varieties.

Let us assume that
$[A, \cL ]=[A', \cL ']$ was not a Heegner
point. Then $\iota :\cO \simeq \End (A)$ would be an isomorphism of rings
such that $c_1(\cL ) = \mu $. We then would have
by Theorem \ref{NeSe} and Corollary \ref{bij}
that $c_1(\cL ) = c_1(\cL ') = \mu = \mu' $
up to multiplication by elements in $F^*$. Since
$\widetilde {X}_{\mu } = \widetilde {X}_{u \mu }$ for all units $u\in R_F^*$,
this would contradict the statement. Since
the set of Heegner points in $\cA _g(\C )$ is discrete, we conclude that
$X_{\mu }$ and $X_{\mu '}$ meet at a finite set of
Heegner points. $\Box $


\begin{proposition}
\label{locus}

\begin{enumerate}

\item The locus $Q_{\cO }$ is the set of complex points $\mathcal Q
_{\cO }(\C )$ of a reduced complete subscheme $\mathcal Q _{\cO }$
of $\mathcal A_g$ defined over $\Q $.

\item Let $\rho (\cO )$ be the number of
absolutely irreducible components of $\mathcal Q _{\cO }$. Then
there exist quaternions $\mu _k\in \cO _0$ with
$\mu _k^2 + u_k D=0$ for $u_k\in R_{F +}^*$, $1\leq k\leq \rho (\cO )$,
such that

$$
\mathcal Q _{\cO } = \bigcup
\widetilde {X}_{\mu _k}.
$$
is the decomposition of $\cQ _{\cO }$ into irreducible components.

\end{enumerate}

\end{proposition}


{\em Proof. } Let $[(A, \cL )]\in Q_{\cO }$ be the isomorphism class of
a complex principally polarized abelian variety such that $\End (A)\simeq \cO
$. As in the proof of Proposition \ref{empty}, we
have that $H_1(A, \Z )\simeq \cI _{\vartheta }$. By
Proposition \ref{polpral}, the
Rosati involution with respect to $\cL $ on $\cO $ must be of
the form $\varrho = \varrho _{\mu }$ for some $\mu \in \cO $ with $\mu ^2 + u D
=0$, $u\in R^*_{F +}$. Thus $[(A, \cL )]\in
\widetilde {X}_{\mu }(\C )$, namely the set
of complex points on a reduced, irreducible, complete and possibly singular scheme
over $\Q $ (cf.\,\cite{Sh63}, \cite{Sh67}). Since the set of Heegner
points $[(A, \cL )] \in \widetilde {X}_{\mu }(\C )$
is a set of isolated points lying on the Zariski
closure of their complement, we conclude that $Q_{\cO }$ is the union
of the Shimura varieties $\widetilde {X}_{\mu }(\C )$ as $\mu $
varies among pure quaternions satisfying the above properties.


Let us now show that $Q_{\cO }$ is actually covered by
finitely many Shimura varieties $\widetilde {X}_{\mu _k}(\C )$. Let
$A/\C $ be an arbitrary abelian variety with quaternionic
multiplication by $\cO $ such that $H_1(A, \Z )\simeq \cI _{\vartheta
}$ and fix an isomorphism $\iota : \cO \simeq \End (A)$.
Let $(\cO , \mu )$ be any principally polarized pair. Since $h_+(F)=1$, there
exists a unit $u\in R_F^*$ such that $u \mu $ is an ample quaternion in the
sense of \cite{Ro2}, \S 5. Let $\cL \in \NS (A)$ be
the line bundle on $A$ such that $c_1(\cL )^{-1}= u \mu $. From
Corollary \ref{bij}, Theorem \ref{index} and Corollary \ref{cor}, it
follows that $\cL $ is a principal
polarization on $A$ such that the isomorphism class of the triplet
$(A, \iota , \cL )$ corresponds to a closed point in $X_{\mu }(\C
)$ and hence $[A, \cL ]\in \widetilde {X}_{\mu }$.
Since, by Proposition \ref{inter},
the intersection points of two different Shimura varieties
$\widetilde {X}_{\mu }(\C )$ and $\widetilde {X}_{\mu '}(\C )$ in
$\cA _{g }(\C )$ are Heegner points, this
shows that for every irreducible component of $Q _{\cO }$ there exists at least
one principal polarization $\cL $ on $A$ such that $[A, \cL ]$ lies on
it. Consequently, the number $\pi _0(A)$ of isomorphism classes of
principal polarizations on $A$ is an upper bound for the number $\rho (\cO
)$ of irreducible components of $Q_{\cO }$. Since, by Theorem \ref{pio}, the
number $\pi _0 (A)$ is a finite number, this yields the
proof of the proposition. $\Box $

In view of Proposition \ref{locus}, it is natural to pose the following

\begin{question}\label{Qirr}

What is the number $\rho (\cO )$ of irreducible components of $\cQ _{\cO
}$? When is $\cQ _{\cO }$ irreducible?

\end{question}

Let us relate Question \ref{Qirr} to the following problem.
In Chapter \ref{GAV}, Theorem \ref{pio}, we computed the number $\pi _0(A)$ of
principal polarizations on an abelian variety $A$ with quaternion
multiplication by $\cO $ as a finite sum of relative class numbers of
suitable orders in the CM-fields $F(\sqrt {- u D})$ for $u\in
R^*_{F +}/R_F^{*2}$. This has the following modular interpretation:

Let $\cL _1$, ..., $\cL _{\pi _0(A)}$ be representatives of the $\pi _0(A)$
distinct principal polarizations on $A$. Then
the pairwise nonisomorphic principally polarized abelian varieties
$[(A, \cL _1)]$, ..., $[(A, \cL _{\pi _0(A)})]$ correspond to all closed points
in $\cQ _{\cO }$ whose underlying abelian variety is isomorphic to
$A$. We then naturally ask the following

\begin{question}\label{Qdistri}

Let $A$ be an abelian variety with quaternionic multiplication by
$\cO $. How are the distinct principal polarizations $[(A, \cL _j)]$ distributed
among the irreducible components $\widetilde {X}_{\mu _k}$ of $\cQ _{\cO }$?

\end{question}

It turns out that the two questions above are very related. The
linking ingredient is provided by the definition below, which
establishes a slightly coarser equivalence relationship on polarizations
than the one considered in Chapter \ref{GAV}. Indeed, in Chapter \ref{GAV}
we agreed to saying that two polarizations $\cL $, $\cL '$ on an abelian variety $A$
with quaternionic multiplication by $\cO $ are
isomorphic if there is an automorphism $\alpha \in \Aut (A)$ such
that $\alpha ^*(\cL ) = \cL '$. Later, we remarked in Section
\ref{ModularAtkin} of Chapter \ref{GSV} that Shimura also considered
weak isomorphisms of polarizations. Namely, in our language,
$\cL $ and $\cL '$ are weakly isomorphic if
$c_1(\cL ) \simeq m c_1(\cL ')\in \NS (A)$ for some $m\in F^*_+$. We shall denote it
$\cL \simeq _{w}\cL '$.


\begin{definition}

Let $A$ be an abelian variety with quaternionic multiplication by $\cO
$. Two principal polarizations $\cL $ and $\cL '$ on $A$ are
{\em Atkin-Lehner isogenous}, denoted by $\cL \sim \cL '$, if there
is an isogeny $\om \in \cO \cap \mathrm{Norm}_{B_+^*}(\cO )$ of $A$ such that

$$
\om ^*(\cL )\simeq _{w}\cL '.
$$


\end{definition}

We note that the above definition bears a closed relationship with the
modular interpretation of the positive Atkin-Lehner group $W^1$ given in
Proposition \ref{Jordan}.

\begin{definition}

Let $A$ be an abelian variety with quaternionic multiplication by $\cO
$. We let $\hat {\Pi }_0(A)$ be the
set of principal polarizations on $A$ up to Atkin-Lehner isogeny and we
let $\hat {\pi }_0(A) = \sharp \hat {\Pi }_0(A)$ denote its
cardinality.

\end{definition}

\begin{theorem} [Distribution of principal polarizations]\label{distri}

Let $A$ be an abelian variety with quaternionic multiplication by
$\cO $ and let $\cL _1$, ..., $\cL _{\pi _0(A)}$ be representatives of
the $\pi _0(A)$
distinct principal polarizations on $A$.

Then, two closed points $[A, \cL _i]$ and
$[A, \cL _j]$ lie on the same irreducible component of $\cQ _{\cO }$
if and only if the
polarizations $\cL _i$ and $\cL _j$ are Atkin-Lehner isogenous.

\end{theorem}

{\em Proof. } We know from Proposition \ref{locus} that any irreducible
component of $\cQ _{\cO }$ is $\widetilde {X}_{\mu } = \widetilde
{X}_{(\cO , \cI _{\vartheta }, \varrho _{\mu })}$ for some $\mu
\in \cO $, $\mu ^2 + u D = 0$, $u\in R^*_{F +}$. We single out and
fix one of them.

Let $\cL $ be a principal polarization on $A$ such that $[(A, \cL
)]$ lies on $\widetilde {X}_{\mu }$ and let
$\cL '$ be a second principal polarization on $A$. We claim that
$[A, \cL ']\in \widetilde {X}_{\mu }$ if and only if there exists $\om \in
\mathrm{Norm}_{B^*_+}(\cO )$ such that $\cL '$ and $\om ^*(\cL )$ are weakly
isomorphic.

Assume first that $\cL '\simeq _{w}\om ^*(\cL )$ for some
$\om \in \cO \cap \mathrm{Norm} _{B^*_+}(\cO )$. This amounts to say
that $\overline {\om } c_1 (\cL ) \om = m c_1(\cL ')$ for some $m\in F^*$.
Since both $\om ^*(\cL )$ and $\cL '$ are polarizations, we deduce from
Theorem \ref{index} that $m\in F^*_+$. Moreover, since
$\cL $ and $\cL '$ are principal, we obtain from
Proposition \ref{degree} that $m = u \n (\om )$ for some $u\in
R^*_{F +}$.

Note that $(A, \iota _{\om }, \cL ')$ is a principally polarized
abelian variety with
quaternionic multiplication such that the
Rosati involution that $\cL '$ induces on $\iota _{\om }(\cO )$
is $\varrho _{\mu }$. Indeed, this follows
because $\iota _{\om }(\beta )^{\circ _{\cL '}}=
\iota ((\om ^{-1}\beta \om ))^{\circ _{\cL '}} =
\iota ((\om ^{-1}\mu \om )^{-1}\\ \overline {\om ^{-1} \beta \om }(\om ^{-1}\mu \om
)) = \iota _{\om }(\mu ^{-1}\overline \beta \mu )$. This shows that,
if $\cL '$ and $\om ^*(\cL )$ are weakly
isomorphic for some $\om \in \Nm _{B^*_+}(\cO )$, then $[A, \cL ']\in
\widetilde {X}_{\mu }$.

Conversely, let us assume that $[A, \cL ']\in \widetilde {X}_{\mu }$. Let
$\iota ':\cO \hookrightarrow \End (A)$ be
such that $[A, \iota ', \cL ']\in X_{(\cO , \cI _{\vartheta
}, \varrho _{\mu })}$. By the Skolem-Noether
Theorem, it holds that $\iota ' = \om ^{-1}\iota \om $ for some
$\om \in \mathrm{Norm}_{B^*}(\cO )$;
we can assume that $\om \in \cO
$ by suitably scaling it. Since it holds that
$\iota _{\om }(\beta )^{\circ _{\cL '}}= \iota _{\om }(\mu ^{-1}\overline \beta
\mu)$ for any $\beta \in \cO $, we have that
$c_1(\cL ') = u \om ^{-1}c_1(\cL )\om
$ for some $u\in R_F^*$ such that $u \n (\om )\in F^*_+$. Since
$\n (\cO ^*)=R_F^*$, we
can find $\alpha \in \cO ^*$ with reduced norm $\n (\alpha )=u^{-1}$ and thus
$\om \alpha \in B^*_+$. The automorphism $\alpha \in \cO ^*=\Aut (A)$
induces an isomorphism between the polarizations $\cL _{\om \alpha }$
and $\cL '$, since $c_1(\alpha ^*(\cL ')) = \overline {\alpha }(u \om ^{-1}c_1(\cL )\om )
\alpha =c_1(\cL _{\om \alpha })$. Since, according to our
Definition \ref{popo}, $\cL _{\om \alpha }$ is weakly equivalent to $\cL $,
this concludes our claim above and also proves the theorem. $\Box $

\begin{corollary}
\label{genus}

The number of irreducible components of $\cQ _{\cO }$ is

$$
\rho (\cO ) = \hat \pi_0(A),
$$
independently of the choice of $A$.

\end{corollary}


For any irreducible
component $\widetilde {X}_{\mu _k}$ of $\mathcal Q _{\cO
}$, let $\Pi _0^{(k)}(A)\subset \Pi _0(A)$ denote the set of
isomorphism classes of the isogeny class of principal polarizations
lying on $\widetilde {X}_{\mu _k}$.

As another immediate consequence of Theorem \ref{distri}, we obtain the
following corollary, which establishes an internal structure
on the set $\Pi _0(A)$. Roughly, it asserts that
$\Pi _0(A) = \bigcup _{k=1}^{\rho (\cO )} \Pi _0^{(k)}(A)$ is
the union of several isogeny classes that are equipped with a free
and transitive action of a 2-torsion finite abelian group.


\begin{corollary}

Let $A$ be an abelian variety with quaternionic multiplication by
$\cO $. Let $\widetilde {X}_{\mu _k}$ be an irreducible component of
$\cQ _{\cO }$ and let $W_0^{(k)}\subseteq W^1$ be the stable subgroup
attached to the polarized order $(\cO , \mu _k)$.

Then there is a free and
transitive action of $W^1/W_0^{(k)}$ on $\Pi _0^{(k)}(A)$.

\end{corollary}

In the case of a non twisting maximal order $\cO $, we have that the
stable group $W_0 (\cO , \mu )$ attached to a principally polarized
pair $(\cO , \mu )$ is $U_0(\cO , \mu )$. The following corollary
follows from Lemma \ref{nontw} in Section \ref{FM}.

\begin{corollary}\label{equi}

Let $\cO $ be a non twisting maximal order in $B$ and assume that, for any
$u\in R^*_{F +}$, all primitive roots of unity of odd order in the
CM-field $F(\sqrt {-u D})$ are contained in the order $R_F[\sqrt {-u D}]$.

Let $A$ be an abelian variety with quaternionic multiplication by $\cO $.
Then the distinct isomorphism
classes of principally
polarized abelian varieties $[(A, \cL _1)]$, ..., $[(A, \cL _{\pi _0(A)})]$ are
{\em equidistributed} among the $\rho (\cO )$ irreducible components
of $\mathcal Q _{\cO }$.

In particular, it then holds that

$$
\pi _0 (A) = \frac {|W^1|}{|W_0|}\cdot \rho (\cO ).
$$

\end{corollary}

\subsection{Shimura curves embedded in Igusa's threefold}
\label{curves}

The whole picture becomes particularly neat when we consider the
case of abelian surfaces and Shimura curves. Let $B$ be an
indefinite quaternion algebra over $\Q $ of discriminant
$D = p_1\cdot ...\cdot p_{2 r}$ and let $\cO $ be a maximal
order in $B$. Since
$h(\Q )=1$, there is a single choice of $\cO $ up to conjugation by
$B^*$. Moreover, all left ideals of $\cO $ are principal and isomorphic
to $\cO $ as left $\cO $-modules.

Let $A$ be a complex abelian surface with quaternionic multiplication
by $\cO $. By Theorem \ref{mainGAV}, $A$ is principally polarizable and
the number of isomorphism classes of principal polarizations on $A$ is

$$
\pi _0(A)= \dfrac {\widetilde {h}(-D)}{2},
$$
where, for any nonzero squarefree integer $d$, we write
$$
\widetilde {h}(d) = \begin{cases}
    h(4 d)+h(d)  & \text{ if }d\equiv 1\quad \mbox{ mod }4, \\
    h(4 d) & \text{otherwise.}
  \end{cases}
$$


For any integral element $\mu \in \cO $ such that $\mu ^2 + D = 0$, let
now $X_{\mu } = X_{(\cO , \cO , \varrho _{\mu })}$ be the Shimura curve
that coarsely represents the functor which classifies
principally polarized abelian surfaces $(A, \iota , \cL )$ with
quaternionic multiplication by $\cO $
such that the Rosati involution with respect to $\cL $ on
$\cO $ is $\varrho _{\mu }$. This is an algebraic curve over $\Q
$ whose isomorphism class does not actually depend on the quaternion
$\mu $, but only on the discriminant $D$ (cf.\,\cite{Sh67}). Hence, it is usual to
simply denote this isomorphism class as $X_D$.

Let $W^1 = W = \{ \omega _m: m\vert D\} \simeq (\Z /2\Z )^{2 r}$
be the Atkin-Lehner group attached to $\cO $ in Section \ref{Atkin}.
From Section \ref{ModularAtkin}, we know that $W\subseteq \Aut _{\Q }(X
_D)$ is a subgroup of the group of automorphisms of the Shimura curve $X
_D$.


Let now $\cA _2$ be the moduli space of principally polarized
abelian surfaces. By the work of Igusa (cf.\,\cite{Ig}), it is
an affine scheme over $\Q $ which
contains the moduli space $\mathcal M
_2$ of curves of genus $2$ as a Zariski open and dense subset, immersed in
$\cA _2$ via the Torelli embedding.

Sitting in $\cA _2$ there is the quaternionic locus $\cQ _{\cO }$ of
isomorphism classes of principally polarized abelian surfaces $[(A, \cL
)]$ such that $\End (A)\supseteq \cO $. Since all maximal orders $\cO $
in $B$ are pairwise conjugate, the quaternionic locus $\cQ _{\cO }$
does not actually depend on the choice of $\cO $ and we may simply
denote it by $\cQ _{\cO } = \cQ _D$.

There are forgetful finite morphisms $\pi :X_{\mu }\rightarrow \cQ _D \subset
\cA _2$ which map the Shimura curve $X_{\mu }$ onto an irreducible
component $\widetilde {X}_{\mu }$ of $\cQ _D$. Although the isomorphism
class of $X_{\mu }/\Q $ does not depend on the choice of $\mu $, the image
$\widetilde {X}_{\mu }\subset \cQ _D$ {\em does depend} on this
choice.


Let us now compare the non twisting and twisting
case, respectively. We firstly assume that

$$
B\not \simeq (\frac {-D, m}{\Q })
$$
for all positive divisors $m\vert D$ of $D$. Then all principally polarized
pairs $(\cO , \mu )$ in $B$ are {\em non twisting} and the stable
subgroup attached to $(\cO , \mu )$ is
$$
W_0 = U_0 = \langle \om _D\rangle \subset W,
$$
independently of the choice of $\mu $. By Theorem \ref{mainGSV}, we
have that any irreducible component $\widetilde {X}_{\mu }$ of
$\cQ _D$ is birationally equivalent to the Atkin-Lehner quotient
$X_D/\langle \om _D \rangle $ and thus the
quaternionic locus $\cQ _D$ in $\cA _2$
is the union of pairwise birationally equivalent
Shimura curves $\widetilde {X}_{\mu _1}$,
...,$\widetilde {X}_{\mu _{\rho (B)}}$, meeting at a finite set of
Heegner points.

Moreover, for any abelian surface $A$ with quaternionic multiplication
by $\cO $, it follows from
Theorem \ref{distri} that the closed points $\{ [(A, \cL _j)]\} _{j=1}^{\pi
_0(A)}$ are equidistributed among the $\rho (\cO )$ irreducible components
of $\cQ _D$. In addition, Corollary \ref{equi} ensures that $|W/W_0| = 2^{2 r-1}
\vert \pi _0(A)$, as genus theory for binary quadratic forms already
ensures. This two facts yield that the number of irreducible
components of the quaternionic locus in the non twisting case is

$$
\rho (\cO ) = \dfrac {\widetilde {h}(-D)}{2^{2 r}}.
$$


On the other hand, let us assume that

$$
B\simeq (\frac {-D, m}{\Q })
$$
for some $m\vert D$. Note that this may hold for several positive
divisors of $D$. In this case, there can be several different
birational classes of irreducible components on $\cQ _D$. Indeed,
the assumption means that there exist pure quaternions $\mu \in
\cO $, $\mu ^2+D=0$, such that $(\cO , \mu )$ is a {\em twisting}
polarized order. Then
$$
W_0 (\cO , \mu ) = \langle \om _m, \om _D\rangle
$$
and $\widetilde {X}_{\mu }$
is birationally equivalent to $X_D/\langle \om _m, \om _D\rangle $.
We may refer to $\widetilde {X}_{\mu }$ as a
twisting irreducible component of $\cQ _D$.

In addition to these, there may exist non twisting
polarized orders $(\cO , \mu )$ such that the
corresponding irreducible components $\widetilde {X}_{\mu }$ of $\cQ _D$
are birationally equivalent to $X_D/\langle \om _D\rangle $. We
may refer to these as the non twisting irreducible components of $\cQ
_D$.

We then have the following lower and upper bounds for the number of
irreducible components of $\cQ _D$:

$$
\dfrac {\widetilde {h}(-D)}{2^{2 r}} <\quad \rho (\cO ) \quad \leq
\dfrac {\widetilde {h}(-D)}{2^{2 r -1}}.
$$

Summing up, we obtain the following

\begin{theorem}\label{pq}

Let $B$ be an indefinite division quaternion algebra over $\Q $
of discriminant $D = p_1\cdot...\cdot p_{2 r}$. Then, the
quaternionic locus $\cQ _D$ in $\cA _2$ is irreducible if and only if

$$
\widetilde {h}(-D) =
  \begin{cases}
    2^{2 r -1} & \mbox{ if }B\simeq (\frac {-D, m}{\Q })\mbox{
    for some }m\vert D, \\
    2^{2 r} & \text{otherwise}.
  \end{cases}
$$

\end{theorem}


{\em Proof. } If $B$ is not a twisting quaternion algebra, we already
know from the above that the number of irreducible components of $\cQ
_D$ is $\dfrac {\widetilde {h}(-D)}{2^{2 r}}$. Hence, in this case, the
quaternionic locus of discriminant $D$ in $\cA _2$ is irreducible if
and only if $\widetilde {h}(-D) = 2^{2 r}$. If on the other hand $B$ is
twisting, it follows from the above inequalities that $\cQ _D$ is
irreducible if and only if $\widetilde {h}(-D) = 2^{2 r - 1}$. $\Box $

In view of Theorem \ref{pq}, there arises a closed relationship
between the irreductibility of the quaternionic locus in Igusa's
threefold and the genus theory of integral binary quadratic forms
and the classical {\em numeri idonei} studied by Euler, Schinzel
and others. We refer the reader to \cite{Ar95} and \cite{Sch59}
for the latter.


\subsection{Hashimoto-Murabayashi's families}\label{Hash}

As the simplest examples to be considered, let $B_6$ and $B_{10}$ be the
rational quaternion algebras of discriminant $D = 2\cdot 3 = 6$ and $2\cdot 5 =
10$, respectively. Hashimoto and Murabayashi \cite{HaMu} exhibited
two families of principally polarized abelian surfaces with
quaternionic multiplication by a maximal order in these quaternion
algebras. Namely, let

$$
C_{(s, t)}^{(6)}: Y^2 = X (X^4 + P X^3 + Q X^2 + R X + 1)
$$
be the family of curves with

$$
P = 2 s + 2 t, \quad Q = \frac {(1+2 t^2) (11-28 t^2 + 8 t^4)}{3 (1-t^2)(1-4
t^2)}, \quad R = 2s -2t
$$
over the base curve

$$
g^{(6)}(t, s) = 4 s^2 t^2 - s^2 + t^2 + 2 = 0.
$$

And let


$$
C_{(s, t)}^{(10)} : Y^2 = X (P^2 X^4 + P^2 (1+R) X^3 + P Q X^2 + P (1-R) X + 1)
$$
be the family of curves with

$$
P = \frac {4 (2 t + 1)(t^2-t-1)}{(t-1)^2},
\quad Q = \frac {(t^2+1)(t^4+8t^3-10t^2-8t+1)}{t (t-1)^2 (t+1)^2}
$$
and
$$
R = \frac {(t-1) s}{t (t+1) (2 t + 1)}
$$
over the base curve

$$
g^{(10)}(t, s) = s^2 - t (t-2) (2 t +1) = 0.
$$

Let $J_{(s, t)}^{(6)} =
\mbox{Jac} ( C_{(s, t)}^{(6)})$ and $J_{(s, t)}^{(10)} =
\mbox{Jac} ( C_{(s, t)}^{(10)})$ be the Jacobian surfaces of the
fibres of the families of curves above respectively. It was proved in \cite{HaMu}
that their ring of endomorphisms contain a maximal order in $B_6$ and
$B_{10}$, respectively.

Both $B_6 = (\frac {-6, 2}{\Q })$ and
$B_{10} = (\frac {-10, 2}{\Q })$ are twisting quaternion algebras. Moreover,
it turns out from our formula for $\pi _0(A)$ above
that any abelian surface $A$ with
quaternionic multiplication by a maximal order in either $B_6$ or
$B_{10}$ admits a {\em single} isomorphism class of principal
polarizations. This implies that
$\rho (B_6)=\hat \pi _0(A) = \pi _0(A) = 1$ and $\rho (B_{10})=\hat \pi _0(A) =
\pi _0(A) = 1$, respectively.

Moreover, the Shimura
curves $X_6/\Q $ and $X_{10}/\Q $ have genus $0$, although they are not
isomorphic to $\PP ^1_{\Q }$ because there are no rational points on
them. However, it is easily seen that $X_6/W_0 = X_6/W \simeq \PP ^1_{\Q
}$ and $X_{10}/W_0 = X_{10}/W \simeq \PP ^1_{\Q }$, respectively.

As it is observed in \cite{HaMu}, the base curves $g^{(6)}$
and $g^{(10)}$ are curves of genus $1$ and not of genus $0$ as it
should be expected. This is explained by the fact that
there are obvious isomorphisms between
the fibres of the families $C^{(6)}$ and $C^{(10)}$, respectively.


Ibukiyama, Katsura and Oort \cite{IbKaOo} proved that the
supersingular locus in $\cA _2/\overline {\F }_p$ is irreducible if and
only if $p\leq 11$. As a corollary to their work, Hashimoto and
Murabayashi obtained that the reduction mod $3$ and $5$ of the
family of Jacobian surfaces with quaternionic multiplication by
$B_6$ and $B_{10}$ respectively yield the single irreducible
component of the supersingular locus in $\cA _2/\overline {\F }_3$ and
$\cA _2/\overline {\F }_5$ respectively. The following statement may be
considered as a lift to characteristic $0$ of these results.

\begin{theorem}\label{uni}

\begin{enumerate}
\item
The quaternionic locus $\cQ _6$ in $\cA _2/\Q $ is absolutely
irreducible and birationally equivalent to $\PP ^1_{\Q }$ over $\Q $.
The generic element of $\cQ _6$ over $\qbar $ is given by
Hashimoto-Murabayashi's family $C^{(6)}$.


\item
The quaternionic locus $\cQ _{10}$ in $\cA _2/\Q $ is absolutely
irreducible and birationally equivalent to $\PP ^1_{\Q }$ over $\Q
$. The generic element of $\cQ _{10}$ over $\qbar $ is given by
Hashimoto-Murabayashi's family $C^{(10)}$.

\end{enumerate}
\end{theorem}

{\em Proof. } This follows from Theorem \ref{pq} and the
discussion above. $\Box $


In particular, we obtain from Theorem \ref{uni} that every principally
polarized abelian surface $(A, \cL)$ over $\qbar $ with quaternionic multiplication
by a maximal order of discriminant $6$ or $10$ is isomorphic over $\qbar $ to
the Jacobian variety of one of the curves $C_{(s, t)}^{(6)}$ or $C_{(s,
t)}^{(10)}$, except for finitely many degenerate cases.

\markright{\ref{FM} The field of moduli of the quaternionic multiplication}
\section{The field of moduli of the quaternionic multiplication on an abelian
variety}
\markright{\ref{FM} The field of moduli of the quaternionic multiplication}
\label{FM}


The aim of this section is to provide an application of the main
results in this chapter to the arithmetic of abelian varieties with
quaternionic multiplication, as it was done in \cite{Ro4}.

Let $\bar \Q $ be a fixed algebraic closure of the field $\Q $ of
rational numbers and let $(A, \cL )/\bar \Q $ be a polarized abelian variety. The
field of moduli of $(A, \cL )/\bar \Q $ is the minimal number field
$k_{A, \cL }\subset \qbar $ such that $(A, \cL )$ is isomorphic (over $\qbar $) to
all its Galois conjugates $(A^{\sigma }, \cL ^{\sg })$, $\sigma \in
\Gal (\qbar /k_{A, \cL })$.

The field of moduli $k_{A, \cL }$ is an essential arithmetic
invariant of the $\qbar $-isomorphism class of $(A, \cL )$. It is
contained in all possible fields of definition of $(A, \cL )$ and,
unless $(A, \cL )$ admits a rational model over $k_{A, \cL }$
itself, there is not a unique minimal field of definition for $(A,
\cL )$. In this regard, we have the following theorem of Shimura \cite{Sh72}.

\begin{theorem}
A generic polarized abelian variety of odd dimension admits a model
over its field of moduli. For a generic polarized abelian variety of even
dimension, the field of moduli is not a field of definition.

\end{theorem}

We will place ourselves in the even dimensional-case since we shall be
interested in abelian varieties with quaternionic multiplication.


\begin{definition}

Let $(A, \cL )/\qbar $ be a polarized abelian variety and let
$S\subseteq \End _{\qbar }(A)$ be a subring of $\End _{\qbar }(A)$. The
field of moduli of $S$ is the minimal number
field $k_S\supseteq k_{A, \cL }$ such that, for any $\sigma \in \Gal
(\qbar /k_S)$,
there is an isomorphism $\varphi _{\sigma } /\qbar :A\ra A^{\sigma
}$, $\varphi _{\sg }^*(\cL ^{\sg }) = \cL $, of polarized abelian varieties
that induces commutative diagrams

$$
\quad A\lra A^{\sg }
$$
$$
\quad \beta \downarrow \quad \quad \quad \downarrow \beta ^{\sg }
$$
$$
\quad A\lra A^{\sg }
$$
for any $\beta \in S$.
\end{definition}

We remark that, as a consequence of the very basic definitions,
the field of moduli of the multiplication-by-$n$ endomorphisms on
$A$ is exactly $k_{\Z } = k_{A, \cL }$. But in the case that $\End
(A)\varsupsetneq \Z $, little is known on the chain of Galois
extensions $k_{\End (A)}\supseteq k_S \supseteq k_{A, \cL }$.

Our main Theorem \ref{mainFM} in this direction concerns the case
of abelian varieties with quaternionic multiplication. It
is a consequence of the results
obtained in Section \ref{theorems} on modular embeddings of Shimura
varieties in Hilbert modular varieties and the moduli spaces of principally
polarized abelian varieties.

Let $(A, \cL )/\qbar $ be a principally polarized abelian variety
with quaternionic multiplication by a maximal order in a
totally indefinite division quaternion algebra $B$ over a totally
real number field $F$ of degree $[F:\Q ]=n$. In Chapter \ref{GAV}, it
was shown that the first Chern class $c_1$ induces a
monomorphism of the N\'{e}ron-Severi group $\NS (A)$ of $A$ into the
additive group $B_0$ of quaternions of $B$ of null reduced trace


$$
c_1: \NS (A) \hookrightarrow  B_0.
$$

Moreover, it was proved that
there exists a left $\cO $-ideal $\cI _{\vartheta }$ in $B$
such that $H_1(A, \Z )\simeq \cI _{\vartheta }$ and
$\n (\cI _{\vartheta }) = \vartheta _{F/\Q }^{-1}$ and that
the discriminant ideal $\disc (B)$ of $B$ is
necessarily principal and totally positive. Moreover, it
can be generated by an element $D\in F^*_+$ such that, as it was shown in
Corollary \ref{bij}, the quaternion $\mu = c_1(\cL )^{-1}\in B_0$ satisfies
that $\mu ^2+D=0$.


Therefore, naturally attached to $(A, \cL )$ there is the
principally polarized order $(\cO , \mu )$.
In turn, as it was shown in Section \ref{ShimuraVarieties},
attached to $(\cO ,\mu )$ there is a Shimura
variety $X_{(\cO , \cI _{\vartheta }, \mu )}/\Q $
which solves the coarse moduli
problem of classifying triplets principally polarized abelian varieties
$(A, \iota , \cL )$ with a monomorphism of rings
$\iota : \cO \hookrightarrow \End (A)$
satisfying a certain compatibility between $\cL $ and $\mu $ (cf.
Section \ref{ShimuraVarieties} for details).

The motivation for recalling this Shimura variety at this point is the
following standard modular interpretation of the field of moduli $k_{\cO }$.


\begin{proposition}

The field of moduli $k_{\cO }$ of the
quaternionic multiplication on the principally polarized abelian variety $(A, \cL
)$ is

$$
k_{\cO } = \Q (P),
$$
the extension over $\Q $ generated
by the coordinates of the point $P=[A, \iota , \cL ]$ on
Shimura's canonical model $X_{(\cO , \cI _{\vartheta }, \mu )}/\Q $ that
represents the $\qbar $-isomorphism class of the triplet.

\end{proposition}

A similar construction holds for the totally real subalgebras of $B$. Indeed,
let $L/F$ be a totally real quadratic extension of $F$ and
$\varphi :L\hookrightarrow B$ an immersion of $L$ in $B$. Then $S =
\varphi ^{-1}(\varphi (L)\cap \cO )$ is an order of $L$ over $R_F$ which is
optimally embedded in $\cO $. By identifying $S$ with a subring of the ring
of endomorphisms of $A$, we again have that

$$
k_S  = \Q (P_{|S})
$$
is the extension over $\Q $ generated by the coordinates of
the point $P_{|S} = [A, \iota _{|S}, \cL ]$ on the
Hilbert variety $\cH _S/\Q $ that solves the coarse moduli
problem of classifying abelian varieties of dimension $2 n$ with
multiplication by $S$.

Along the same lines, the field of moduli $k_{R_F}$ of the central
endomorphisms of $A$ is the extension $\Q (P_{|{R_F}})$ of $\Q $
generated by the coordinates of the point $P_{|{R_F}} = [A, \iota
_{|{R_F}}, \cL ]$ on the Hilbert variety $\cH _F/\Q $
which solves the coarse moduli problem of classifying abelian
varieties of dimension $2 n$ with multiplication by $R_F$.

Note that there are infinitely many choices of totally real
quadratic orders $S$ in $\cO $ over $R_F$.

The tool for studying the Galois extensions $k_{\cO }/k_{S}/k_{R_F}$ is
provided by the forgetful modular maps


$$
\begin{matrix}
\pi _F: & X_{(\cO , \cI _{\vartheta }, \mu )} &
\stackrel {\pi _{S, \varphi }}{\longrightarrow }&
\cH _S & \longrightarrow  & \cH _F \\
&  P  &\mapsto & P_{|S} & \mapsto & P_{|R_F}.\\
\end{matrix}
$$

It was shown in Theorem \ref{mainGSV} that $\pi _F$ and $\pi _{S, \varphi }$
are quasifinite maps over $\Q $. Let us identify $S$ with its image $\varphi
(S)$ in $\cO $. Let $V_0(S)$ denote the Atkin-Lehner subgroup of
twisting involutions in $S$ and let $W_0$ denote the Atkin-Lehner
stable subgroup introduced in Section \ref{twstable}. Both are $2$-torsion finite
abelian subgroups of the positive Atkin-Lehner group $W^1$. Furthermore, it was
proved that, up to a
birational equivalence over $\Q $  which
is biregular away of a finite set of Heegner points, these
maps are the natural projections

$$
X_{(\cO , \cI _{\vartheta }, \mu )}
\ra X_{(\cO , \cI _{\vartheta }, \mu )}/V_0(S)\ra X_{(\cO , \cI _{\vartheta },
\mu )}/W_0.
$$

In consequence, the
Galois group $G=\Gal (k_{\cO }/k_{R_F})$ of the extension of
fields of moduli $k_{\cO }/k_{R_F}$ is naturally embedded in
$W_0$: any $\sigma \in G$ acts on a principally polarized abelian
variety with quaternionic multiplication
$(A, \iota , \cL )$ by leaving the $\qbar $-isomorphism class of
$\pi _F(A, \iota , \cL )=(A, \iota _{|R_F}: R_F\hookrightarrow
\End _{\qbar }(A), \cL )$ invariant. Similarly, $\Gal (k_{\cO
}/k_{S})$ embeds in $V_0(S)$ for any totally real order $S$
embedded in $\cO $.

As a corollary of Theorem \ref{finitedeg}, the
above discussion yields the main result of this section.

\begin{theorem}
\label{mainFM}

Let $(A, \cL )/\qbar $ be a principally polarized abelian variety
of dimension $2 n$ such that $\End (A)\simeq \cO $ is a maximal
order in a totally indefinite division quaternion algebra $B$ over
a totally real number field $F$ of degree $[F:\Q ]=n$.

\begin{enumerate}
\item[(i)]
If $(\cO , \mu)$ is non twisting, then
$k_{\cO }=k_S$ for any totally real quadratic order $S\subset \cO $ over $R_F$
and

$$
\Gal (k_{\cO }/k_{R_F})\subseteq C_2^{\omega _{odd}}.
$$

\item[(ii)]
If $(\cO , \mu)$ is twisting, let
$\{ \chi _1, ..., \chi _{s_0}\} $ be representatives of the finite set of twists of
$(\cO , \mu )$ up to multiplicative elements in $F^*$.
Then, for any real quadratic order $S\subset \cO $, $S \not \subset F(\chi
_i)$, $1\leq i \leq s_0$,

$$
k_{\cO }=k_S.
$$

On the other hand, $k_{\cO }/k_{S_i}$ is (at most) a quadratic extension
for any totally real order $S\subset F[\chi _i]$.

Moreover, $k_{\cO }= k_{S_1}\cdot ...\cdot k_{S_{s_0}}$ and

$$
\Gal (k_{\cO }/k_{R_F})\subseteq C_2^{2 \omega _{odd}}.
$$

\end{enumerate}
\end{theorem}

{\em Proof. } The non twisting claim $(i)$ follows from Theorem
\ref{finitedeg} and the discussion previous to it. Let us assume that $(\cO , \mu
)$ is a twisting pair. If $S\not \subset F(\chi _i)$ for any twist $\chi _i$,
$i=1, ..., s_0$, of $(\cO , \mu )$, $V_0(S)$ is trivial and hence, by Theorem
\ref{finitedeg}, $\Gal (k_{\cO }/k_S)$ is also trivial. If, on the
other hand, $S=R_F[\chi _i]$ (or any other order in $F(\chi
_i)\cap \cO $), then $V_0(S)\simeq C_2$ is generated by the
twisting involution associated to $\chi _i$. Again, we deduce that
in this case $k_{\cO }/k_S$ is at most a quadratic extension.

With regard to the last statement, note that $U_0\supseteq \langle
[\mu ]\rangle $ is at least of order $2$. It then follows from Lemma \ref{nontw}
that there exist two noncommuting twists $\chi $, $\chi '\in \cO
$. Then $R_F[\chi , \chi ']$ is a suborder of $\cO $ and, since
they both generate $B$ over $\Q $, the fields of moduli $k_{\cO }$
and $k_{R_F[\chi , \chi ']}$ coincide. This shows that $k_{\cO
}\subseteq k_{S_1}\cdot ...\cdot k_{S_{s_0}}$. The converse
inclusion is obvious.

Finally, we deduce from Theorem
\ref{finitedeg} and Lemma \ref{lemmatw} that $k_{\cO }/k_{R_F}$ is a $(2, ...,
2)$-extension of degree at most $2^{2 \omega _{odd}}$. $\Box $

Theorem \ref{mainFM} combine with the
following theorem of Shimura \cite{Sh75} to determine in many cases the Galois
groups $\Gal (k_{\cO }/k_{R_F})$ and $\Gal (k_{\cO }/k_{S})$
completely.


\begin{theorem}\label{Sh75}
The field of moduli $k_{\cO }$ of totally indefinite qua\-ter\-nio\-nic
mul\-tiplication
on an abelian variety is totally imaginary.

\end{theorem}

\begin{corollary}
\label{quad}

Let $(\cO , \mu )$ be a non twisting principally polarized maximal order and
assume that $F(\sqrt {-D})$ is a CM-field with no purely imaginary roots of
unity. Then, for any real quadratic order $S$ over $R_F$,
$k_{\cO }/k_{R_F}=k_{S}/k_{R_F}$ is at most a quadratic extension.

If, in addition, $k_{R_F}$ admits a real embedding, then $k_{\cO }$
is indeed a quadratic extension of $k_{R_F}$.

\end{corollary}


\begin{remark} In the twisting case, the field of moduli of quaternion
multiplication is already generated by the field of moduli of any
maximal real commutative multiplication but for finitely many
exceptional cases. This homogeneity does not occur in the
non twisting case. \end{remark}

In view of Theorem \ref{mainFM}, the shape of the
fields of moduli of the endomorphisms of the polarized abelian
variety $(A, \cL )$ differs considerably depending on whether it
gives rise to a twisting polarized order $(\cO , \mu )$ or not. For more details on
this question, we
refer the reader to the discussion at the end of Section \ref{theorems}
and Section \ref{distance}


\chapter{Arithmetic
of abelian surfaces with quaternionic multiplication} \label{AAV}

\section*{Introduction}\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{AAV}. Arithmetic of abelian surfaces with QM}{\ref{Intro3}
Introduction}\label{Intro3}


In this chapter we provide an in\-sight to the arithmetic of
abelian surfaces with quaternionic multiplication over number
fields. Whereas a lot of research has been carried out in the last
century on abelian varieties with complex or totally real
multiplication, it is remarkable that there is very little
knowledge on the arithmetic of abelian varieties with QM.
Consequently, we will be concerned with some of the fundamental
questions that firstly arise in this regard.

In Section \ref{FD}, we review some general results due to
Silverberg and Ribet and we investigate the Galois theoretical
properties of the minimal field of definition $K_B$ of the
quaternionic multiplication on an abelian surface $A/K$ over a
number field $K$. In Theorem \ref{CF} of Section \ref{end}, we
firstly prove a very general result that holds for arbitrary
abelian varieties whose ring of endomorphisms is an order in an
arbitrary quaternion algebra over a field $F$. Subsequently, in
Theorems \ref{LL} and \ref{L} of Section \ref{nes}, we prove a
more precise statement for abelian surfaces. In order to
accomplish that, we study Galois representations on the ring of
endomorphisms $\End _{\qbar }(A)$ and on the N\'{e}ron-Severi group
$\NS (A_{\qbar })$ of these abelian varieties.

In Section
\ref{JacSurf}, we combine our results with those of Mestre and Jordan to
make the relationship between the field of moduli and field of
definition of the quaternionic multiplication
explicit for the case of Jacobian surfaces.

Finally, we consider in Section \ref{examplesHashi} several concrete examples of
curves $C/K$ of genus two over a number field such that their Jacobian
varieties $J(C)/K$ have quaternionic multiplication over $\bar K$ due
to Hashimoto, Mu\-ra\-ba\-ya\-shi and Tsunogai and we explore their diophantine
properties.

The results presented in this chapter are partly contained in
\cite{Ro4}, \cite{DiRo1} and \cite{DiRo2}.


\markright{\ref{FD} The field of definition of the quaternionic multiplication}
\section{The field of definition of the quaternionic multiplication on an abelian
surface}
\markright{\ref{FD} The field of definition of the quaternionic multiplication}
\label{FD}

Fix $\qbar \subset \C $ an algebraic closure of the field $\Q $ of
rational numbers and let $K\subset \qbar $ be a number field. The
following result is due to Ribet \cite{Ri75}.

\begin{proposition}
\label{teo:SiRi}
Let $A/K$ be an abelian variety over $K$ and let
$S \subseteq \End _{\qbar }(A)$ be a subring of endomorphisms of
$A$. Then there is a {\em unique minimal extension} $K_S/K$ such
that $S \subseteq \End _{K_S}(A)$.

The extension $K_S/K$ is normal and unramified at the prime ideals
of $K$ of semistable reduction of $A$.

\end{proposition}


We remark that the minimal field of definition of a subring $S$ of endomorphisms of
$A$ only depends on the algebra $L=S\otimes \Q \subseteq \End _{\qbar }(A)\otimes
\Q $ and not on
the particular choice of the integral order sitting in $L\cap \End _{\qbar
}(A)$. Hence, we will often denote it simply by $K_L=K_S$.

Let now $F$ be a totally real number field and let $R_F$ denote
its ring of integers. Let $\cO $ be a maximal order over $R_F$ in
a totally indefinite quaternion algebra $B$ over $F$. Let $A/K$ be
an abelian variety of dimension $2[F:\Q ]$ and assume that there
is an isomorphism $\iota : \cO \stackrel {\sim }{\rightarrow }
\End _{\qbar }(A)$.

According to Definition
\ref{defquat}, we say that $A/K$ is an abelian variety with quaternionic
multiplication by $\cO $, although attention must be paid to the fact
that the quaternion multiplication need not be defined over the field
$K$ itself.

In regard to Proposition \ref{teo:SiRi}, we remark that $A/K$ has
potential good reduction and therefore no places of $K$ of bad
reduction of $A$ are semistable. This is a consequence of
Grothendieck's Potential Good Reduction Theorem.

For any algebra $L\subseteq B$ over $\Q $ and in accordance to the above, we let
$K_L/K$ denote the minimal field of definition of the ring of endomorphisms
$S=L\cap \End _{\qbar }(A)$ on $A$ over $K$. Note that we have $K_{\Q }=K$.

With particular interest on abelian surfaces, it is the aim of
this section to study

\begin{enumerate}
\item
The field extension $K_B/K$ given by the minimal field of definition $K_B$ of
the quaternionic multiplication on $A$ over $K$.

\item
The filtration of intermediate endomorphism algebras $F\subseteq \End
_{\tilde K}(A)\otimes \Q \subseteq B$ for any subfield $K_F\subseteq
\tilde {K}\subseteq K_B$.
\end{enumerate}

The first question has been studied in a more general context by
several authors. In \cite{Si}, Silverberg also gave an explicit
upper bound for the degree $[K_B:K]$ in terms of certain
combinatorial numbers. In the particular case of abelian surfaces
with quaternionic multiplication, \cite{Si}, Proposition 4.3,
predicts that $[K_B:K]\leq 48$. As our results will show, these
bounds are not sharp: see Theorem \ref{CF} for arbitrary orders
$\cO $ and Theorem \ref{LL} for maximal orders.

The study of the Galois extension $K_B/K$ can be divided into that of the
extensions $K_F/K$ and $K_B/K_F$. The former have been
studied by several authors in the
totally real case. See \cite{Ri80} and \cite{Ri94}) for details. Here
there are some known general facts.

\begin{proposition}

Let $A/K$ be an abelian variety over a number field $K$ and assume
that the algebra of endomorphisms $\End _{\qbar }(A)\otimes \Q $
of $A$ over $\qbar $ contains a totally real number field $F$ of
arbitrary degree $[F:\Q ]=n$. Then there is a monomorphism of
groups $\Gal (K_F/K)\hookrightarrow \Aut (F/\Q )$ and, in
particular, $[K_F:K]\leq n$.

\end{proposition}

Therefore, we will focus ourselves on the extension $K_B/K_F$ given by
the field of definition of the quaternionic multiplication on
an abelian variety $A$ over $K_F$.


We devote next two
subsections to carefully state and prove our results. This is accomplished
by studying the absolute Galois action
on the ring of endomorphisms and on the N\'{e}ron-Severi group of an abelian
variety, respectively.

\subsection{The action of $\Gal ({\qbar /K})$ on the endomorphism ring.}
\label{end}

In this section we use Chinburg-Friedman's recent classification
of the finite subgroups of maximal arithmetic Kleinian groups
in \cite{ChFr00F} to describe the field of definition of the
quaternionic multiplication on an abelian variety. We will be able to
present a result that holds true for
abelian varieties whose ring of endomorphisms is an arbitrary order $\cO $ in
an arbitrary quaternion algebra $B$ over an arbitrary field $F$.

Let $A/K$ be an abelian variety over a number field $K$ and assume that
$\End _{\qbar }(A)\simeq \cO $ is an order in a quaternion algebra $B$ over a
field $F$. By Theorem \ref{Albert}, we know
that $F$ is either a totally real field or a CM-field. Moreover, if $F$
is totally real, $B$ is either totally definite or totally indefinite over $F$ and
$2 [F:\Q ]\vert \dim (A)$.

Let $K_F$ be the minimal field of definition of
the central endomorphisms on $A$. The
absolute Galois group $G_{K_F}=\Gal (\qbar /K_F)$ acts in a
natural way on the full ring of endomorphisms $\End _{\qbar
}(A)=\cO $ and induces a Galois representation

$$
\gamma :G_{K_F}\lra \Aut (\cO ).
$$

By Skolem-Noether's Theorem \ref{Sk}, for any $\tau \in G_{K_F}$ there exists
an automorphism $[\gamma _{\tau }]:B\ra B$ of $B$ such that $\beta ^{\tau } =
\gamma _{\tau }^{-1}\beta \gamma _{\tau }$ for any $\beta \in \End
_{\qbar }(A)=\cO $. We obtain an exact sequence of groups

$$
1\ra G_{K_B}\ra G_{K_F}\ra \Nm _{B^*}(\cO )/F^*
$$
and thus a monomorphism $\Gal (K_B/K_F)\hookrightarrow \Nm _{B^*}(\cO )/F^*$.

The next theorem easily follows from the work of Chinburg anf Friedman
\cite{ChFr00F}.

\begin{theorem}
\label{CF}

Let $A/K$ be an abelian variety such that $\End _{\qbar }(A)\simeq
\cO $ is a non necessarily maximal order in an arbitrary
quaternion algebra $B$ over a field $F$. Let $K_B$ (respectively
$K_F$) be the minimal extension of $K$ over which all
(respectively central) endomorphisms of $A$ are defined.

Then $K_B/K_F$ is a Galois extension with $\Gal (K_B/K_F)$ isomorphic to
either a
cyclic group, a dihedral group, the alternating groups
$A_4$, $A_5$ or the symmetric
group $S_4$.

\begin{enumerate}

\item
If $\Gal (K_B/K_F)\simeq C_n$ or $D_n$ is either cyclic of order $n$ or a
dihedral group of order $2 n$, then there exists
a $n$-th primitive root of unity $\zeta _n\in B^*$.


\item
If $\Gal (K_B/K_F)\simeq A_4$ or $S_4$, then $B\simeq (\frac {-1,
-1}{F})$ is the totally definite Hamilton's quaternion algebra.


\item
If $\Gal (K_B/K_F)\simeq A_5$, then $B\simeq (\frac {-1,
-1}{F})$ and $\sqrt {5}\in F$.

\end{enumerate}


\end{theorem}

{\em Proof. } As it was already observed, $\Gal (K_B/K_F)$ is a finite
subgroup of $\Nm _{B^*}(\cO )/F^*\subset B^*/F^*$.
Chinburg and Friedman \cite{ChFr00F} proved that the only
possible finite subgroups of $B^*/F^*$ are the cyclic groups
$C_n$, the dihedral groups $D_n$ and $S_4$, $A_4$ and $A_5$. By
\cite{ChFr00F}, Lemma 2.8, a necessary condition for $B^*/F^*$
to contain either $S_4$ or $A_4$ is that $B=(\frac {-1,
-1}{F})$. Also, if $B^*/F^*$ contains a finite subgroup isomorphic
to the alternating group $A_5$, then $B=(\frac {-1, -1}{F})$ and
$F\supseteq \Q (\sqrt {5})$.

Moreover, Lemma 2.1 in \cite{ChFr00F} shows that $B^*/F^*$ contains a cyclic
group of order $n>2$ if and only if there exists $\zeta _n\in B^*$
satisfying $\zeta _n ^n = 1$, $\zeta _n ^{n/d}\not =1$ for any
proper divisor $d$ of $n$. In this case, any subgroup
$C_n\subseteq B^*/F^*$ is conjugate to $\langle [1+\zeta
_n]\rangle $.
Moreover, Chinburg and Friedman prove that $B^*/F^*$ contains a
dihedral subgroup $D_n$, $n\geq 2$, if and only if it contains a
cyclic group $C_n$ (cf.\,\cite{ChFr00F}, Lemma 2.3). If $n=2$, any
subgroup of $B^*/\Q ^*$ isomorphic to $D_2=C_2\times C_2$ is of
the form $\langle [x], [y]\rangle \subset B^*/F^*$ with $x, y\in
B^*$, $x^2=d$, $y^2=m$, $x y = -y x$ for some $d, m\in F^*$. This
yields the proof of the theorem.
$\Box $

\begin{corollary}
\label{absurf}

Let $A/K$ be an abelian surface with quaternionic multiplication
by a non necessarily maximal order $\cO $ in a quaternion algebra
$B$. Then $\Gal (K_B/K)\simeq C_n$ or $D_n$ with $n=1, 2, 3, 4$ or
$6$.

\end{corollary}


{\em Proof.} Since $B$ is an indefinite quaternion algebra over
$\Q $, we have that $F=\Q $ and $K_{\Q } = K$. By the theorem
above, $\Gal (K_B/K)\simeq C_n$ or $D_n$ for some $n\geq 1$ such
that there exists a $n$-th primitive root of unity $\zeta _n\in
B^*$. However, any $\zeta _n\in B^*$ generates either a trivial or
a quadratic field extension $\Q (\zeta _n)/\Q $ and this is only
possible for $n=1, 2, 3, 4$ and $6$. $\Box $

\subsection{The action of $\Gal (\qbar /K)$ on the N\'{e}ron-Severi group
of abelian surfaces}\label{nes}

In this subsection we prove a more accurate result on the field of
definition of the quaternionic multiplication on an abelian surface.
Moreover, we show that there are very
restrictive conditions for a quadratic field
$\Q (\sqrt{d})$ to be realized as the algebra of endomorphims
$\End _K^0(A)$ of an abelian surface
$A/K$ over a number field with quaternionic multiplication over $\qbar $.



Let $\cO $ be a maximal order in an indefinite division quaternion
algebra $B$ over $\Q $. Assume that $A$ is an abelian surface
defined over a number field $K$ together with an isomorphism of
rings $\iota :\cO \stackrel {\sim } {\ra }\End _{\qbar }(A)$. In
Chapter \ref{GAV}, the absolute N\'{e}ron-Severi group $\NS (A_{\C
})\simeq \NS (A_{\qbar })$ was largely studied: it was seen that
the first Chern class allows us to regard $\NS (A_{\C })$ as a
sub-lattice of the $3$-dimensional vector space $B_0$ of pure
quaternions of $B$ in a way that fundamental properties of
invertible sheaves $\cL $ on $A$ can be interpreted in terms of
the arithmetic of $B$. This is the case of the degree $\deg (\cL
)$, the behaviour under pull-backs by endomorphisms, the index
$i(\cL )$ and the ampleness of an invertible sheaf. We summarize
these results in the following statement; see Theorem \ref{NeSe},
Proposition \ref{degree} and Theorem \ref{index} for more details.

\begin{theorem}
\label{Ro2}

Let $A/\qbar $ be an abelian surface with $\End
_{\qbar }(A) \stackrel {\iota }{\simeq } \cO $ a maximal order
in a quaternion algebra of discriminant $D$. Then $A$ is principally polarizable and
there is an isomorphism of additive groups

$$
\begin{matrix}
c_1: & \NS (A_{\qbar }) & \ra & \cO _0^{\sharp } \\
     & \cL             & \mapsto & c_1(\cL ) \\
\end{matrix}
$$
such that
\begin{enumerate}
\item
$\deg (\cL ) = D\cdot \n (c_1(\cL ))$.

\item
For any endomorphism $\alpha \in \cO =\End _{\qbar }(A)$,
$c_1(\alpha ^*(\cL )) = \overline {\alpha }c_1(\cL )\alpha $.

\item
An invertible sheaf $\cL \in \NS (A_{\qbar })$ is a polarization
if and only if $\n (c_1(\cL ))>0$ and $\det (\nu _{\cL })>0$ where
$\nu _{\cL }\in \GL _2(\R )$ is any matrix such that $\nu _{\cL
}^{-1}\cdot c_1(\cL )\cdot \nu _{\cL }\in \Q ^* \begin{pmatrix} 0
& 1 \\ -1 & 0 \end{pmatrix}$.

\end{enumerate}
\end{theorem}

We recall from Section \ref{QA} that
$\cO ^{\sharp }=\{ \beta \in B: \tr (\cO \beta )\subseteq \Z
\}$ denotes the codifferent ideal of $\cO $ in $B$. By $\cO _0
^{\sharp }$ we mean the subgroup $\cO ^{\sharp }\cap B_0$ of pure
quaternions of $\cO ^{\sharp }$. For our purposes in this section, we
only need to know that it is a lattice in $B_0$ and in particular
the Picard number $\rho (A_{\qbar })=3$.

Since $A_K$ is always attached with a nonnecessarily principal
polarization, we have that $1\leq \rho (A_K)\leq 3$. Both three cases are
possible and each possibility has a direct translation in terms of the algebra of
endomorphisms:

\begin{itemize}

\item
$\rho (A_K)=1$ if and only if $\End _K^0(A)$ is $\Q $ or an imaginary quadratic field,

\item
$\rho (A_K)=2$ if and only if $\End _K^0(A)$ is a real quadratic field
and

\item
$\rho (A_K)=3$ if and only if $\End _K^0(A)=\End _L(A)=B$.

\end{itemize}

The proof of these facts follows from \cite{Mu} and \cite{LaBi}.

We now consider the action of the Galois group $G_K = \Gal (\qbar
/K)$ on $\NS (A_{\qbar })$ given by $\cL (\Theta )^{\tau } = \cL
(\Theta ^{\tau })$ for any invertible sheaf $\cL $ on $A$ represented by a Weil
divisor $\Theta $ and $\tau \in G_K$. From Theorem \ref{Ro2}, any
automorphism of $\NS (A_{\qbar })$ can be regarded as a linear
automorphism of $B_0$. Moreover, since the Galois action conserves
the degree of invertible sheaves  and the first Chern class is a
monomorphism of quadratic modules $c_1:(\NS (A_\qbar ), \deg
)\hookrightarrow (B_0, D\cdot \n )$, we obtain a Galois
representation

$$
\begin{matrix}
\eta : & G_K & \lra & \Aut (\NS (A_{\qbar }),
\deg )&\subset &\Aut (B_0, D\cdot \n )\\
       & \tau &\mapsto & & &\eta _{\tau }\\
\end{matrix}
$$

We have that

\begin{enumerate}
\item
For any $\alpha \in \cO =\End _L(A)$, $(\alpha ^*(\cL )^{\tau }) =
(\alpha ^{\tau })^*(\cL ^{\tau })$.

\item
The index $i(\cL )$ only depends on the $G_K$-orbit of $\cL $,
that is, $i(\cL ^{\tau})=i(\cL )$, for any $\tau \in G_K$. In
particular $\cL ^{\tau }$ is a polarization if and only if $\cL $
is.

\end{enumerate}

The following relates the Galois actions on $\End _{\qbar }(A)$
and on the N\'{e}ron-Severi group $\NS (A_{\qbar })$ by means of a
reciprocity law.

\begin{theorem}
\label{law}

Let $A/K$ be an abelian surface with quaternionic multiplication by a maximal order
$\cO $ of discriminant $D$ in a quaternion algebra $B$.

Let

$$
\begin{matrix}
\gamma :& G_K & \ra & \Aut (\End _{\qbar }(A))\\
        &\tau  & \mapsto &   \\
        &     &     &  {\begin{matrix} [\gtau ]: &\cO &\ra &\cO \\ &\beta
        &\mapsto &\gtau ^{-1}\beta \gtau \end{matrix}}
\end{matrix}
$$
be the action of $\Gal (\qbar /K)$ on the ring of endomorphisms of
$A$. Define $\etau  = \sign (\n (\gtau ))\in \{ \pm 1\} $. Then

$$
c_1(\cL ^{\tau }) = \etau \cdot \gtau ^{-1} c_1(\cL ) \gtau
$$
for any invertible sheaf $\cL \in \NS (A_{\qbar })$ and any $\tau
\in G_K$.


\end{theorem}

{\em Proof. } Fix $\tau \in G_K$. We firstly claim that $\eta
_{\tau }:$ $B_0\ra $ $B_0$ is given by $\mu \mapsto \et \gt
^{-1}\mu \gt $ for some $\et =\pm 1$, $\gt \in B^*$. Indeed, any
linear endomorphism of $B_0$ extends uniquely to an endomorphism
of $B$ and $\End (B)\simeq B\otimes B$ with $\gt _1\otimes \gt
_2:B\ra B$, $\beta \mapsto \gt _1\beta \gt _2$. We must have in
addition that $\tr (\gt _1 \mu \gt _2)=\tr (\gt _2 \gt _1 \mu )=0$
for any $\mu \in B_0$. This automatically implies that $\gt _2 \gt
_1\in \Q $.

Since the action of $G_K$ on $\NS (A_{\qbar })$ conserves the
degree of invertible sheaves , we deduce from Theorem \ref{Ro2}
that
$$
\n (\eta _{\tau }(\mu )) = \n (\gt _1 \mu \gt _2) =\n (\gt _1) \n (\mu
) \n (\gt _2) = \n (\mu )
$$
for all $\mu \in B_0$. Hence $\n (\gt
_2)= \n (\gt _1)^{-1}$ and thus $\gt :=\gt _2 = \et \gt _1^{-1}$
for some $\et =\pm 1$. This proves the claim.

We now show that $\gt = \gtau \in B^*/\Q ^*$ and $\et =\etau $. We
know that $(\alpha ^*(\cL )^{\tau }) = (\alpha ^{\tau })^*(\cL
^{\tau })$ for any $\alpha \in \cO $. Taking Theorem \ref{Ro2}
into account, this implies that $\eta _{\tau }(\overline {\alpha }\mu
\alpha ) = \overline {[\gtau ](\alpha )} \eta _{\tau }(\mu ) [\gtau
](\alpha )$ and thus $\et \gt ^{-1}(\overline {\alpha } \mu \alpha )
\gt  = \et (\overline {\gtau ^{-1}\alpha \gtau }) \gt ^{-1}\mu \gt
(\gtau ^{-1}\alpha \gtau ) $ for any $\alpha \in B$, $\mu \in $
$B_0$. Choosing $\alpha = \mu $ and bearing in mind that $\gtau
^{-1}=\overline {\gtau }\n (\gtau )^{-1}$, this says that $\gt ^{-1}
\mu \gt = \gtau ^{-1} $ $\mu ^{-1} \gtau \gt ^{-1} \mu \gt $ $\gtau
^{-1} \mu \gtau $ and thus $\mu (\om \mu \om ^{-1})=(\om \mu \om
^{-1}) \mu $, where we write $\om = \gtau \gt ^{-1}$. The
centralizer of $\Q (\mu )$ in $B$ is $\Q (\mu )$ itself and
therefore $(\om \mu \om ^{-1})\in \Q (\mu )$. But $\tr (\mu )=\tr
(\om \mu \om ^{-1})=0, \n(\mu )=\n (\om \mu \om ^{-1})$ and this
implies that $\mu = \pm \om \mu \om ^{-1}$. Since this must hold
for any $\mu \in B_0$, it follows that $\om \in \Q ^*$ and thus
$\gt = \gtau \in B^*/\Q ^*$ as we wished.

We then already have that $\eta _{\tau }:$ $B_0\ra B_0$ is given
by $\mu \mapsto \et \gtau ^{-1}\mu \gtau $ for some $\et =\pm 1$.
If $\mu = c_1(\cL )$ for a polarization $\cL $ on $A$, this means
that $c_1(\cL ^{\tau })= \et \gtau ^{-1}\mu \gtau $. Since $\cL
^{\tau }$ is still an ample invertible sheaf we have, according to
Theorem \ref{Ro2}, \S 3, that $\et = \sign (\n (\gtau ))$. $\Box $


In Section \ref{twstable}, we introduced the {\em twists} and {\em principal twists}
of a polarized order in a quaternion algebra. In the case of a rational
indefinite quaternion algebra, let us recall and refine our definitions as
follows.

\begin{definition}

Let $\cO$ be a maximal order in an indefinite
quaternion algebra $B$ over $\Q $ and
let $D=\disc (B)$. We say that $\cO $ admits a {\em twist of
degree} $\delta \geq 1$ and {\em norm} $m\in \Z $, $m\vert D$,
if

$$
B = \Q +\Q i + \Q j + \Q i j = (\dfrac {-D \delta , m}{\Q })
$$
with $i$, $j\in \cO $, $i^2=-D \delta $, $j^2=m$ and $i j = -j i$.

\end{definition}

For any positive integer $\delta \geq 1$, let
$\mathcal {N}_{\delta }=\{ m_1, ..., m_t\} $,
$0<m_i\vert D$, denote the (possibly empty) set of norms of the
twists of degree $\delta
$ on $\cO $. It is easy to show that the set $\mathcal {N}_1$ of norms of
principal twists is either $\mathcal {N}_1=\emptyset $ or
$\mathcal {N}_1=\{ m, D/m\} $ for some
$m\vert D$. In other cases, $\mathcal {N}_{\delta }$ can be larger.
Indeed, if $\delta =D$ for instance, then $\mathcal {N}_D$ is either
empty or equal to the set of sums of two squares $m=m_1^2+m_2^2$ that divide $D$. We
finally note that a
quaternion order $\cO $ can very well admit twists of several different
degrees.

In practice, the
computation of a finite number of Hilbert symbols suffices to decide whether a
given indefinite order is
twisting of certain degree $\delta $. Let us just
quote that a necessary and sufficient condition for $B$ to contain
a maximal order $\cO $ admitting a twist of degree $\delta
$ and norm $m$ is that $m>0$, $m\vert D=\disc (\cO )=\disc (B)$ and
that for any odd prime $p\vert D$: $m\not \in \F _p^{*^2}$ if $p\nmid m$
($D/m\not \in \F _p^{*^2}$ if $p\vert m$ respectively).

Examples of quaternion algebras with principally twisting maximal
orders are $B = (\frac {-6, 2}{\Q }  )$ and $B = (\frac {-10,
2}{\Q } )$ of discriminant $D = 6$ and $D = 10$, respectively.


We can now prove the following result.

\begin{theorem}
\label{LL}

Let $A/K$ be an abelian surface defined over a number field $K$
such that $\End _{\qbar }\simeq \cO $ is a maximal order $\cO $ in a
quaternion algebra $B$  of
discriminant $D$ and let $K_B/K$ be the
minimal extension of $K$ such that $\End _{K_B}(A)\simeq \cO $. Fix a
polarization $\cL _0$ on $A_K$ and let $\delta =\deg (\cL _0)$ be
its degree.

\begin{enumerate}
\item[{\bfseries A}]
\begin{enumerate}
\item
If $\delta $ is not equal to $D$ neither to $3 D$ up to squares,
then $\Gal (K_B/K) \simeq \{ 1 \}$, $C_2$ or $D_2 = C_2\times C_2$.

\item
If $\delta = D k^2$ for some $k\in \Z $, then $\Gal (K_B/K)\simeq
C_n$ or $D_n$ with $n = 1$, $2$ or $4$.

\item
If $\delta = \frac {D k^2}{3}$ for some $k\in \Z $, then $\Gal
(K_B/K)\simeq C_n$ or $D_n$ for $n = 1$, $2$, $3$ or $6$.


\end{enumerate}

\item[{\bfseries B}] In any of the cases above, if $\cO $ does not admit
any twist of degree $\delta $, then $\Gal (K_B/K)$ is necessarily
cyclic.

\item[{\bfseries C}]
\begin{enumerate}
\item
If $\Gal (K_B/K)\simeq C_2$, then $\End _K^0(A)\simeq \Q (\sqrt {-D
\delta })$ or $\Q (\sqrt {m_i})$ for $m_i\in \mathcal
{N}_{\delta }$ a norm of a twist of degree $\delta $ on $\cO $.

\item
If $\Gal (K_B/K)=C_3$ or $C_6$, then $\End _K^0(A)\simeq \Q (\sqrt {-3})$.

\item
If $\Gal (K_B/K)=C_4$, then $\End _K^0(A)\simeq \Q (\sqrt {-1})$.

\item
If $\Gal (K_B/K)=D_n$, then $\End _K(A) \simeq \Q $.

\end{enumerate}
\end{enumerate}
\end{theorem}

{\em Proof. } Recall that, according to Theorem \ref{CF}, $\Gal
(K_B/K) \simeq C_n$ or $D_n$ with $n=2$, $3$, $4$ or $6$. Let $\cL
_0$ be a polarization on $A_K$ and let $\mu = c_1(\cL _0)\in \cO
_0$. It satisfies that $\mu ^2 + D \delta  = 0$ by Theorem
\ref{Ro2}, \S 1. Since $\cL _0\in \NS (A_{\qbar })^{G_K}$, it is
invariant under the action of $\Gal (K_B/K)$. Fix $\tau \in \Gal
(K_B/K)$ and let $\gtau \in B^*$ the quaternion associated to
$\tau $ in Section \ref{end}. Suitably scaling it, we can (and we
do) choose a representative in $\Nm _{B^*}(\cO )/\Q ^*$ such that
$\gtau \in \cO $ and $\n (\gtau )$ is a square-free integer. Then,
since $\gtau $ must normalize $\cO $, we know that $\n (\gtau
)\vert D$.

Since $\cL _0$ in $\NS (A_{\qbar })$ is $G_K$-invariant, it
follows from Theorem \ref{law} that $\mu =c_1(\cL _0) = c_1(\cL
_0^{\tau }) = \etau \gtau  ^{-1} \mu \gtau $.

If $\etau =-1$, then the above expression implies that $\mu \gtau
= -\gtau \mu $. Since $\tr (\mu \gtau )=\mu \gtau - \overline {\gtau
}\mu =-\tr (\gtau ) \mu \in \Q ^*$, we deduce that $\tr (\gtau
)=0$. This means that $\gtau ^2=m$ for some $m\vert D$ and $B = \Q
+ \Q \mu + \Q \gtau + \Q \mu \gtau = (\frac {-D \delta , m}{\Q
})$. The indefiniteness of $B$ forces $m$ to be positive. We
obtain that in this case $\langle [\gtau ] \rangle \simeq C_2$.

On the other hand, if $\etau =1$, then $\gtau \in \Q (\mu )\simeq
\Q (\sqrt {-D \delta })=\Q (\sqrt {-\underline {D \delta }})$,
where we let $\underline {D \delta }$ denote the square-free part
of $D \delta $. In this case, and bearing in mind that $\gtau $
must generate a finite subgroup of $B^*/\Q ^*$, we deduce that
either

\begin{itemize}
\item
$\gtau = \pm \sqrt {\underline {D \delta }/D \delta } \cdot \mu $
and hence $\underline {D \delta }\vert D $ and $\langle [\gtau
]\rangle \simeq C_2$,

\item
$\gtau = 1+\zeta _n$ for some $n^{\mbox{th}}$-primitive root of
unity $\zeta _n\in B^*$, $n=3$ or $6$, and hence $\underline {D
\delta } = 3$ and $\langle [\gtau ]\rangle \simeq C_n$ or

\item
$\gtau = 1+\zeta _4$ for some $4^{\mbox{th}}$-primitive root of
unity $\zeta _4\in B^*$ and hence $\underline {D \delta } = 1$ and
$\langle [\gtau ]\rangle \simeq C_4$.

\end{itemize}
We conclude that a necessary condition for $\Gal (K_B/K)$ to contain
a cyclic subgroup of order $n\geq 3$ is $\underline{D \delta } =
1$ or $3$ which amounts to saying that $\deg (\cL _0)=\delta $ is $D$
or $3 D$ up to squares respectively. Also, if $\deg (\cL _0)= D
k^2$, then necessarily $\Gal (K_B/K)\simeq C_n$ or $D_n$ with $n=1$,
$2$ or $4$ and an analogous statement holds if $\deg (\cL _0)= 3
D$ up to squares. Further, if $B\not \simeq (\frac {-D \delta ,
m}{\Q })$ for any $0<m\vert D$, then it follows from the discussion
above that $\epsilon _{\tau }=1$ for any $\tau \in \Gal (K_B/K)$
and, as a consequence, $\Gal (K_B/K)\subset \Q (\mu )^*/\Q ^*$.
Since the only finite subgroups of $\Q (\mu )^*/\Q ^*$ are cyclic,
the proof of parts {\bfseries A} and {\bfseries B} is completed.

As for part {\bfseries C}, assume first that $\Gal (K_B/K)=\langle
[\gtau ] \rangle \simeq C_2$. Then $\gtau \in B^*$ satisfies
$\gtau ^2=-\n (\gtau )\in \Q ^*$ and we already saw that the only
possibilities are, up to squares, $\n (\gtau ) = D \delta $ or
$m\in \mathcal {N}_ {\delta }$. In any of these cases, $\End
_K^0(A)=\{ \beta \in \End _{K_B}(A): \beta ^{\tau }=\beta \} = \{
\beta \in \End _{K_B}(A): \beta \gtau =\gtau \beta \} = \Q (\gtau )$
and this implies our first assertion of part {\bfseries C}.
Similarly, if $\Gal (K_B/K)=\langle 1+\zeta _n\rangle \simeq C_n$
with $n=3$, $4$ or $6$ then $\End _K^0(A)=\Q (1+\zeta _n)\simeq \Q
(\sqrt {-1})$ or $\Q (\sqrt {-3})$ depending on the cases.
Finally, if $\Gal (K_B/K)=\langle \gtau , \gamma _{\tau '}\rangle
\simeq D_n$ with $\langle \gtau \rangle \simeq C_n$ and $\langle
\gamma _{\tau '}\rangle \simeq C_2$, then $\End _K^0(A)=\{ \beta
\in \Q (\gtau ): \beta ^{\tau '}=\beta \} = \Q $. Here, the last
equality holds because it is not possible that $\gtau $ and
$\gamma _{\tau ' }$ commute. $\Box $

The following lemma may be useful in many situations in order to
apply Theorem \ref{LL}. It easily follows from Theorem \ref{Ro2}.

\begin{lemma}

Let $A/\qbar $ be an abelian surface with $\End (A)$ a maximal
order in a quaternion algebra of discriminant $D$. If there exist
prime numbers $p, q\vert D$ such that $p$ splits in $\Q (\sqrt
{-1})$ and $q$ splits in $\Q (\sqrt {-3})$, then no polarizations
on $A$ have degree $D k^2$ or $3 D k^2$ for any $k\in \Z $.

\end{lemma}


When particularized to the Jacobian variety of a curve,
Theorem \ref{LL} asserts the following.


\begin{theorem}
\label{L}

Let $C/K$ be a curve of genus $2$ defined over a number field $K$
and let $J(C)$ be its Jacobian variety. Assume that
$\End _{\qbar }(J(C))=\cO $ is a maximal order
in a division quaternion algebra $B$ of discriminant $D$.

Let $K_B/K$ be the minimal extension of $K$ over which all
endomorphisms of $J(C)$ are defined. Then

\begin{enumerate}

\item
$K_B/K$ is an abelian extension with $G = \Gal (K_B/K)\simeq (1)$,
$C_2$ or $D_2 = C_2\times C_2$, where $C_2$ denotes the cyclic
group of order two.


\item
If $B \not \simeq (\dfrac {-D, m}{\Q })$ for any $m\vert D$, then
$K_B/K$ is at most a quadratic extension of $K$. In this case, $\End
_K(A)\simeq \Q (\sqrt {-D})$.

\item
If $B = (\dfrac {-D, m}{\Q })$ for some $m\vert D$, then $\End
_K(A)$ is isomorphic to either $\cO $, an order in $\Q (\sqrt
{-D})$, $\Q (\sqrt {m} )$ or $\Q (\sqrt {D/m})$, or $\Z $. In each
case, we respectively have $\Gal (K_B/K)\simeq (1)$, $C_2$ and
$D_2$.


\end{enumerate}
\end{theorem}



\section{Fields of moduli and fields of definition of
Jacobian varieties of genus two curves}
\markright{\ref{JacSurf} Fields of moduli versus fields of definition}
\label{JacSurf}

Let $C/\qbar $ be
a smooth irreducible curve of genus $2$ and let $(J(C), \Theta
_C)$ denote its principally polarized Jacobian variety. Assume
that $\End _{\qbar }(J(C))=\cO $ is a maximal order in an
indefinite quaternion algebra $B$ over $\Q $ of reduced
discriminant $D=p_1\cdot ... \cdot p_{2 r}$. Recall that $\cO $ is
unique up to conjugation or, equivalently by the Skolem-Noether
Theorem, up to isomorphism.

Attached to $(J(C), \Theta _C)$ is the polarized order $(\cO , \mu
)$, where $\mu =c_1(\Theta _C)\in \cO $ is a pure quaternion of
reduced norm $D$. As we have seen, a necessary condition for $(\cO
, \mu )$ to be twisting is that $B\simeq (\frac {-D,m}{\Q })$ for
some $m\vert D$. The isomorphism occurs if and only if for any $p
\vert D$, the integer $m$ (respectively $D/m$) is not a
square mod $p$ if $p\nmid m$ (if $p
\vert m$, respectively).

In the rational case, the Atkin-Lehner and the positive
Atkin-Lehner groups coincide and $W=W^1=\{ \om _d: d\vert D\}
\simeq C_2^{2 r}$ is generated by elements $\om _d\in \cO $, $\n
(\om _d)=d\vert D$. Moreover, $U_0=\langle \om _D\rangle \simeq
C_2$.

If $(\cO , \mu )$ is a non twisting polarized order, then the
field of moduli of quaternionic multiplication $k_{\cO }$ is at most a
quadratic extension over the field of moduli $k_C$ of the curve $C$
by Theorem \ref{mainFM}.

On the other hand, if $(\cO , \mu )$ is twisting and $B=(\frac {-D, m}{\Q })$
for $m\vert D$, then $V_0=\{ 1, \om
_m, \om _{D/m},\om _D\} \simeq C_2^2$, where we can choose
representatives $\om _m$, $\om _{D/m}$ in $\cO $ such that $\mu
\om _m=-\om _m \mu $ and $\mu \om _{D/m}=-\om _{D/m} \mu $. Note
that, up to multiplication by non zero rational numbers, $\om _m$ and
$\om _{D/m}$ are the only
twists of $(\cO , \mu )$. We obtain by Theorem \ref{mainFM} that

\begin{itemize}

\item
$k_{\cO
}/k_C$ is at most a quartic abelian extension.

\item
$k_{\cO }=k_S$ for any real quadratic order $S\not \subset \Q (\om
_m)\simeq \Q (\sqrt {m})$ or $\Q (\om _{D/m})\simeq \Q (\sqrt
{D/m})$.

\item
$k_{\Z [\om _m]}$ and $k_{\Z [\om _{D/m}]}$ are at most
quadratic extensions of $k_C$ and these are such that $k_{\cO
}=k_{\Z [\om _m]}\cdot k_{\Z [\om _{D/m}]}$.
\end{itemize}


Mestre \cite{Me} studied the relation between the field of
moduli $k_C = k_{(J(C), \Theta _C)}$ of a curve of genus $2$ and its
possible fields of definition, under the hypothesis that the
hyperelliptic involution is the only automorphism on the curve.
Mestre constructed an obstruction in $\mathrm{Br }_2(k_C)$ for $C$
to be defined over its field of moduli. On identifying this
obstruction with a quaternion algebra $H_C$ over $k_C$, he showed
that $C$ admits a model over a number field $K$ such that $k_C\subseteq K$
if and only if $H_C\otimes K\simeq \M _2(K)$.

If $\Aut (C)\varsupsetneq C_2$, Cardona and Quer \cite{CaQu} have recently proved
that $C$ always admits a model over its field of moduli $k_C$.

Assume now, as before, that $\End _{\qbar}(J(C))\simeq \cO $ is a
maximal order in an indefinite division quaternion algebra $B$
over $\Q $. Let $K$ be a field of definition of $C$; note that,
since $\End _{\qbar }(J(C))\otimes \Q = B$ is division, $\Aut
(C)\simeq C_2$ and therefore $k_C$ may not be a
field of definition of the curve. Having made the choice of a
model $C/K$, there is a minimal (Galois) field extension $L/K$ of
$K$ such that $\End _L(J(C))\simeq \cO $. This gives rise to a
diagram of Galois extensions

\vspace{0.5cm}
\begin{picture}(160,60)

\put(300,10){\makebox(0,0){\bfseries }}
\put(122,60){\makebox(0,0){$K_B$}}
\put(100,40){\makebox(0,0){$k_{\cO }$}}
\put(140, 0){\makebox(0,0){$k_C$}}
\put(162,22){\makebox(0,0){$K$}}


\thicklines
\put(128,52){\line(1,-1){22}}
\put(107,47){\line(1,1){10}}
\put(110,30){\line(1,-1){22}}
\put(147,7){\line(1,1){10}}
\end{picture}
\vspace{0.5cm}

The nature of the Galois extensions $K_B/K$ was studied in
Section \ref{FD}, while the relation between the field of moduli
$k_{\cO }$ and the possible fields of definition $K_B$ of the
quaternionic multiplication was investigated by Jordan in
\cite{Jo86}. The combination of all these facts yields the following
statement.

\begin{proposition}

Let $C/K$ be a smooth curve of genus $2$ over a number field $K$
and assume that $\End _{\qbar }(J(C))$ is a maximal quaternion
order $\cO $. Let $K_B/K$ the minimal extension of $K$ over which
all endomorphisms of $J(C)$ are defined.

Then $\Gal (K_B/K)\simeq \Gal (k_{\cO }/k_C)\simeq \{ 1\}$, $C_2$ or
$D_2=C_2\times C_2$.


\end{proposition}

{\em Proof. } Assume first that $k_C=K$ is a field of definition
of the curve. Then $k_{\cO }=K_B$ is a field of definition of all
endomorphisms of $J(C)$: if it was not, there would be infinitely
many pairwise different extensions $L_{\alpha }/K$ such that $\End
_{L_{\alpha }}(J(C))=\cO $. This would contradict Silverberg's
result that states that such extensions are unique (cf.\,\cite{Si}).

If, on the contrary,
Mestre's obstruction $H_C$ is non trivial in $\mathrm{Br }_2(k_C)$,
then $C$ admits a model over any quadratic extension $K/k_C$ that splits $H_C$
but not over $k_C$ itself. We then have that any field of definition $K_B$
of all endomorphisms of $J(C)$ must be a quadratic extension of $k_C$
that strictly contains it. Indeed, by \cite{Jo81}, we know that $[K_B:k_C]\leq
2$. Suppose that $K_B=k_C$. Then, $K_B$ would contain all fields of
definition $H$ of $C$ and this is not possible.

In either case, the possibilities for $\Gal (K_B/K)$ are $\{1\}$,
$C_2$ and $D_2$, by Theorem \ref{mainFM}. $\Box  $

\begin{remark} The above argument actually yields more than this: it either holds
that $k_C\varsubsetneq K\varsubsetneq k_{\cO }\varsubsetneq K_B$ is
a chain of quadratic extensions or $K_B = k_{\cO }\cdot K$.
Moreover, the first of these possibilities only arises when
$[K_B:K]=4$ and for the finitely many subextensions $K/k_C$ of
$k_{\cO }$ such that $H_C\otimes _{k_C} K$ is trivial.
\end{remark}



\begin{example} Let $C$ be the smooth projective curve of hyperelliptic
model

$$
Y^2 = \frac{1}{48}X(9075X^4+3025(3+2 \sqrt{-3}) X^3 -6875 X^2 +220
(-3+ 2 \sqrt{-3}) X + 48).
$$

Let $A=J(C)/K$ be the Jacobian variety of $C$ over $K=\Q (\sqrt
{-3})$. By \cite{HaMu}, $A$ is an abelian surface with quaternionic
multiplication by a maximal order in the quaternion algebra of
discriminant $6$. See also Section \ref{Hash}. As it is explicitly
shown in \cite{HaMu}, there is an isomorphism between $C$ and the conjugate curve
$C^{\tau}$ over $\Q $. Hence, the field of moduli $k_C=\Q $ is the
field of rational numbers.



In addition, it was shown in \cite{DiRo1} that $K_B= \Q(
\sqrt{-3}, \sqrt{-11})$ is the minimal field of definition of the
quaternionic multiplication on $A$. By our last proposition and
remark, we must have that $K_B = K\cdot k_{\cO }$ with $[k_{\cO
}:k_C]=2$. In addition, by Shimura's Theorem \ref{Sh75}, $k_{\cO
}$ must be an imaginary quadratic extension of $\Q $. This shows
that $k_{\cO }=\Q (\sqrt {-11})$ and the picture of fields of
moduli and definition of $A$ is completed:

\vspace{3cm}

\sk

\sk

$\\ $

$\\ $

\begin{picture}(160,60)

\put(300,10){\makebox(0,0){\bfseries }}
\put(122,70){\makebox(0,0){$\Q(\sqrt{-3}, \sqrt{-11})$}}
\put(100,40){\makebox(0,0){$\Q (\sqrt {-11})$}}
\put(140, 0){\makebox(0,0){$\Q $}}
\put(162,32){\makebox(0,0){$\Q (\sqrt {-3})$}}


\thicklines
\put(128,60){\line(1,-1){20}}
\put(100,55){\line(1,1){10}}
\put(110,30){\line(1,-1){22}}
\put(147,7){\line(1,1){10}}
\end{picture}
\vspace{0.5cm}

\end{example}

\markright{\ref{examplesHashi} Examples of Jacobian varieties of genus two curves}
\section[Examples of Jacobian varieties
of genus two curves]{Explicit examples of Jacobian varieties
of genus two curves}
\markright{\ref{examplesHashi} Examples of Jacobian varieties of genus two curves}
\label{examplesHashi}

Let $K\subset \qbar $ be a number field in an algebraic closure $\qbar $ of the field
$\Q $ of rational numbers and let
$C/K$ be an irreducible smooth algebraic curve of genus $2$ over
$K$. Assume that $\End _{\qbar }(J(C))=\cO $ is a maximal order in an
indefinite division quaternion algebra $B$ of discriminant $D$. As it
was shown in Theorem
\ref{L}, the minimal field of definition $K_B$ of all endomorphisms of
$J(C)$ is either a trivial, a quadratic or a biquadratic extension of $K$.
Moreover, we have that

\begin{itemize}
\item
(Trivial case) If $K_B=K$, then $\End _K(J(C))=\cO $,

\item
(Quadratic case)
If $[K_B:K]=2$, then $\End _K(J(C))\otimes \Q = \Q (\sqrt {-D})$, $\Q (\sqrt
{m})$ or $\Q (\sqrt {D/m})$, for some positive proper divisor $m\vert D$,

\item
(Biquadratic case) If $[K_B:K]=4$, then $\End _K (J(C)) = \Z $.
\end{itemize}

In this section, we show that Theorem \ref{L} is sharp by exhibiting
several explicit examples which attain all the possibilities allowed by
the theorem. We obtain them by considering several particular fibres
of Hashimoto-Murabayashi's and Hashimoto-Tsunogai's
families of curves of genus two whose Jacobian varieties are abelian
surfaces with
quaternionic multiplication (cf.\,Section \ref{curves}, \cite{HaMu} and
\cite{HaTs}).

As a by-product, we also show that all computations performed in
\cite{HaTs} supporting an analogue of the Sato-Tate conjecture for
these abelian surfaces are unconditionally correct (cf.\,\cite{HaTs} for
details).


Being these examples part of a joint work with L. Dieulefait, we
omit the details of the proofs. Instead, we refer the reader to
\cite{DiRo2}.

\begin{example}[Biquadratic case]

\sk
{\bfseries I.} Let $C_1/\Q (\sqrt {2})$ be a smooth projective
model of the curve

$$
Y^2 = (X^2 +5) ( (-1/6 + \sqrt{2}) X^4 + 20 X^3 - 490/6 X^2 + 100
X + 25 (-1/6 - \sqrt{2})).
$$

Then, the Jacobian variety $J_1=J(C_1)/\Q (\sqrt {2})$ of $C_1$
has multiplication by a maximal order $\cO $ in the quaternion
algebra $B_6$ of discriminant $6$ over the quartic extension
$K_B=\Q(\sqrt{-2}, \sqrt{-1}, \sqrt{-5})$ of $\Q (\sqrt {2})$.
Moreover,

\begin{itemize}
\item
$\End _{\Q(\sqrt{2}, \sqrt{-5})}(J_1)\otimes \Q = \Q(\sqrt{2})$,


\item
$\End _{\Q(\sqrt{2}, \sqrt{5})} (J_1)\otimes \Q =\Q (\sqrt {3})$,

\item
$\End _{\Q(\sqrt{2}, \sqrt{-1})}(J_1)\otimes \Q = \Q (\sqrt
{-6})$ and

\item
$\End _{\Q(\sqrt{2})}(J_1)=\Z $.

\end{itemize}

\sk
{\bfseries II.} Let $C_2/\Q $ be a smooth projective model of the
curve

$$
Y^2 = (X^2 + 7/2) (83/30 X^4 + 14 X^3 - 1519/30 X^2 + 49 X -
1813/120)
$$
and let $J_2=J(C_2)/\Q $ be its Jacobian variety. Then,

\begin{itemize}
\item
$\End _{L}(J_2)=\cO $ is a maximal order in $B_6$ for
$L=\Q(\sqrt{-6}, \sqrt{-14})$,


\item
$\End _{\Q (\sqrt{-14})}(J_2)\otimes \Q = \Q(\sqrt{2})$,


\item
$\End _{\Q (\sqrt{21})} (J_2)\otimes \Q =\Q (\sqrt {3})$,

\item
$\End _{\Q (\sqrt{-6})}(J_2)\otimes \Q = \Q (\sqrt {-6})$ and

\item
$\End _{\Q }(J_2)=\Z $.

\end{itemize}

\end{example}

Note that $B_6 = (\frac {-6, 2}{\Q }) = (\frac {-6, 3}{\Q })$ is
a twisting quaternion algebra. Hence, the above two examples are in
perfect concordance with Theorem \ref{L}.


\begin{example}[Quadratic case]

Let $C_3$ be the smooth algebraic curve of genus $2$ of hyperelliptic model

$$
Y^2 = \frac{1}{48}X(9075X^4+3025(3+2
\sqrt{-3}) X^3 -6875 X^2 +220 (-3+ 2 \sqrt{-3}) X + 48).
$$

Let $J_3/K$ be the Jacobian variety of $C_3$ over $K=\Q (\sqrt
{-3})$. By \cite{HaMu}, the ring of endomorphisms of $J_3$ over
$\qbar $ is a maximal order in the quaternion algebra of
discriminant $D= 10$ over $\Q $. As is shown in \cite{DiRo2}, $L=
\Q( \sqrt{-3}, \sqrt{-11})$ is the minimal field of definition of
the quaternion endomorphisms of $J_3$ and

$$
\End_K(J_3)\otimes_\Z \Q = \Q(\sqrt{5}).
$$

\end{example}


Finally, let us conclude with a pair of examples of abelian surfaces
$A/K$ with quaternionic multiplication over a quadratic
imaginary field $K$ such that all
quaternion endomorphisms of $A$ are defined over $K$ itself. Obvious
examples can be
obtained by suitably extending the base field of the Jacobian varieties
of the curves in the above examples. Indeed, $J_3/\Q( \sqrt{-3},
\sqrt{-11})$ is an example of this phenomenon.

Below, we present two non trivial examples of abelian surfaces $A/K$
with $\End _K(A)$ a maximal quaternion order that cannot be obtained
by base extension from a subfield of $K$.


\begin{example} [Trivial case]


Let $C_4/K_4$, $C_5/K_5$ be the fibres of the Hashimoto-Tsunogai's
family $S_6(t,s)$ at the values $t_4= 2$ and $t_5=2/3$ over $K_4 =
\Q(\sqrt{-379})$ and $K_5=\Q(\sqrt{-19})$, respectively. See Section
\ref{Hash} for details. Then,
$\End _K(J(C_i))\simeq \cO $ is a maximal order in $B_6$.



The computation of the absolute Igusa invariants of
these curves show that there does not exist any curve $C/\Q $ such
that $C\simeq C_3$ nor $C\simeq C_4$ over $\overline \Q $. See
\cite{CaQu} for details.

\end{example}


\begin{remark} The above two examples are defined over completely
imaginary fields $L=K$. This is not a coincidence since, by a
result of Shimura (cf.\,\cite{Sh75}), there are not curves $C/K$ of
genus $2$ over a number field $K$ admitting a real archimedean
place such that $\End _{K}(J(C))$ is a quaternion order.
\end{remark}


To close this section devoted to present some examples, we
wish to illustrate several modularity questions. We recall that an
abelian variety $A/\Q $ of $\GL _2$-type over the field $\Q $
of rational numbers is an abelian variety whose algebra of
endomorphisms $\End _{\Q }(A)\otimes \Q =F$ is a number field $F$
of degree $[F:\Q ]=\dim (A)$. The generalized
Shimura-Taniyama-Weil conjecture predicts that any abelian variety
of $\GL _2$-type over $\Q $ is modular: it is isogenous over $\Q $
to a factor of the Jacobian variety $J_1(N)$ of the modular curve
$X_1(N)$ for some positive integer $N$. As is well known, as a
consequence of the work of Wiles, Taylor, Diamond, Conrad, Breuil
and others, any elliptic curve $E/\Q $ is modular (cf.\,\cite{Wi95}). In dimension
two, Ellenberg \cite{Ell} has shown that any abelian surface $A/\Q $ of $\GL
_2$-type over $\Q $ with sufficiently good reduction at $3$ and
$5$ is modular.

We note that any elliptic curve $E/K$ gives rise to an abelian
surface with multiplication by the split quaternion order $\M
_2(\Z )$ by considering the power $A=E^2$. In the literature,
abelian surfaces with quaternionic multiplication are often called
{\em fake elliptic curves}. The motivation for this terminology is
the strong analogy between the arithmetic of elliptic curves and
fake elliptic curves. We refer the reader to \cite{Bu},
\cite{Jo81}, \cite{Jo86}, \cite{JoLi85}, \cite{DiRo1},
\cite{DiRo2}, \cite{Oh}, \cite{Se} for these analogies, but also
for some interesting differences. In this regard and in view of
Wiles et al.'s Theorem, it is natural to wonder:

\begin{question}
Is any abelian surface $A/\Q $ with quaternionic multiplication modular?
\end{question}

A necessary condition for $A/\Q $ to be modular is
that $A/\Q $ be of $\GL _2$-type. Moreover, according to Shimura-Taniyama-Weil's
conjecture, this is also a sufficient condition. Then,
the above question translates to
ask whether any abelian surface $A/\Q $ with quaternionic multiplication is of $\GL
_2$-type over $\Q $. It has recently been suggested that this could be the
case. However, our previous examples show that there exist fake
elliptic curve over $\Q $ that fail to be modular. Indeed, we obtain
the following

\begin{corollary}

The Jacobian variety $J_2/\Q $ is a nonmodular abelian surface with
quaternionic multiplication.

\end{corollary}


\chapter{Diophantine properties of Shimura curves}\label{ASV}

\section*{Introduction}\addcontentsline{toc}{section}{Introduction}
\markboth{Chapter \ref{ASV}. Diophantine properties of Shimura curves}{\ref{Intro4}
Introduction}\label{Intro4}


\sk\noindent

In this chapter, we explore the diophantine properties of
Shimura curves.

Let $B$ be an indefinite rational quaternion algebra and choose a maximal order
$\mathcal O$ of $B$. Let $D=p_1\cdot ... \cdot p_{2r}$,
$p_i$ prime numbers, be the discriminant of $B$. We can view
$\Gamma =\{ \gamma \in \cO : n(\gamma )=1\} $ as an arithmetic subgroup of
$\SL _2(\R )$ through an identification $\Psi :B\otimes \R \simeq M_2(\R )$ and
consider the Riemann surface $\Gamma \backslash \mathcal H $, where $\mathcal H $
denotes the upper-half plane of Poincar\'e. As it was pointed out in Chapter
\ref{GSV}, Shimura showed in \cite{Sh67} that this is
the set of complex points of an algebraic curve $X_B = X_D/\Q $ over
$\Q $ which
parametrizes abelian surfaces with quaternionic multiplication by $\cO $.

The classical modular case arises when we consider the split quaternion algebra
$B=M_2(\Q )$ of discriminant $D=1$. In this case, $X_1={\bf A} _{\Q }^1$ is the
$j$-line that classifies elliptic curves or, by squaring, abelian surfaces with
multiplication by $M_2(\Z )$. Throughout, we limit ourselves to {\em non split}
quaternion algebras, that is, $D\not =1$. In this case, $\Gamma $ has no parabolic
elements and  $\Gamma \backslash \mathcal H $ is already compact so there are no
cusps and the automorphic forms on $X_D$ do not admit Fourier expansions. In this
regard, see \cite{Mor} and \cite{Ba}.

As it was introduced in Chapter \ref{GSV}, the elements of the Atkin-Lehner group
$W^1=W=\{\om _m : m\vert D \} \simeq C_2^{2r}$, where $C_2$ is the cyclic group of
order two, act as rational involutions on the Shimura curve $X_D$ and there
is a natural inclusion $W\subseteq \Aut _{\Q }(X_D)$. It is the aim of
Section \ref{secaut} to
examine the full group of automorphisms $\Aut (X_D\otimes\C )$ of these
curves. We firstly study the fields of
definition of the automorphisms $\om \in \Aut (X_D\otimes \C )$ and describe the
possible group structures of $\Aut (X_D\otimes \C )$. In many cases,
we prove that $\Aut (X_D\otimes \C )=W$.

Let us recall that Ogg \cite{Ogg77} studied
the group of automorphisms of the modular
curves $X_0(N)$ for square-free level $N$. There, the action of
$\Aut (X_0(N))$ on the set of cusps played a fundamental role.
When $D>1$, the difficulty lies precisely in the absence of cusps on $X_D$.

Next, in Section \ref{secBi}, the family of Shimura curves $X_D$ that
admit bielliptic involutions
is determined. The hyperelliptic problem was already settled by Michon and Ogg
independently in \cite{Mic}, \cite{Ogg74} and \cite{Ogg83}. Also,
the family of bielliptic modular curves $X_0(N)$ was given in \cite{Bar}. Since
$\Aut (X_0(N))$ is largely understood (cf.\,\cite{Ogg77} and
\cite{KeMo}), the main point in \cite{Bar} was to
count the number of fixed points of the non
Atkin-Lehner involutions that appear when $4\vert N$ or $9\vert N$. In our case this
difficulty does not arise, but on the other hand the automorphism groups of the
Shimura curves $X_D$ are less known.

In Section \ref{secQu}, we derive
some arithmetical consequences from the above results
concerning the set of rational points on $X_D$ over quadratic fields. Recall that
by a fundamental theorem of Shimura, there are no real points on Shimura curves and
therefore quadratic imaginary fields are the simplest fields over which these
curves may have rational points. Our main theorem in this section
completely solves a question posed and studied by Kamienny in \cite{Ka}:

\begin{question}\label{question}
Which Shimura curves $X_D$ of genus $g\geq 2$ admit infinitely
many quadratic points?
\end{question}

This question is motivated
by Faltings' Theorem on Mordell's conjecture. In Theorem \ref{Kamienny}
and Table \ref{tableQuadraticPoints}, we
give a complete and explicit
answer to question \ref{question} by listing the finitely many
discriminants $D$ such that

$$
\sharp \{ P\in X_D(\bar \Q ): [ \Q (P):\Q ] \leq 2\}=+\infty .
$$

Our method is based upon ideas of
Abramovich, Harris and Silverman (see \cite{AbHa} and \cite{HaSi}). We
also note that our Theorem \ref{Kamienny} can be used to derive interesting
consequences on several questions concerning modular abelian surfaces
with extra-twist.
%%%%%%%%%%%%%%%%%%
%We refer the reader to the work in progress of Clark \cite{Cl}.

Finally, in Section \ref{secEx}, we use the theory of
\v{C}erednik-Drinfeld to compute equations of elliptic
Atkin-Lehner quotients of Shimura curves. Table
\ref{tableequations} in Section \ref{secEx} gives a Weierstrass
equation of {\em all} elliptic curves of the form $X_D/\langle
w\rangle $ where $w\in \Aut(X_D)$ is any $\Q $-bielliptic
involution on the curve. Some examples were already given in
\cite{Rob}.

The main tools used in this chapter come from the reduction of
Shimura curves at both good and bad places. Drinfeld constructed a
projective model $M_D$ over $\Z $ of the Shimura curve $X_D$ which
extends the moduli interpretation given by Shimura to abelian
schemes over arbitrary bases (cf.\,\cite{Dr}, \cite{Bu} and
\cite{BoCa}). Morita \cite{Mo} showed that $M_D$ has good
reduction at all primes $p\nmid D$ and Shimura determined in
\cite{Sh67} the zeta function of the special fibre of $M_D$ at
$p$. Moreover, the \v{C}erednik-Drinfeld theory (cf.\,\cite{BoCa},
\cite{Ce}, \cite{Dr} and \cite{JoLi85}) provides a good account of
the behaviour of the reduction of $M_D \, $ (mod $p$) when $p\vert
D$.

Our results in this chapter were presented in \cite{Ro02}.

\markright{\ref{secaut} The group of automorphisms of Shimura curves}
\section{The group of automorphisms of Shimura curves}
\markright{\ref{secaut} The group of automorphisms of Shimura curves}
\label{secaut}
\sk\noindent

Throughout, $X_D$ will denote the canonical model over $\Q $ of the
Shimura curve of
discriminant $D=p_1\cdot ...\cdot p_{2r}\not =1$. It is a proper
smooth scheme over $\Q $ of dimension 1. Let $\Aut _{\Q }(X_D)$ be the group of
$\Q $-automorphisms of $X_D$ that sits inside the full group of geometric
automorphisms $\Aut _{\C }(X_D \otimes \C )$ of the complex algebraic
curve $X_D \otimes \C $.

\begin{proposition}\label{pipi}

If $g(X_D)\geq 2$, then
\begin{enumerate}
\item
All automorphisms of $X_D \otimes \C $ are defined over $\Q $.
That is: $\Aut _{\C }(X_D\otimes \C )=\Aut _{\Q }(X_D)$.

\item
$\Aut _{\Q }(X_D)\simeq C_2^s$, $s\geq 2r$.

\end{enumerate}
\end{proposition}


{\em Proof.} Ribet \cite{Ri75} proved that all the endomorphisms of an abelian
variety $A/K$ with semistable reduction over a number field $K$ are defined over an
unramified extension of $K$. The Jacobian variety $J_D/\Q $ of $X_D$ has good
reduction at primes $p\nmid D$ and, from \cite{JoLi86}, we know that the
identity's connected
component of the reduction mod $p$, $p\vert D$, of the N\'eron model of $J_D$ is a
torus. Hence, $J_D$ has semistable reduction over $\Q $ and all its endomorphisms
are rational because $\Q $ has no nontrivial unramified extensions.

Since, by Hurwitz Theorem, $\Aut _{\C }(X_D\otimes \C )$ is a
finite group, all automorphisms of
$X_D\otimes \C $ are defined over $\bar \Q $. Moreover, the natural map
$\Aut _{\bar \Q }(X_D\otimes \bar \Q )\rightarrow \Aut _{\bar \Q }
(J_D\otimes \bar \Q )$ is injective and $\Gal (\bar \Q/\Q )$-equivariant
and therefore we conclude from above that all automorphisms of $X_D\otimes \C $
are rational.

For the second part, let $X_0(D)/\Q $ be the modular curve of level $D$ and consider
the new part $J_0(D)^{new}/\Q $ of its Jacobian variety $J_0(D)$. It is well known
(cf.\,\cite{Ri75}) that $\End^0_{\Q }(J_0(D)^{new})\simeq \TT \otimes \Q \simeq
\prod_{i=1}^tK_i$, where $\TT $ denotes the Hecke algebra of level $D$ and $K_i$ are
totally real number fields. Ribet's Isogeny Theorem (cf.\,\cite{Ri80}) states the
existence of an isogeny

$$
\varphi : J_D \longrightarrow J_0(D)^{new}
$$
between $J_D$ and $J_0(D)^{new}$. This isogeny is Hecke invariant
(but sign\-inter\-changing for the Atkin-Lehner action) and
defined over $\Q $. Hence, the ring of endomorphisms $\End _{\Q
}(J_D) $ is an order in $\prod_{i=1}^t K_i$. An automorphism of
the curve $X_D$ induces an automorphism of finite order on $J_D$.
Moreover, the group of integral units in $\prod_{i=1}^t K_i$ is
isomorphic to $C_2^t$. Thus $\Aut _{\Q }(X_D)\simeq C_2^s$ with
$2r\leq s\leq t$, the first inequality holding just because
$W\subseteq \Aut _{\Q }(X_D)$. $\Box $

\vspace{0.5cm}

We conclude that any automorphism of $X_D$ acts as a rational
involution on it. In view of the above proposition, we will simply denote the group
$\Aut _{\C }(X_D\otimes \C )=\Aut _{\Q }(X_D)$ by
$\Aut (X_D)$. Naturally we ask
whether the Atkin-Lehner group is the full group of automorphisms of the curve,
provided that $g(X_D)\geq 2$. This is the case for modular curves
$X_0(N)$ of square free level $N$, $N\not =37$ (cf.\,\cite{Ogg77} and \cite{KeMo}).

Recall that an elliptic point on the curve $X_D$ is a
branched point of the natural projection

$$
\mathcal H \rightarrow \Gamma \backslash \mathcal H\simeq X_D(\C ).
$$

The stabilizers of those elliptic points in $\Gamma/\{ \pm 1\} $ are of
order $2$ or $3$. $2$-elliptic points (respectively $3$-elliptic points) correspond
to $\Gamma $-conjugacy classes of embeddings of the quadratic order
$\Z [i]$, $i^2=-1$ (respectively $\Z [\rho ]$, $\rho ^3=1$) in the quaternion
order $\cO$. Their cardinality is given by

$$
e_2=\prod_{\ell \vert D} (1-{\underline {-4}\choose \ell }),
$$
$$
e_3=\prod_{\ell \vert D} (1-{\underline{-3} \choose \ell }),
$$
where $(\frac{\cdot }{\cdot })$ denotes the Kronecker symbol.

\begin{theorem}
\label{ellpoints}

Let $X_D$ be the Shimura curve of discriminant $D$. If it has no elliptic points,
then $\Aut (X_D)=W$.

\end{theorem}

{\em Proof.} If there are no elliptic points on $X_D(\C )$, then the natural
projection $\mathcal H \rightarrow \Gamma \backslash \mathcal H \simeq
X_D(\C )$ is
the universal cover of the Riemann surface $X_D(\C )$ so $\Aut (X
_D)\simeq \Norm _{\PGL _2^+(\R )}(\Gamma )/\Gamma $. Here the
superindex $+$ denotes
matrices with positive determinant. It is known that
$W\simeq \Nm _{B^{\times }}(\Gamma )/\Q ^{\times }\Gamma \simeq C_2^{2r}$
(cf.\,\cite{Mic} and \cite{Ogg83}). We observe now that
the $\Q $-vector space spanned by $\Gamma $ is $\langle  \Gamma  \rangle _{\Q } =
B$. Indeed, since the reduced norm $\n $ is indefinite on the space of
pure quaternions $B_0$, we can find linearly
independent elements $\omega _1$, $\omega _2$,
$\omega _3\in B_0$ such that $\Z [\omega _i]\subset B$ is a real quadratic order in
$B$. Then, by solving  the corresponding Pell equations, we find units $\gamma _i
\in \Z [\omega _i ]\cap \Gamma $, $\gamma _i \not =\pm 1$ such that $\{1, \gamma _1,
\gamma _2, \gamma _3\} $ is a $\Q $-basis of $B$.

Hence, any $\alpha \in \Nm _{\GL _2^+(\R )}(\Gamma )$ will actually normalize
$B^*$. By the Skolem-Noether Theorem, $\alpha $ induces an inner
automorphism of $B$ so that $\alpha \in \R ^*\Nm _{B^*}
(\Gamma )$. This shows that $\Aut (X_D)=W$. $\Box $


\begin{remark} In proving the above theorem, we have also
shown that the Atkin-Lehner group $W$ of an arbitrary Shimura curve
$X_D$ is exactly the subgroup of automorphisms that lift to a M\"{o}bius
transformation on $\mathcal H$ through the natural uniformization $\mathcal H
\rightarrow \Gamma \backslash \mathcal H \simeq X_D(\C )$.

The proof remains valid for Eichler orders of square-free
level $N$ and therefore it generalizes an analogous result of Lehner and Newman for
discriminant $D=1$ (cf.\,\cite{LeNe}).
\end{remark}

The next theorem is similar in spirit to Theorem \ref{ellpoints} and
requires a previous lemma due to Ogg \cite{Ogg77}.

\begin{lemma}\label{oogg}

Let $K$ be a field and $\mu (K)$ its group of roots of unity. Let
$p=\mathrm{max }(1, \mathrm{char }K)$ the characteristic exponent of $K$. Let $C$ be
an irreducible curve defined over $K$ and $P\in C(K)$ a regular point on it. Let $G$
be a finite group of $K$-automorphisms acting on $C$ and fixing the point $P$. Then
there is a homomorphism $f: G \rightarrow \mu (K)$ whose kernel is a $p$-group.

\end{lemma}

\begin{theorem}\label{2p}

Let $D=2p$, $3p$; $p$ a prime number. If $g(X_D)\geq 2$, then
$\mathrm{Aut} (X_D) = W\simeq C_2\times C_2$.

\end{theorem}

{\em Proof.} Suppose first that $D=2p$ with $p\equiv 3 \, $ (mod 4). In this case,
the fixed points on $X_D$ of the Atkin-Lehner involution $\om _2$ are {\em Heegner
points} (see e.g. \cite{Al} for a
general account). Their coordinates on Shimura's canonical model $X_D$ generate
certain class fields. More precisely, if the genus $g(X_D)$ is even, then
$\om _2$
exactly fixes two points $P$, $P'$ with complex multiplication by the quadratic
order $\Z [i]$ and hence (cf.\,\cite{ShTa}) $P$, $P'\in X_D(\Q (i))$. If
$g(X_D)$ is odd, then, besides $P$ and $P'$, $\om _2$ fixes two more points $Q$,
$Q'\in X_D(\Q (\sqrt{ -2}))$  which have complex multiplication
by $\Z [\sqrt{ -2}]$. As we have seen,
$\Aut (X_D)$ is an abelian group so it acts on the set of fixed points of
$\om _2$ on $X_D$. Since all automorphisms are rational, they must keep
the field of rationality
of these points so that $\Aut (X_D)$ actually acts on $\{ P, P'\} $. It follows
from the previous lemma that the order of the stabilizer of $P$ or $P'$ in
$\Aut  (X_D)$ is at most 2. Hence, $\# \Aut (X_D)\leq 4$ and $\Aut (X_D)=W$.

Suppose now that $D=2p$, $p\equiv 1\, $ (mod $4$) or $D=3p$,
$p\equiv 1\, $ (mod 3). By the theory of \v{C}erednik-Drinfeld,
the special fibre $M_D \otimes \F _p$ of the reduction mod $p$ of
the integral model $M_D$ of our Shimura curve consists of two
rational irreducible components $Z$, $Z'$ defined over $\F
_{p^2}$. The complete local rings of the intersection points of
$Z$ and $Z'$ over the maximal unramified extension $\Z _p^{unr}$
of $\Z _p$ are isomorphic to $\Z _p^{unr}[x, y]/(x y - p^{\ell })$
for some length $\ell \geq 1$. The reduction mod $p$ of the
Atkin-Lehner involution $\om _p$ switches $Z$ and $Z'$, fixing the
double points of intersection. Among these double points, there is
exactly one, say $\widetilde Q$, which has {\em length} 2, as it
follows from \cite{Ku}. Thus $\Aut (X_D)$ acting on $M_D\otimes \F
_p$ must fix $\widetilde Q$. Recalling now that $\Aut (X_D)\simeq
C_2^s$, we again apply Ogg's Lemma \ref{oogg} to the curve $Z/\F
_{p^2}$ ($p\not =2$) and the point $\widetilde Q$ to obtain that
$\# \Aut (X_D)\leq 4$. Therefore $\Aut (X_D)=W$.

In the remaining case, namely when $D=3p$, $p\equiv -1 \, $ (mod 3), we observe the
curious phenomenon that $\ell =109$ is a prime of good reduction for the Shimura
curve $X_D$ that yields

$$
\# M_D\otimes \F _{109} (\F _{109}) \not \equiv 0 \quad (\mbox{ mod } 4)
$$
except for the two exceptional cases $D=3\cdot 89$ and $D=3\cdot 137$. In any case,
we check that $\# M_{3\cdot 89}\otimes \F _{67} (\F _{67})=94$ and
$\# M_{3\cdot 137}\otimes \F _{103} (\F _{103})=98$. This is carried out by using
the explicit formula for the number of rational points over finite fields of the
reduction of Shimura curves at good places given by Jordan and Livn\'e in
\cite{JoLi85}. From this
we proceed as above: since all automorphisms of $X_D$ are defined over $\Q $, their
reduction mod $\ell $ must preserve the $\F _{\ell }$-rational points on
$M_D\otimes \F _{\ell }$ and we apply Ogg's Lemma \ref{oogg} to the  regular curve
$M_D\otimes \F _{\ell }$ to conclude that $\Aut (X_D)=W$. $\Box $

%\vspace{0.5cm}
\begin{remark} The first argument can be adapted for more general discriminants in an
obvious way. For instance, if $D=p\delta $ where $p$ is a prime integer,
$p\equiv 3 \, $ (mod 8), and ($\frac {-p}{\ell })=-1$ for any $\ell \vert \delta $,
then we again obtain that $\Aut (X_D)=W$ because the Hilbert class field of
$\Q (\sqrt{-p})$ is strictly contained in the ring class field of conductor 2 and,
by genus theory, $h(-p)$ is odd.
\end{remark}

%\vspace{0.5cm}
\begin{example} Shimura curve quotient $X_{291}^+ = X_{291}/W$ has
genus 2 and therefore it is hyperelliptic. However, the hyperelliptic
involution on $X_{291}^+$ is exceptional: it does not lift to a M\"{o}bius
transformation on $\mathcal H$
through $\pi :\mathcal H \rightarrow
X_{291}^+(\C )=\Gamma \cdot W\backslash \mathcal H$, while all
automorphisms of $X_{291}$ are of Atkin-Lehner type by Theorem \ref{2p}. This
is caused by the fact that $\pi $ is not the universal cover of
$X_{291}^+$.
\end{example}

\markright{\ref{secBi} Bielliptic Shimura curves}
\section{Bielliptic Shimura curves}
\markright{\ref{secBi} Bielliptic Shimura curves}
\label{secBi}
\sk\noindent


Recall that an algebraic curve $C$ of genus $g\geq 2$ is
bielliptic if it admits a degree 2 map $\varphi : C \rightarrow E$ onto a curve $E$
of genus 1. We will ignore fields of rationality until the next section.
Alternatively, $C$ is bielliptic if and only if
there is an involutive automorphism acting on
it with $2g-2$ fixed points. We present now some facts about bielliptic curves $C$
such that, like Shimura curves, $\mathrm{Aut} (C)\simeq C_2^s$.

\begin{lemma}\label{lin}
Let $C/K$, $\mbox{char }K \not =2$, be a bielliptic curve of genus
$g$ with $\mathrm{Aut} (C)\simeq C_2^s$ for some $s\geq 1$. For any $w\in \mathrm{Aut} (C)$, let $n(w)$
denote the number of fixed points of $w$ on $C$. Let $v$ be a bielliptic involution
on $C$ and for any $w\in \mathrm{Aut} (C)$, $w\not =1 \mbox{ or } v$, denote
$w'=v\cdot w$.

\begin{enumerate}
\item
If $g$ is even, then $n(w)=2$ and $n(w')=6$, or viceversa. If $g$
is odd, then $\{ n(w), n(w')\} =\{ 0, 0\}, \{0, 8\} $ or $\{ 4,
4\} $ as non ordered pairs.

\item
If $g$ is even, then $s\leq 3$. If $g$ is odd, then $s\leq 4$.

\item
If $g\geq 6$, then the bielliptic involution $v$ is unique.

\end{enumerate}
\end{lemma}

{\em Proof. } It follows from Hurwitz's Theorem applied to the
projection of the curve $C$ onto its quotient by suitable groups
generated by involutions acting on it. $\Box $

%\vspace{0.5cm}
\begin{remark}
Observe that if $D$ is odd, then $g(X_D)$ is always odd, as we check
from Eichler's formula for the genus (see e.g. \cite{Ogg83}).
\end{remark}

%\vspace{0.5cm}
Obviously, the main source for possible bielliptic involutions on the curves $X_D$
is the Atkin-Lehner group. From Eichler's formula for $n(w)$, $w\in W$
(see \cite{Ogg83}), it
is a routine exercise to check whether $X_D$ has bielliptic involutions of
Atkin-Lehner type. An alternative way to compute $n(w)$ is to read {\em backwards}
the last column of Table 5 in \cite{Ant}. This is because Ribet's isogeny
$\varphi : J_D \rightarrow J_0(D)^{new}$ switches the sign of the Atkin-Lehner
action. But, first, we should focus on possible extra involutions and also bound the
bielliptic discriminants $D$. Following Ogg's method in \cite{Ogg74}, we give such an
upper bound in the next

\begin{proposition}
If $D>547$, $X_D$ is not bielliptic.

\end{proposition}

{\em Proof. } Suppose that the curve $X_D$ is bielliptic: there is a degree 2 map
$\varphi : X_D\rightarrow E$ onto a curve $E$ of genus 1. By
Proposition \ref{pipi}, both
$\varphi $ and $E$ are defined over $\Q $ although $E$ may not be an elliptic curve
over $\Q $ since it may fail to have rational points. See Section \ref{secQu} for
examples. Choose a prime of good reduction $\ell \nmid D$ of $X_D$, let
$K_{\ell }$ be the
quadratic unramified extension of $\Q _{\ell }$ and let $R_{\ell }$ denote its
ring of integers. As follows from \cite{JoLi85},
$X_D(K_{\ell })\not =\emptyset $ and hence $E$ is an elliptic curve over
$K_{\ell }$. Moreover, due to Ribet's Isogeny Theorem,
$E$ also has good reduction over $\ell $. By the universal property of the N\'eron
model of $E$ over $R_{\ell }$, $\varphi $ extends to the minimal smooth model
$M_D\otimes R_{\ell }$ of $X_D$  and we can reduce the bielliptic structure
mod $\ell $ to obtain a $2:1$
map $\tilde \varphi : M_D\otimes \F _{\ell ^2} \rightarrow \widetilde E$. From
Weil's estimate, $N_{\ell ^2}=\# M_D\otimes \F _{\ell ^2}(\F _{\ell ^2})\leq 2\cdot
\# \widetilde E(\F _{\ell ^2})\leq 2(\ell +1)^2$.  Besides, we obtain from
\cite{JoLi85} that $\frac {(\ell -1)}{12}\prod _{p\vert D}{(p-1)}\leq N_{\ell ^2}$
so $N_{\ell ^2}$ grows as $D$ tends to infinity. Applying these inequalities for
$\ell =2, 3, 5, 7, 11$, we conclude that, if $2\cdot 3\cdot 5\cdot 7\cdot 11\nmid D$,
then $D\leq 546$. But, if $2\cdot 3\cdot 5\cdot 7\cdot 11\vert D$,
then $s\geq 5$ and therefore, by Lemma \ref{lin}, $X_D$ cannot be
bielliptic. $\Box $

\vspace{0.5cm}
We are now able to prove

\begin{theorem}

There are exactly thirty-two values of $D$ for which $X_D$ is bielliptic. In each
case, the bielliptic involutions are of Atkin-Lehner type. These values, together
with the genus $g=g(X_D)$ and the bielliptic involutions are given in Table
\ref{bielliptic} below.

\newpage
\vspace{1cm}
\begin{table}\label{bielliptic}

\begin{tabular}{|r|r|l|r|r|r|l|r|r|r|l|} \hline
\multicolumn{11}{|c|}{\bfseries Table \ref{bielliptic}:
Bielliptic Shimura curves}\\

\hline
D&g&$\om _m$&   &D&g&$\om _m$&   &D&g&$\om _m$ \\
\hline \hline
$26$&$2$&$\om _2$,$\om _{13}$&  &$82$&$3$&$\om _{82}$&   &$210$&$5$&$\om _{30}$,$\om _{42}$, \\
$35$&$3$&$\om _7$&   &$85$&$5$&$\om _{17}$&   &   &   &$\om _{70}$,$\om _{105}$, \\
$38$&$2$&$\om _2$,$\om _{19}$&   &$94$&$3$&$\om _2$&   &   &   &$\om _{210}$ \\
$39$&$3$&$\om _{13}$&   &$106$&$4$&$\om _{53}$,$\om _{106}$&  &$215$&$15$&$\om _{215}$  \\
$51$&$3$&$\om _3$&   &$115$&$6$&$\om _{23}$&   &$314$&$14$&$\om _{314}$   \\
$55$&$3$&$\om _5$&   &$118$&$4$&$\om _{59}$,$\om _{118}$&   &$330$&$5$&$\om _3$,$\om _{22}$  \\
$57$&$3$&$\om _{57}$&    &$122$&$6$&$\om _{122}$&   &   &   &$\om _{33}$,$\om _{165}$, \\
$58$&$2$&$\om _2$,$\om _{58}$&    &$129$&$7$&$\om _{129}$&    &    &    &$\om _{330}$ \\
$62$&$3$&$\om _2$&   &$143$&$12$&$\om _{143}$&   &$390$&$9$&$\om _{390}$ \\
$65$&$5$&$\om _{65}$&   &$166$&$6$&$\om _{166}$&   &$462$&$9$&$\om _{154}$ \\
$69$&$3$&$\om _3$&    &$178$&$7$&$\om _{89}$&   &$510$&$9$&$\om _{510}$ \\
$77$&$5$&$\om _{11}$,$\om _{77}$&   &$202$&$8$&$\om _{101}$&   &$546$&$13$&$\om _{546}$\\
\hline
\end{tabular}
\end{table}
\end{theorem}

\newpage

{\em Proof. } Since we need only consider discriminants $D\leq 546$, we can first
use any programming package to build up the list of Atkin-Lehner bielliptic
involutions on Shimura curves $X_D$. These computations yield
Table \ref{bielliptic} above.
In order to ensure that no extra bielliptic involutions arise, we observe that the
above results, and particularly Theorem \ref{2p}, imply that any
bielliptic involution on
$X_D$, for most of the discriminants $D\leq 546$, must be of Atkin-Lehner type.
There are exactly three cases, namely $D=55$, $D=85$ and $D=145$, for which none of
the previous results and their obvious generalizations seem to apply.

{\em Ad hoc} arguments can be worked out for them. Firstly, the Jacobian varieties
of the curves $X_{55}$ and $X_{85}$ have just one $\Q $-isogeny class of sub-abelian
varieties of dimension 1, so there can be at most one bielliptic involution on these
curves. But $\om _5$ (respectively $\om _{17}$) is already a bielliptic involution on $X_{55}$
(respectively $X_{85}$).

More interesting is the curve $X_{145}$ of genus 9. It is not bielliptic by any
Atkin-Lehner involution although $J_{145}\sim _{\Q }E\times S\times A_3\times A'_3$,
where each factor has dimension 1, 2, 3 and 3 respectively. We check that
$n(\om _5)=n(\om _{29})=0$ and $n(\om _{145})=8$, so if there was a bielliptic involution $v$
on $X_{145}$ then $n(w'_{145})=0$,  by Lemma 5.1. It follows from Lefschetz's fixed
point formula (see \cite{LaBi}) that the rational traces of these three involutions
on the Jacobian $J_{145}$ would be $\tr (\om _5)=\tr (\om _{29})=
\tr (w'_{145})=2$.  Moreover, involutions on $J_{145}$ must be of the form
$\{ \pm 1_{E}\}\times \{ \pm 1_S\} \times \{ \pm 1_{A_3}\} \times
\{ \pm 1_{A'_3}\} $, up to conjugation by $\varphi $, thus $\tr =2$ can only be
attained by two different involutions. Therefore $v$ cannot exist and $X_{145}$ is
not bielliptic.

It can be showed that actually $\Aut (X_{145})=W$: from the decomposition of
$J_{145}$ we know that $W\simeq C_2^2\subseteq \Aut (X_{145})\subseteq C_2^4$. Since
$X_{145}$ is neither hyperelliptic (cf.\,\cite{Ogg83}) nor bielliptic
(as we have just seen), it follows that the involutions
$\{ -1_{E}\}\times \{ -1_S\} \times \{ -1_{A_3}\} \times \{ -1_{A'_3}\} $ and
$\{ +1_{E}\}\times \{ -1_S\} \times \{ -1_{A_3}\} \times \{ -1_{A'_3}\} $ cannot be
induced from $\Aut (X_{145})$. Thus it is a subgroup of index at least 4 in $C_2^4$
and $\Aut (X_{145})=W$. $\Box $

\markright{\ref{secQu} Infinitely many quadratic points on Shimura curves}
\section{Infinitely many quadratic points on Shi\-mura
curves}
\markright{\ref{secQu} Infinitely many quadratic points on Shimura curves}
\label{secQu}
\sk\noindent

Shimura \cite{Sh75} proved that  $X_D(\R )=\emptyset $ and in particular there
are no $\Q $-rational points on Shimura curves $X_D$. Jordan and
Livn\'e \cite{JoLi85} gave
explicit criteria for deciding whether the curves $X_D$ do have
rational points over the $p$-adic fields $\Q _p$ for any finite prime $p$.

Less is known about rational points over global fields. Jordan \cite{Jo86}
proved that for a fixed quadratic imaginary field $K$, with class number
$h_K\not =1$, there are only finitely many discriminants $D$ for which $K$ splits
the quaternion algebra $B$ of discriminant $D$ and $X_D(K)\not =
\emptyset $. In this
section we solve a question that is to an extent reciprocal: which Shimura curves
$X_D$, $g(X_D)\geq 2$, have infinitely many quadratic points over $\Q $?

That is,
$$
\# X_D(\Q , 2)=\# \{ P\in X_D(\bar \Q ): [ \Q (P):\Q ] \leq
2\}=+\infty .
$$
We will say that an
algebraic curve $C/K$ of genus $g\geq 2$ is {\em hyperelliptic} over $K$
(respectively {\em bielliptic} over $K$) if there is an involution $v\in \Aut
_K(C)$ such that the quotient curve $C/\langle v\rangle
$ is $K$-isomorphic to $\PP _{K }^1$ (respectively an elliptic curve $E/K$).
Notice that in both cases $C/\langle v\rangle (K)\not =\emptyset $
while it perfectly well happen that $C(K)=\emptyset $.

The following theorem of Abramovich and Harris \cite{AbHa} shows that the
question above is closely related to the diophantine
problem of determining the family of
hyperelliptic and bielliptic Shimura curves over $\Q $.

\begin{theorem}

Let $C$ be an algebraic curve of genus greater than or equal to $2$, defined over a
number field $K$. Then $C(K , 2)=\# \infty $ if and only if $C$ is either
hyperelliptic over $K $ or bielliptic over $K $ mapping to an elliptic curve $E$
of $\rank _{K }(E)\geq 1$.

\end{theorem}

Ogg \cite{Ogg83}, \cite{Ogg84}
gave the list of hyperelliptic Shimura curves over
$\Q $. In what follows, we will determine which bielliptic Shimura curves from
Table \ref{bielliptic} are bielliptic over $\Q $.

We first observe that the map $X_D \rightarrow X_D/\langle w\rangle $ is always
defined over $\Q $ since we showed in Proposition \ref{pipi} that
$\Aut _{\C }(X_D\otimes \C )=\Aut _{\Q }(X_D)$. In order to
check whether $X_D/\langle w\rangle (\Q )\not =\emptyset $ for each pair $(D$,$w)$
in Table \ref{bielliptic}, we can
disregard those in which $X_D/\langle w\rangle $ fails to have
rational points over some completion $\Q _v$ of $\Q $. This is done by using
the precise results in that direction given by Jordan and Livn\'e in  \cite{JoLi85}
and Ogg in \cite{Ogg83} and \cite{Ogg84}; the conclusions are compiled in the
following table. Let us say that a field $L$ is {\em deficient}
for an algebraic curve $C$
defined over a subfield $K\subset  L$ if $C(L)=\emptyset $.



\newpage

\vspace{1cm}
\begin{table}\label{deficient}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{11}{|c|}{\bfseries Table \ref{deficient}: Deficient completions $L$
of $\Q $ for $X_D/\langle \om _m\rangle $}\\

\hline
D&$\om _m$&L&&D&$\om _m$&L&&D&$\om _m$&L \\
\hline \hline

$35$&$\om _7$&$\Q _5$& &$115$&$\om _{23}$&$\Q _5$& &$330$&$\om _3$&$\R $,$\Q _2$\\

$39$&$\om _{13}$&$\R $,$\Q _3$& &$178$&$\om _{89}$&$\R $,$\Q _2$& &  &  &$\Q _5$,$\Q _{11}$\\

$51$&$\om _{3}$&$\Q _{17}$& &$210$&$\om _{30}$&$\R $,$\Q _3$& &$330$&$\om _{22}$&$\R $,$\Q _2$,$\Q _3$\\

$55$&$\om _5$&$\R $,$\Q _{11}$& &$210$&$\om _{42}$&$\R $,$\Q _2$,$\Q _3$& &  &  &$\Q _5$,$\Q _{11}$\\

$62$&$\om _2$&$\R $,$\Q _{31}$& & & &$\Q _5$,$\Q _7$& &$330$&$\om _{33}$&$\R $,$\Q _2$\\

$69$&$\om _3$&$\R $,$\Q _{23}$& &$210$&$\om _{70}$&$\R $,$\Q _2$&  &&&$\Q _3$,$\Q _5$\\

$77$&$\om _{11}$&$\R $,$\Q _7$& &  & &$\Q _3$,$\Q _5$ &  &$330$&$\om _{165}$&$\Q _2$,$\Q _3$\\

$85$&$\om _{17}$&$\Q _5$& &$210$&$\om _{105}$&$\R $,$\Q _2$& & & &$\Q _5$,$\Q _{11}$\\

$94$&$\om _2$&$\R $,$\Q _{47}$& & & &$\Q _7$& &$462$&$\om _{154}$&$\R $,$\Q _{11}$\\

\hline
\end{tabular}
\end{table}

\newpage


On the genus 1 Atkin-Lehner quotients $X_D/\langle \om _m\rangle $  that do have
rational points over all completions of $\Q $, we can try to construct a
$\Q $-rational point by means of the theory of complex multiplication.
That is, a Heegner point $P\in X_D(K)$ with CM by a quadratic imaginary order $R$,
$R\otimes \Q =K$, $h(R)=1$, will project onto a $\Q $-rational point on
$X_D/\langle \om _m\rangle $ if and only if $\om _m(P)=\overline P$, where $\overline P$ is the
complex
conjugate of $P$ on $X_D(K)$. From \cite{Jo81}, 3.1.4, we deduce that $\om _m(P)=\overline P$
if $m$ is the product of the primes $p\vert D$ that are inert in $K$.

Performing the necessary computations, it follows that among those pairs $(D, w)$
that $X_D/\langle w\rangle (\Q _v)\not =\emptyset $ for every completion $\Q _v$ of
$\Q $, it is always possible to produce a $\Q $-rational point on
$X_D/\langle w\rangle $ by the above means, except for two interesting cases:
$(X_{26}$, $\om _2)$ and $(X_{58}$, $\om _2)$.

Since $g(X_{26})=g(X_{58})=2$, we may apply a result of
Kuhn (cf.\,\cite{Kuhn}) to deduce that there are also rational points on the
quotients $X_{26}/\langle \om _2\rangle $ and $X_{58}/\langle \om _{2}\rangle
$. Therefore, the Hasse-Minkowsky principle is never violated for the
Atkin-Lehner quotients from Table \ref{bielliptic} and those pairs
$X_D/\langle \om _m\rangle
$ that do not appear in Table \ref{deficient}
are bielliptic curves over $\Q $. There
are only eighteen values of $D$ for which $X_D$ admits a bielliptic involution over
$\Q $.

It still remains to compute the Mordell-Weil rank of the elliptic curves
$X_D/\langle w\rangle $ over $\Q $. Using Cremona's tables \cite{Cre}, switching the
sign of the Atkin-Lehner action as explained above, we can determine the
$\Q $-isogeny class of these elliptic curves.

This is enough to compute their Mordell-Weil rank but we can use a
beautiful idea of Roberts (cf.\,\cite{Rob}) to compute the $\Q
$-isomorphism class and hence a Weierstrass equation for them as
follows: Cremona's tables give the Kodaira symbols of the
reduction of elliptic curves $E$ at the primes $p\vert
\mathrm{cond } (E)$. This is done by using Tate's algorithm which
makes use of a Weierstrass equation of the curve. This is not
available in our case, but we can instead use
\v{C}erednik-Drinfeld theory to compute the Kodaira symbols for
the reduction mod $p$, $p\vert D$, of $X_D/\langle w\rangle $ and
contrast them with Cremona's tables. This procedure uniquely
determines the $\Q $-isomorphism class of the curves.

%\vspace{0.5cm}
\begin{example} Curve $X_{210}$ has genus 5 and is bielliptic by the Atkin-Lehner
involution $\om _{210}$. From Eichler's Theorem \ref{Eic1}, the
quadratic order $\Z [\sqrt{-43} ]$ embeds in the quaternion
algebra $B$ of discriminant 210. Such an embedding produces a point
$P\in X_{210}(\Q (\sqrt{-43} ))$. From the above, it follows that
$\om _{210}(P)=\overline P$. Therefore, $X_{210}/\langle \om _{210}\rangle (\Q )\not
=\emptyset $ and we obtain that $(X_{210}, \om _{210})$ is a bielliptic pair over
$\Q $. A glance at Cremona's Table 3, p.\,249-250, shows that the elliptic curve
$X_{210}/\langle \om _{210}\rangle $ falls in the $\Q $-isogeny class $210D$ because it
is the only one that corresponds to a newform $f\in H^0(\Omega ^1,J_{210})$ such
that $\om _{210}^*(f)=f$ (recall that the sign for the Atkin-Lehner action is opposite
to the classical modular case!). Therefore, from Cremona's Table 4,
$\rank _{\Q }(X_{210}/\langle \om _{210}\rangle )=1$.

In order to determine a Weierstrass equation for $X_{210}/\langle
\om _{210}\rangle $ we may compute the Kodaira symbols of its
reduction mod $p$, $p\vert 210$. It suffices to study the
reduction at $p=3$. The \v{C}erednik-Drinfeld theory asserts that
$M_{210}\otimes \F _3$ is reduced and its irreducible components
are all rational and defined over $\F _9$. Moreover,
$M_{210}\otimes \Z _3$ is a (minimal) regular model over $\Z _3$.
This is because over the quadratic unramified integral extension
$R_3$ of $\Z _3$, $M_{210}\otimes R_3$ is a Mumford curve
uniformized by a (torsion-free) Schottky group, as one checks from
\v{C}erednik-Drinfeld's explicit description of this group and the
congruences $5\equiv -1\quad (\mbox{mod } 3)$ and $7\equiv -1\quad
(\mbox{mod } 4)$.

In a way, \v{C}erednik-Drinfeld's description of the reduction of
Shimura curves at $p\vert D$ is not so different from
Deligne-Rappoport's for the modular curves $X_0(N)$ at $p\parallel
N$ because $M_{p\delta }\otimes \F _p$ is again the union of two
copies of the Shimura curve -also called Gross curve- $M_{\delta
}\otimes \F _p$, defined in terms of a {\em definite} quaternion
algebra.

Let $h(\delta ,\nu )$ denote the class number of an (arbitrary) Eichler order of
level $\nu $ in the quaternion algebra of discriminant $\delta $. The dual graph
$\mathcal G$ of $M_{210}\otimes \F _3$ has as vertices the irreducible components of
$M_{210}\otimes \F _3$. There are $2h(\frac{210}{3}, 1)=2h(70, 1)=4$ of them. Two
vertices $v$, $\tilde v$ in $\mathcal G$ are joined by as many edges as there are
intersection points between the corresponding components $Z$, $\widetilde Z$ in
$M_{210}\otimes \F _3$. In our case, there are $h(\frac{210}{3}, 3)=8$ edges in
$\mathcal G$, that is, 8 double points in $M_{210}\otimes \F _3$.

We may label the 4 vertices $v_1$, $v_1'$, $v_2$, $v_2'$ so that $\om _3(v_i)=v_i'$,
where $\om _3$ still denotes the involution $\om _3$ now acting on $\mathcal G$. There are
no edges joining $v_1$ and $v_2$, and the same holds for $v_1'$ and $v_2'$. The
total number of edges joining $v_1$ with $v_1'$ and $v_2$ with $v_2'$ is 4, as
Kurihara (cf.\,\cite{Ku}) deduced from trace formulae of Brandt matrices. Since there
must also be $p+1=4$ edges at the star of any vertex, it turns out that the dual
graph $\mathcal G$ must be

\quad
\quad
\quad
\quad
\quad
\quad
\quad



\begin{picture}(350,80)

\put(300,10){\makebox(0,0){\bfseries Dual graph of $M_{210}\otimes \F _3$  }}
\put(120,20){\circle*{2}}
\put(180,20){\circle*{2}}
\put(120,60){\circle*{2}}
\put(180,60){\circle*{2}}

\put(110,10){\makebox(0,0){$v_2'$}}
\put(190,10){\makebox(0,0){$v_1$}}
\put(110,70){\makebox(0,0){$v_2$}}
\put(190,70){\makebox(0,0){$v_1'$}}

\thicklines\qbezier(120,20)(110,40)(120,60)
\qbezier(120,20)(130,40)(120,60)
\qbezier(120,20)(150,10)(180,20)
\qbezier(120,20)(150,30)(180,20)
\qbezier(180,60)(150,70)(120,60)
\qbezier(180,60)(150,50)(120,60)
\qbezier(180,60)(170,40)(180,20)
\qbezier(180,60)(190,40)(180,20)


\end{picture}

Since $3\vert 210$, $\om _{210}(\{ v_1, v_2\} )=\{v_1', v_2'\} $ and therefore
$\mathcal G/\langle \om _{210}\rangle $ is a graph with two vertices joined by two
edges, which corresponds to the Kodaira symbol $I_2$. The only elliptic curve in the
$\Q $-isogeny class $210D$ whose reduction type at $p=3$ is $I_2$ is $210D2$. Hence,
a Weierstrass equation for $X_{210}/\langle \om _{210}\rangle $ is
$y^2+xy=x^3+x^2-23x+33$.
\end{example}


Performing similar computations, we obtain the list of bielliptic Shimura curves
$(X_D$, $w)$ over $\Q $ such that the genus 1 Atkin-Lehner quotient
$X_D/\langle w\rangle $ is an elliptic curve with infinitely many rational points.
With this procedure, we also give a Weierstrass equation for the elliptic curves
$X_D/\langle w\rangle $. Together with the hyperelliptic Shimura curves over $\Q $
given by Ogg, we obtain the family of Shimura curves of genus $g(X_D)\geq 2$ with
infinitely many quadratic points. Summing up, we obtain the following

\begin{theorem}\label{Kamienny}
There are only finitely many $D$
for which $X_D$ has infinitely many quadratic
points over $\Q $. These curves, together with their rational
or elliptic quotients, are listed below.
\newpage
%\pagebreak
\vspace{1cm}

\begin{table}
\label{tableQuadraticPoints}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{11}{|c|}{\bfseries Table \ref{tableQuadraticPoints}:
Shimura curves $X_D$, $g(X_D)\geq 2$, with $\# X_D(\Q, 2)=+\infty $}\\
\hline
D&$\om _m$&$g(X_D/\langle \om _m\rangle )$&
&D&$\om _m$&$g(X_D/\langle \om _m\rangle )$&   &D&$\om _m$&$g(X_D/\langle
\om _m\rangle )$ \\
\hline \hline
$26$&$\om _{26}$& $0$ &  &$77$&$\om _{77}$& $1$ &   &$143$&$\om _{143}$ &$1$ \\
$35$&$\om _{35}$ &$0$&   &$82$&$\om _{82}$& $1$ &   &$146$&$\om _{146}$&$0$ \\
$38$&$\om _{38}$ &$0$&   &$86$ & $\om _{86}$ &$0$ &   &$159$&$\om _{159}$&$0$ \\
$39$&$\om _{39}$ &$0$ &   &$87$&$\om _{87}$ & $0$&  &$166$&$\om _{166}$&$1$  \\
$51$&$\om _{51}$ &$0$&   &$94$&$\om _{94}$& $0$&   &$194$&$\om _{194}$&$0$   \\
$55$&$\om _{55}$ & $0$&   &$95$&$\om _{95}$ &$0$ &   &$206$&$\om _{206}$ &$0$  \\
$57$&$\om _{57}$ &$1$&     &$106$&$\om _{106}$& $1$&   &$210$&$\om _{210}$ &$1$ \\
$58$&$\om _{29}$& $0$&    &$111$&$\om _{111}$&$0$&    &$215$&$\om _{215}$&$1$ \\
   &$\om _{58}$&$1$&   &$118$&$\om _{118}$ & $1$ &   &$314$&$\om _{314}$&$1$ \\
$62$&$\om _{62}$ & $0$&    &$119$&$\om _{119}$ & $0$&   &$330$&$\om _{330}$ & $1$ \\
$65$&$\om _{65}$ &$1$&    &$122$&$\om _{122}$ & $1$&   &$390$&$\om _{390}$& $1$ \\
$69$&$\om _{69}$ & $0$&    &$129$&$\om _{129}$ & $1$&    &$510$&$\om _{510}$& $1$ \\
$74$&$\om _{74}$ & $0$&   &$134$&$\om _{134}$&$0$ &   &$546$&$\om _{546}$& $1$   \\
\hline
\end{tabular}

\end{table}


\end{theorem}

\markright{\ref{secEx} Equations of elliptic quotients of Shimura curves}
\section{Equations of elliptic quotients of Shimura curves}
\markright{\ref{secEx} Equations of elliptic quotients of Shimura curves}
\label{secEx}

By using the method explained in the preceding section, we obtain the
following complete list of Weierstrass equations of quotients $X_D/\langle
\om \rangle $ of degree $2$ of a Shimura curve $X_D$
which are elliptic curves over $\Q $. The instances corresponding
to the discriminants $D\leq 60$
were already given in \cite{Rob}. In the second
column, we quote the isomorphism class of the elliptic curve in
Cremona's notation. Note that in the table below there are missing some
of the Atkin-Lehner quotients $X_D/\langle \om _m\rangle /\Q $ of genus
$1$ compiled in Table \ref{bielliptic}. These are
exactly those genus $1$ quotients which fail to have
rational points (see Table \ref{deficient}). Let us also remark that there are
several examples of elliptic quotients $X_D/\langle
\om _m\rangle $ of Mordell-Weil rank $0$ over $\Q $. This explains why
they do not appear in Table \ref{tableQuadraticPoints}.

\begin{table}
\label{tableequations}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c|}{\bfseries Table \ref{tableequations}:
Equations of all elliptic quotients of Shimura curves of degree $2$}\\
\hline
$X_D/\langle \om _m\rangle $&Cremona symbol& Weierstrass equation   \\
\hline \hline

$X_{26}/\langle \om _{2}\rangle $ &$26B_2$ & $y^2 + x y + y = x^3 - x^2 -213 x -1257$\\
$X_{26}/\langle \om _{13}\rangle $ &$26A_1$ & $y^2 + x y + y = x^3 -5 x -8$\\
$X_{38}/\langle \om _{2}\rangle $ &$38B_2$ & $y^2 + x y + y = x^3 + x^2 -70 x -279$\\
$X_{38}/\langle \om _{19}\rangle $ &$38A_3$ & $y^2 + x y + y = x^3 -16 x + 22$\\
$X_{57}/\langle \om _{57}\rangle $ &$57A_1$ & $y^2 + y = x^3 - x^2 -2 x + 2$\\
$X_{58}/\langle \om _{2}\rangle $&$58B_2$ & $y^2 + x y + y = x^3 + x^2 -455 x -3951$\\
$X_{58}/\langle \om _{58}\rangle $&$58A_1$ & $y^2 + x y = x^3 - x^2 - x + 1$\\
$X_{65}/\langle \om _{65}\rangle $ &$65A_1$ & $y^2 + x y = x^3 - x $\\
$X_{77}/\langle \om _{77}\rangle $& $77A_1$ & $y^2 + y = x^3 + 2 x $\\
$X_{82}/\langle \om _{82}\rangle $& $82A_1$  & $y^2 + x y + y = x^3 -2 x $\\
$X_{106}/\langle \om _{53}\rangle $& $106D_1$ & $y^2 + x y = x^3 + x^2 -27 x -67$\\
$X_{106}/\langle \om _{106}\rangle $& $106B_1$ & $y^2 + x y  = x^3 + x^2 -7 x + 5$\\
$X_{118}/\langle \om _{59}\rangle $ & $118D_1$  & $y^2 + x y = x^3 + x^2 + 56 x -192$\\
$X_{118}/\langle \om _{118}\rangle $ & $118A_1$  & $y^2 + x y = x^3 + x^2 + x + 1$\\
$X_{122}/\langle \om _{122}\rangle $ & $122A_1$ & $y^2 + x y + y = x^3 + 2 x$\\
$X_{129}/\langle \om _{129}\rangle $ & $129A_1$ & $y^2 + y = x^3 - x^2 -19 x + 39$\\
$X_{143}/\langle \om _{143}\rangle $ &$143A_1$ & $y^2 + y = x^3 - x^2 - x -2$\\
$X_{166}/\langle \om _{166}\rangle $&$166A_1$  & $y^2 + x y = x^3 + x^2 -6 x + 4$\\
$X_{202}/\langle \om _{101}\rangle $ & $202A_1$  & $y^2 + x y = x^3 - x^2 + 4 x -176$\\
$X_{210}/\langle \om _{210}\rangle $ &$210D_2$  & $y^2 + x y = x^3 + x^2 -23 x + 33$\\
$X_{215}/\langle \om _{215}\rangle $&$215A_1$ & $y^2 + y = x^3 -8 x -12$\\
$X_{314}/\langle \om _{314}\rangle $&$314A_1$ & $y^2 + x y = x^3 - x^2 + 13 x - 11 $\\
$X_{330}/\langle \om _{330}\rangle $ & $330E_2$ & $y^2 + x y = x^3 + x^2 -102 x + 324$\\
$X_{390}/\langle \om _{390}\rangle $& $390A_2$ & $y^2 + x y = x^3 + x^2 -33 x -63 $\\
$X_{510}/\langle \om _{510}\rangle $& $510D_2$ & $y^2 + x y + y = x^3 + x^2 -421 x - 3157$\\
$X_{546}/\langle \om _{546}\rangle $& $546C_2$  & $y^2 + x y + y = x^3 -137 x + 380$\\
\hline
\end{tabular}
\end{table}

\chapter{Resum en catal\`{a}}\label{catala}

\section*{Introducci\'{o}}
\addcontentsline{toc}{section}{Introducci\'{o}} \markboth{Chapter
\ref{catala}. Resum en catal\`{a}} {\ref{IntroC1}
Introducci\'{o}}\label{IntroC1}


En aquest treball estudiem diferents q\"{u}estions sobre la
geometria i l'aritm\`{e}tica de les \`{a}lgebres de quaternions, les varietats
abelianes i les varietats de Shimura, amb l'objectiu d'investigar les
estretes relacions existents entres elles.

M\'{e}s concretament, l'estudi se centra en varietats abelianes $A$ tals
que el seu anell d'endomorfismes $\End (A)$ \'{e}s un ordre maximal en una
\`{a}lgebra de quaternions $B$ totalment indefinida sobre un cos de nombres
$F$ totalment real i en les varietats de Shimura $X_B/\Q $ que
sorgeixen de manera natural com als seus espais de moduli. Tal i com
pretenem mostrar, moltes de les propietats aritm\`{e}tiques i geom\`{e}triques
d'aquestes varietats abelianes estan codificades o b\'{e} en l'\`{a}lgebra de
quaternions $B$ o b\'{e} en les varietats de Shimura $X_B$. Alhora, no \'{e}s
possible portar a terme un estudi d'aquestes varietats de
Shimura sense un bon coneixement dels objectes que parametritzen.


Del treball de Shimura \cite{Sh63} i
la classificaci\'{o} d'\`{a}lgebres de divisi\'{o} involutives deguda a Albert
(cf.\,\cite{Mu}), se'n segueix que hi
ha un ventall limitat d'anells que es realitzen com l'anell
d'endomorfismes d'una varietat abeliana. En efecte, si $A$ \'{e}s una
varietat abeliana simple sobre un cos algebraicament tancat, aleshores $\End
(A)$ \'{e}s un ordre en un cos totalment real, una \`{a}lgebra de quaternions
sobre un cos totalment real o una \`{a}lgebra de divisi\'{o} sobre un cos de
multiplicaci\'{o} complexa. Molts dels aspectes de la geometria i
l'aritm\`{e}tica de les varietats abelianes es poden interpretar en els
seus anells d'endomorfismes. De fet, \'{e}s remarcable que, en molts sentits,
varietats abelianes amb anells d'endomorfismes diferents tenen
comportaments diferents.

Hi ha un nombre considerable de treballs sobre varietats abelianes amb
multiplicaci\'{o} complexa. Sense cap \`{a}nim de donar-ne un llistat
exhaustiu, algunes de les aportacions m\'{e}s rellevants es deuen a
Shimura i Taniyama \cite{ShTa}, Lang \cite{La} o Mumford
\cite{Mu}. L'impacte d'aquests treballs en branques cabdals
de la teoria de nombres, com ara la teoria de cossos de classes o la conjectura
de Birch i Swinnerton-Dyer, ha estat enorme.

Tamb\'{e} hi ha literatura abundant sobre varietats abelianes amb
multiplicaci\'{o} totalment real. Destaquem els treballs de Humbert
\cite{Hu}, Ribet \cite{Ri80}, \cite{Ri94}, Lange \cite{La} i Wilson \cite{Wi}. En
aquest cas, el desenvolupament d'aquesta teoria ha
estat tradicionalment motivat per la seva relaci\'{o} evident amb les
conjectures generalitzades de Shimura-Taniyama-Weil. En efecte, les
varietats abelianes modulars $A_f/\Q $ associades a una forma modular
cuspidal $f\in S_2(\Gamma _0(N))$ de pes $2$ i car\`{a}cter trivial
tenen multiplicaci\'{o} real sobre $\Q $. El lector pot consultar
\cite{HaHaMo} i \cite{Ri90} per a m\'{e}s detalls.

En canvi, l'estudi de les varietats abelianes amb multiplicaci\'{o}
quaterni\`{o}ni\-ca ha estat portat a terme per menys autors. En
aquest cas, l'aritm\`{e}tica de les \`{a}lgebres d'endomorfismes
d'aquestes varietats abelianes \'{e}s m\'{e}s complexa que en el cas
commutatiu i aquest fet t\'{e} com a conseq\"{u}\`{e}ncia que el grup de
N\'{e}ron-Severi de les varietats abelianes amb multiplicaci\'{o}
quaterni\`{o}nica \'{e}s m\'{e}s inaccessible. Sigui com sigui, adrecem el
lector a \cite{No}, \cite{JoMo}, \cite{HaMu}, \cite{HaHaMo},
\cite{Oh}, \cite{DiRo1} i \cite{DiRo2} per a algunes contribucions
recents.

Shimura \cite{Sh63}, \cite{Sh67} va considerar els espais grollers de
m\`{o}duli de varie\-tats abelianes amb multiplicaci\'{o} quaterni\`{o}nica i va
provar que admeten un model can\`{o}nic $X_B/\Q $ sobre el cos $\Q $ dels nombres
racionals. Com a varietats anal\'{\i}tiques, les varietats $X_B(\C )$
es poden descriure per mitj\`{a} de quocients compactes de certs dominis
sim\`{e}trics
i afitats per grups aritm\`{e}tics que actuen en ells. Shimura va explorar
les propietats diofantines de les varietats $X_B$ i va demostrar que
les coordenades dels anomenats {\em punts de Heegner} sobre $X_B$
generen cossos de classes tals que l'acci\'{o} galoisiana sobre ells pot
ser descrita mitjan\c{c}ant lleis de reciprocitat expl\'{\i}cites.

En els darrers anys, hi ha hagut un inter\`{e}s creixent en l'estudi de
les varietats de Shimura que ha estat cruicial en molts aspectes de la
teoria dels nombres.

En efecte, pel que fa a les conjectures modulars en el seu sentit mes
ampli, les corbes de Shimura juguen un paper fonamental en la
demostraci\'{o} de Ribet de la conjectura Epsilon que, a la vegada, implica
que el darrer teorema de Fermat \'{e}s una conseq\"{u}\`{e}ncia de la conjectura de
Shimura-Taniyama-Weil (cf.\,\cite{Ri89}, \cite{Ri90} i \cite{Pr}).

En relaci\'{o} a les conjectures de Birch i Swinnerton-Dyer, Vatsal
\cite{Va} i Cornut \cite{Cor} han provat recentment i de manera
independent unes conjectures de Mazur sobre el comportament dels
punts de Heegner sobre cossos anticiclot\`{o}mics en corbes
el$\cdot $l\'{\i}ptiques, tot fent \'{u}s de corbes modulars, corbes de Shimura,
corbes de Gross i les teories erg\`{o}diques de Ratner. D'altra banda,
Bertolini i Darmon \cite{BeDa96}, \cite{BeDa98}, \cite{BeDa99} han
explotat la teoria de \v{C}erednik-Drinfeld sobre les fibres
singulars dels models enters de Morita de les corbes de Shimura
per provar versions anti-ciclot\`{o}miques de les conjectures de
Mazur, Tate i Teitelbaum sobre variants p-\`{a}diques de la conjectura
de Birch i Swinnerton-Dyer.

Pel que fa a les conjectures de finitud i de quadratura del grup de
Shafare\-vich-Tate d'una varietat abeliana sobre un cos de nombres,
Poonen i Stoll \cite{PoSt} recentment han fet un estudi de
l'aparellament de Cassels-Tate en aquest context i han donat criteris
expl\'{\i}cits per a la quadratura de la part de torsi\'{o} del grup de
Shafare\-vich-Tate de les varietats Jacobianes de les corbes
algebraiques. Al seu torn, i basant-se en aquests resultats, Jordan
i Livn\'{e} \cite{JoLi99} han mostrat quocients Atkin-Lehner de corbes de Shimura
tals que el nombre d'elements de la part finita del grup de
Shafare\-vich-Tate de les seves varietats Jacobianes
no \'{e}s un quadrat perfecte sin\'{o} el doble d'un quadrat perfecte.

En un treball recent, Stein \cite{St} ha proporcionat exemples
expl\'{\i}cits de varietats abelianes $A/\Q $ tals que
$\sharp \mathrm{Sha }(A/\Q ) = p\cdot n^2$, $n\in \Z $, per tot nombre
primer senar $p<10000$, $p\not = 37$.

A banda de les seves moltes aplicacions, molts autors han
investigat les propietats geom\`{e}triques i diofantines de les corbes
de Shimura, que s\'{o}n interessants per elles mateixes. D'aquesta
manera, els models enters d'aquestes corbes i les seves fibres
especials han estat considerats per Morita \cite{Mo}, Boutot i
Carayol \cite{BoCa}, Buzzard \cite{Bu}, \v{C}erednik \cite{Ce},
Drinfeld \cite{Dr} i Zink \cite{Zi}, entre molts d'altres. De gran
inter\`{e}s tamb\'{e} \'{e}s la s\`{e}rie d'articles de Kudla i Kudla-Rapoport
sobre els aparellaments d'altura en les corbes de Shimurs, nombres
d'intersecci\'{o} de $0$-cicles especials i els valors de les funcions
derivades de certes s\`{e}ries d'Einsestein al centre del seu puntt de
simetria, en la l\'{\i}nea dels treballs cl\`{a}ssics de Gross-Zagier i
Hirzebruch-Zagier. Vegeu \cite{Kud} i \cite{KudRa}, per exemple.

Resultats de caire efectiu i computacional sobre les corbes de Shimura
han estat portats a terme per Kurihara \cite{Ku}, Elkies
\cite{El}, Alsina \cite{Al} i Bayer \cite{Ba}, entre d'altres.
Aquests treballs tenen una v\`{a}lua particular, degut a l'abs\`{e}ncia de punts
cuspidals en aquestes corbes, que fa el tractament d'aquestes q\"{u}estions
m\'{e}s dif\'{\i}cil que en el cas modular cl\`{a}ssic.

Un altre resultat remarcable i molt recent \'{e}s la demostraci\'{o}
d'Edix\-ho\-ven i Ya\-fa\-ev \cite{EdYa} de la conjectura
d'An\-dr\'{e}-Oort sobre la dis\-tri\-bu\-ci\'{o} de punts espe\-cials
en les varietats de Shi\-mu\-ra.

En una altra direcci\'{o}, I\-ha\-ra \cite{Ih}, Jor\-dan-Liv\-n\'{e}
\cite{JoLi85}, \cite{Jo86}, \cite{JoLi86}, Ogg \cite{Ogg83}, \cite{Ogg84},
Mil\-ne \cite{Mi} i Ka\-mienny \cite{Ka} han estudiat, sota diferents
punts de vista, els conjunts de punts racionals de les corbes de
Shi\-mu\-ra, els seus quocients At\-kin-Leh\-ner i les seves varietats
Ja\-co\-bianes sobre cossos glo\-bals, lo\-cals i fi\-nits. Finalment,
adrecem el lector a \cite{Gr} per a una
introducci\'{o} espec\'{\i}fica a les superf\'{\i}cies de
Shi\-mura.




\section{Resultats}
\markright{\ref{resultats} Resultats}
\label{resultats}

\subsection*{Introducci\'{o}}\addcontentsline{toc}{subsection}{Introducci\'{o}}
\sk\noindent

L'objectiu d'aquest apartat \'{e}s exposar els resultats fonamentals
d'aquesta tesi sobre varietats
abelianes amb multiplicaci\'{o} quaterni\`{o}nica de dimensi\'{o} arbitr\`{a}ria i
les varietats de Shimura que sorgeixen com als seus espais de m\`{o}duli.

\subsection[Varietats abelianes amb multiplicaci\'{o} quaterni\`{o}\-ni\-ca]
{Varietats abelianes amb multiplicaci\'{o} quaterni\`{o}\-ni\-ca}

En aquesta secci\'{o} exposem els resultats continguts en el cap\'{\i}tol \ref{GAV}
d'aquesta mem\`{o}ria.

\'{E}s ben conegut que les corbes el$\cdot $l\'{\i}ptiques sobre un cos algebraicament tancat
qualsevol admeten exactament una
polaritzaci\'{o} principal llevat de translacions. En general, les
varietats abelianes de dimensi\'{o} superior no comparteixen aquesta
propietat. Si $A$ \'{e}s una varietat abeliana, no \'{e}s una q\"{u}esti\'{o} trivial
decidir si $A$ \'{e}s principalment polarizable i, en aquest cas, \'{e}s un
problema interessant descriure el conjunt $\Pi _0(A)$ de classes d'isomorfisme
de polaritzacions principals de $A$. El teorema de Narasimhan-Nori
\cite{NaNo} assegura que $\Pi _0(A)$ \'{e}s un conjunt finit i nosaltres
denotem per $\pi _0(A)$ el seu cardinal.

Una varietat abeliana principalment polaritzable gen\`{e}rica admet una \'{u}nica
classe de polaritzacions principals. Humbert \cite{Hu} va exhibir
superf\'{\i}cies abelianes complexes i simples amb dues polaritzacions principals
no isomorfes. M\'{e}s tard, Hayashida i Nishi
(cf.\,\cite{HaNi} i \cite{Ha}) van calcular el nombre
$\pi _0(E_1\times E_2)$ per parelles de corbes el$\cdot $l\'{\i}ptiques is\`{o}genes
$E_1/\C $ i $E_2/\C $ amb multiplicaci\'{o} complexa. En caracter\'{\i}stica positiva,
Ibukiyama, Katsura i Oort \cite{IbKaOo} van relacionar el nombre de polaritzacions
principals en la pot\`{e}ncia $E^n$ d'una corba el$\cdot $l\'{\i}ptica supersingular amb
el nombre de classes de certes formes hermitianes. Lange \cite{La}
va traduir aquest problema al llenguatge de la teoria de nombres en termes de
l'aritm\`{e}tica de l'anell $\End (A)$ i va produir exemples de varietats
abelianes simples de dimensi\'{o} superior amb diverses polaritzacions
principals. Al mateix temps, per\`{o}, va mostrar que per a les varietats abelianes amb
\`{a}lgebra d'endomorfismes $\End (A)\otimes \Q
=F$ commutativa i totalment real, el nombre $\pi _0(A)$ \'{e}s
uniformement afitat en termes de la dimensi\'{o} de $A$: $\pi _0(A)\le
2^{\dim (A)-1}$. En altres paraules: les varietats abelianes amb
multiplicaci\'{o} real poden admetre diverses polaritzacions principals
per\`{o} no un nombre {\em arbitr\`{a}riament gran}.

Es podria considerar natural que la fita de Lange, o alguna altra fita
per a $\pi _0(A)$, fos v\`{a}lida per a {\em qualsevol} varietat abeliana simple.
\'{E}s doncs natural plantejat la seg\"{u}ent

$\\ $
{\bfseries Q\"{u}esti\'{o}.} {\em Sigui $g\ge 1$ un enter positiu. Existeixen
varietats abelianes
simples de dimensi\'{o} $g$ amb un nombre arbi\-tr\`{a}riament gran de polaritzacions
principals no isomorfes?}
$\\ $

Tal i com ja hem observat, en dimensi\'{o} $1$ la resposta a aquesta
q\"{u}esti\'{o} \'{e}s
negativa. Quan $g=2$, tans sols es coneixien superf\'{\i}cies abelianes simples amb
$\pi _0(A)\leq 2$, degut al treball de Humbert. Un dels nostres
resultats principals, enunciat en un cas particular, \'{e}s el
seg\"{u}ent.

$\\ $
{\bfseries Teorema.}
{\em Sigui $F$ un cos de nombres totalment real de grau $[F:\Q ]=n$, sigui
$R_F$ el seu anell d'enters i sigui $\vartheta _{F/\Q }$
la diferent de $F$ sobre $\Q $. Sigui $A$ una varietat abeliana complexa
de dimensi\'{o} $2 n$ i tal que el seu anell d'endomorfismes $\End (A)\simeq \cO$ \'{e}s
un ordre maximal en una \`{a}lgebra de divisi\'{o} de quaternions totalment indefinida
$B$ sobre $F$.}

$\\ $
{\em Suposem que el nombre restringit de classes $h_+(F)$ de $F$ \'{e}s $1$ i que
$\vartheta _{F/\Q }$ i $\disc (B)$ s\'{o}n ideals coprimers de $F$.
Aleshores,}

\begin{enumerate}
\item
{\em $A$ \'{e}s principalment polaritzable.}

\item
{\em El nombre de classes d'isomorfisme de polaritzacions principals de $A$ \'{e}s

$$
\pi _0(A)= \frac {1}{2} \sum _S h(S),
$$
on $S$ recorre el conjunt finit d'ordres en el cos de multiplicaci\'{o} complexa
$F(\sqrt {-D})$ que contenen $R_F[\sqrt {-D}]$, $D\in F^*_+$ \'{e}s un
generador totalment positiu del discriminant redu\"{\i}t $\cD $ de
$B$, i $h(S)$ denota el nombre de classes de $S$.}

\end{enumerate}

En particular, si $A$ \'{e}s una superf\'{\i}cie abeliana,

$$
\pi _0(A)=
  \begin{cases}
    \dfrac {h(-4 D)+h(-D)}{2} &
    \text{ if }D\equiv 3\quad \mbox{ mod } 4, \\
    \dfrac {h(-4 D)}{2} & \text{altrament.}
  \end{cases}
$$

Per tal d'acometre la demostraci\'{o} del teorema anterior, presentem una
aproximaci\'{o} al problema que t\'{e} els seus or\'{\i}gens en el
treball cl\`{a}ssic de Shimura \cite{Sh63} sobre fam\'{\i}lies anal\'{\i}tiques de
varietats abelianes
amb anell d'endomorfismes prescrit.

La nostra proposta \'{e}s essencialment diferent de la presa per Lange en \cite{La} o
Ibukiyama-Katsura-Oort's en \cite{IbKaOo}. En efecte, mentre que a \cite{La}
i \cite{IbKaOo} s'explota la interpretaci\'{o} dels
fibrats de l\'{\i}nea com a endomorfismes sim\`{e}trics, nosaltres tradu\"{\i}m les
q\"{u}estions que ens ocupen al llenguatge d'immersions optimals d'ordres quadr\`{a}tics
d'Eichler. Aix\`{o} ens porta a resoldre un problema que t\'{e} les seves
arrels en el treball de O'Connor, Pall i Pollack (cf.\,\cite{Po}).

En relaci\'{o} a la q\"{u}esti\'{o} anterior, el nostre segon resultat principal del cap\'{\i}tol
\ref{GAV} \'{e}s el seg\"{u}ent.

$\\ $
{\bfseries Teorema.}
{\em Sigui $g$ un enter positiu. Aleshores,}
\begin{enumerate}
\item
{\em Si $g$ \'{e}s parell, existeixen varietats abelianes simples $A$ de
dimensi\'{o} $g$ tals que $\pi _0(A)$ \'{e}s arbitr\`{a}riament gran.}
\item
{\em Si $g$ \'{e}s senar i lliure de quadrats, es t\'{e} que
$\pi _0(A)\leq 2^{g-1}$, per tota
varietat abeliana simple $A$ de dimensi\'{o} $g$ sobre $\C $.}

\end{enumerate}

$\\ $
El creixement indefinit de $\pi _0(A)$ quan $g$ \'{e}s parell se segueix
del teorema enunciat anteriorment, en la seva versi\'{o} precisa enunciada
com a Teorema \ref{pio} al cap\'{\i}tol \ref{GAV} combinat amb resultats  anal\'{\i}tics
sobre el comportament assimpt\`{o}tic dels nombres de classes relatius de
cossos CM deguts a Horie-Horie \cite{HoHo} i Louboutin \cite{Lo00},
\cite{Lo}. La segona part del teorema se segueix de les idees de Lange a
\cite{La}.

Del teorema anterior i del fet que tota superf\'{\i}cie abeliana
simple i principalment polaritzada \'{e}s la varietat Jacobiana d'una corba llisa
de g\`{e}nere $2$ que, pel teorema de Torelli, \'{e}s \'{u}nica llevat d'isomorfisme.
$\\ $

{\bfseries Corol$\cdot $lari.} {\em Existeixen conjunts
$\{ C_1$, ..., $C_N\} $ amb un nombre arbitr\`{a}ria\-ment
gran de corbes de g\`{e}nere $2$ no isomorfes que posseeixen
varietats Jacobianes simples i
isomorfes $J(C_1)\simeq J(C_2)\simeq ... \simeq J(C_N)$.}
$\\ $

Notem que l'enunciat del darrer teorema no t\'{e} en consideraci\'{o} les
varietats abelianes de dimensi\'{o} senar i no lliure de quadrats. La
conjectura seg\"{u}ent est\`{a} motivada pel fet que, quan $g$ no \'{e}s lliure
de quadrats, existeixen varietats abelianes tals que el seu anell
d'endomorfismes \'{e}s un ordre en una \`{a}lgebra de divisi\'{o} no commutativa
sobre un cos de multiplicaci\'{o} complexa i hi ha una estreta similitud
entre l'aritm\`{e}tica dels grups de N\'{e}ron-Severi d'aquestes varietats
abelianes i de les varietats abelianes amb multiplicaci\'{o} quaterni\`{o}nica.

$\\ $
{\bfseries Conjectura. } {\em
Sigui $g$ un enter positiu no lliure de quadrats. Aleshores, existeixen
varietats abelianes simples de dimensi\'{o} $g$ tals que $\pi _0(A)$ \'{e}s
arbitr\`{a}riament gran.}

\subsection{Varietats de Shimura i morfismes d'oblit}

Sigui $F$ un cos de nombres totalment real de grau $[F:\Q ]=n$ i sigui
$B$ una \`{a}lgebra de divisi\'{o} de quaternions totalment indefinida sobre $F$. En el
cap\'{\i}tol \ref{GSV} d'aquesta mem\`{o}ria estudiem certes varietats de Shimura
$X_B$ associades a l'\`{a}lgebra $B$ i certs morfismes d'oblit que ocorren
entre elles.

Per simplicitat en l'exposici\'{o}, suposarem al llarg de la secci\'{o} que
$h_+(F)=1$. Sigui $D\in F^*_+$ un generador totalment positiu de $\disc (B)$.

$\\ $
{\bfseries Definici\'{o}. } Un {\em ordre maximal principalment
polaritzat} de $B$ \'{e}s un parell $(\cO , \mu )$ tal que $\cO \subset B$
\'{e}s un ordre maximal i $\mu \in \cO $ \'{e}s un quaterni\'{o} pur tal que
$\mu ^2 + u D = 0$ per alguna unitat $u\in R_{F +}^*$.

$\\ $
Associat a un ordre maximal principalment polaritzat $(\cO , \mu )$,
podem consi\-derar el problema seg\"{u}ent de m\`{o}duli sobre $\Q $: classificar
les classes d'isomorfis\-me de tripletes $(A, \iota , \cL )$ donades per

\begin{itemize}
\item
Una varietat abeliana $A$ de dimensi\'{o} $g=2n$.

\item
Un homomorfisme d'anells $\iota :\cO \hookrightarrow \End (A)$.

\item
Una polaritzaci\'{o} principal $\cL $ en $A$ tal que
$$
\iota (\beta )^{\circ } = \iota (\mu ^{-1}\bar {\beta }\mu )
$$
per tot $\beta \in \cO $, on
$\circ :\End (A)\ra \End (A)$ denota la involuci\'{o} de Rosati respecte $\cL $.

\end{itemize}

Shimura \cite{Sh63}, \cite{Sh67} va demostrar que el corresponent
functor de moduli es pot representar grollerament per un esquema $X _{\mu }/\Q $
redu\"{\i}t, irreductible i quasi-projectiu sobre $\Q $ de dimensi\'{o} $n=[F: \Q ]$.
A m\'{e}s a m\'{e}s, degut al fet que
$B$ \'{e}s una \`{a}lgebra de divisi\'{o}, la varietat de
Shimura $X _{\mu }$ \'{e}s completa.

Sigui $\mathfrak H = \{ z\in \C : \mathrm{Im }(z)>0 \}$ l'hiperpl\`{a} superior de
Poincar\'{e}. Com a varietat anal\'{\i}tica, $X _{\mu }(\C )$ es pot
descriure, independentment de la tria de $\mu $, com el quocient
$$
\cO ^1\backslash \mathfrak H^n\simeq X _{\mu }(\C )
$$
de l'espai sim\`{e}tric $\mathfrak H^n$ per l'acci\'{o} del grup
$\cO ^1$, ent\`{e}s com a subgrup discontinu de
$\mbox{SL}_2(\R )^n$.

A m\'{e}s de les varietats de Shimura que tot just hem introdu\"{\i}t, sigui $\cH _F/\Q $
l'esquema modular de Hilbert que representa de forma grollera el functor
associat al problema de m\`{o}duli de classificar varietats abelianes principalment
polaritzades $(A, \cL )$ de dimensi\'{o} $g$ prove\"{\i}des d'un
morfisme d'anells $R_F\hookrightarrow \End (A)$. La varietat modular de
Hilbert $\cH _F$ t\'{e} dimensi\'{o} $3 n$ i el conjunt de punts complexos
$\cH _F(\C )$ \'{e}s el quocient de $\mathfrak H_2^n$, on $\mathfrak H_2$
denota l'espai de Siegel de dimensi\'{o} $3$, per un cert grup discontinu
(cf.\,\cite{Sh63}, \cite{LaBi}).

Observem que quan $F=\Q $, aleshores $\cH _F = \cA _2$ \'{e}s la
$3$-varietat d'Igusa, l'espai de m\`{o}duli de les superf\'{\i}cies
abelianes principament polaritzades.

Existeixen uns morfismes naturals

$$
\begin{matrix}
 \pi : & X _{\mu } & \stackrel{\pi _F}{\longrightarrow }&
 \cH _F &\longrightarrow  & \cA _g  \\
       &(A, \iota , \cL )& \mapsto & (A, \iota |_{R_F}, \cL ) &\mapsto & (A, \cL )
\end{matrix}
$$
de la varietat de Shimura $X _{\mu }$ en la varietat modular de
Hilbert $\cH _F$ i l'espai de m\`{o}duli $\cA _g$ que
consisteixen en oblidar de manera gradual l'estructura d'endomorfismes
quaterni\`{o}nics. Aquests morfismes s\'{o}n representables, propis i estan definits
sobre el cos $\Q$ dels nombres racionals.

$\\ $
{\bfseries Definici\'{o}.} El {\em grup d'Atkin-Lehner} $W$
d'un ordre maximal $\cO $ de $B$ \'{e}s

$$
W = \Nm _{B_+^*}(\cO )/(F^*\cdot \cO ^1).
$$

Es satisf\`{a} que

$$
W\simeq \Z /2 \Z \times
\stackrel{2 r}{...}\times \Z /2 \Z ,
$$
on $2 r\geq 2$ \'{e}s el nombre de
ideals primers de $F$ que ramifiquen en $B$.

$\\ $
{\bfseries Definici\'{o}.} Sigui $(\cO , \mu )$ un ordre maximal principalment
polaritzat en $B$. Un {\em tor\c{c}ament} de $(\cO ,\mu )$ \'{e}s un element
$\chi \in \cO \cap \Nm _{B^*}(\cO )$ tal que
$\chi ^2+\n (\chi )=0$ i $\mu \chi =-\chi \mu $.

$\\ $
En altres paraules, un tor\c{c}ament de $(\cO , \mu )$ \'{e}s un quaterni\'{o} pur
$\chi \in \cO \cap \Nm _{B^*}(\cO )$ tal que
$$
B = F+F\mu +F\chi +F \mu \chi = (\frac {-u D, -\n (\chi )}{F}).
$$

Diem que un ordre maximal principalment polaritzat $(\cO , \mu )$ en $B$ \'{e}s
{\em tor\c{c}at} si admet un tor\c{c}ament $\chi $ en $\cO $. Diem que un ordre
maximal $\cO $ \'{e}s {\em tor\c{c}at} si existeix $\mu \in \cO
$ tal que $(\cO , \mu )$ \'{e}s tor\c{c}at. Finalment, diem que
$B$ \'{e}s {\em tor\c{c}ada} si existeix un ordre maximal tor\c{c}at en $B$. Observem que
$B$ \'{e}s tor\c{c}ada si, i nom\'{e}s si,
$B\simeq (\frac {-u D, m}{F})$ per alguna unitat $u\in R^*_{F +}$ i $m\in
F^*$ tal que $m\vert D$.

$\\ $
{\bfseries Definici\'{o}.} Una {\em involuci\'{o} tor\c{c}ada}
$\om \in W$ de $(\cO , \mu )$
\'{e}s una involuci\'{o} d'Atkin-Lehner tal que $[\om ]=[\chi ]\in
W$ es pot representar en $B^*$ per un tor\c{c}ament $\chi $ de $(\cO , \mu )$.

$\\ $

Per tot ordre maximal principalment polaritzat $(\cO , \mu )$, sigui
$R_{\mu }=F(\mu )\cap \cO $ i sigui $\Omega =
\Omega (R_{\mu })=\{ \xi \in R_{\mu }: \xi ^f=1, f\geq 1\} $ el grup finit
d'arrels de la unitat en l'ordre de multiplicaci\'{o} complexa $R_{\mu }$ sobre $R_F$.

$\\ $
{\bfseries Definici\'{o}. } El {\em grup estable} de $(\cO , \mu )$ \'{e}s el subgrup

$$
W_0=U_0\cdot V_0
$$
de $W$ generat per

$$
U_0 = U_0(\cO , \mu )=\Nm _{F(\mu )^*}(\cO )/(F^*\cdot \Omega (R_{\mu })),
$$
i el grup $V_0$ generat per les involucions tor\c{c}ades de $(\cO , \mu )$.

$\\ $
Disposem dels seg\"{u}ents monomorfismes naturals de grups
$V_0\subseteq W_0\subseteq W\subseteq \Aut _{\Q }(X _{\mu })\subseteq
\Aut _{\qbar }(X _{\mu }\otimes \qbar )$.

$\\ $
{\bfseries Teorema. } {\em Sigui $(\cO , \mu )$ un ordre maximal principalment
polaritzat en $B$ i sigui $X _{\mu }$ la varietat de
Shimura associada a $(\cO , \mu )$. Aleshores existeix un diagrama
commutatiu de morfismes de fibres finites}
$$
\begin{matrix}
X_{\mu } &  & \stackrel {\pi _F}{\longrightarrow } &  & \cH _F \\
       &  \searrow &                 & \stackrel {b_F}{\nearrow  } &  \\
       &          & X _{\mu }/W_0,  &  &
\end{matrix}
$$
{\em on $X _{\mu }\rightarrow X _{\mu }/W_0$ \'{e}s la projecci\'{o}
natural i $b_F: X _{\mu }/W_0\rightarrow \pi _F(X _{\mu })$ \'{e}s una
equival\`{e}ncia birracional entre $X _{\mu }/W_0$ i la imatge de
$X_{\mu }$ en $\cH _F$.

El domini de definici\'{o} de $b_F^{-1}$ \'{e}s $\pi _F(X_{\mu
})\setminus \mathcal T _F$, on $\mathcal T _F$ \'{e}s un conjunt finit de punts de
Heegner.}

$\\ $
Concloem aquesta exposici\'{o} dels resultats de la tesi en dimensi\'{o}
arbitr\`{a}ria tot indicant dues aplicacions diferents dels resultats que hem
presentat.

La primera aplicaci\'{o} ha estat desenvolupada a la secci\'{o}
\ref{qlocus} i es refereix a la geometria del {\em lloc quaterni\`{o}nic}
$\cQ _{\cO }$ de les varietats abelianes que admeten multiplicaci\'{o} per un
ordre maximal $\cO $ en l'espai de m\`{o}duli $\cA _g$ de varietats abelianes
principalment polaritzades de dimensi\'{o} parell $g$.

Mitjan\c{c}ant el teorema anterior
i la teoria d'Eichler en immersions optimals, el nombre de components
irreductibles de $\cQ _{\cO }$ es poden relacionar amb certs nombres de classes i es
pot estudiar la seva irreductibilitat.

En segon lloc, el
teorema anterior tamb\'{e} es pot utilitzar per a explorar
l'aritm\`{e}tica de les varietats abelianes amb multiplicaci\'{o}
quaterni\`{o}nica. Efectivament, a la secci\'{o} \ref{FM} combinem el
teorema anterior amb la seva interpretaci\'{o} modular per tal d'obtenir
resultats sobre el cos de m\`{o}duli dels endomorfismes en aquestes
varietats abelianes.


\markright{\ref{ccl} Conclusions}
\section{Conclusions}
\markright{\ref{ccl} Conclusions}
\label{ccl}


\subsection*{Introducci\'{o}}\addcontentsline{toc}{subsection}{Introducci\'{o}}
\sk\noindent

En aquest apartat exposem els resultats, de caire m\'{e}s acurat,
que hem obtingut en els cap\'{\i}tols \ref{AAV} i \ref{ASV} en l'estudi de les
varietats abelianes amb multiplicaci\'{o} quaternionica de dimensi\'{o} $g=2$ i
les varietats de Shimura de dimensi\'{o} $n=1$. Aquests resultats s\'{o}n fruit
de la combinaci\'{o} de la teoria desenvolupada en els cap\'{\i}tols anteriors i
d'eines noves.


\subsection{Superf\'{\i}cies abelianes amb multiplicaci\'{o} quaterni\`{o}\-nica}

Fixem una clausura algebraica $\qbar \subset \C $ del cos
$\Q $ dels nombres racionals i sigui $K\subset \qbar $ un cos de nombres.

$\\ $
{\bfseries Proposici\'{o}.} (Silverberg) {\em
Sigui $A/K$ una varietat abeliana sobre $K$ i sigui
$S \subseteq \End _{\qbar }(A)$ un subanell d'endomorfismes de $A$. Aleshores
hi ha una \'{u}nica extensi\'{o} minimal $K_S/K$ tal que $S \subseteq \End _{K_S}(A)$.

L'extensi\'{o} $K_S/K$ \'{e}s normal i no ramificada en els ideals primers de
$K$ de bona reducci\'{o} de $A$.}

$\\ $
{\bfseries Teorema.}
Sigui $A/K$ una superf\'{\i}cie abeliana amb multiplicaci\'{o} quaterni\`{o}nica per un
ordre $\cO $ en una \`{a}lgebra de quaternions $B$. Aleshores,
$\Gal (K_B/K)$ \'{e}s isomorf a un grup c\'{\i}clic $C_n$ o un grup dihedral
$D_n$ amb $n=1, 2, 3, 4$ o $6$.

$\\ $
{\bfseries Teorema.}
{\em Sigui $C/K$ una corba de g\`{e}nere $2$ i definida sobre $K$. Sigui
$J(C)$ la varietat Jacobiana de $C$ i suposem que
$\End _{\qbar }(J(C))=\cO $ \'{e}s un ordre maximal en
una \`{a}lgebra de divisi\'{o} de quaternions $B$ sobre $\Q $ de discriminant $D$.

Sigui $K_B/K$ l'extensi\'{o} minimal de $K$ sobre la qual tots els
endomorfismes de $J(C)$ estan definits. Aleshores}

\begin{enumerate}

\item
{\em $K_B/K$ \'{e}s una extensi\'{o} abeliana amb $G = \Gal (K_B/K)\simeq (1)$,
$C_2$ o $D_2 = C_2\times C_2$.}

\item
{\em Si $B \not \simeq (\dfrac {-D, m}{\Q })$ per a qualsevol $m\vert D$,
aleshores $K_B/K$ \'{e}s una extensi\'{o} trivial o
quadr\`{a}tica de $K$. En aquest darrer cas, $\End _K(A)\simeq \Q (\sqrt
{-D})$.}

\item
{\em Si $B = (\dfrac {-D, m}{\Q })$ per algun enter positiu $m\vert D$, aleshores
$\End _K(A)$ \'{e}s isomorf o b\'{e} a $\cO $, a un ordre de $\Q (\sqrt
{-D})$, $\Q (\sqrt {m} )$ o $\Q (\sqrt {D/m})$, o $\Z $. En cada
cas, tenim respectivement que $\Gal (K_B/K)\simeq (1)$, $C_2$ i $D_2$.}


\end{enumerate}


A continuaci\'{o} presentem alguns exemples que il$\cdot $lustren els nostres
resultats.

$\\ $
{\bfseries Exemple I.} Sigui $C_I/\Q $ el model projectiu i llis de la
corba
$$
Y^2 = (X^2 + 7/2) (83/30 X^4 + 14 X^3 - 1519/30 X^2 + 49 X -
1813/120)
$$
i sigui $J_I=J(C_I)/\Q $ la seva varietat Jacobiana. Aleshores,

\begin{itemize}
\item
$\End _{L}(J_I)=\cO $ \'{e}s un ordre maximal de l'\`{a}lgebra de quaternions
$B_6$ de discriminant $6$ i $L=\Q(\sqrt{-6}, \sqrt{-14})$,


\item
$\End _{\Q (\sqrt{-14})}(J_I)\otimes \Q = \Q(\sqrt{2})$,


\item
$\End _{\Q (\sqrt{21})} (J_I)\otimes \Q =\Q (\sqrt {3})$,

\item
$\End _{\Q (\sqrt{-6})}(J_I)\otimes \Q = \Q (\sqrt {-6})$ i
\item
$\End _{\Q }(J_I)=\Z $.

\end{itemize}

$\\ $
Si tenim en compte que $B_6 = (\frac {-6, 2}{\Q }) = (\frac {-6, 3}{\Q })$
\'{e}s una \`{a}lgebra de quaternions tor\c{c}ada, l'exemple anterior
est\`{a} d'acord amb el
teorema anterior.

$\\ $
{\bfseries Exemple II.}
Sigui $C_{II}$ el model projectiu i llis de la
corba

$$
Y^2 = \frac{1}{48}X(9075X^4+3025(3+2
\sqrt{-3}) X^3 -6875 X^2 +220 (-3+ 2 \sqrt{-3}) X + 48).
$$

Sigui $J_{II}/K$ la varietat Jacobiana de $C_{II}$ sobre $K=\Q (\sqrt
{-3})$.

\begin{itemize}

\item
$\End _{L}(J_{II})=\cO $ \'{e}s un ordre maximal de l'\`{a}lgebra de quaternions
$B_{10}$ de discriminant $D= 10$ i $L= \Q( \sqrt{-3}, \sqrt{-11})$
\'{e}s el cos minimal de definici\'{o} dels endomorfismes quaterni\`{o}nics de
$J_{II}$ i

\item
$\End_K(J_3)\otimes_\Z \Q = \Q(\sqrt{5})$.
\end{itemize}
$\\ $

{\bfseries Exemple III.}
Sigui $C_{III}/K$ la fibra de la fam\'{\i}lia $S_6(t,s)$ de Hashimoto-Tsunogai \cite{HaTs}
en el par\`{a}metre $t = 2$ sobre $K =\Q(\sqrt{-379})$.  Aleshores
$\End _{}(J_{III})=\cO $ \'{e}s un ordre maximal de l'\`{a}lgebra de quaternions
$B_{6}$ de discriminant $D = 6$ i tots els endomorfismes quaterni\`{o}nics de
$J_{III}$ estan definits sobre el mateix cos $K$.


\subsection{Propietats diofantines de les corbes de Shimura}


Sigui $B$ una \`{a}lgebra de divisi\'{o} de quaternions indefinida sobre
$\Q $ i sigui $\mathcal O$ un ordre maximal de $B$. Sigui
$D=p_1\cdot ... \cdot p_{2r}$ el discriminant de $B$. Podem veure
el grup $\Gamma =\{ \gamma \in \cO : n(\gamma )=1\} $ com un
subgrup aritm\`{e}tic de $\SL _2(\R )$ mitjan\c{c}ant una identificaci\'{o}
$\Psi :B\otimes \R \simeq M_2(\R )$ i considerar la superf\'{\i}cie de
Riemann $\Gamma \backslash \mathcal H $. Shimura \cite{Sh67} va
demostrar que aquest \'{e}s el conjunt de punts complexos d'una corba
algebraica $X_D = X_D/\Q $ sobre $\Q $ que parametritza
superf\'{\i}cies abelianes amb multiplicaci\'{o} quaterni\`{o}nica per $\cO $.


En el cap\'{\i}tol \ref{ASV} d'aquesta mem\`{o}ria hem estudiat
l'estructura del grup d'automorfismes de les corbes de Shimura.
L'eina principal per a obtenir els nostres resultats ha estat la
teoria de \v{C}erednik-Drinfeld.

$\\ $
{\bfseries Teorema. }
{\em Sigui $X_D$ la corba de Shimura de discriminant $D$. Si $g(X_D)\geq 2$,
aleshores}

\begin{enumerate}
\item [(i)]
{\em Tots els automorfismes de $X_D \otimes \C $ estan definits sobre $\Q
$.}

\item [(ii)]
{\em $\Aut _{\Q }(X_D)\simeq C_2^s$ per algun enter $s\geq 2r$.}

\item [(iii)]
{\em Si $X_D$ no t\'{e} punts el$\cdot $l\'{\i}ptics, $\Aut (X_D)\simeq C_2^{2 r}$.}

\item [(iv)]
{\em Si $D=2p$ o $3p$ amb $p$ un nombre primer,
$\Aut (X_D)\simeq C_2\times C_2$.}

\end{enumerate}

$\\ $ Com a conseq\"{u}\`{e}ncia d'aquest teorema, en el cap\'{\i}tol \ref{ASV}
determinem quines corbes de Shimura s\'{o}n biel$\cdot $l\'{\i}ptiques, \'{e}s
a dir, admeten un morfisme de grau $2$ en una corba de g\`{e}nere $1$.

$\\ $
{\bfseries Teorema.} {\em Existeixen exactament trenta-dos valors de
$D$ tals que la corba $X_D$ \'{e}s biel$\cdot $l\'{\i}ptica. En cada cas, les
involucions biel$\cdot $l\'{\i}ptiques s\'{o}n de tipus Atkin-Lehner. Aquests valors, juntament
amb el g\`{e}nere $g=g(X_D)$ i el llistat de les involucions
biel$\cdot $l\'{\i}ptiques de $X_D$,
es proporcionen en la taula seg\"{u}ent.}

\newpage
\vspace{1cm}
\begin{table}

\begin{tabular}{|r|r|l|r|r|r|l|r|r|r|l|} \hline
\multicolumn{11}{|c|}{\bfseries Corbes de Shimura biel$\cdot $l\'{\i}ptiques}\\

\hline
D&g&$\om _m$&   &D&g&$\om _m$&   &D&g&$\om _m$ \\
\hline \hline
$26$&$2$&$\om _2$,$\om _{13}$&  &$82$&$3$&$\om _{82}$&   &$210$&$5$&$\om _{30}$,$\om _{42}$, \\
$35$&$3$&$\om _7$&   &$85$&$5$&$\om _{17}$&   &   &   &$\om _{70}$,$\om _{105}$, \\
$38$&$2$&$\om _2$,$\om _{19}$&   &$94$&$3$&$\om _2$&   &   &   &$\om _{210}$ \\
$39$&$3$&$\om _{13}$&   &$106$&$4$&$\om _{53}$,$\om _{106}$&  &$215$&$15$&$\om _{215}$  \\
$51$&$3$&$\om _3$&   &$115$&$6$&$\om _{23}$&   &$314$&$14$&$\om _{314}$   \\
$55$&$3$&$\om _5$&   &$118$&$4$&$\om _{59}$,$\om _{118}$&   &$330$&$5$&$\om _3$,$\om _{22}$  \\
$57$&$3$&$\om _{57}$&    &$122$&$6$&$\om _{122}$&   &   &   &$\om _{33}$,$\om _{165}$, \\
$58$&$2$&$\om _2$,$\om _{58}$&    &$129$&$7$&$\om _{129}$&    &    &    &$\om _{330}$ \\
$62$&$3$&$\om _2$&   &$143$&$12$&$\om _{143}$&   &$390$&$9$&$\om _{390}$ \\
$65$&$5$&$\om _{65}$&   &$166$&$6$&$\om _{166}$&   &$462$&$9$&$\om _{154}$ \\
$69$&$3$&$\om _3$&    &$178$&$7$&$\om _{89}$&   &$510$&$9$&$\om _{510}$ \\
$77$&$5$&$\om _{11}$,$\om _{77}$&   &$202$&$8$&$\om _{101}$&   &$546$&$13$&$\om _{546}$\\
\hline
\end{tabular}
\end{table}

Com a conseq\"{u}\`{e}ncia dels resultats anteriors, ens trobem en disposici\'{o}
de respondre una q\"{u}esti\'{o} de Kamienny amb tota completitud:

$\\ $
{\bfseries Q\"{u}esti\'{o}. }(Kamienny) Quines corbes de Shimura admeten
infinits punts quadr\`{a}tics sobre $\Q $?

$\\ $
Aquesta \'{e}s la pregunta diofantina m\'{e}s natural i senzilla que hom es pot
plantejar a la llum del teorema de Shimura \cite{Sh75}, que assegura que
el conjunt $X_D(\R )$ de punts reals de les corbes de Shimura \'{e}s buit.

$\\ $
Tan sols existeixen un nombre finit de valors de $D$
tals que la corba de Shimura $X_D$ admet infinits punts quadr\`{a}tics
sobre $\Q $. El llistat d'aquestes corbes, juntament amb els seus quocients
racionals o el$\cdot $l\'{\i}ptics es proporcionat en la taula seg\"{u}ent.
\newpage
%\pagebreak
\vspace{1cm}

\begin{table}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{11}{|c|}{\bfseries
Corbes de Shimura $X_D$ de g\`{e}nere $\geq 2$ amb infinits punts
quadr\`{a}tics }\\
\hline
D&$\om _m$&$g(X_D/\langle \om _m\rangle )$&
&D&$\om _m$&$g(X_D/\langle \om _m\rangle )$&   &D&$\om _m$&$g(X_D/\langle
\om _m\rangle )$ \\
\hline \hline
$26$&$\om _{29}$& $0$ &  &$77$&$\om _{77}$& $1$ &   &$143$&$\om _{143}$ &$1$ \\
$35$&$\om _{35}$ &$0$&   &$82$&$\om _{82}$& $1$ &   &$146$&$\om _{146}$&$0$ \\
$38$&$\om _{38}$ &$0$&   &$86$ & $\om _{86}$ &$0$ &   &$159$&$\om _{159}$&$0$ \\
$39$&$\om _{39}$ &$0$ &   &$87$&$\om _{87}$ & $0$&  &$166$&$\om _{166}$&$1$  \\
$51$&$\om _{51}$ &$0$&   &$94$&$\om _{94}$& $0$&   &$194$&$\om _{194}$&$0$   \\
$55$&$\om _{55}$ & $0$&   &$95$&$\om _{95}$ &$0$ &   &$206$&$\om _{206}$ &$0$  \\
$57$&$\om _{57}$ &$1$&     &$106$&$\om _{106}$& $1$&   &$210$&$\om _{210}$ &$1$ \\
$58$&$\om _{29}$& $0$&    &$111$&$\om _{111}$&$0$&    &$215$&$\om _{215}$&$1$ \\
   &$\om _{58}$&$1$&   &$118$&$\om _{118}$ & $1$ &   &$314$&$\om _{314}$&$1$ \\
$62$&$\om _{62}$ & $0$&    &$119$&$\om _{119}$ & $0$&   &$330$&$\om _{330}$ & $1$ \\
$65$&$\om _{65}$ &$1$&    &$122$&$\om _{122}$ & $1$&   &$390$&$\om _{390}$& $1$ \\
$69$&$\om _{69}$ & $0$&    &$129$&$\om _{129}$ & $1$&    &$510$&$\om _{510}$& $1$ \\
$74$&$\om _{74}$ & $0$&   &$134$&$\om _{134}$&$0$ &   &$546$&$\om _{546}$& $1$   \\
\hline
\end{tabular}

\end{table}







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\end{thebibliography}


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