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\begin{document}

\title[Oda's period relations]
{The Birch and Swinnerton-Dyer conjecture
for $\Q$-curves \\ and Oda' s period relations}

\author{Henri Darmon, Victor Rotger and Yu Zhao}

\thanks{The research of the second author is financially supported by DGICYT Grant MTM2009-13060-C02-01 and the Grup de recerca consolidat de Catalunya 2009 SGR 1220.}



\address{H. D.:  Department of Mathematics and Statistics,
McGill University, 805 Sherbooke Street West, H3A-2K6 Montreal, Canada}
\email{darmon@math.mcgill.ca}
\address{V. R.: Departament de Matem\`{a}tica Aplicada II, Universitat Polit\`{e}cnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain}
\email{victor.rotger@upc.edu}
\address{Y. Z.:  Department of Mathematics and Statistics,
McGill University, 805 Sherbooke Street West, H3A-2K6 Montreal, Canada}
\email{zhao@math.mcgill.ca}

%\subjclass{11G18, 14G35}


\maketitle

\begin{center}
{\em To Takayuki Oda on his 60th birthday.}
\end{center}

\medskip\medskip


\tableofcontents

\section{Introduction}

Let $E$ be an elliptic curve over a number field $F$
 and let $L(E/F,s)$ denote its  Hasse-Weil $L$-series.
It is widely believed that
the Shafarevich-Tate group $\sha(E/F)$ is finite and that
$L(E/F,s)$ extends to an entire function of the complex variable
$s$. The order of vanishing of this function at $s=1$,
 denoted by $r_{\an}(E/F)$,
 is commonly referred to as the
{\em analytic rank} of $E$ over $F$, a terminology  justified by
the Birch and Swinnerton-Dyer conjecture which asserts that
\begin{equation}
\label{eqn:bsd}
\rank(E(F)) \stackrel{?}{=}  r_{\an}(E/F).
% \ord_{s=1} L(E/F,s).
\end{equation}
 The most convincing evidence for the Birch and Swinnerton-Dyer conjecture is the fact that
it is proved when $F=\Q$ and $L(E,s) :=L(E/\Q,s)$ has at most  a simple zero
at $s=1$:
\begin{theorem}[Gross-Zagier, Kolyvagin]
\label{thm:gzk}
If $r_{\an}(E/\Q)\le 1$, then $(\ref{eqn:bsd})$ holds for $E/\Q$, and
  $\sha(E/\Q)$ is finite.
\end{theorem}
\noindent
The proof of Theorem \ref{thm:gzk},
which is briefly recalled in Section \ref{sec:gzkz},
rests  on two key ingredients. The first is
the modularity of $E$, in the strong geometric form which asserts that
$E$ is  a quotient  of the Jacobian of a modular curve over $\Q$. The second is
 the collection of Heegner points on this modular curve, which
satisfies the axioms of an ``Euler system" and
provides a valuable bridge between the arithmetic of $E$  and the analytic
behaviour of its $L$-series.

Both these   ingredients are  available  in greater generality,
most notably when $F$ is a {\em totally real field}. In this setting, a modular elliptic curve $E$ over $F$ is said to satisfy the {\em Jacquet-Langlands
hypothesis} (JL)  if either $[F:\Q]$ is odd, or there is at least one prime of $F$ at which the
automorphic form on $\GL_2(\A_F)$ attached to $E$ is not in the  principal series. Here, $\A_F$
stands for the ring of ad{\`e}les of $F$. The meaning of
condition (JL), which only fail to hold for certain
 elliptic curves of square conductor, is
described more concretely in Section \ref{sec:gzkz}.

Most importantly for the proof of (\ref{eqn:bsd}),
the Jacquet-Langlands  hypothesis implies that
$E$ is the quotient of the Jacobian of  a suitable Shimura curve over $F$.
Shimura curves are equipped with a  plentiful supply of CM points,
which have been parlayed into the proof of the
following number field generalisation of
Theorem \ref{thm:gzk}.

\begin{theorem}[Zhang]
\label{thm:zhang}
Let $E$ be  a modular elliptic curve over a totally
real field $F$ satisfying $($JL$)$.
If $r_{\an}(E/F)\le 1$, then $\sha(E/F)$ is finite and $(\ref{eqn:bsd})$ holds for $E/F$.
\end{theorem}

\noindent
Denote by $\zfrakN$ the conductor of $E/F$.   If either $[F:\Q]$ is odd or there exists a prime $\fp$ of $F$
for which $\ord_\fp(\mathfrak N)=1$, this is  Theorem A of \cite{zhang}.
The  full result in which $E$ is only assumed to satisfy
(JL) follows from the subsequent strengthening of the Gross-Zagier
formula proven in \cite{yuan-zhang-zhang}.  Both results are discussed further
in Section \ref{sec:gzkz}.

In analytic rank zero,
 the Jacquet-Langlands hypothesis can be dispensed with:
\begin{theorem}[Longo]
\label{thm:longo}
Let $E$ be a modular elliptic curve  over a totally real field $F$. If $L(E/F,1)\ne 0$, then
$E(F)$ and $\sha(E/F)$ are finite.
\end{theorem}
\noindent
Longo's proof \cite{longo}, building on  the approach of \cite{bd-mc}, exploits  the theory
of congruences between modular forms to realise the Galois representation $E[p^n]$ in the Jacobian of a
Shimura curve $X_n$ whose level may (and indeed does) depend on $n$.
The Euler system of CM points
on $X_n$ then gives rise to a collection of  $p^n$-torsion  cohomology classes  which is  used to bound the $p^n$-Selmer group
of $E $ over $F$ independently of $n$, and thereby obtain the finiteness of $E(F)$ and $\sha(E/F)$.

The problem of removing the Jacquet-Langlands hypothesis from Theorem \ref{thm:zhang}---or equivalently,
 of  extending Theorem  \ref{thm:longo}
to the case where $L(E/F,s)$ has a simple zero at $s=1$---is still very much open.


To better understand  the difficulty which arises,
it is instructive to examine
the simplest  setting where the Jacquet-Langlands hypothesis
fails to hold. Assume for the rest of the introduction that $F$ is a real quadratic field, and
consider for now  the  case where $E/F$ is an elliptic curve of conductor $1$.

Assuming $E$ is modular,
the $L$-series  $L(E/F,s)$ is  known to have
 a functional equation  relating its
values at $s$ and $2-s$, and the sign  $w_E\in \{-1,1\}$    in this
functional equation is always equal to $1$ in this case.

Let $M$ be any quadratic extension of $F$,  let
$$
\chi_M:G_F\lra \pm 1
$$
be its associated Galois character, and denote by $E_M$ the twist of $E$ over $F$ by $\chi_M$, so that the $L$-series $L(E/M,s)$ factors as
$$L(E/M,s) = L(E/F,s) L(E/F,\chi_M,s) = L(E/F,s) L(E_M/F,s).
$$

Since $E$ has conductor $1$, the sign $w_{E_M}$ of the twisted $L$-series is controlled by  the local signs attached to the archimedean places $\infty_1$
and $\infty_2$ of $F$, which are equal to $\chi_M(\infty_1)$ and $\chi_M(\infty_2)$ respectively.
It follows that $w_{E_M}=1$ if  $M$ is either totally real or CM. In particular,
the elliptic curve $E$ is always of {\em even} analytic rank over  such  $M$.
Since an Euler system of Heegner points attached to a quadratic  CM extension $M/F$ is only expected to be available
when $E$ has odd analytic  rank over $M$, this suggests that the mathematical objects
so crucial in Kolyvagin's descent method may be unavailable for elliptic curves of
conductor $1$.

A similar expectation can be derived more generally for all elliptic curves
which do not satisfy (JL). Indeed, if $E/F$ is an elliptic curve of square conductor $\mathfrak N$
and $M$ is a quadratic extension of $F$ which is unramified at the primes dividing $\mathfrak N$, then
the same analysis as above reveals that
\begin{equation}
\label{eqn:rank-parity}
r_{\an}
(E/M) \equiv  \left\{\begin{array}{ll}
0 \pmod{2},  & \mbox{ if $M$ is CM or totally real}; \\
1 \pmod{2}  & \mbox{ otherwise}.
\end{array} \right.
\end{equation}
A quadratic extension $M$ of $F$ which is neither
CM nor totally real
 is called an {\em ATR extension} of $F$.
An ATR extension of $F$ thus has two real places and one complex place.
(The acronym ``ATR" stands for
``Almost Totally Real", and is used more generally  in \cite{DaLo} to designate
quadratic extensions of a totally real field having exactly one complex place.)

The present  article is motivated by the
following specific instance of the Birch and Swinnerton-Dyer conjecture   which emerges naturally from the discussion above.
\begin{conjecture}
\label{conj:main}
Let $E$ be a (modular) elliptic curve over a real quadratic field  $F$ of square conductor $\mathfrak N$ for which $w_E=1$,  and
 let $M/F$ be an ATR extension of $F$ of discriminant prime to $\mathfrak N$.
If $L'(E_M/F,1)\ne 0$, then $E_M(F)$ has rank one and $\sha(E_M/F)$ is finite.
\end{conjecture}
Although it seems  tantalisingly  close to the setting of Theorem \ref{thm:zhang},
Conjecture \ref{conj:main} presents  a real mystery and
appears to lie    beyond the reach of known methods.
The difficulty is that, in the absence of the Jacquet-Langlands hypothesis,
 no natural  ``modular" method presents itself  in general for constructing the point of
infinite order on $E_M(F)$ whose existence is predicted by the Birch and Swinnerton--Dyer conjecture.

One of the original motivations for singling out Conjecture \ref{conj:main} for special  study
lies in the conjectural construction of a so-called
{\em Stark-Heegner point} $P_M^?\in E(M)$ described in
 \cite{DaLo}.
This construction, which is recalled briefly in Section
\ref{sec:atr-cycles},
involves the images  under a complex Abel-Jacobi map
attached to the Hilbert modular form associated to $E/F$
 of  certain
 ``ATR cycles"  indexed by ideals of $M$.
The ATR cycles
are null-homologous cycles of real dimension one on the corresponding  Hilbert modular surface.
It is conjectured in \cite{DaLo} that the
point $P_M^?$ is of infinite order precisely when $L'(E/M,1)\ne 0$, and that $P_M^{?}$ is part of a
norm-coherent collection of points defined over abelian extensions of $M$ satisfying
Euler-System-like properties.
However, progress on Conjecture \ref{conj:main} through the theory of ATR cycles
is thwarted by our inability to provide much  theoretical evidence for the conjectures of \cite{DaLo}
at present.


\sk
The first aim of this note is to  study Conjecture \ref{conj:main}
 for the class of elliptic curves $E/F$ which are isogenous over $F$  to their Galois conjugate.
Following  a terminology that was first introduced by Ribet in \cite{Ri}, these
elliptic curves are called {\em $\Q$-curves}.  Their basic
 properties are reviewed
in Section \ref{sec:bsdQ}.  As explained in that section,
the case of $\Q$-curves is ultimately  made tractable by the existence of a classical
elliptic cusp form $f$ (with {\em non-trivial} nebentypus character in general)
satisfying
$$ L(E/F,s) = L(f,s) L(\bar f,s),$$
leading to a modular parametrisation of $E$ by a classical modular curve
$X_1(N)$ for a suitable $N\ge 1$.
The main  theorem of Section \ref{sec:bsdQ}
 is a proof of Conjecture \ref{conj:main} for $\Q$-curves:
 \begin{theorem}
\label{thm:main}
Let $E/F$ be a $\Q$-curve of square conductor $\mathfrak N$, and let  $M/F$ be an ATR extension of $F$
of discriminant prime to  $\mathfrak N$.
If $L'(E_M/F,1)\ne 0$, then $E_M(F)$ has rank one and $\sha(E_M/F)$ is finite.
\end{theorem}
\noindent
The key ingredients in the proof of Theorem \ref{thm:main}
are a strikingly general recent extension of the theorem of Gross-Zagier
obtained by Xinyi Yuan, Shouwu Zhang and Wei Zhang \cite{yuan-zhang-zhang} covering
cusp forms with possibly non-trivial nebentypus characters,
and a  strengthening of Kolyvagin's descent method to cover abelian variety quotients
of $J_1(N)$, as worked out in the forthcoming book of Ye Tian and Shouwu Zhang \cite{tian-zhang}.
Section \ref{sec:bsdQ} explains how Theorem \ref{thm:main} follows
from these results  and the {\em Artin formalism} for certain Rankin $L$-series.

The second part of the article focuses on the special case where the $\Q$-curve $E$ is of conductor $1$.
Such elliptic curves, which were first systematically studied  by Shimura
 \cite{shimura-book}, are essentially in bijection with newforms $f$  in   $S_2(\Gamma_0(N),\varepsilon_N)$
 with quadratic Fourier coefficients, where
   $N$ is the discriminant of the real quadratic field  $F$,
 and
$$\varepsilon_N: (\Z/N\Z)^\times \lra \pm 1$$
 is  the corresponding even Dirichlet character.




Section \ref{sec:heegner}
describes the explicit
construction, for all quadratic ATR extensions $M$ of $F$,
of a canonical point $P_M\in E(M)$ arising from suitable CM divisors on $X_1(N)$.
The trace to $E(F)$ of $P_M$  is  shown to vanish, so that $P_M$ can also be
viewed as an $F$-rational point on the twisted curve $E_M$.

After explaining how  the points $P_M$  can be computed
complex analytically by integrating the elliptic modular form $f$, we tabulate these points  for  a few ATR
extensions $M$  of small discriminant.
One expects that the height of the point $P_M$ is related in a simple way to $L'(E_M/F,1)$.

Finally,  Conjecture \ref{conj:main-oda}
 spells out a precise conjectural  relationship between  the {\em classical} Heegner point $P_M$
and the Stark-Heegner point $P_M^?$ arising from ATR cycles on the Hilbert modular variety.
This conjecture, which relates certain complex analytic invariants attached to an
elliptic modular form $f$ and its Doi-Naganuma lift, can be viewed
as an analogue  for  Abel-Jacobi maps  of Oda's period relations which
are formulated in \cite{oda-book}. It is therefore a pleasure to dedicate this article to Takayuki Oda
whose work on periods of Hilbert modular surfaces
was a major source of inspiration for the conjectures of \cite{DaLo}.

It is also a pleasure to thank Xavier Guitart, Ariel Pacetti and David Rohrlich for their comments on a previous version of this manuscript.





\section{Background}
\label{sec:background}


\subsection{The Birch and Swinnerton-Dyer conjecture in low analytic rank}
\label{sec:gzkz}
We begin by recalling in greater detail the main ideas behind the proofs of
Theorems \ref{thm:gzk} and \ref{thm:zhang}.
We start with the  assumption  that $F$ is a  number field
 and write $\cO_F$ for its ring of integers.
Let $\mathfrak N\subset \cO_F$ denote the  conductor of the elliptic curve $E/F$.

The proofs of Theorems \ref{thm:gzk} and \ref{thm:zhang}
can be  broken up into five steps:


\sk\noindent
(i) {\em Modularity}:  When $F=\Q$, the main results of   \cite{wiles} and
 \cite{taylor-wiles} (as completed in \cite{bcdt}) imply
that there is a normalised newform  $f$ of weight $2$ on $\Gamma_0(N)$ satisfying
 $ L(E,s)=L(f,s)$. In particular, $L(E,s)$
has an analytic continuation to the left of
 the half plane
 $\mathrm{Re}(s)>3/2$, and its order of vanishing at $s=1$ is therefore well defined.
For general $F$,
the  modularity of $E/F$ is just the assertion that $L(E/F,s)$ is
the $L$-series attached to an automorphic representation of $\GL_2(\A_F)$.
Such a property is predicted to hold, as a (very special) case of the Langlands functoriality
conjectures. In spite of the powerful ideas introduced into the subject building on Wiles'
breakthrough, a proof in the general number field setting still seems a long way off.
When $F$ is totally real, modularity
 can be phrased in terms of modular forms much as in the case $F=\Q$.
Namely, $E/F$ is modular whenever there is a normalised Hilbert modular eigenform $f$
 of parallel weight $2$ on the congruence group $\Gamma_0(\zfrakN)\subseteq \SL_2(\cO_F)$ satisfying
$L(E/F,s)=L(f,s)$.
The methods originating from Wiles' work seem well suited to
 yield a proof of modularity of all elliptic curves over  totally real fields.
(See for example the works of Skinner-Wiles \cite{SkWi}, Fujiwara \cite{Fu}, Jarvis-Manoharmayum \cite{JM} and
the references therein for
an overview of the significant progress that has been achieved in this direction.)
Currently, the case which offers
most difficulties arises when the residual Galois representation at $3$ is {\em reducible}.

 \sk \noindent
(ii) {\em Geometric modularity}: Thanks to the geometric construction of Eichler-Shimura and to
Faltings' proof of the Tate conjecture for abelian varieties
 over number fields, the modularity of $E$ in the case where $F=\Q$
 can be recast as the statement that $E$ is a quotient of the jacobian
$J_0(N)$ of the modular curve $X_0(N)$ over $\Q$, where $\mathfrak N=(N)$, $N\geq 1$.
A non-constant morphism
\begin{equation}
\label{eqn:mod-par-Q}
\pi_E: J_0(N) \lra E
\end{equation}
of abelian varieties over $\Q$  is called a {\em modular parametrisation} attached to $E$.

When $F$ is a totally real field and $E/F$ is known to be modular, the modular parametrisation  arising
from Eichler-Shimura theory admits no counterpart in general. However, such a modular parametrisation can be obtained when the Jacquet-Langlands
hypothesis formulated in the introduction holds. More precisely, as it is explained in \cite[\S 3]{zhang},  hypothesis (JL) implies that $E$ is a quotient of the Jacobian
of a suitable Shimura curve $X$ attached to an order in a quaternion algebra over $F$ which splits at exactly one archimedean place of $F$. That is, there is a non-constant map
\begin{equation}
\label{eqn:mod-par-F}
\pi_E: J(X) \lra E
\end{equation}
of abelian varieties over $F$ generalising
(\ref{eqn:mod-par-Q}).
The condition that the automorphic form $\pi = \otimes \pi_v$
 attached to $E$ be a principal series representation at a place
$v$ of $F$  is  satisfied precisely when $E$ acquires good reduction over
an abelian extension of $F_v$. For $v\nmid 2$, the meanings of  various conditions on the local
representations $\pi_v$ in terms of the behaviour of $E$  over $F_v$
are summarised in the table below.

\begin{equation}
\begin{array}{c|c|c}
\pi_v & E/F_v &  \ord_v(\mathfrak N) \\  \hline
\mbox{Unramified principal series} & \mbox{Good reduction over  $F_v$} & 0       \\
\mbox{Principal series} & \mbox{Good reduction over an abelian extension of $F_v$} & \mbox{even}       \\
\mbox{Steinberg} & \mbox{Potentially multiplicative reduction over $F_v$} & 1 \mbox{ or } 2 \\
\mbox{Supercuspidal} & \mbox{Otherwise} &  \geq 2
\end{array}
\end{equation}

 \vspace{0.3cm}


We refer the reader to \cite[p. 73]{Ge}, \cite{Pa}, \cite[Prop. 2]{Ro}, \cite[Prop. 2 and 3]{Ro2}
for proofs of these statements. (Note that, although in the latter
article the ground field is assumed to be $F=\Q$, the results remain valid for
arbitrary $F$ as the questions at issue are purely local).  See \cite{Pa} for the behaviour at places $v$ above $2$.

In particular, an
elliptic curve which fails to satisfy hypothesis (JL) is necessarily of square
conductor. The converse is not true, but note that it also follows from the
table that all elliptic curves of conductor $1$ over a totally real number field of even degree fail to satisfy (JL). We
will often restrict our attention to elliptic curves of square conductor,
thus  encompassing all elliptic curves that do not satisfy
hypothesis (JL).



\sk\noindent
(iii) {\em Heegner points and $L$-series}:
Suppose first that $F=\Q$ and let
 $K$ be an imaginary quadratic field of discriminant relatively prime to $N$ and satisfying the

 \vspace{0.3cm}

\noindent
 {\bf Heegner hypothesis:} $\cO_K$ has an ideal $\fN$ of norm $N$ satisfying $\cO_K/\fN \simeq \Z/N\Z$.


  \vspace{0.3cm}

\noindent
An ideal $\fN$ of $K$ with  this property is sometimes called a {\em cyclic ideal} of norm $N$.
When the Heegner hypothesis is satisfied, it can be shown
that the $L$-function $L(E/K,s)$ has sign $-1$ in its functional equation, and
therefore  vanishes at $s=1$.
The CM points on $X_0(N)$ attached to the moduli of elliptic curves with complex
 multiplication by $\cO_K$, and their images under $\pi_E$, can be used to construct a
 canonical point $P_K\in E(K)$:  the so-called {\em Heegner point}  on $E$ attached
 to $K$.  The main result of
 \cite{gross-zagier} expresses
 $L'(E/K,1)$ as a multiple by a simple non-zero scalar of the
 N\'eron-Tate height of $P_K$.
 In particular, the point $P_K$ is of infinite order if and only if $L'(E/K,1)\ne 0$.

In the setting where $F$ is a totally real field,
 the Shimura curve $X$
is equipped with an infinite supply of CM points  enjoying properties similar to their
counterparts on modular curves. The auxiliary field  $K$ is now
 a totally complex quadratic extension of $F$ satisfying a  suitable
Heegner hypothesis relative to $X$.
The CM points attached to $K$ can be used
 to construct a canonical point $P_K\in E(K)$ as in the case $F=\Q$. A general extension of the Gross-Zagier
theorem (\cite[Theorem C]{zhang}) to this context relates the height of $P_K$  to the  derivative
 $L'(E/K,1)$. In particular, the point $P_K$ is of infinite order precisely
 when $L(E/K,s)$ has a simple zero at $s=1$. We emphasise that this more general
Heegner point construction relies crucially on
$E/F$ satisfying hypothesis (JL).



 \sk\noindent
(iv)  {\em The Euler system argument}: The Heegner point $P_K$ does not come alone, but can
be related to the norms of algebraic points on $E$ defined over abelian extensions of $K$.
Using this fact, it is shown in \cite{kolyvagin} in the case $F=\Q$
 that the point $P_K$, when it is of infinite order,
necessarily generates $E(K)\otimes\Q$.
Koyvagin's argument extends without essential difficulties to the context of
Shimura curves over totally real fields (cf.\,\cite{KL2}, \cite[\S 7.2]{zhang}, or
the forthcoming book \cite{tian-zhang}).



\sk\noindent
(v) {\em Descending from $K$ to $F$}:
 Assume first that $F=\Q$.
If $\ord_{s=1}(L(E,s))\le 1$, the analytic non-vanishing results of \cite{bfh} or
\cite{murty-murty} produce  an imaginary quadratic field $K$ satisfying
the Heegner hypothesis, and  for which $L(E/K,s)$ has a simple zero
at $s=1$.
By the Gross-Zagier theorem, the Heegner point $P_K$ generates $E(K)$, and its
trace therefore  generates $E(\Q)$. The known properties of the Heegner point
$P_K$ imply in particular that its trace to $\Q$  vanishes
when $L(E,1)\ne 0$, and is of infinite order when $L(E,1)=0$. Theorem \ref{thm:gzk}
  for $E/\Q$ follows from this.
The proof of Theorem \ref{thm:zhang} is deduced similarly,  by noting that
the analytic non-vanishing results of \cite{bfh} or \cite{Wa} generalize to any number field
and again produce a totally complex imaginary quadratic extension $K/F$ satisfying the Heegner hypothesis
for which $\ord_{s=1}(L(E/K,s)) =1$.



\subsection{Oda's period relations and ATR points}
\label{sec:atr-cycles}.
This section briefly recalls the main construction of \cite{DaLo}
which to any ATR extension $M$ of $F$ (satisfying a suitable Heegner condition)
associates a point $P_M\in E(\C)$
belonging conjecturally to $E(M)$.
The points $P_M$  arise by
considering  the images of certain {\em non-algebraic} cycles on Hilbert modular varieties
under a map which is formally analogous to the Griffiths-Weil
 Abel-Jacobi maps on null-homologous algebraic cycles.

\sk\noindent
{\bf The general setting}.
We begin by treating a more general context where
 $F$ is  a totally real field of degree $r+1$.
(This extra generality does not unduly complicate the notations,
and  may even clarify some of  the key features of the construction.)
Fix
an ordering $v_0, v_1,\ldots, v_r$ of the $\rp$ distinct real embeddings of $F$.
Let $E$ be an elliptic curve over $F$, and let
$$E_j := E\otimes_{v_j} \R  \qquad
(0\le j \le r)$$
 be the $\rp$ elliptic curves over $\R$ obtained by taking
the base change of $E$ to $\R$ via  the embedding
$v_j$.
To ease the exposition, we will make the following inessential assumptions:
\begin{enumerate}
\item  The field $F$ has narrow class number one;
\item the conductor of $E/F$ is equal to $1$ (i.e., $E$ has everywhere good reduction).
\end{enumerate}
(For a more general treatment where these assumptions are significantly relaxed, see
for instance the
 forthcoming PhD thesis \cite{gaertner}.)


The  Hilbert modular form $G$ on $\SL_2(\cO_F)$ attached to $E$ is
 a holomorphic function on the
product $\cH_0\times\cH_1\times\cdots\times\cH_r$
of $\rp$ copies of the complex upper half plane, which
is of parallel weight $(2,2,\ldots,2)$ under the action
of the Hilbert modular group $\SL_2(\cO_F)$.
The latter group acts discretely
on $\cH_0\times\cdots\times\cH_r$
by M\"obius transformations
via the embedding
$$(v_0,\ldots,v_r):\SL_2(\cO_F)\lra \SL_2(\R)^{r+1}.$$
Because of this transformation property,
the Hilbert modular form $G$  can be interpreted
 geometrically as a holomorphic
differential $(\rp)$-form on the
 complex analytic quotient
\begin{equation}
\label{eqn:XC}
X(\C) := \SL_2(\cO_F)\backslash (\cH_0\times\cH_1\times\cdots \times \cH_r),
\end{equation}
by setting
$$\omega_G^{\hol}  := (2\pi i)^{r+1}G(\tau_0,\ldots,\tau_r) d\tau_0\cdots d\tau_r. $$
It is important to replace
 $\omega_G^{\hol}$ by a closed, but non-holomorphic differential $(\rp)$-form
$\omega_G$ on $X(\C)$.
When $r=1$, the differential $\omega_G$ is defined by
choosing a unit $\epsilon\in \cO_F^\times$ of norm $-1$ satisfying
$$\epsilon_0:= v_0(\epsilon)>0, \qquad \epsilon_1:=v_1(\epsilon) <0, $$
and setting
$$ \omega_G = (2\pi i)^2 \left( G(\tau_0,\tau_1) d\tau_0 d\tau_1 -
G(\epsilon_0 \tau_0,\epsilon_1 \bar \tau_1) d\tau_0 d\bar \tau_1 \right).$$
For general $r$, one defines $\omega_G$ similarly, but this time summing over the
subgroup of $\cO_F^\times/(\cO_F^+)^\times$  of cardinality $2^r$
consisting of units $\epsilon$ with $v_0(\epsilon)>0$.
Note that the closed $(\rp)$-form $\omega_G$  is holomorphic in $\tau_0$, but only harmonic in
the remaining variables $\tau_1,\ldots,\tau_r$.
The justification for working with $\omega_G$ rather than
$\omega_G^{\hol}$ lies in the following statement which is a reformulation of a conjecture of
Oda \cite{oda-book} in the special case of modular forms with rational fourier coefficients:
\begin{conjecture}[Oda]
\label{conj:oda}
Let
$$ \Lambda_G := \left\{ \int_\gamma \omega_G, \quad \gamma\in H_{r+1}(X(\C),\Z) \right\}.$$
Then $\Lambda_G$ is a lattice in $\C$ and the elliptic curve
$\C/\Lambda_G$ is isogenous to $E_0$.
\end{conjecture}
In \cite{oda-book}, this
 conjecture is shown to hold for Hilbert modular forms which are base change lifts of
classical elliptic modular forms, which
 corresponds  to the case where the associated
elliptic curve $E$ is a $\Q$-curve. But it should be emphasised that
no $\Q$-curve hypothesis on $E$ is necessary in
 Conjecture \ref{conj:oda}.



Let
$$
\cZ_r(X(\C)):=
\left\{ \begin{array}{c}
 \mbox{Null-homologous cycles}\\
\mbox{of real dimension $r$ } \\
\mbox{on } X(\C)
\end{array}\right\}.
$$
Conjecture \ref{conj:oda} makes it possible to define an
``Abel-Jacobi map"
\begin{equation}
\label{eqn:AJf}
 \AJ_G:
\cZ_r(X(\C))
\lra
E_0(\C),
\end{equation}
by choosing an
isogeny $\iota: \C/\Lambda_G \lra E_0(\C)$, and setting
\begin{equation}
\label{eqn:defAJf}
 \AJ_G(\Delta) := \iota\left(\int_{\tilde\Delta} \omega_G\right),
\qquad \mbox{(for any } \tilde\Delta \mbox{ with } \partial\tilde \Delta = \Delta).
\end{equation}
Note that the domain $\cZ_r(X(\C))$ of $\AJ_G$ has no natural algebraic structure, and
that the map $\AJ_G$ bears no simple relation (beyond an analogy
in its definition)  with the Griffiths-Weil Abel-Jacobi map on the Hilbert modular variety $X$.


\sk\noindent
{\bf ATR Cycles}.  Generalising slightly the definitions given in the
Introduction to the case $r>1$, a
quadratic extension $M$ of $F$ is called an ATR extension
if
$$ M\otimes_{F,v_0} \R \simeq \C,\qquad
M\otimes_{F,v_j}\R \simeq \R\oplus \R,  \ \ (1\le j \le r). $$
The acronym ATR stands for ``Almost Totally Real", since an ATR extension of $F$  is
``as far as possible" from being a CM extension, without being totally real.


Fix an ATR extension $M$ of $F$, and let
$\Psi:M\lra M_2(F)$ be an $F$-algebra embedding.
Then
\begin{enumerate}
\item
Since $M\otimes_{F,v_0}\R \simeq \C$, the torus $\Psi(M^\times)$ has a unique
fixed point $\tau_0\in \cH_0$.
\item
For each $1\le j \le r$, the fact that $M\otimes_{F,v_j} \R\simeq \R\oplus \R$
shows that $\Psi(M^\times)$ has two fixed points $\tau_j$ and $\tau_j'$
on  the boundary of $\cH_j$. Let $\varUpsilon_j\subset \cH_j$ be the hyperbolic geodesic
joining $\tau_j$ to $\tau_j'$.
\end{enumerate}
An embedding $\Psi:M\lra M_2(F)$ has a {\em conductor}, which is defined to be the  $\cO_F$-ideal
$c_\Psi$ for which
$$ \Psi(M) \cap M_2(\cO_F) = \Psi(\cO_F + c_\Psi \cO_M).$$
The $\cO_F$-order $\cO_\Psi := \cO_F + c_\Psi \cO_M$ is called the
{\em order associated to }  $\Psi$.
It can be shown that there are finitely many distinct
$\SL_2(\cO_F)$-conjugacy classes of
embeddings of  $M$  into $M_2(F)$ associated to a fixed order $\cO\subset \cO_M$, and that the
Picard group (in a narrow sense) of $\cO$ acts simply transitively on the set of such
conjugacy classes of embeddings.

By the Dirichlet unit theorem, the group
$$\Gamma_\Psi := \Psi((\cO_\Psi^+)^\times)\subset \SL_2(\cO_F) $$
is of rank $r$ and  preserves the region
$$ R_\Psi := \{\tau_0\} \times \varUpsilon_1\times\cdots\times \varUpsilon_r. $$
The ATR cycle associated to the embedding  $\Psi$ is defined to be
the quotient
$$ \Delta_\Psi :=  \Gamma_\Psi \backslash R_\Psi.$$
It is a closed cycle on $X(\C)$ which is topologically isomorphic to an
$r$-dimensional  real torus.
In many cases, one can show that  $\Delta_\Psi$ is null-homologous, at least
after tensoring with $\Q$ to avoid the delicate issues arising from the possible
presence of torsion in integral homology.
(The homological triviality of $\Delta_\Psi$ always holds, for instance, when $r=1$,
and  follows from the fact that the group
cohomology $H^1(\SL_2(\cO_F),\C)$ is trivial.)
Assume from now on that $\Delta_\Psi$ is homologically trivial, and therefore
that it belongs to $\cZ_r(X(\C))$.

The following conjecture  lends arithmetic meaning to the
Abel-Jacobi map $\AJ_G$ and to the ATR cycles $\Delta_\Psi$.
\begin{conjecture}
\label{conj:atr}
Let $\Psi:M\lra M_2(F)$ be an $F$-algebra embedding of an ATR extension $M$ of $F$.
Then the complex point $\AJ_G(\Delta_\Psi)\in E_0(\C)$ is algebraic.
More precisely,  the isogeny $\iota$ in the definition
(\ref{eqn:defAJf})  of
$\AJ_G$ can be chosen so that, for all  $\Psi$,
$$ \AJ_G(\Delta_\Psi) \mbox{ belongs to } E(H_{c_\Psi}),$$
where $H_{c_\Psi}$ is the ring class field of $M$ of conductor $c_\Psi$.
Furthermore, if $\Psi_1,\ldots,\Psi_h$ is a
 complete system of representatives for
the  $\SL_2(\cO_F)$-conjugacy classes of embeddings of $M$ in $M_2(\cO_F)$ of
a given conductor $c$, then the Galois group
 $\gal(H_c/M)$ acts (transitively) on the set
$\{\AJ_G(\Delta_{\Psi_1}),\ldots,\AJ_G(\Delta_{\psi_h})\}$.
\end{conjecture}
 Conjecture \ref{conj:atr} is
poorly understood at present.
For instance, it is not clear whether the Tate conjecture
sheds any light on it. On the positive side, the ATR points that are
 produced by  Conjecture \ref{conj:atr}
 are ``genuinely new" and  go beyond what can be obtained
using only CM points on
Shimura curves.
Indeed, the former are  defined over abelian extensions of
ATR extensions of totally real fields, while
the latter
 are defined over abelian extensions of CM fields.

Most germane to the concerns of this paper,
Conjecture \ref{conj:atr} can  be used as a basis for the construction of a point
$P_M^? \stackrel{?}{\in} E(M)$, by letting
$\Psi_1,\ldots, \Psi_h$ be a complete system of representatives for
the  $\SL_2(\cO_F)$-conjugacy classes of embeddings of $M$ in $M_2(\cO_F)$
of conductor $1$
and setting
\begin{equation}
P_M^{?} := \AJ_G(\Delta_{\Psi_1}) + \cdots + \AJ_G(\Delta_{\Psi_h}).
\end{equation}



\section{The Birch and Swinnerton-Dyer conjecture for $\Q$-curves}
\label{sec:bsdQ}

\subsection{Review of $\Q$-curves}\label{Qcurves}
The first
 goal of the present work is to study Conjecture
\ref{conj:atr} for $\Q$-curves, which are defined as follows:
\begin{definition}
Let $F$ be a number field and fix an algebraic closure $\qbar $ of
$\Q$ containing $F$. We say that an elliptic curve $E/F$ is a {\em $\Q$-curve} if it is
{\em isogenous over $F$} to all its Galois conjugates over $\Q$.
\end{definition}
In the literature, these curves are sometimes referred as $\Q$-curves {\em completely defined over $F$},
reserving the term $\Q$-curve for the wider class of elliptic curves over $F$ which are isogenous
{\em over $\qbar$} to all their Galois conjugates over $\Q$.

$\Q$-curves are known to be modular, thanks to the work of Ellenberg and Skinner [ES]
(who proved (geometric) modularity of $\Q$-curves under local conditions at $3$), now vastly superseded by \cite{KW},
which implies modularity of all $\Q$-curves as a very particular case. Combining this with the older work
 of Ribet (cf.\,\cite{Ri} for a survey), it follows that $\Q$-curves $E/F$ are arithmetically uniformisable
{\em over $\qbar$} by the classical modular curves $X_1(N)$. By this, we mean that
there exists a non-constant morphism of curves
\begin{equation}
\label{eqn:mod-par-Qcurves}
\pi_E: X_1(N)_{\qbar} \lra E_{\qbar}
\end{equation}
over $\qbar$, for some $N\geq 1$.

For simplicity, in this article we shall restrict our attention to $\Q$-curves
 over a {\em quadratic} field $F$, which represents
 the simplest non-trivial scenario.
However, we believe that the ideas present in this note should allow, with some more effort,
to treat more general cases; see the forthcoming Ph.D thesis \cite{Zhao} of the third author.

Let $F$ be a quadratic field with ring of integers $\cO_F$ and write $\Gal(F/\Q)=\{1,\tau\}$.
Let $E$ be a $\Q$-curve over $F$ of conductor $\mathfrak N\subset \cO_F$.

Given a Dirichlet character $\varepsilon$ of conductor $N$, let
 $$\Gamma_{\varepsilon}(N) = \{\begin{smallmat} abcd \end{smallmat} \zspace \big | \zspace N| c, \,\varepsilon(a)=1\} \subseteq \SL_2(\Z)$$
and let  $X_{\varepsilon}(N)$ be the modular curve associated to this congruence subgroup.
The curve $X_\varepsilon(N)$
 admits a canonical model over $\Q$, and coarsely represents the moduli problem of parametrizing triples
$(A,C,t)$ where $A$ is a generalised elliptic curve, $C$ is a cyclic subgroup of
order $N$ of $A(\qbar)$ and $t$ is an orbit in $C\setminus \{ O\}$ for the
action of $\ker(\varepsilon)\subset (\Z/N\Z)^{\times }$.
Note that the group  $(\Z/N\Z)^{\times}$ acts on $X_\varepsilon(N)$  via the
diamond operators,
and that the subgroup
$\ker(\varepsilon)$ fixes it.

As discussed for example  in \cite{GQ} or \cite{Ri}, the modular parametrisation
\eqref{eqn:mod-par-Qcurves} is particularly well-behaved when $F$ is quadratic.
More precisely, there exists a positive integer $N\geq 1$, an even Dirichlet
character $\varepsilon:(\Z/N\Z)^{\times}\ra \{\pm 1\} \subset \C^{\times}$,
and a pair
$f_E, f'_E\in S_2(\Gamma_{\varepsilon}(N))\subseteq S_2(\Gamma_0(N), \varepsilon)$
of
normalised {\em newforms}
 of weight $2$,
level $N$ and nebentypus $\varepsilon$, such that
\begin{equation}\label{Ldecomposition}
L(E/F,s)= L(f_E,s)\cdot L(f'_E,s).
\end{equation}
In this case, the uniformisation in  \eqref{eqn:mod-par-Qcurves} factors through a modular parametrisation
\begin{equation}
\label{parametrisation}
\pi_E: X_{\varepsilon}(N)_{F} \lra E_{F}
\end{equation}
defined over $F$.


Let $K_f$ denote the field generated by the Fourier coefficients of $f_E$. It is either $\Q$ or a
quadratic field. When $K_f=\Q$, the elliptic curve
 $E$ is in fact isogenous to the base change  of an elliptic curve defined over $\Q$
and question \eqref{eqn:bsd} can rather be tackled with the classical techniques reviewed in \S \ref{sec:background}; we
assume throughout that this is not the case. Hence $[K_f:\Q]=2$ and, letting $\sigma$ denote
the single nontrivial automorphism of $K_f$,  we have
$$f'_E={}^{\sigma}\!\!f_E.$$

Weil's restriction of scalars $A:=\mathrm{Res}_{F/\Q}(E)$ is an abelian surface of $\GL_2$-type over  $\Q$ such that
\begin{equation}\label{Kfmodule}
\End_{\Q}(A)\otimes \Q\simeq K_f \quad \text{ (and thus is simple over $\Q$)},
\end{equation}
\begin{equation}
A_{/F} \simeq E\times {}^{\tau}E, \text{ and}
\end{equation}
\begin{equation}
L(A/\Q,s)=L(E/F,s)=L(f_E,s)\cdot L(f'_E,s).
\end{equation}

Moreover, for any field extension $L/\Q$, there is a canonical isomorphism
\begin{equation}\label{Weil}
A(L)\simeq E(F\otimes_{\Q}L)
\end{equation}
and in particular $A(\Q)\simeq E(F)$. As shown by Carayol, the conductor of $A$ over $\Q $ is $N^2$, and it follows from \cite[Prop. 1]{Mi} (see also \cite[Remark 9]{GG} for a more detailed discussion) that the conductor of $E/F$ is $\mathfrak N=N_0\cdot \cO_F$, where $N_0\in \Z$ satisfies
\begin{equation}\label{conductor}
N = N_0 \cdot |\disc(F)|.
\end{equation}

As we shall now explain, when $F$ is imaginary the problem can be reduced to the classical setting
considered by Gross-Zagier and Kolyvagin-Logachev, and presents no mysteries.
 It is the case of $F$ real that deserves more attention, and to which the main bulk of this note will be devoted.


If $\varepsilon$ is trivial, then $K_f$ is real and $F$ can be either real or imaginary (and indeed both cases occur in examples). As a direct consequence of  \eqref{Ldecomposition}, \eqref{Weil} and the generalization \cite{KL} of the work of Kolyvagin to higher dimensional quotients of $J_0(N)$ over $\Q$, \eqref{eqn:bsd} also holds for $E/F$ provided $\ord_{s=1} L(f_E,s)\leq 1$.


{\em Assume $\ep\ne 1$ for the rest of this article.} Now $K_f$ is an imaginary quadratic field. Besides, it follows from an observation of Serre (cf.\,\cite[Proposition 7.2]{Ri}) that $F$ is necessarily {\em real}. In fact, $F$ can be computed explicitly from $f_E$ as $F=\qbar^{\ker(\ep)}$. In particular it follows that $\varepsilon$ is the quadratic Dirichlet character associated with $F$.

%Equation \eqref{Ldecomposition} is a manifestation of the fact that the Hilbert modular form $h_E\in S_2(\Gamma_0(\mathfrak N))$ associated with $E$
%is the image, under the Doi-Naganuma lift, of the elliptic cuspform $f_E$.



Let $\om_N$ denote the Fricke involution of $X_{\varepsilon}(N)_F$ defined on the underlying Riemann surface by the rule $\tau \mapsto -\frac{1}{N\tau}$. It induces an involution on the jacobian $J_{\varepsilon}(N)_F$ of $X_{\varepsilon}(N)_{F}$ which leaves $A_F$ stable. We have
\begin{equation}
A_F \sim (1+\om_N)A_F \times (1-\om_N)A_F,
\end{equation}
where both factors on the right have dimension $1$, are isogenous over $F$ and conjugate one each other over $\Q$ (cf.\,\cite[\S 5]{Cremona}). By replacing $E$ by its conjugate if necessary, we shall assume throughout that $E=(1+\om_N)A_F$.


It then follows that \eqref{parametrisation} factors through the following commutative diagram:
\begin{equation}
\label{factorisation}
\begin{matrix}
X_{\varepsilon}(N)_F                    & \lra & J_{\varepsilon}(N)_F           & \lra & A_F        \\
\downarrow   \pi_N                                  &    &  \downarrow                  &    & \downarrow \\
X^+_{\varepsilon}(N)      & \lra & J^+_{\varepsilon}(N)    & \overset{\varphi_E}{\lra} & E,
\end{matrix}
\end{equation}
where we set $X^+_{\varepsilon}(N):=X_{\varepsilon}(N)_F/\langle \om_N \rangle$ and $J^+_{\varepsilon}(N) := (1+\om_N)J_{\varepsilon}(N)_F $. The reader should keep in mind that both are varieties over $F$, not over $\Q$.

\subsection{The main result}
The goal of this section is to prove Theorem \ref{thm:main} of the Introduction.
Let  $M$ be a quadratic ATR extension of $F$. Since $M$ has two real places and
one complex place, it
 is not Galois over $\Q$. Let $M'$ denote its
Galois conjugate over $\zQ$, and let $\fM$ be the Galois closure
of $M$ over $\Q$. It is not hard to see that $\fM$ is
the compositum over $F$ of $M$ and $M'$ and that
$\gal(\fM/\Q)$ is isomorphic to the dihedral group of
order $8$. The subgroup $V_F :=\gal(\fM/F)$ is
isomorphic to a Klein $4$-group.
The dihedral group of order $8$
contains two distinct, non-conjugate subgroups which are isomorphic
to the Klein $4$-group.
This is most easily seen by
viewing $D_8$ as the symmetry group of a square, as in the figure below, in which
$V_F$ is identified with the  subgroup generated by the
reflections
about the two  diagonals.
These two reflections can be labeled as $\tau_M$ and $\tau_M'$, in such a way
that
$$
 \fM^{V_F} = F, \qquad
\fM^{\tau_M} = M, \qquad \fM^{\tau_M'} = M'.$$
The
second Klein four-group, which shall be denoted $V_K$, is
 generated by the  reflections about the vertical and horizontal axes of symmetry of the square.
We label these reflections as
 $\tau_L$ and $\tau_L'$,  as
shown  in the figure below.
\begin{equation}
\label{eqn:plain-square}
\xymatrix{
\ar@{.}[ddddrrrr]|-<{\tau_M'}|-<<<<<<{\circlearrowright} &
& \ar@{.}[dddd]|-<{\tau_L}|-<<<<<{\circlearrowright} &
& \ar@{.}[lllldddd]|-<{\tau_M}|-<<<<<<{\circlearrowleft} \\
 & \bullet \ar@{-}[rr] \ar@{-}[dd]
& \circ  &
\bullet \ar@{-}[dd] &   \\
\ar@{.}[rrrr]|->{\phantom{pppp}\tau_L'}|->>>{\phantom{}\circlearrowleft}
 & \circ  & & \circ  &  \\
& \bullet \ar@{-}[rr] & \circ & \bullet \\
&    &   &   & \\  }
\end{equation}


\noindent
Now let
$$ K := \fM^{V_K},\qquad L:= \cM^{\tau_L},\qquad L' = \fM^{\tau_L'}.$$
These fields fit into the following diagram of field extensions,
where each unbroken line indicates an extension of degree $2$:
\begin{equation}
\label{eqn:field-diag}
\xymatrix{
   &   &  \cM &   &  \\
M \ar@{-}[rru] & M' \ar@{-}[ru]   && L' \ar@{-}[lu] & L \ar@{-}[llu] \\
 & F\ar@{-}[lu] \ar@{-}[u] &  &  K  \ar@{-}[u] \ar@{-}[ru] &  \\
 & & \Q \ar@{-}[lu]  \ar@{-}[ru] & &  }
\end{equation}
Let
$$\chi_M, \chi_M': G_F\lra \{\pm  1\} $$
denote the  Galois characters of the real quadratic
field $F$ which cut out the extensions $M$
and $M'$, and let
$$\chi_L, \chi_L': G_K\lra \{ \pm  1\} $$
be the quadratic characters of the imaginary quadratic field $K$
which cut out the extensions $L$ and $L'$.
We will often view these characters as id\`ele class characters defined on
$\A_F^\times$ and on $\A_K^\times$ respectively.
Finally, let $\varepsilon_F$ and $\varepsilon_K$ denote the
quadratic Dirichlet characters attached to $F$ and $K$, and let
$\N^F_\Q:\A_F^\times\lra\A_\Q^\times$ and $\N^K_\Q:\A_K^\times\lra \A_\Q^\times$
denote the norms on ad\`eles.
\begin{proposition}
\label{prop:all-about-M}
\begin{enumerate}
\item The field $K$ is a quadratic imaginary field.
\item The characters
$\chi_M$, $\chi_M'$, $\chi_L$  and $\chi_L'$,
 viewed as id\`ele class characters of
$F$ and $K$ respectively, satisfy
$$ \chi_M \chi_M' =  \varepsilon_K \circ N^F_\Q; \qquad
 \chi_L \chi_L' =  \varepsilon_F \circ N^K_\Q.$$
\item The central character of $\chi_M$ and $\chi_M'$ is
$\varepsilon_K$, and the central character
of $\chi_L$ and $\chi_L'$ is $\varepsilon_F$.
\item The following two-dimensional representations of $G_\Q$ are isomorphic:
$$\Ind_F^\Q \chi_M = \Ind_F^\Q \chi_M' = \Ind_K^\Q \chi_L = \Ind_K^\Q \chi_L'.$$
\end{enumerate}
\end{proposition}
\begin{proof}
The quadratic field $K$ is of the form $\Q(\sqrt{-d})$,
where $-d$ is defined (modulo squares in $\Q^\times$) by
$$ -d = \Norm_{F/\Q}(\alpha), \qquad
\mbox{ with } \quad M = F(\sqrt{\alpha}).$$
The fact that $M$ is ATR implies that $-d$
is a negative rational number, and therefore that
$K$ is an imaginary quadratic field.
The second part follows directly from the field diagram
(\ref{eqn:field-diag}) above. To prove the third part,
note that part (2) implies
 that the central character of
$L$ restricted to the group of norms from $K$  is equal to $\varepsilon_F$.
(This is because $\chi_L(x) = \chi_L'(\bar x)$, where $x\mapsto\bar x$ is
complex conjugation). Class field theory implies that  this central
character differs from $\varepsilon_F$ by a power of $\varepsilon_K$.
But the central character of $\chi_L$  cannot  be $\varepsilon_F \varepsilon_K$ since this is an odd Dirichlet
character and the central character of a finite order
Hecke character of an imaginary quadratic field is necessarily even,
because the map from the group of components of
$\R^\times$  to the group of components of $\C^\times$ is trivial.
Finally, the proof of part (4) is a simple exercise in
representation theory: the four representations that are listed in (4) are all isomorphic to the unique
irreducible two-dimensional representation of $\gal(\fM/\Q)$.
\end{proof}

As at the end of the previous section \S \ref{Qcurves}, let $E$
 be a $\Q$-curve over a real quadratic field $F$ and $A=\mathrm{Res}_{F/\Q}(E)$,
let $f=f_E\in S_2(\Gamma_{\varepsilon}(N))$ denote the modular form associated to it
and let $K_f$ denote the imaginary quadratic field generated over $\Q$ by the Fourier coefficients of $f$.
\begin{theorem}[Tian, Yuan, Zhang and Zhang]
\label{thm:yuan-zhang-zhang}
Let $K$ be a quadratic imaginary field satisfying the Heegner hypothesis, and let
$\chi:\A_K^\times\lra \C^\times$ be a finite order Hecke character of
$K$ satisfying
\begin{equation}\label{trivialproductcharacter}
 \chi|_{\A_\Q^\times} = \varepsilon_f^{-1}.
\end{equation}
Then
\begin{enumerate}
\item[(i)]
The $L$-function $L(f/K,\chi,s)$ vanishes to odd order at $s=1$;
\item[(ii)]
If $L'(f/K,\chi,1)\ne 0$, then
$(A(K^{\rm ab})\otimes\C)^{\chi}$ has rank one over $K_f\otimes_\Q \C$,
and $\sha(A/K^{\rm ab})^\chi$ is finite.
\end{enumerate}
\end{theorem}

\begin{proof}
The modular form $f$ gives rise to a cuspidal automorphic representation $\pi$ of $\GL_2(\mathbb A_{\Q})$ whose central character is $\omega_{\pi}=\varepsilon_f$, the nebentypus of $f$. Condition \eqref{trivialproductcharacter} ensures that the tensor product of the motives attached to $\pi$ and $\chi$ is self-dual, and therefore the $L$-function
$$L(f/K,\chi,s)=L(\pi,\chi,s-\frac{1}{2})$$
satisfies a functional equation whose central critical point is $s=1$; since the discriminant of $K$
is relatively prime to $N$, the sign of this functional equation is $(-1)^{\sharp \Sigma}$, where
$$\Sigma=\{\text{primes } \ell \text{ inert in } K \text{ such that } \ord_{\ell}(N) \text{ is odd}\}\cup \{\infty\}.$$
 The Heegner hypothesis satisfied by $K$ says that $\Sigma=\{\infty\}$ and thus the sign is $-1$; this implies (i).

As a consequence of \eqref{Kfmodule}, the complex vector space
 $(A(K^{\rm ab})\otimes\C)^{\chi}$ is
naturally a $K_f\otimes_\Q \C$-module. Part (ii) is a theorem of Tian-Zhang \cite{tian-zhang}
which follows as a corollary of \cite[Theorem 1.3.1]{yuan-zhang-zhang} by applying Kolyvagin's method.
Since \cite{tian-zhang} is not currently available, the reader may consult \cite[Theorem 1.4.1]{yuan-zhang-zhang}
and, for the precise statement quoted here, \cite[Theorem 4.3.1]{zhang2}.
\end{proof}


We are now ready to prove Theorem \ref{thm:main} of the introduction.
 \begin{theorem}
\label{thm:main-bis}
Let $E$ be a $\Q$-curve over $F$  of square conductor $\mathfrak N$, and let  $M/F$ be an ATR extension of $F$
of discriminant prime to $\mathfrak N$. If $L'(E_M/F,1)\ne 0$, then $E_M(F)$ has rank one and $\sha(E_M/F)$ is finite.
\end{theorem}
\begin{proof}
By (\ref{Ldecomposition})  and the Artin formalism for $L$-series,
\begin{equation}
\label{eqn:fromEtof}
L(E_M/F,s)  = L(E,\chi_M,s) = L(f\otimes \chi_M/F,s) = L(f\otimes \Ind_F^\Q \chi_M,s).
\end{equation}
By part 4 of Proposition \ref{prop:all-about-M},
\begin{equation}
\label{eqn:fromFtoK}
L(f\otimes \Ind_F^\Q\chi_M,s) = L(f\otimes \Ind_K^\Q \chi_L,s) = L(f\otimes \chi_L/K,s).
\end{equation}
It follows from (\ref{eqn:fromEtof}) and (\ref{eqn:fromFtoK}) that
\begin{equation} \label{eqn:prodOfL} L'(E_M/F,1) = L'(f\otimes\chi_L/K,1) = L'(f^\sigma\otimes\chi_L/K,1). \end{equation}
  Therefore the two rightmost expressions in \eqref{eqn:prodOfL} are non-zero by assumption, so that
the product
$$L(f\otimes\chi_L/K,s) L(f^\sigma \otimes\chi_L/K,s) =
L(A/K,\chi_L,s)$$
vanishes to order exactly  $2=[K_f:\Q]$ at $s=1$.
By Theorem \ref{thm:yuan-zhang-zhang},
it follows that $A(L)^-$ is of rank two, where $A(L) ^-$
denotes the subgroup of the Mordell-Weil group of
$A(L)$ of points whose trace to $K$ is trivial.
In particular, the Galois representation $\Ind_K^\Q \chi_L$ occurs in $A(\overline\Q)\otimes \C$ with multiplicity $2$.
Hence, invoking once again   part  4 of  Proposition \ref{prop:all-about-M},
we find that
$$ \rank(A(M)^-) = 2.$$
But since $M$ contains $F$ and since $A$ is isogenous over $F$ to $E\times E$, it follows that
$$ \rank(E(M)^-) =\rank(E_M(F)) = 1.$$
The result about the ranks follows. The result about the finiteness of $\sha(E_M/F)$ follows in the same way
from part (ii) of Theorem
\ref{thm:yuan-zhang-zhang}.
\end{proof}





\section{Heegner points on Shimura's elliptic curves}
\label{sec:heegner}

Implicit in the proof of Theorem \ref{thm:main-bis} (via the use that is made of it in the proof of Theorem
\ref{thm:yuan-zhang-zhang}) is  the construction of a Heegner point $P_M\in E_M(F)$ arising from the image of
certain CM divisors on $X_\varepsilon(M)$ via the modular parametrisation (\ref{eqn:mod-par-Qcurves}).
We  now wish to  make this construction explicit in the case where the $\Q$-curve $E$
has everywhere good reduction
over the real quadratic field
$F$.  
The $\Q$-curves  with this property
are sometimes called   {\em Shimura elliptic curves} because they were  first systematically
considered by
Shimura. More precisely,  in  \cite{shimura-book} it is shown
 how to associate a Shimura elliptic curve over $F=\Q(\sqrt{N})$
to any classical elliptic modular form
$f\in S_2(\Gamma_0(N),\varepsilon_N)$ with quadratic fourier coefficients.
(Cf. also \eqref{conductor}.)

 It will be assumed throughout this chapter that $E/F$ is  a Shimura elliptic curve,
and that $f$
is the corresponding elliptic modular form.
We also assume for simplicity that {\em $N$ is odd}, and thus square-free.

\begin{remark}
According to  calculations performed by
the third author using PARI \cite{PARI} (extending the
 data gathered in \cite[\S 6]{Cremona}, \cite{pinch}  in the range $N\le 1000$),
there exists an eigenform $f\in S_2(\Gamma_0(N),\varepsilon_N)$
of prime level $1<N<5000$,
with  fourier coefficients
in a quadratic imaginary extension
 $K_f =\zQ(\sqrt{-d})$
 for precisely the values of $N$ and $d$  listed in the following table.

\begin{center}
%\begin{tiny}

\begin{longtable}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}


\hline
$N$   & $29$  &   $37$  &   $41$  & $109$  & $157$  & $229$ & $257$ & $337$ & $349$ & $373$ & $397$ & $421$ \\

\hline

$d$   & $5$  &   $1$  &   $2$  & $3$ & $1$ & $5$ & $2$ & $2$ & $5$ & $1$  & $1$ & $7$
\\ \hline
$N$ & $461$ & $509$ & $877$ & $881$ &$997$ & $1069$  &  $1709$  & $1861$ & $2657$ & $4481$ & $4597$ &

\\ \hline
%\\ \cline{1-6}

$d$    & $5$ & $5$ & $1$  & $2$ & $3$   & $1$    & $5$ & $5$ & $2$ & $11$ & $1$ &
%\\ \cline{1-6}
\\ \hline
\end{longtable}
%\end{tiny}
\end{center}
Associated to each such eigenform there is a Shimura elliptic curve over $F=\Q(\sqrt{N})$.
Furthermore,
according to computations due to Cremona, Dembel\'e, Elkies and Pinch, there are only
 four primes $N$ in the range $[1,1000]$ for which there exists an elliptic curve with good
reduction everywhere over $F=\Q(\sqrt{N})$ which is {\em not} a $\Q$-curve, namely,
 $N=509$, $853$, $929$ and $997$. It is hard to predict whether the preponderance of Shimura elliptic curves
among elliptic curves of conductor one  will persist or
 is merely an artefact of the relatively low ranges in which numerical
data has been gathered. Note that it is not even known whether there exist infinitely many Shimura elliptic curves over real quadratic fields, while it is a theorem of S. Comalada \cite{Co} that there are infinitely many elliptic curves over real quadratic fields with good reduction everywhere.
\end{remark}




\subsection{An explicit Heegner point construction}\label{sec:explicit}

Let us recall the diagram of field extensions introduced in \eqref{eqn:field-diag}:

\begin{equation}
\label{eqn:field-diag-complete}
\xymatrix{
   &   &  \cM &   &  \\
M \ar@{-}[rru] & M' \ar@{-}[ru] & \Q(\sqrt{N},\sqrt{-d}) \ar@{-}[u] & L' \ar@{-}[lu] & L \ar@{-}[llu] \\
 & F\ar@{-}[lu] \ar@{-}[u] \ar@{-}[ru] & \Q(\sqrt{-Nd}) \ar@{-}[u] &  K \ar@{-}[lu] \ar@{-}[u] \ar@{-}[ru] &  \\
 & & \Q \ar@{-}[lu] \ar@{-}[u] \ar@{-}[ru] & &  }
\end{equation}


The following lemma is crucial in constructing
the point $P_M\in E(M)^- = E_M(F)$ explicitly.
\begin{lemma}
Let $M$ be an ATR extension of $F$ and let $K$ be the quadratic
imaginary field attached to $M$ as in the diagram
$(\ref{eqn:field-diag-complete})$.
Then $K$ has a  $($canonical$)$ ideal $\fN$ of $K$ of norm $N$. In
particular, all the prime divisors of $N$ are either split or ramified in $K$.
\end{lemma}
\begin{proof}
The conductor-discriminant formula combined with part 4 of Proposition
\ref{prop:all-about-M} show that
$$ \disc(F) \zNm_{F/\Q}(\disc(M/F)) = \disc(K) \zNm_{K/\Q}(\disc(L/K)).$$
Therefore, after setting
\begin{eqnarray*}
  N_{\ram} = \gcd(N,\disc(K)),   \quad  & &  N_{\splt} = N/N_{\ram}, \\
\fN_{\ram} = (N_{\ram}, \sqrt{\disc(K)}), \quad  & &
\fN_{\splt} = (N_{\splt}, \disc(L/K)),
\end{eqnarray*}
we find that $\fN := \fN_{\ram} \fN_{\splt}$ gives the desired ideal of norm $N$.
\end{proof}





Let $\A_K$ denote the ring of ad\`eles of $K$, and let
$$ {\hat\cO}_K^\times := \prod_{v} \cO_v^\times$$
denote the maximal compact subgroup of the group
$\A_{K,\fin}^\times$ of  finite id\`eles of $K$. Given
a rational integer $c\geq 1$, $(c,N)=1$, we define
$$
U_c = {\hat \Z}^\times (1+c{\hat \cO_K}) \C^\times \subset \A_K^\times.
$$


By class field theory, the quotient $G_c:=\A_K^\times/(K^\times U_c)$ is identified with
$\gal(H_c/K)$, where $H_c$ is the {\em ring class
field} of $K$  of conductor $c$.

As a piece of notation, we shall write $H_c$ for the ring class field attached to the order in $K$ of
conductor $c\geq 1$ and write $K_{\mathfrak a}$ for the ray class field of conductor $\mathfrak a$.

Define
$$
U^+_c =\{  \beta\in U_c \mbox{ such that } (\beta)_{\fN}  \in \ker(\varepsilon)\subset (\Z/N\Z)^{\times}\},
$$
$$
U^-_c =\{  \beta\in U_c \mbox{ such that } (\beta)_{\overline{\fN}}  \in \ker(\varepsilon)\subset (\Z/N\Z)^{\times}\},
$$
and $\tilde U_c=U_c^+\cap U_c^-$. Here $(\beta)_{\fN}$ denotes
the image of the local term of the id\` ele $\beta$ at
$\fN$ in the quotient $\cO_{\fN}^{\times}/(1+\fN\cdot
\cO_{\fN})\simeq (\Z/N\Z)^{\times}$. Similarly for $\overline\fN$. This way we can regard the character $\varepsilon$ as having source either $\cO_{\fN}^{\times}$ or $\cO_{\overline{\fN}}^{\times}$.

Set
$$ {\tilde G_c} := \A_K^\times/(K^\times \tilde U_c) = \gal({\tilde H_c}/K),$$
where ${\tilde H_c}$ is a biquadratic extension of the
ring class field $H_c$. It can be written as $\tilde{H}_c
= L_c L'_c$, where $L_c$ (resp. $L'_c$) is the class field attached to $U^+_c$ (resp. $U^-_c$).

%Let us assume, only for simplicity, that $d_0\ne 1, 3$ so that $\cO_K^{\times}=\{\pm 1\}$.

\begin{proposition}\label{c} The relative discriminant of $L/K$ factors as
$d(L/K)=c \cdot \fN$, where $c$ is a positive integer such that $L\subset L_{c}$
and  $L'\subset L'_{c}$ and thus $\zmathcalM \subset \tilde H_{c}$.
\end{proposition}

More precisely, we have $c=2^t\cdot c_0$ where $0\leq t\leq 3$ and $c_0$ is odd and square-free.
The proof of this proposition is an exercise in class field theory, which we relegate to \S \ref{Sc} for the convenience of the reader.


We now explain how to construct a degree zero
divisor on $X_{\varepsilon}(N)$ defined over $\tilde H_c$. To do this, let $A_c$ be an elliptic curve
satisfying
$$ \End(A_c) = \cO_c,$$
where $\cO_c:= \Z + c\cO_K$ is the order in $K$ of
conductor $c$. Such a curve, along with its
endomorphisms, may be defined over the ring class
field $H_c$. The module $A_c[\fN]$ of $\fN$-torsion
points is therefore defined over $H_c$, yielding a point $P_c:=[A_c,A_c[\fN]]\in
X_0(N)(H_c)$.

The action of $G_{H_c}:= \gal({\overline \zQ}/H_c)$ on the points of
this group scheme gives a Galois representation
$$ \rho_{\fN}: G_{H_c} \lra (\Z/N\Z)^\times.$$

The composition of $\rho_{\fN}$ with the nebentypus
character $\varepsilon$ is a quadratic character of
$G_{H_c}$, which cuts out the quadratic extension $L_c$
of $H_c$. The point $P_c$ lifts to two points $P_c^+$ and $P_c^-$  in
$X_{\varepsilon}(N)(L_c)$ which are interchanged by the action of
$\gal({L_c}/H_c)$; we do not specify the order in
which these points are to be taken. Similarly, we can replace the module $A_c[\fN]$ by $A_c[\overline{\fN}]$, mimic the
 above construction and obtain points $P_c^{\prime +}$ and  $P_c^{\prime -}$ defined  over $L'_c$.


\begin{definition}
\label{CM}
Let
$$\CM(c)= \bigcup \{ P_c^+, P_c^-, P_c^{\prime +}, P_c^{\prime -}\}
\subset X_{\varepsilon}(N)(\tilde H_c)$$
 be the set of Heegner points on
$X_{\varepsilon}(N)$ obtained by letting $A_c$ run over all isomorphism
classes of elliptic curves with CM by $\cO_c$.
\end{definition}

If we let $h(O_c)$ denote the cardinality of the group  $\Pic(\cO_c)$ of classes of locally free ideals of $\cO_c$, the cardinality of $\CM(c)$ is $4 h(\cO_c)$. In fact, $\CM(c)$ is naturally the disjoint union of the two subsets $CM(c)\cap X_{\varepsilon}(N)(L_c)$ and $CM(c)\cap X_{\varepsilon}(N)(L'_c)$, each of cardinality $2 h(\cO_c)$.

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This part is based on Darmon's suggestion. The two approaches are essentially
% the same.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A Heegner point $P\in \CM(c)$ of conductor $c$ may be described by a triple $([\zfraka], \zfrakn, t)$, where

\begin{itemize}

\item $[\zfraka]\in \zPic(\zmathcalO_{c})$ is the class of an invertible $\zmathcalO_{c}$-module of $K$,

\item  $\zfrakn$ is an integral ideal of $\zmathcalO_{c}$ such that the quotient $\zmathcalO_{c}/\zfrakn$ is cyclic of order $N$,

\item $t$ is an orbit for the action of $\zker(\varepsilon)$ of an element of order $N$ in $\zfraka \zfrakn^{-1}/\zfraka \cong \zZ/N\zZ$.

\end{itemize}

 Let $\tilde{C}$ be the quotient of the ray class
  group of $K$ of conductor $c\fN$ for which Artin's
  reciprocity map of global class field theory furnishes
  a canonical isomorphism
  \[
  \mathrm{rec}: \tilde{C}\overset{\sim}{\lra }
  \zGal(\tilde H_{c}/K).
  \]

  Let $\cO=\cO_{c}$ denote the order of conductor $c$ in $K$. There are natural exact sequences, sitting
   in the commutative diagram
 $$
\xymatrix{
  1  \ar[r]  &  \gal(\tilde H_{c}/H_{c}) \ar [r]
\ar[d]^{\mathrm{rec}}
 &  \gal(\tilde H_{c}/K)
\ar[d]^{\mathrm{rec}}
\ar[r]^{\mathrm{res}_{\tilde H_{c}/H_{c}}}
 &  \gal(H_{c}/K)
\ar[r]
\ar[d]^{\mathrm{rec}} & 1   \\
  1
\ar[r]
 &  \langle [\beta_0], [\beta'_0] \rangle \ar[r]
&  \tilde{C}  \ar[r]    &   \Pic(\cO) \ar[r]    &   1,}
$$
  where the vertical arrows are isomorphisms.
Here, $\beta_0 \in \cO^{\times}_{\fN}$ and $\beta'_0 \in \cO^{\times}_{\overline{\fN}}$ are
elements   such that $\varepsilon(\beta_0)=-1$ and  $\varepsilon(\beta'_0)=-1$.
Artin's reciprocity map induces an isomorphism
$$ \gal(\tilde H_{c}/H_{c})\simeq \cO^{\times}_{\fN}/\ker(\varepsilon)\times
\cO^{\times}_{\overline{\fN}}/\ker(\varepsilon)\simeq \Z/2\Z \times \Z/2\Z. $$
 We thus can formally write elements of $\tilde{C}$  as classes of {\em enhanced ideals},
which are defined as elements of the form
$\underline{\mathfrak b}:=\beta_{\fN} \beta_{\overline{\fN}}\prod_{\wp \nmid N} \wp^{n_{\wp}}$,
taken up to principal ideals $(b)$ with $b\in K^{\times}$.
Here $\beta_{\fN}$ and
$\beta_{\overline\fN}$
belong to $K_{\fN}^{\times}/\ker(\varepsilon)$  and $K_{\overline{\fN}}^{\times}/\ker(\varepsilon)$
respectively,   $\wp$ runs over all prime invertible ideals of $\cO$ not dividing $N$, and the exponents
 $n_{\wp}$ are integers which are almost all zero.
We say an enhanced ideal is {\em integral}
 if $\beta_{\fN}$
and
$\beta_{\overline \fN}$ have representatives in
$\cO^{\times}_{\fN}$ and
$\cO_{\overline {\fN}}^{\times}$
respectively,
and $n_{\wp}\geq 0$ for all $\wp$.
The image of the class
$\underline{\mathfrak b}$ in $\Pic(\cO)$ is simply the class of the
ideal $\mathfrak b=\fN^{\ord_{\fN}(\beta_{\fN})} \overline{\fN}^{\ord_{\overline{\fN}}(\beta_{\overline \fN})}
\prod_{\wp \nmid N} \wp^{n_{\wp}}$ generated by it.


  By Shimura's reciprocity law,
  \begin{equation}\label{reclaw}
  \mathrm{rec}(\underline{\mathfrak b})(D) = \underline{\mathfrak b}^{-1}\star D
 \end{equation}
 for all $\underline{\mathfrak b}\in \tilde{C}$ and all divisors $D\in J_\varepsilon(N)(\tilde H_{c})$ supported on $\mathrm{CM}(c)$.

 On the left hand side we make use of the natural
 Galois action of $\gal(\tilde H_{c}/K)$ on $J_\varepsilon(N)(\tilde H_{c})$, via Artin's reciprocity isomorphism. On the right hand side, a class $[\underline{\mathfrak b}]\in \tilde{C}$ acts on $\CM(c)$ by the rule
 \begin{equation}\label{f1}
 \underline{\mathfrak b}\star P = ([\zfraka \mathfrak b^{-1}], \zfrakn, \varphi_{\mathfrak b}(\beta_{\fN} t)),
 \end{equation}
 where $P=([\zfraka], \zfrakn, t)\in \CM(c)$, $\underline{\mathfrak b}=\beta_{\fN} \beta_{\overline{\fN}}\prod_{\wp \nmid N} \wp^{n_{\wp}}$ is an integral representative of its class and $\varphi_{\mathfrak b}: \zC/\zfraka \to \zC/\zfraka \mathfrak b^{-1}$ is the natural projection map. Writing $P=[\tau]\in X_{\varepsilon}(N)(\C)$ for some $\tau\in \mathcal H$, let $\gamma_{\underline{\mathfrak b}}\in \GL_2^+(\Q)$ be such that $\underline{\mathfrak b}\star P = [\gamma_{\underline{\mathfrak b}} \tau]$.

 Besides this action, there is also the diamond involution $W_{\varepsilon}$, acting on $P=[\tau]\in X_{\varepsilon}(N)(\C)$ as $W_{\varepsilon}([\tau])=[\gamma_{\epsilon}\tau]$ and on $P=([\zfraka], \zfrakn, t)\in \CM(c)$ as
  \begin{equation}
\label{f3}
 W_{\varepsilon}(P)=([\zfraka], \zfrakn, d t),
\quad \mbox{ for }  \gamma_{\epsilon}=\zsmatrix{a & b \\ Nc & d}\in \Gamma_0(N)\setminus \Gamma_{\varepsilon}(N).
 \end{equation}
The cardinality of $\CM(c)$ is $4 h(\cO)$ and it is acted on freely and transitively by the group $\langle W_N, W_{\varepsilon} \rangle \times \tilde{C}_{\fM}$, where we let $\tilde{C}_{\fM}:=\mathrm{rec}^{-1}(\gal(\tilde H_{c}/\fM))\subset \tilde{C}_K$. Note that the restriction map $\mathrm{res}_{\tilde H_{c}/H_{c}}$ induces an isomorphism $\tilde{C}_{\fM}\cong \Pic(\cO)\cong \gal(H_{c}/K)$.


%% Now, according to \eqref{DM_1},
%% \begin{equation}\label{DM}
%% D_{\fM}= \sum_{\underline{\mathfrak b}\in \tilde{C}_{\fM}} ( \underline{\mathfrak b}^{-1} + W_N\underline{\mathfrak b}^{-1} - W_{\varepsilon}\underline{\mathfrak b}^{-1} -W_{\varepsilon} W_N \underline{\mathfrak b}^{-1}) \star P_c^+\in J^+_{\varepsilon}(N)(\fM).
%% \end{equation}

%% It follows from \eqref{cc} and the subsequent discussion that
%% \begin{equation}\label{invarianceundercc}
%% \varrho(D_{\fM})=\sum_{\underline{\mathfrak b}\in \tilde{C}_{\fM}} ( \underline{\mathfrak b} + W_N\underline{\mathfrak b} - W_{\varepsilon}\underline{\mathfrak b} -W_{\varepsilon} W_N \underline{\mathfrak b}) \star \varrho(P_c^+)=(-1)^s D_{\fM}.
%% \end{equation}

It is our aim now to define a point $P_M\in E(M)$ (and thus also, by conjugation over $F$, a point $P_{M'}\in E(M')$) on the elliptic curve $E$, rational over the ATR extension $M/F$. We shall construct $P_M$ as a suitable
linear combination of certain  points $P_L\in A(L)$ and
 $P_{L'}\in A(L')$ on the abelian surface $A=\mathrm{Res}_{F/\Q}(E)$. These
points are defined as the trace to $L$ of the projection of $P_c^+\in X_\varepsilon(N)(L_c)$
(respectively of $P_c^{'+}\in X_\varepsilon(N)(L'_c)$) on $A$.

Before doing so, we first observe that choosing $P_c^-=W_{\varepsilon}(P_c^+)$ instead of $P_c^+$ (and similarly $P_c'^{-}$ instead of $P_c'^{+}$) is unimportant for our construction, as the
next lemma shows that both lead to the same point on $A$ up to sign and torsion.
Recall the canonical projection $\pi_f: J_\varepsilon(N) \to A$ defined
over $\Q$ and reviewed in \eqref{factorisation}, which can be composed with the
natural embedding of $X_\varepsilon(N)$ into its jacobian $J_\varepsilon(N)$
given by the map $P \mapsto P-i\infty$. By an abuse of notation, we continue to denote by $\pi_f$ this
composition.
\begin{lemma}
\label{equal}
For any  $P\in X_\varepsilon(N)(\overline\Q)$, the point
$\pi_f(P)+\pi_f(W_\varepsilon(P))$ belongs to $A(F)_{\mathrm{tors}}.$
\end{lemma}
\begin{proof} There is a natural decomposition $S_2(\Gamma_\varepsilon(N)) =
S_2(\Gamma_0(N))\oplus S_2(\Gamma_0(N), \varepsilon)$ corresponding to the eigenspaces
of eigenvalue $\pm 1$ with respect to the action of the involution $W_{\varepsilon}$.
The rule $f(z)\mapsto f(z)dz$ yields an identification of $S_2(\Gamma_\varepsilon(N))$
with the space of holomorphic differentials on $X_\varepsilon(N)_\C$. Via
  this isomorphism, $\pi_f^* H^0(\Omega^1_A)$ is contained in $S_2(\Gamma_0(N), \varepsilon)$.
Consequently, $\pi_f(P-i\infty)=-\pi_f (W_\varepsilon(P-i\infty))$ and
\begin{eqnarray*}
\pi_f(P)+\pi_f(W_\varepsilon(P)) =  \pi_f(P-i\infty) + \pi_f(W_\varepsilon(P)-i\infty)  \\
=  \pi_f(P-i\infty)+\pi_f(W_\varepsilon(P)-W_\varepsilon(i\infty)) + \pi_f(W_\varepsilon(i\infty)-i\infty)
 =   \pi_f(W_\varepsilon(i\infty)-i\infty).
\end{eqnarray*}
This last expression is a torsion point on $A(F)$ by the  Manin-Drinfeld theorem which asserts that
degree zero cuspidal divisors on a modular curve give rise to torsion elements in its Jacobian.
\end{proof}
We now set
$$  P_L = \zTr_{L_c/L}(\pi_f(P_c^+))\in A(L). $$
Note that $\tau_M(P_L^+)$ is either equal to  $\zTr_{L_c'/L'}(\pi_f({P_c'}^+))$ or  to
$\zTr_{L_c'/L'}(\pi_f({P_c'}^-))$. Without loss of generality, assume
that  $\tau_M(P_L^+) =\zTr_{L_c'/L'}(\pi_f({P_c'}^+))$ and denote it by $P_L'$.

Set
$$ u = \left\{
\begin{array}{ll}
2 & \mbox{ if }  K=\zQ(\sqrt{-1}) \mbox{  and }  c=1;  \\
3 & \mbox{ if }   K=\zQ(\sqrt{-3}) \mbox{ and }  c=1; \\
1 & \mbox{   otherwise,}
\end{array} \right.
$$
and define
$$ P_{A,M} := \frac{1}{u} (P_L + P_{L}'),
\qquad  P_{A,M'} := \frac{1}{u} (P_L - P_L'). $$
The construction of the point $P_{A,M}$ is illustrated
in the figure below.
$$
\xymatrix{
\ar@{.}[ddddrrrr]|-<{\tau_M'}|-<<<<<<{\circlearrowright} &
& \ar@{.}[dddd]|-<{\tau_L}|-<<<<<{\circlearrowright} &
& \ar@{.}[lllldddd]|-<{\tau_M}|-<<<<<<{\circlearrowleft} \\
 & \bullet \ar@{-}[rr]^<{P_{A,M'}}^>{P_{A,M}}_{\phantom{ppp}P_L} \ar@{-}[dd]
& \circ  &
\bullet \ar@{-}[dd] &   \\
\ar@{.}[rrrr]_<<<<<<<<{-P_L'}|->{\phantom{pppp}\tau_L'}|->>>{\phantom{}\circlearrowleft}_>>>>>>>>>{P_L'}
 & \circ  & & \circ  &  \\
& \bullet \ar@{-}[rr]_<{-P_{A,M}}_<<<<<<<<<<<<{-P_L}_>{-P_{A,M'}} & \circ & \bullet \\
&    &   &   & \\  }
$$
This figure suggests--and it is indeed easy to check--that
$$P_{A,M} \in A(M),
\qquad  P_{A,M'}\in A(M').$$
Recall that the morphism $\varphi_F: A_F \lra E$ introduced in \eqref{factorisation}
is defined over $F\subset M$, and therefore that the point
$$P_M:=\varphi_F(P_{A,M})$$
belongs to $E(M)$.
As a by-product of our explicit construction we obtain the following analytic formula for calculating
the point $P_M$.
\begin{theorem}\label{thm:1}
Let $\tau_{c},\tau_c' \in \mathcal H$ be elements representing the Heegner points $P_c^+,P_c^{'+}\in X_\varepsilon(N)(\tilde H_c)$. Set
\begin{equation}\label{zM}
z_M = \sum_{\underline{\mathfrak b}\in \tilde{C}_{\fM}}\!\!
\left [\int_{i \infty}^{\gamma_{\underline{\mathfrak b}} \tau_{c}} \,(f_E(\tau) +
f_E\big |_{W_N}(\tau))\, d\tau +\int_{i \infty}^{\gamma_{\underline{\mathfrak b}} {\tau_{c}'}} \,(f_E(\tau) +
f_E\big |_{W_N}(\tau))\, d\tau  \right ].
\end{equation}
Then $P_M=\eta(z_M)$ where $\eta$ is the Weierstrass parametrization
\begin{equation}\label{Wpar}
\eta: \C/\Lambda_E \lra E(\C), \quad
\eta(z)=(\wp(z),\wp'(z)).
\end{equation}
Here, $\wp$ is the
Weierstrass function associated with the lattice of
periods  \newform{ \Lambda_E: = \left \{
\int_\delta (f_E + f_E\big |_{W_N})d\tau \right \} }
where $\delta \in H_1(X_\varepsilon(\zC),\zZ)$ runs over the cycles of $X_\varepsilon(\zC)$ such that $\int_\delta (f_E -f_E\big |_{W_N})d\tau =0$.
\end{theorem}


\subsection{Heegner points and ATR cycles}
The main conjecture that will be formulated in this section
relates the Heegner point $P_M$
with the Stark Heegner point $P_M^?$ arising from  ATR cycles.
 Recall that  $\mathrm{Gal}(F/\Q)=\{1,\tau\}$ and $D_F=\disc(F)$. Let also $c_{E/F}$ (resp. $c_{E^\tau/F}$) denote either the real period or twice the real period of $E/\R$ (resp. of $E^\tau/\R$) depending on whether $E(\zR)$ (resp. $E^\tau(\zR)$) is connected or not.

\begin{conjecture}
\label{conj:main-oda}
The ATR point $P_M^?$ is of infinite order if and only if
$P_M$ is of infinite order and $L(E/F,1)\ne 0$. More precisely,
\begin{equation}
\label{eqn:main-conj}
P_M^? = 2^s \ell \cdot P_M
\end{equation}
where $\ell\in \Q^\times$, which depends only on $(E,F)$ and not on $M$, satisfies
$$\ell^2 = \frac{L(E/F,1)}{\Omega_{E/F}}, \quad \mbox{ with }
\Omega_{E/F}=\frac{c_{E/F}\cdot c_{E^\tau/F}}{D_F^{1/2}\cdot |E_{\mathrm{tor}}(F)|^2},$$
 and $s \in \zZ$ depends on $M$.
\end{conjecture}
Below we collect some numerical data in support of conjecture
\ref{conj:main-oda}.
Besides the numerical evidence, Conjecture \ref{conj:main-oda} is also motivated by the equality
$$L'(E/M,1) =  L(E/F,1)L'(E/F,\chi_M,1)$$
 as recalled in the proof of Theorem \ref{thm:main-bis},
 and by  the facts that
\begin{enumerate}
\item the
N\'eron-Tate height of the
Stark-Heegner point
 $P_M^?$ is
expected (cf.~\cite{DaLo})  to be related in a simple way to  $L'(E/M,1)$.
\item the extension of the Gross-Zagier formula proved in
\cite{yuan-zhang-zhang} should in principle lead to an analogous relationship
between  the Heegner point
 $P_M$ and the derivative $L'(f/K, \chi_L, 1)=L'(f/F,\chi_M,1) = L'(E/F,\chi_M,1)$.
\end{enumerate}
It would be interesting to formulate a precise recipe predicting the power of $2$ that arises
as a fudge factor in \eqref{eqn:main-conj}. The authors have not made a serious attempt to
do this.



\subsection{Numerical examples}

For $N=29, 37, 41$ it is known
(cf.~\cite{Shiota} and \cite[\S \,6]{Cremona})
 that there is a unique Shimura elliptic curve defined
over $F=\zQ(\sqrt{N})$ up to isogeny over $F$.

The aim of this section is to provide numerical evidence for conjecture \ref{conj:main-oda},
which we have gathered by explicitly computing the points $P_M$ and $P_M^?$ for several ATR
extensions $M/F$ on each of these three elliptic curves.
The computation of the Heegner point $P_M$ was performed with the software package PARI \cite{PARI}
by exploiting formula \eqref{zM} and the material in \cite{Shiota} and \cite[\S \,6]{Cremona} to produce
a complex invariant $z_M\in \C/\Lambda_E$ mapping to $P_M$ under the  Weierstrass uniformisation.
Similarly, the ATR point $P_M^?$
 was computed by following the method explained in \cite{DaLo}.

In fact, for our experiments it was sufficient to compute the
element $z_M^?\in \C/\Lambda_E$ mapping to $P_M^?$ under the Weierstrass uniformisation.
For several values of  $M$,
the invariants $z_M$ and $z_M^?$ were calculated to roughly $50$ digits of decimal
accuracy, and the constants $s$ and $\ell$ in  (\ref{eqn:main-conj}) could then be
 obtained by picking a basis $(e_1, e_2)$ for
$\Lambda_E$ and searching for a
linear dependence relation with small integer coefficients
between the four complex
numbers $z_M$, $z_M^?$, $e_1$ and $e_2$,
 using Pari's {\tt lindep} command.

This approach represents a dramatic improvement over the one that had to be followed in
\cite{DaLo}, in which only the point $P_M^?$ was computed. In practice,
recognizing $P_M^?$ as an algebraic point using standard rational recognition
programs is  difficult  once the
height of $P_M^?$ becomes large. The authors of \cite{DaLo}
were forced instead  to perform an independent search
for a generator of   $E(M)$--a computationally difficult and time-consuming
task--in order to check that
$P_M^?$ indeed agreed with a point of small height on $E(M)$
 to within the calculated decimal accuracy.
The new approach based on the Heegner point $P_M$ makes
 the experimental verifications  of
\cite{DaLo}   much more systematic and efficient, and allows them to be carried out
for much further ranges.

In the tables below, we have  followed almost the same notations
 as in \cite[\S 3]{DaLo}. In particular, we have written
 $M=F(\beta)$ with  $\beta^2\in F$, and denoted
by
$D_M=\zNm_{F/\Q}(\disc(M/F))$ the
 absolute discriminant. (Note that
$K$ is used in \cite{DaLo} to denote the field
that we call  $M$ in the present work.)
As before,  $L/K$ denotes the quadratic extension sitting in the Galois closure of $M$ as
 in the field diagram \eqref{eqn:field-diag-complete}, and we denote
 $D_K=\disc(K/\Q)$ and $D_L=\zNm_{K/\Q}(\disc(L/K))$ and $\ell^2 \in \Q^{\times}$ is the value we found numerically for the constant alluded to in Conjecture \ref{conj:main-oda}. Finally note  that $\ell$ and $s$ uniquely determine $P_M$ up to sign and $E(M)_{\mathrm{tor}}$.


\vspace{0.5cm}
\noindent
{\bf The case  $N=29$.}
Let $\delta = 2+\omega= (5+\sqrt{29})/2$. Shiota's Weierstrass equation for $E_N$ is
given by
\newform{
E_{29} : y^2 + xy + \delta^2y = x^3,
}
whose discriminant is $\Delta_{29} = -\delta^{10}$. Our calculations
convincingly suggest that
$$\ell^2 = \frac{L(E_{29}/F,1)}{\Omega_{E_{29}/F}} = 1 $$
 and that the point $P_M^?$ and $s$ are given in the following table.  The table suggests that
$s=-2$ in all cases that have been calculated for this particular curve.

\sk
{\begin{tiny}
\begin{center}
\begin{longtable}{|r|c|c|c|c|c|} \label{table1}
$D_M= D_K\cdot c^2$ & $ \beta^2 $  & $D_L$ & $|\mathrm{Pic}(\cO_c)|$  & $P^?_M$  & $s$  \\
\hline
$-7 = -7 \cdot 1$       & $-1+\omega$           & $29$                  & $1$   & $(\beta^2 + 3, \, -\frac{5}{2}\beta^3-3\beta^2-8\beta-\frac{19}{2})$ & $-2$ \\
$-16 = -4\cdot 2^2$     & $2+\omega$            & $2^2\cdot 29$         & $1$   & $(\frac{\beta^2}{2}, \, -\frac{5}{4}\beta^3-\frac{11}{4}\beta^2-\frac{\beta}{4}-\frac{ 1}{2})$ & $-2$ \\
$-23 = -23\cdot 1 $     & $17+8\omega$          & $29$                  & $3$    & $(\frac{1}{8}(11\beta^2 + 5), \, -\frac{13}{8}\beta^3-\beta^2-\frac{7}{8}\beta-\frac{1}{2})$ & $-2$ \\
$-35_1=-35\cdot 1$      & $19+9\omega$          & $29$                  & $2$   & $(\frac{1}{5}(2\beta^2 + 1),\, -\frac{59}{225}\beta^3-\frac{ 43}{90}\beta^2- \frac{89}{450}\beta-\frac{ 29}{90})$ & $-2$ \\
$-35_2=-35 \cdot 1 $    & $ 4+3\omega$          & $29$                  & $2$   & $(-\frac{1}{15}(4\beta^2+11),\, -\frac{1}{150} (17\beta^3+ 105\beta^2+43\beta+270))$& $-2$ \\
$-59 = -59\cdot 1$      & $ 61+28\omega $       & $ 29 $                & $3$   &  $(-\frac{1}{9},\, -\frac{11}{1512}\beta^3- \frac{5}{56}\beta^2-\frac{1}{1512}\beta+ \frac{1}{504})$& $-2$ \\
$-63  = -7 \cdot 3^2$    & $ 3\omega $           & $ 3^2\cdot 29 $       & $4$   & $(\frac{7}{9}\beta^2+5, \frac{26}{27}\beta^3 -\frac{11}{9}\beta^2+\frac{57}{9}\beta -8)$ & $-2$ \\
$-64  = -4\cdot 4^2$    & $ 4+2\omega    $      & $ 2^4\cdot 29 $       & $2$   & $(-\frac{1}{4}, \, -\frac{3}{8}\beta^3-\frac{5}{4}\beta^2-\frac{\beta}{4}-\frac{3}{8})$ & $-2$ \\
$-80 = -20 \cdot 2^2$   & $ 1+\omega     $      & $ 2^2\cdot 29 $       & $4$   & $(\frac{1}{10}(43\beta^2 + 51),\, -\frac{517}{50}\beta^3 - \frac{93}{20}\beta^2 -\frac{ 1233}{100}\beta- \frac{ 111}{20})$ & $-2$ \\
$-91 = -91 \cdot 1 $    & $ 7+5\omega  $        & $ 29 $                & $2$   & $(\frac{1}{13}(98\beta^2+387), \, -\frac{18939}{845}\beta^3- \frac{111}{26}\beta^2-\frac{150109}{1690}\beta-\frac{439}{26})$ & $-2$ \\
$ -175 = -7 \cdot 5^2$  & $ -5+5\omega $        & $ 5^2\cdot 29 $       & $6$   & $(-\frac{6}{50}\beta^2-2, \frac{1}{10}\beta^3-\frac{11}{25}\beta^2+\frac{98}{100}\beta-\frac{45}{10})$ & $-2$ \\
\hline
\caption{ATR extensions of $\Q(\sqrt{29})$ and ATR points on $E_{29}$}
\end{longtable}
\end{center}
\end{tiny}}

\vspace{0.5cm}
\noindent
{\bf The case
 $N=37$}. Letting $\omega= \frac{1+\sqrt{37}}{2}$,  Shiota's Weierstrass equation for $E_{37}$ is given by
\newform{
E_{37} : y^2 + y = x^3+2x^2-(19+8\omega)x+(28+11\omega),
}
and its discriminant is $\Delta_{37} = (5+2\omega)^6$.
Note that $5+2\omega$ is a fundamental unit of $F$. Our calculations are consistent with the fact that
$$\ell^2 = \frac{L(E_{37}/F,1)}{\Omega_{E_{37}/F}} = 1. $$
More precisely, the Stark-Heegner point $P_M^?$ and $s$ are given in the tables below.

\sk
{\begin{tiny}
\begin{center}
\begin{longtable}{|r|c|c|c|c|c|} \label{table3}
$D_M= D_K\cdot c^2$ & $ \beta^2 $  & $D_L$ & $|\mathrm{Pic}(\cO_c)|$ & $P^?_M$ & $s$  \\
\hline
$-3 = -3 \cdot 1$           & $-3+\omega$           & $37$  & $1$  & $(-\frac{2}{3}\beta- \frac{ 13}{3}, \, -\frac{61}{18}\beta^3-\frac{ 169}{9}\beta-\frac{1}{2})$ & $-1$  \\
$-7 = -7\cdot 1$            & $1+\omega$            & $37$                  & $1$  & $(\frac{2}{7}\beta-\frac{3}{7},\  -\frac{57}{98}\beta^3-\frac{ 44}{49}\beta-\frac{1}{2})$ & $-1$ \\
$-11 = -11\cdot 1 $         & $38+15\omega$         & $37$                  & $1$  &  $(-\frac{2}{165}\beta^2-\frac{ 104}{165},\, -\frac{17}{1210}\beta^3- \frac{2}{605}\beta-\frac{ 1}{2})$ & $-1$ \\
$-16 = -4\cdot 2^2$         & $5+2\omega$           & $2^2\cdot 37$         & $1$  &  $(\frac{\beta^2}{8}-\frac{ 5}{8},\, \frac{\beta^3}{8}- \frac{1}{2})$ & $-2$\\
$-48= -3\cdot 4^2 $         & $2+\omega$            & $4^2\cdot 37$         & $3$  &  $(\frac{115}{588}\beta^2-\frac{ 80}{147}, \  -\frac{11225}{24696}\beta^3- \frac{1529}{6174}\beta-\frac{ 1}{2})$& $-1$\\
$-64 = -4\cdot 4^2$         & $10+4\omega $         & $4^2\cdot 37$         & $2$  &  $(-\frac{\beta^2}{8}-\frac{ 3}{4},\  -\frac{\beta^3}{8}-\frac{ 1}{2})$ & $-2$ \\
$-67 = -67\cdot 1$          & $193+76\omega $       & $67$                  & $1$   & $(-1, -\frac{1}{2}+\frac{1}{2}\beta)$& $-2$\\
$-75 = -3 \cdot 5^2$        & $-15+5\omega $        & $5^2\cdot 37$         & $3$   & $(\frac{196}{775}\beta^2+\frac{136}{27}, -\frac{1559}{12150}\beta^3-\frac{25732}{6075}\beta-1/2)$& $-1$\\
$-192 =-3 \cdot 8^2$        & $18+8\omega$          & $8^2\cdot 37$         & $6$   & $(\frac{7}{3}+\frac{7}{6}\omega, -\frac{1}{2}+\frac{1}{36}(\frac{85}{3}+\frac{14}{3}\sqrt{37})\beta)$& $-2$\\
$-275 =-11 \cdot 5^2$        & $445+180\omega$          & $5^2\cdot 37$       & $4$   & $(\frac{2}{11}+\frac{4}{11}\omega, -\frac{1}{2}+\frac{1}{242}(\frac{62}{7}+\frac{9}{7}\sqrt{37})\beta)$& $-2$\\
$-448 = -7 \cdot 8^2$       & $2+2\omega$           & $8^2\cdot 37$         & $4$   & $(\frac{45}{7}+\frac{39}{14}\omega, -\frac{1}{2} +\frac{1}{196}(\frac{689}{2}\sqrt{37}+\frac{4191}{2})\beta)$ & $-2$
\\
\hline
\caption{ATR extensions of $\Q(\sqrt{37})$ and ATR points on $E_{37}$}
\end{longtable}
\end{center}
\end{tiny}}





\vspace{0.5cm}


\noindent
{\bf The case  $N=41$}. Shiota's Weierstrass equation for $E_{41}$ is
\newform{
E_{41} : y^2=  x^3 -\frac{17}{48}x +
 \Bigl ( -\frac{5}{32}+ \frac{1}{27}\sqrt{41} \Bigr )
}

In their computations, Darmon and Logan used instead curve $E'_{41}: y^2+xy=x^3-(32+5\sqrt{41}) x$.
This Weierstrass equation was first found by Oort, and there is an explicit isogeny
 $\psi: E'_{41}\ra E_{41}$ of degree $2$. Following Darmon-Logan's approach, points $P_M^?$ listed below
 are points on $E'_{41}$. Since the isogeny $\psi$ is explicit, it is an easy task to transfer them
to points on $E_{41}$, and this is what we did in order to compare the Heegner points $P_M\in E_{41}(M)$
with the Stark-Heegner points $\psi(P_M^?)\in E_{41}(\C)$.
In this case, calculations suggest once again that
$$\ell^2 = \frac{L(E_{41}/F,1)}{\Omega_{E_{41}/F}} = 1,$$
while the values of the exponent $s$ also appear in the table below.
\sk
{\begin{tiny}
\begin{center}
\begin{longtable}{|r|c|c|c|c|c|} \label{table5}
$D_M= D_K\cdot c^2$ & $ \beta^2 $  & $D_L$ & $|\mathrm{Pic}(\cO_c)|$ & $P^?_M$  &  $s$ \\
\hline
$-4 = -4 \cdot 1$           & $27+10\omega$         & $41$                  & $1$  &  $(-\frac{1}{4},\  -\frac{\beta}{2}+ \frac{1}{8})$  & $1$ \\
$-8 = -8 \cdot 1$           & $-248+67\omega$ & $41$                  & $1$   & $ \bigl(-\frac{1}{268}(3\beta^2 +1481),\  \frac{1}{536}(-254\beta^3 + 3\beta^2 -108954\beta+ 1481)\bigr )$ & $0$\\
$-20 = -20\cdot 1$          & $697+258\omega$       & $41$                  & $2$   & $(\frac{1}{43}(\beta^2 - 9), \  \frac{1}{258}(-\beta^3 -3\beta^2 + 181\beta+ 27))$& $0$ \\
$-23 = -23\cdot 1 $         & $398+144\omega$       & $41$                  & $3$   & $\bigl (\frac{-71027\beta^2 - 1271153}{
9884736}, \ \frac{-1095348\beta^3 + 9304537\beta^2 + 16459332\beta + 166521043}{
2589800832}\bigr )$& $0$ \\
$-32 = -8\cdot 2^2$         & $1+\omega$            & $2^2\cdot 41$         & $2$   & $\bigl(\frac{29\beta^2 + 49}{4}, \  \frac{1}{16}(-359\beta^3 -58\beta^2 - 611\beta-98)\bigr )$ & $0$ \\
$-36 = -4\cdot 3^2$         & $6+3\omega$           & $3^2\cdot 41$         & $4$  & $\bigl (-8+2\omega, (\frac{7}{2}-\frac{1}{2}\sqrt{41})(1+5\beta) \bigr)$  & $-1$ \\
$-40 = -40\cdot 1$          & $35+13\omega$         & $41$                  & $2$  & $\bigl (9+\frac{27}{8}\omega, -\frac{171}{32}-\frac{27}{32}\sqrt{41}+\frac{3}{32}(\frac{109}{2}+\frac{17}{2}\sqrt{41})\beta \bigr )$ & $-1$ \\
$-100 = -4 \cdot 5^2$       & $10+5\omega$          & $5^2\cdot 41$         & $2$   & $\bigl (\frac{9}{2}+\frac{7}{4}\omega, -\frac{43}{16}-\frac{7}{16}\sqrt{41} +(\frac{3}{8}\sqrt{41}+\frac{19}{8})\beta \bigr )$ & $-2$\\
$-115=-115 \cdot 1$         & $177+68\omega$        & $41$                  & $2$   & $(-\frac{31}{9}-\frac{11}{9}\omega, \frac{73}{36}+\frac{11}{36}\sqrt{41}+\frac{1}{108}(\frac{59}{5}+\frac{9}{5}\sqrt{41})\beta)$ & $-1$ \\
$-160 = -40 \cdot 2^2$       & $4\omega$            & $2^2\cdot 41$         & $4$   & $(32+12\omega, -19-3\sqrt{41}+(\frac{173}{2}+\frac{27}{2}\sqrt{41})\beta)$ & $-2$ \\
$-368=-23 \cdot 4^2$        & $43+16\omega$         & $4^2 \cdot 41$        & $6$   & $(\frac{29}{4}+\frac{11}{4}\omega, -\frac{69}{16}-\frac{11}{16}\sqrt{41}+(\frac{13}{4}+\frac{1}{2}\sqrt{41})\beta)$ & $-2$ \\ \hline
\caption{ATR extensions of $\Q(\sqrt{41})$ and ATR points on $E'_{41}$}
\end{longtable}
\end{center}
\end{tiny}}





\subsection{Proof of Proposition \ref{c}.}\label{Sc}

The aim of this section is proving Proposition \ref{c}, which was left unproved in \S \ref{sec:explicit} and asserts that the relative discriminant of $L/K$ factors as
$d(L/K)=c \cdot \fN$, where $c$ is a positive integer such that $L\subset L_{c}$ (and similarly $L'\subset L'_{c}$).

Recall our assumption on $N=\disc(F)$ to be odd, and thus square-free. Here we shall assume for notational simplicity that $K \not = \zQ(\sqrt{-1})$ and $\zQ(\sqrt{-3})$, so that $\cO_K^\times=\{\pm 1\}$; we leave to the reader the task of filling the details for the two excluded fields.

Let us recall first the following classical lemma on Kummer extensions of local fields, which applies in particular to our quadratic extension $L/K$.

\begin{lemma}\cite[\S 38-39]{Hecke}, \cite{Daberkow}\label{disc}
Let $k$ be a local field containing all $p$-th roots of unity for some prime $p$ and let $v_k:k^\times \ra \Z$ denote the valuation map of $k$, normalized so that $v_k(k^{\times})=\Z$. Let $K/k$ be a Kummer extension of degree $p$ with discriminant $\zfrakd_{K/k}$. Then $K = k(\sqrt[p]{\vartheta})$ for some $\vartheta \in k$ such that $v_k(\vartheta) \in \{0, 1\}$. Moreover,
\begin{itemize}
\item[(i)] If $v_k(\vartheta)=1$, $v_k(\zfrakd_{K/k})=pv_k(p)+(p-1)$.
\item[(ii)] Assume $v_k(\vartheta)=0$. If $v_k(p)=0$, then $v_k(\zfrakd_{K/k}) =0$. Otherwise, write $\zfrakp_k$ for the unique maximal ideal in $k$. We have:
\begin{itemize}
\item[(a)]  If equation $x^p\equiv \vartheta (\zmod \zfrakp_k^{pv_k(p)/(p-1)})$ can be solved in $k$, then $v_k(\zfrakd_{K/k})=0$.
\item[(b)] If not, $v_k(\zfrakd_{K/k}) = pv_k(p)+(p-1)(1-\eta)$, where $\eta = \underset {\ell} \max\{0\le \ell < pv_k(p)/(p-1) \zspace \big | \zspace x^p \equiv \vartheta(\zmod \zfrakp_k^\ell) \text{ can be solved in } \zmathcalO_k \}$.
\end{itemize}
\end{itemize}
\end{lemma}

We use the above result in order to deduce several lemmas which shall allow us to reduce the proof of Proposition \ref{c} to the case in which $L/K$ is unramified at dyadic primes.

\begin{lemma} \label{c-decrease}Let $p\nmid \disc(K)$ be a prime and put $p^*=8$ if $p=2$, $p^* = p$ if $p \equiv 1(\zmod 4)$ and $p^*=-p$ if $p \equiv -1(\zmod 4)$. Then $K(\sqrt{p^*})$ is contained in the {\em ring class field} $H_c$ of $K$ associated to the order $\cO_c$ of conductor $c=|p^*|$.
\end{lemma}
\begin{proof}
Suppose first that $p$ is split in $K$ and fix a prime $\zfrakp | p$ in $K$. Let
\newform{
U = K_\zfrakp^\times \bigcap
K^{\times}\prod_v \cO_{{c},v}^{\times},
}
where the intersection is computed by regarding $K_\zfrakp^\times$ as a subgroup of  $\prod_v K_v^{\times}$ by means of the usual embedding
$x_\zfrakp\mapsto (1,...,1,x_\zfrakp,1,...,1)$.

Since the map $K_\zfrakp^\times/U \to \mathbb{I}_K/\left ( K^{\times}\prod_v \cO_{c,v}^{\times}  \right )$ is injective by \cite[p.\,173, Prop. 5.2]{Mi2}, it follows that
$U \subset K_\zfrakp^\times\simeq \zQ_p^\times$ corresponds to $H_{{c},\zfrakP}/K_\zfrakp$ by local class field theory for any prime $\zfrakP$ of $H_c$ above $\zfrakp$.


Write $c=p^r$ with $r=3$ if $p=2$, $r=1$ if $p$ is odd. Since $1+p^r\Z_p\subseteq U$, $1+p^{r-1}\Z_p\not\subset U$ by \cite[p. 197]{Cox}, an easy calculation shows that
\newform{
U = \{ \frac{\alpha}{\overline \alpha} \zspace \big | \zspace \alpha \in V \} \cdot (1+\zfrakp^r),
} where $V = \{ \alpha \in K^\times \zspace \big | \zspace  \zord_v(\alpha) =0 \ \forall v \ne \zfrakp\}$. Note that $V=\{\pm \alpha_0^{n},
n\in \Z\}$ for some  $\alpha_0\in K^{\times}$  such
that $\ord_v(\alpha_0)=0$ for all $v\ne \zfrakp$ and
$\ord_{\zfrakp}(\alpha_0)=n_0\geq 1$ is minimal. With this notation we have
\newindform{ \label{openGroup1}
U=\{ \bigl (\frac{\alpha_0}{\overline{\alpha}_0} \bigr )^n, n\in \Z \}
\cdot (1 + \zfrakp^r).
}

Suppose now that $p$ remains inert in $\cO_K$. Arguing similarly as before we obtain that the open subgroup $U \subset
K_{p}^\times$ corresponding to $
H_{p^r,\zfrakP}/K_{p}$ by local
class field theory is $U=K_{p}^\times \cap \left
(K^\times \prod_v \zmathcalO_{p^r, v}^\times \right )$, i.e. \newindform{\label{openGroup2} U =\{\alpha
\zspace \big | \zspace \alpha \in K^\times, \zspace
\zord_v(\alpha) =0, \zspace \forall v \ne p\} \cdot
(1+ p^r\cO_{K_p}) =\{(\pm \alpha_0)^n \zspace \big |\zspace n \in \zZ\} \cdot (1+p^r\cO_{K_p}), } where $\alpha_0 \in K^\times $ is chosen such that
$\zord_v(\alpha_0)=0$ for all $v \ne p$ and $\zord_p(\alpha_0)\ge 1$ is minimal. We can thus take $\alpha_0=p^{n_0}$ for some $n_0\geq 1$.

Put $K'=K(\sqrt{p^*})$. Any prime $\zfrakp$ in $K$ above $p$ ramifies in $K'$. Fix one such prime $\zfrakp$ and put $\zfrakp = \wp^2$ in $K'$ so that $K'_\wp=K_\zfrakp(\sqrt{p^*})$. By class field theory, in order to prove that $K' \subset H_c$ it is enough to show that $U \subset  \zNm_{K'_\zfrakp/K_\zfrakp}(K_\zfrakp^{'\times})$. Since $d(K'/K) =p^*$ by Lemma \ref{disc}, $K'$ is contained in the {\em ray class field} $K_c$ of conductor $c$ of $K$ and it thus suffices to verify that $\alpha_0 \overline{\alpha}_0$ (resp. $\pm \alpha_0$) lies in $\zNm_{K'_\zfrakp/K_\zfrakp}(K_\zfrakp^{'\times})$ if $p$ splits (resp. remains inert) in $K$.

Assume $p=2$. Then $\pm 1,\pm 2 \in \zNm_{K'_\wp/K_\zfrakp}(K_{\wp}^{'\times})$ because $-1=\zNm(1+\sqrt{2})$ and $-2=\zNm(\sqrt{2})$. The lemma thus follows automatically if $2$ is inert in $K$, while if $2$ splits, it follows because  $\alpha_0 \overline \alpha_0$ is a power of $2$, hence $\alpha_0 \overline \alpha_0$ lies in either $\pm \zQ_2^{\times^2}$ or $\pm 2\zQ_2^{\times^2}$.

Assume $p$ is odd. Then $-p^*=\zNm(\sqrt{p^*})\in \zNm_{K'_\zfrakp/K_\zfrakp}(K_\zfrakp^{'\times})$. Suppose first $p$ splits in $K$: as before, it is enough to show that
$p\in \zNm_{K'_\zfrakp/K_\zfrakp}(K^{'\times})$, which we already did if $p^*=-p$. That the same holds when $p^*=p$ follows because $p\equiv 1(\zmod 4)$ implies that $-1\in \zNm_{K'_\zfrakp/K_\zfrakp}(K_\zfrakp^{'\times})$.
Suppose now $p$ remains inert in $K$; we must show that $\pm p\in  \zNm_{K'_\zfrakp/K_\zfrakp}(K_\zfrakp^{'\times})$. If $p^*=p$ this follows by the same reason as above; if $p^*=-p$, then $p \equiv 3\, (\zmod 4)$, $K_p = \zQ_p(\sqrt{-1})$ and thus $-1 \in K_p^{\times^2}$, which allows us to conclude.
\end{proof}

Note that a direct consequence of the previous lemma is that  for any odd square free integer $m$ relatively coprime with $\disc(K)$ either $K(\sqrt{m})$ or $K(\sqrt{-m})$ is contained in $H_{m}$.

\begin{lemma} \label{c-firstpart}
$d(L/K)=2^tc_0\fN$ for some integer $0\le t \le 3$ and some positive integer $c_0\geq 1$ relatively coprime to $2$ and $\fN$. If further $2$ is ramified in $K$, $0 \le t \le 2$.
\end{lemma}
\begin{proof}


Write $K=\zQ(\sqrt{-d_0})$ for some square free integer $d_0>0$ and
$L=K(\sqrt{\beta})$ for some $\beta\in \zZ+\zZ\sqrt{-d_0}$ and square free in $K$. Without loss of generality, $N$ can be written as $\fN \overline \fN$ where $\fN$ divides the square free part $\zfrakB$ of $(\beta)$ in $K$ and $ \overline \fN$ is relatively coprime to $\zfrakB$.

Write $\zfrakB_2$ for the largest ideal which divides $\zfrakB$ and is relatively coprime to any prime of $K$ above $2$.  Since $v_{K_{\zfrakp'}}(2)=0$ and $v_{K_{\zfrakp'}}(\zfrakB_2)=1$ for any prime $\zfrakp'\mid \zfrakB_2$, Lemma \ref{disc} shows that $v_{K_{\zfrakp'}}(\zfrakd_{L_{\zfrakP'}/K_{\zfrakp'}})=1$, where $\zfrakP'$ is the prime in $L$ above $\zfrakp'$, thus the prime-to-$2$ part of $d(L/K)$ is $\zfrakB_2$. Besides, $\zfrakB_2 = \fN \cdot \zfrakC$ with $(\zfrakC, \fN)=1$.  Since ${\zNm_{K/\zQ}(\beta)}/{N}$ is a perfect square in $\zZ$, $\zfrakC$ is principal and can be written as $\zfrakC=(c_0)$ for some integer $c_0 >0$. Hence $\fN c_0 \mid d(L/K) \mid 2^t \fN c_0$ for some integer $t\ge 0$.

If $2$ is unramified in $K$ we have  $v_{K_\zfrakp}(2)=1$ for any prime $\zfrakp \mid 2$ in $K$ and it follows from Lemma \ref{disc} that $d(L/K)=\fN c_0 2^t$ with $0\le t \le 3$.

Suppose now that $2$ ramifies in $K$ with $(2)=\zfrakp^2$ in $K$. Then, since $v_{K_\zfrakp}(2)=2$ and $\zNm_{K/\zQ}(\beta)/N$ is a  perfect square in $\zZ$, we fall into case (ii) of Lemma \ref{disc}: for any prime $\zfrakP$ in $L$ above $\zfrakp$, $L_\zfrakP$ can be written as $K_\zfrakp(\sqrt \vartheta)$ for some $\vartheta \in K_\zfrakp^\times $ such that $v_{k_\zfrakp}(\vartheta)=0$. Suppose $v_{K_\zfrakp}(\zfrakd_{L_\zfrakP/K_\zfrakp}) \ne 0$. Then Lemma \ref{disc} (b) asserts that $$v_{K_\zfrakp}(\zfrakd_{L_\zfrakP/K_\zfrakp})=5-\eta,  $$ where $$\eta = \max \{0 \le \ell < 4 \zspace \big | \zspace \exists \iota \in \zmathcalO_{K_\zfrakp},\zspace \iota^2 \equiv \vartheta(\zmod \zfrakp^\ell)\}. $$ A classical result of Hilbert (cf.\,\cite{HSW}, \cite{Hilbert}) implies that $v_{K_\zfrakp}(\zfrakd_{L_\zfrakP/K_\zfrakp})$ is even. Hence $d(L/K)=\fN2^tc_0$ with $0\le t \le 2$.
% Similar argument and using Hensel's lemman, one can show for any odd prime $p$, at least one of $K(\sqrt{\pm p}) \subset H_{p}$. This further implies for any odd square free integer $m$, at least one of $K(\sqrt{\pm m})$ resides in $H_{m}$.
\end{proof}


\begin{lemma}\label{enough}
It is enough to prove Proposition $\ref{c}$ when $d(L/K)=2^t\fN$ and $0 \le t \le 2$.
\end{lemma}

\begin{proof}
Lemma \ref{c-firstpart} shows in general $d(L/K) =2^tc_0\fN$, where $0 \le t \le 3$ and $c_0 \ge 1$. Suppose first that $t=3$, then $2$ is unramified in $K$ by the same lemma. Let $\zfrakP$ and $\zfrakp$ be  prime ideals in $L$ and $K$ respectively such that $\zfrakP | \zfrakp | 2$. Then $L_\zfrakP$ can be written as $K_\zfrakp(\sqrt \vartheta)$ for some $\vartheta \in K$ with $v_{K_\zfrakp}(\vartheta) \in \{0, 1\}$.  Define $L'=K(\sqrt{\vartheta'})$, where $\vartheta'$ is defined as $$
\vartheta'= \begin{cases} \vartheta/2  & \textrm{ if } v_{K_\zfrakp}(\vartheta) =1; \\
                    \vartheta   &  \textrm{ if } v_{K_\zfrakp}(\vartheta) =0.
\end{cases}
$$
Hence $v_{K_\zfrakp}(\vartheta')=0$. Let $\zfrakP'$ be a prime in $L'$ above $\zfrakp$. Then either case (a) or (b) of Lemma \ref{disc} applies. In case (a), $v_{K_\zfrakp}(\zfrakd_{L'_{\zfrakP'}/K_\zfrakp}) =0$. In case (b),  $v_{K_\zfrakp}(\zfrakd_{L'_{\zfrakP'}/K_\zfrakp}) = 3-\eta$,
where $0 \le \eta \le 2$, hence $\zfrakp$ is ramified in $L'$, so residue field of $L'_{\zfrakP'}$ is equal to that of $K_\zfrakp$ and consequently $ \eta \ge 1$. We conclude that $v_{K_\zfrakp}(\zfrakd_{L'_{\zfrakP'}/K_\zfrakp}) \le 2$. By Lemma \ref{c-decrease}, $L\subset K(\sqrt{2}) L' \subset H_8 L'$ with $d(L'/K)= 2^{t'}c_0\fN$ for some integer $0 \le t' \le 2$.

Suppose now $c_0>1$. Setting $L''=K(\sqrt{\delta \vartheta'/c_0})$ we have $d(L''/K)=2^{t'} \fN$, where $\delta \in  \{\pm 1\}$ such that $K(\sqrt{\delta c_0}) \subset H_{c_0}$ as described in Lemma \ref{c-decrease}. By  the same lemma, $L' \subset K(\sqrt{\delta c_0})L'' \subset H_{c_0} L''$.

So $L \subset H_8 L' \subset H_8 H_{c_0} L'' =H_{8c_0}L''$ such that $L''/K$ is a quadratic extension and $d(L''/K)=2^{t'} \fN$ for some integer $0 \le t' \le 2$. This justifies we only need to prove proposition \ref{c} when $0 \le t \le 2$ and $d(L/K)=2^t\fN$.
\end{proof}

Thanks to Lemma \ref{enough} we can assume in what follows that $c_0=1$ and $0 \le t\leq 2$.

\begin{lemma}
\label{uniqueq}
There is a unique quadratic extension $\zmathcalL_{2^t}/K_{2^t}$ contained in $K_{2^t\fN}$ such that the set of primes in $K_{2^t}$ which ramify in $\zmathcalL_{2^t}$ is the set of primes above $\fN$. We have $L \subset \zmathcalL_{2^t}$.
\end{lemma}
\begin{proof}
Assume first $t=0$ or $1$. Then $\zGal(K_{2^t\fN}/K_{2^t}) \cong (\prod_{\zfrakp\mid \fN} (\zmathcalO_K/\zfrakp)^\times)/\{\pm 1\} \cong (\zZ/N\zZ)^\times/\{\pm 1\}$. This is obvious for $t=0$, and holds for $t=1$ because $K_{2}K_\fN = K_{2\fN}$. Extension $\zmathcalL_{2^t}/K_{2^t}$ corresponds by Galois theory to the unique primitive even quadratic Dirichlet character $\varepsilon$ of conductor $N$.

Suppose now $t=2$. Then \newform{
\zGal(K_{2^t\fN}/K_{2^t}) \cong G:=\bigl ( \{\pm 1\} \times (\zZ/N\zZ)^\times \bigr )/\{\pm \mathbbm{1}\},
}
where
%$\{\pm 1\}$ is a subgroup of $(\zmathcalO_K/2^t)^\times$ and
$\mathbbm{1}=(1, \mathbf 1)$ is the identity element of $\{\pm 1\} \times (\zZ/N\zZ)^\times$ . Again, any extension $\zmathcalL_{2^t}/K_{2^t}$ as in the statement corresponds to a non-trivial character $\varepsilon': G\ra \{\pm 1\}$ which is trivial on $\{\pm 1\}\times \{\mathbf 1\}$ and is even and primitive on $\{1\}\times (\zZ/N\zZ)^\times$. As above, the only such character is $\varepsilon'=1\times \varepsilon$.

Finally, note that $LK_{2^t}/K_{2^t}$ is a quadratic extension contained in $K_{2^t\fN}$. Since $\disc(LK_{2^t}/K_{2^t})=\mathcal N$ it follows that $\zmathcalL_{2^t}=LK_{2^t}$ and thus $L\subset \zmathcalL_{2^t}$.
\end{proof}

Recall the quadratic extension $L_c$ of the ring class field $H_c$ introduced
 in \S \ref{sec:explicit}, over which the  Heegner points
 $P^+_c$ and  $P_c^-\in \CM(c)$ are rational.
Lemma \ref{uniqueq}
 reduces the proof of Proposition \ref{c}
to showing that $L_c=\mathcal L_c$. Since $L_c$ was defined as the quadratic extension of $H_c$ cut out by the kernel of the single even primitive character $\varepsilon$ of conductor $N$, it suffices to show that $H_{2^t}=K_{2^t}$ for $0\leq t\leq 2$.

When $t=0$ and we obviously have $H_1 =K_1$. If $t=1$ or $2$, the ratio of the ray class number $h_{2^t}$
by the ring class number $h(\mathcal O_{2^t})$ is
(cf.\,\cite[p.154]{Mi2} for this and the remaining
notations):
\newindform{\label{rayisring}
\begin{split} \frac{h_{2^t}}{h(\mathcal O_{2^t})} &
= \frac{[U:U_{2^t, 1}]^{-1} \text{Nm}(2^t)
\prod_{\zfrakp \mid
2^t}(1-\frac{1}{\zNm(\zfrakp)})}{\frac{2^t}{[\mathcal
O_K^\times: \mathcal O_{2^t}^\times]}\prod_{p\mid
2^t}(1-\bigl (\frac{d_K}{p} \bigr )\frac{1}{p})} \\ & =
\begin{cases}
\frac{[\mathcal O_K^\times: \mathcal
O_{2^t}^\times]}{[U: U_{2^t , 1}]} \cdot
\frac{2^{2t}\bigl (1-\frac{1}{4} \bigr )}{2^t \bigl (1-\bigl (\frac{d_K}{2}\bigr ) \frac{1}{2}\bigr) }
 & \text{ if $2$  is inert
in  $K$},
\\
\frac{[\mathcal O_K^\times: \mathcal
O_{2^t}^\times]}{[U: U_{2^t , 1}]} \cdot
\frac{2^{2t} \bigl (1-\frac{1}{2} \bigr )^2} {2^t
\bigl (1-\bigl (\frac{d_K}{2}\bigr )\frac{1}{2} \bigr ) }
&  \text{if $2$  is split in  $K$}, \\
\frac{[\mathcal O_K^\times: \mathcal
O_{2^t}^\times]}{[U: U_{2^t , 1}]} \cdot
\frac{2^{2t}\bigl (1-\frac{1}{2} \bigr )}{2^t }  & \text{ if
$2$ is ramified in } K.
\end{cases} \\ & = \frac{[\mathcal O_K^\times: \mathcal
O_{2^t}^\times]}{[U:U_{2^t, 1}]} \cdot 2^{t-1}.
\end{split}
}
Since $K \ne \zQ(\sqrt{-1})$ and $\zQ(\sqrt{-3})$, $[\mathcal O_K^\times: \mathcal
O_{2^t}^\times]=1$.   If $t=1$, then $U=U_{2^t ,1}$, so $K_2=H_2$. If $t=2$, then
$[U:U_{2^t, 1}]=2$, and therefore $K_{2^2} =H_{2^2}$.

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\end{document}
