Some remarks on Heegner point computations
Mark Watkins
We give an overview of the theory of Heegner points for elliptic curves, and then describe various new ideas that can be used in the computation of rational points on rank 1 elliptic curves. In particular, we discuss the idea of Cremona (following Silverman) regarding recovery a rational point via knowledge of its height, the idea of Delaunay regarding the use of Atkin-Lehner involutions in the selection of auxiliary parameters, and the idea of Elkies regarding descent and lattice reduction that can result in a large reduction in the needed amount of real-number precision used in the computation.
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\begin{document}
\title{Some remarks on Heegner point computations}
\alttitle{Une partie remarque sur des calculs de point de Heegner}
\author{Mark Watkins}
\address
{School of Mathematics, University Walk,
University of Bristol, Bristol BS8 1TW, ENGLAND}
\email{watkins@maths.usyd.edu.au}
\begin{abstract}
We give an overview of the theory of Heegner points for elliptic curves,
and then describe various new ideas that can be used in the computation
of rational points on rank 1 elliptic curves.
In particular, we discuss the idea of Cremona (following Silverman)
regarding recovery a rational point via knowledge of its height, the idea of
Delaunay regarding the use of Atkin-Lehner involutions in the selection of
auxiliary parameters, and the idea of Elkies regarding descent and lattice
reduction that can result in a large reduction in the needed amount of
real-number precision used in the computation.
\end{abstract}
\kill{
\begin{altabstract}
Nous donnons une vue d'ensemble de la th\'eorie de points de Heegner
pour les courbes elliptiques, et puis d\'ecrivons les diverses
nouvelles id\'ees qui peuvent \^etre empoly\`ees
dans le calcul des points raisonnables sur les
courbes elliptiques du grade 1.
En particulier, nous discutons l'id\'ee de Cremona
(apr\`es Silverman) concernant le r\'etablissement par point
raisonnable par l'interm\'ediaire de la connaissance de sa taille,
l'id\'ee de Delaunay concernant l'utilisation des involutions d'Atkin-Lehner
dans le choix des param\`etres auxiliaires, et l'id\'ee 'Elkies
concernant la descente et la r\`eduction de trellis qui peuvent avoir comme
cons\`equence une grande r\'eduction de la quantit\'e n\'ecessaire
de pr\'ecision de vrai-nombre utilis\'ee dans le calcul.
\end{altabstract}}
\subjclass{11G05, 11G40, 14G05}
\keywords{elliptic curves, Heegner points, descent, lattice reduction}
\altkeywords{elliptic curves, Heegner points, descent, lattice reduction}
\maketitle

\section{Introduction}
We make some remarks concerning Heegner point computations.
One of our goals shall be to give an algorithm (perhaps conditional
on various conjectures) to find a non-torsion rational point
on a given rank 1 elliptic curve.
Much of this is taken from a section in Henri Cohen's
latest book~\cite{Cohen}, and owes a great debt to Christophe Delaunay.
The ideas in the section about lattice reduction
are largely due to Noam Elkies.
We do not delve deeply into the theory of Heegner points, but simply
give references where appropriate; the recent MSRI book
``Heegner points and Rankin $L$-series'' \cite{MSRI} contains many
good articles which consider Heegner points and their generalisations
from the standpoint of representation theory.

The author thanks the Institut Henri Poincar\'e for its hospitality
and the Centre National de la Recherche Scientifique for financial support.
The algorithms described here have been implemented in the Magma computer
algebra system \cite{magma};
the author thanks the Magma computer algebra group at
the University of Sydney for their hospitality and financial support.
The author was also partially funded by an NSF VIGRE Postdoctoral
Fellowship during part of this work.

\section{Definitions and Outline of Theory}
Let $\tau$ be a quadratic surd in the upper-half-plane~$\HH$.
Let $f_\tau=(A,B,C)$ be the associated integral primitive
positive-definite binary quadratic form,
so that \hbox{$A\tau^2+B\tau+C=0$} with $A>0$
and~$\gcd(A,B,C)=1$. The discriminant $\Delta(\tau)$
is $\Delta(f_\tau)=B^2-4AC$, which is negative.
For simplicity we take $\Delta(\tau)$ to be fundamental,
though much of the theory can be made to work when it is not.
However, consideration of positive discriminant does not follow
in the same manner.
\begin{definition}
A Heegner point of level~$N$ and discriminant~$D$ is a quadratic surd
in the upper-half-plane with $\Delta(\tau)=D=\Delta(N\tau)$.
We let $\H_N^D$ be the set of Heegner points of level~$N$ and
discriminant~$D$.
\end{definition}
\begin{proposition}
Let $\tau\in\HH$ be a quadratic surd with discriminant~$D$
and~$f_\tau=(A,B,C)$. Then $\tau\in\H_N^D$ iff $N|A$ and $\gcd(A/N,B,CN)=1$.
\end{proposition}
\begin{proof}
Note that $\tau={-B+\sqrt D\over 2A}$ and $N\tau={-NB+N\sqrt D\over 2A}$.
For $\Delta(\tau)=\Delta(N\tau)$ we need $N\tau={-B'+\sqrt D\over 2A'}$
and by equating imaginary and real parts we get $A=NA'$ and~$B=B'$,
so that $N|A$. Also note that $(A/N)(N\tau)^2+B(N\tau)+(CN)=0$,
from which we get the rest of the lemma.
\end{proof}
Note that $\H_N^D$ will be empty unless $D=B^2-4N(A/N)C$
is a square modulo~$4N$.
\begin{lemma}
The set $\H_N^D$ is closed under $\Gamma_0(N)$-action.
\end{lemma}
\begin{proof}
If $\gamma\in SL_2(\Z)\supseteq \Gamma_0(N)$ then
$\Delta\bigl(\gamma(\tau)\bigr)=\Delta(\tau)$ since
the discriminant is fixed. A computation shows that
$\gamma\in\Gamma_0(N)$ and $\tau\in\H_N^D$ imply~$\gamma(\tau)\in\H_N^D$.
\end{proof}
\begin{lemma}
The set $\H_N^D$ is closed under the $W_N$-action that
sends $\tau\rightarrow -1/N\tau$.
\end{lemma}
\begin{proof}
Follows from the above proposition since $f_{(-1/N\tau)}=(CN,-B,A/N)$.
\end{proof}
\begin{definition}
We let $\SS(D,N)$ be the set of square roots mod~$2N$ of $D$ mod~$4N$.
\end{definition}
\begin{theorem}
The sets $\H_N^D/\Gamma_0(N)$ and
$\SS(D,N)\times \Cl\bigl(\Q(\sqrt D)\bigr)$ are in bijection.
\end{theorem}
\begin{proof}
This can be shown by chasing definitions.
Essentially, $[\tau]\in\H_N^D/\Gamma_0(N)$ gets mapped to
$(B$ mod $2N)\times [\Z+\tau\Z]$ where $f_\tau=(A,B,C)$,
and in the other direction,
when we are given $\beta\times l\in \SS(D,N)\times \Cl\bigl(\Q(\sqrt D)\bigr)$
we take $(A,B,C)\in l$ with $N|A$ and $B\equiv\beta$ (mod~$2N$),
and then~$\tau={-B+\sqrt D\over 2A}$.
\end{proof}
From now on we let $E$ be a global minimal model of a rational elliptic curve
of conductor~$N$, and take $D$ to be a negative fundamental discriminant such
that $D$ is a square mod $4N$.
We let $\HH^\star$ be the union of $\HH$ with the rationals and~$i\infty$.
We let $\WP(z)$ be the function that sends $z\in\C/\Lambda$ to the point
$\bigl(\wp(z),\wp^\prime(z)\bigr)$ on~$E$.
\begin{theorem}
There is a surjective map \hbox{$\hat\phi:X_0(N)\rightarrow E$}
(the modular parametrisation)
where $X_0(N)=\HH^\star/\Gamma_0(N)$ and $E$ can be viewed
as $\C/\Lambda$ for some lattice~$\Lambda$.
This map can be defined over the rationals.
\end{theorem}
\begin{proof}
This is due to Wiles and others \cite{wiles,TW,diamond,CDT,BCDT}.
We let $\phi$ be the associated map from $\HH^\star/\Gamma_0(N)$
to~$\C/\Lambda$.
Explicitly, we have that $\tau\in\HH^\star$ gets mapped to the complex point
$\phi(\tau)=2\pi i\int_{i\infty}^\tau \psi_E=\sum_n (a_n/n)e^{2\pi i n\tau}$,
where $\psi_E$ is the modular form of weight 2 and level~$N$ associated to~$E$.
The lattice~$\Lambda$ is generated by the
real and imaginary periods,\footnote{
 Our convention is that the imaginary period is purely imaginary
 when the discriminant of~$E$ is positive, and in the negative discriminant
 case the real part of the imaginary period is~$\Omega_{\rm re}/2$.
 The fundamental volume~$\Omega_{\rm vol}$
 is the area of the period parallelogram.}
which we denote by $\Omega_{\rm re}$ and~$\Omega_{\rm im}$.
We assume that the Manin constant is~1,
which is conjectured always to be the case for curves of positive rank
(see~\cite{Stevens} and~\cite{SteinWatkins}).
\end{proof}
\begin{theorem}\label{thm:cm}
Let $\tau=\beta\times l\in\H_N^D$.
Then $\WP\bigl(\phi(\tau)\bigr)$ has its coordinates
in the Hilbert class field of~$\Q(\sqrt D)$.
Also we have
\begin{enumerate}
\item $\BAR{\phi(\beta\times l)}=\phi\bigl(-\beta\times l^{-1}\bigr)$,
in~$\C/\Lambda$.
\item $\phi\bigl(W_N(\beta\times l)\bigr)=
\phi\bigl(-\beta\times ln^{-1}\bigr)$ in $\C/\Lambda$ where
$n=\bigl[ N\Z+{\beta+\sqrt D\over 2}\Z\bigr]$,
\item
$\WP\bigl(\phi(\beta\times l)\bigr)^{{\rm Artin}(m)}
=\WP\bigl(\phi\bigl(\beta\times lm^{-1}\bigr)\bigr)$
for all $m\in\Cl\bigl(\Q(\sqrt D)\bigr)$.
\end{enumerate}
\end{theorem}
\begin{proof}
This is the theorem of complex multiplication
of Shimura~\cite{ShimuraTaniyama,Shimura}.
We outline the proof of the first statement,
for which we work via the modular $j$-function.
We have that $j(\tau)$ is in the Hilbert class field $H$
(see \cite[II, 4.3]{silverman2}) and similarly with~$j(N\tau)$.
Thus we get that $X_0(N)$ over $H$ contains the moduli point
corresponding to the isogeny between curves with these $j$-invariants.
Since the modular parametrisation map $\hat\phi$ can be defined over the
rationals, the image of the moduli point under $\hat\phi$ has its
coordinates in the Hilbert class field.
\end{proof}
Note that $\WP\bigl(\overline{\phi(\tau)}\bigr)=
\overline{\WP\bigl(\phi(\tau)\bigr)}$, so that there
is no danger of confusing complex conjugation in~$\C/\Lambda$
with complex conjugation of the coordinates of the point on~$E$.
Using the third fact of Theorem~\ref{thm:cm}, we can take the trace
of~$\WP\bigl(\phi(\tau)\bigr)$
and get a point that has coordinates in~$\Q(\sqrt D)$.
Indeed, writing $H$ for the Hilbert class field and $K$ for
the imaginary quadratic field $\Q(\sqrt D)$ we get that
\begin{align*}
P={\rm Trace}_{H/K}\bigl(\WP\bigl(\phi(\tau)\bigr)\bigr)
&=\sum_{\sigma\in{\rm Gal}(H/K)} \WP\bigl(\phi(\tau)\bigr)^\sigma=
\sum_{m\in\Cl(K)} \WP\bigl(\phi(\beta\times l)\bigr)^{{\rm Artin}(m)}\\
&\qquad=\sum_{m\in\Cl(K)} \WP\bigl(\phi\bigl(\beta\times lm^{-1}\bigr)\bigr)=
\sum_{m\in\Cl(K)} \WP\bigl(\phi(\beta\times m)\bigr)
\end{align*}
has coordinates in~$\Q(\sqrt D)$. When $E$ has odd functional equation,
we can use the first two facts of Theorem~\ref{thm:cm} to show that~$P=\BAR P$,
so that $P$ has coordinates in~$\Q$. In this case we have
$\psi_E=\psi_E \circ W_N$ which implies~$\phi=\phi\circ W_N$,
so that in $\C/\Lambda$ we have
$$\BAR{\phi(\beta\times m)}=\BAR{\phi\bigl(W_N(\beta\times m)\bigr)}
=\BAR{\phi\bigl(-\beta\times mn^{-1}\bigr)}=
\phi\bigl(\beta\times m^{-1}n\bigr),$$
which gives us that
$$\BAR P=\sum_{m\in\Cl(K)} \WP\bigl(\BAR{\phi(\beta\times m)}\bigr)
=\sum_{m\in\Cl(K)} \WP\bigl(\phi\bigl(\beta\times m^{-1}n\bigr)\bigr)
=\sum_{m\in\Cl(K)} \WP\bigl(\phi(\beta\times m)\bigr)=P.$$
We can rewrite some of this by introducing some new notation.
\begin{definition}
We write $\H_N^D(\beta)$ for subset of $\tau\in\H_N^D$ such that the
associated form $f_\tau=(A,B,C)$ has $B\equiv\beta$ (mod~$2N$).
We write $\hat\H_N^D=\H_N^D/\Gamma_0(N)$, and noting that $\Gamma_0(N)$ acts
on $\H_N^D(\beta)$, we write $\hat\H_N^D(\beta)=\H_N^D(\beta)/\Gamma_0(N)$.
\end{definition}
Since $\hat\H_N^D(\beta)$ is in 1-1 correspondence
with $\Cl\bigl(\Q(\sqrt D)\bigr)$, we get that
$$P=\sum_{m\in\Cl(Q(\sqrt D))} \WP\bigl(\phi(\beta\times m)\bigr)=
\sum_{\tau\in\hat\H_N^D(\beta)} \WP\bigl(\phi(\tau)\bigr).$$

\section{The Gross-Zagier theorem and an algorithm}
We now have a plan of how to find a non-torsion point
on a curve of analytic rank~1.
We select an auxiliary negative fundamental discriminant~$D$ such that
$D$ is a square modulo~$4N$, choose~$\beta\in \SS(D,N)$,
find $\tau$-representatives for~$\hat\H_N^D(\beta)$,
compute $\phi(\tau)$ for each, sum these in $\C/\Lambda$,
map the resulting point to $E$ via the Weierstrass parametrization,
and try to recognize the result as a rational point. One problem
is that we might get a torsion point. Another problem is that we won't
necessarily get a generator, and thus the point might have inflated height,
which would increase our requirements on real-number precision.
The Gross-Zagier Theorem tells us what height to expect, and combined with
the Birch--Swinnerton-Dyer Conjecture, we get a prediction of what height
a generator should have. Our heights will be the ``larger'' ones, and are
thus twice those chosen by some authors.
\begin{theorem}
Suppose $D<-3$ is a fundamental discriminant
with $D$ a square modulo~$4N$ and $\gcd(D,2N)=1$. Then
$$h(P)={\sqrt{|D|}\over 4\Omega_{\rm vol}}L'(E,1)L(E_D,1)\times
2^{\omega(\gcd(D,N))}\biggl({w(D)\over 2}\biggr)^2.$$
\end{theorem}
\begin{proof}
This is due to Gross and Zagier~\cite{GrossZagier}.
Here $E_D$ is the quadratic twist of~$E$ by~$D$,
while $w(D)$ is the number of units in $\Q(\sqrt D)$
and $\omega(n)$ is the number of distinct prime factors of~$n$.
\end{proof}
Calculations of Gross and Hayashi~\cite{Hayashi} indicate that
this height formula is likely to be true for all negative fundamental
discriminants~$D$ that are square mod~$4N$.

We now write $P=lG+T$ where $G$ is a generator\footnote
{Note that we will actually get $\sqrt{\#\Sha}$ times a generator
 from our method, since we cannot disassociate $\Sha$ from the
 regulator in the Birch--Swinnerton-Dyer formula.}
and $T$ is a torsion point,
so that $h(P)=l^2h(G)$. Then we replace $L'(E,1)$ through use of the
Birch--Swinnerton-Dyer conjecture~\cite{BSD} to get the following:\footnote{
 We use the convention that the Tamagawa number at infinity is equal to the
 number of connected components of~$E$ over~$\R$ --- thus it is 1 for curves
 with negative discriminant and 2 for curves with positive discriminant.}
\begin{conjecture}
With notations as above we have that
$$l^2={\Omega_{\rm re}\over 4\Omega_{\rm vol}}
\biggl({\prod_{p|N\infty} c_p\cdot\#\Sha}\biggr)
{\sqrt{|D|}\over \#E(\Q)_{\rm tors}^2}L(E_D,1)\cdot
\biggl({w(D)\over 2}\biggr)^2 2^{\omega(\gcd(D,N))}.$$
\end{conjecture}
In particular, we note that we should use a quadratic twist~$E_D$
that has rank zero, so that $L(E_D,1)$ does not vanish.
The existence of such a twist is proven in~\cite{BumpFriedbergHoffstein}.
Thus we have the following algorithm, which we shall work on improving.
\begin{algorithm}
Given a rational elliptic curve~$E$ of conductor~$N$ of analytic rank~1,
find a non-torsion rational point.
\begin{enumerate}
\item Compute $L'(E,1)$ and find a fundamental discriminant $D<0$ with $D$
a square modulo~$4N$ and $L(E_D,1)\neq 0$, so that the index~$l$ is nonzero.
\item Choose $\beta\in \SS(D,N)$ and compute (to sufficient precision)
the complex number
$$z=\sum_{\tau\in\hat\H_N^D(\beta)}\phi(\tau)=
\sum_{\tau\in\hat\H_N^D}\sum_{n=1}^\infty {a_n\over n}e^{2\pi i n\tau}.$$
\item Let $m$ be the gcd of~$l$ and the exponent of the torsion group of~$E$.
If the discriminant of $E$ is positive, check if $\WP(\dot z)$
is close to a rational point on~$E$ for $u=1,\dots,lm$ for both
$$\dot z=(m\Re(z)+u\Omega_{\rm re})/ml\quad{\rm and}\quad
\dot z=(m\Re(z)+u\Omega_{\rm re})/ml+\Omega_{\rm im}/2.$$
\vskip0pt\noindent
If the discriminant of $E$ is negative,
let $o=\Im(z)/\Im(\Omega_{\rm im})$ and check $\WP(\dot z)$ for
$\dot z=(m\Re(z)+u\Omega_{\rm re})/ml+o\Omega_{\rm re}/2$
over the same \hbox{$u$-range}.
\end{enumerate}
\end{algorithm}
One can compute the index~$l$ in parallel with the~$\phi(\tau)$,
since both involve computing the $a_n$ of the elliptic curve~$E$.
However, this can cause problems if the index turns out to be zero
(that is, if $E_D$ has positive rank).

\subsection{Step 2 of the Algorithm}
We now discuss how to do the second step efficiently.
First note that we can sometimes pair $\phi(\tau)$ with its complex conjugate;
recalling that $\phi=\phi\circ W_N$,
by Theorem~\ref{thm:cm} in $\C/\Lambda$ we have
\begin{equation}\label{eq:chap3}
\overline{\phi(\beta\times l)}=\phi(-\beta\times l^{-1})=
\phi\bigl(W_N(-\beta\times l^{-1})\bigr)=
\phi\bigl(\beta\times(ln)^{-1}\bigr).
\end{equation}
For $f=(A,B,C)\in\hat\H_N^D$ we write~$\BAR f=(A/N,-B,CN)$,
so that when~$g\sim\BAR f$ in the class group we have $\phi(f)=\BAR{\phi(g)}$
and thus $\phi(f)+\phi(g)=2\,\Re\,\phi(f)$ in~$\C/\Lambda$.
We refer to this as {\bf pairing} the forms.

For $\sum_n (a_n/n)e^{2\pi i n\tau}$ to converge rapidly, we wish for
the imaginary parts of our representative~$\tau$'s to be large.
It turns out the best we can do is essentially have the smallest
imaginary part be about $1/N$ in size. We can achieve this via
a trick of Delaunay, which introduces more Atkin-Lehner involutions.
\begin{definition}
Let $Q|N$ with $\gcd(Q,N/Q)=1$, and let $u,v\in\Z$ be such that $uQ^2-vN=Q$.
The Atkin-Lehner involution $W_Q$ sends $\tau$ to ${uQ\tau+v\over N\tau+Q}$.
\end{definition}
This defines $W_Q$ up to transformations by elements in~$\Gamma_0(N)$.
One can check that $W_Q(W_Q(\tau))$ is in the $\Gamma_0(N)$-orbit of~$\tau$,
and that the $W_Q$ form an elementary abelian 2-group~$W$
of order~$2^{\omega(N)}$.
The important fact about the $W_Q$'s that we shall use is that
\hbox{$\psi_E=\pm\psi_E\circ W_Q$}, so that
$\phi(\tau)=\pm\phi(W_Q(\tau))+\phi\bigl(W_Q(i\infty)\bigr)$.
The sign can be computed as
$\epsilon_Q=\prod_{p|Q} \epsilon_p$ where $\epsilon_p$
is the local root number of~$E$ at~$p$. Delaunay's idea is to maximise
the imaginary part of~$\tau$ over $\Gamma_0(N)$ and $W$ rather than
just $\Gamma_0(N)$; the difficulty is that the action of $W_Q$
need not preserve~$\beta$. However, we still have that
$$P=\sum_{\tau\in\hat\H_N^D(\beta)} \phi(\tau)=
\sum_{\tau\in\hat\H_N^D(\beta)} \epsilon_Q\phi\bigl(W_Q(\tau)\bigr)
+({\rm torsion\>point}).$$
For the analogue of the second part of Theorem~\ref{thm:cm},
we need to consider what happens with to~$\beta$.
We define $\beta_Q$ as follows.
We make $\beta_Q$ and $\beta$ have opposite signs mod~$p^k$ for prime
powers~$p^k$ with $p^k\|Q$ and~$\gcd(p,D)=1$, and else $\beta=\beta_Q$.
In particular, we have that~$Q=\gcd(\beta-\beta_Q,N)$ when~$N$ is odd.
The desired analogue is now that
$$\phi\bigl(W_Q(\beta\times l)\bigr)=
\epsilon_Q\phi(\beta_Q\times lq^{-1})+
\phi\bigl(W_Q(i\infty)\bigr)\quad{\rm with}
\quad q=\biggl[Q\Z+{-\beta_Q+\sqrt D\over 2}\Z\biggr].$$

The primes $p$ which divide $D$ are different since there is
only one square root of~$D$ mod~$p$; thus $\beta$ is preserved upon
application of~$W_Q$ for $Q$ that are products of such primes.
For such~$Q$, we can note the following with respect to complex conjugation.
Suppose we have that $m\sim (ln)^{-1}$ so that by \eqref{eq:chap3} we have
$\BAR{\phi(\beta\times l)}=\phi(\beta\times m)$.
Then, using the fact that $q^{-1}=q$ in this case,
in $\C/\Lambda$ we have, up to torsion, that
\begin{align*}
\BAR{\phi\bigl(W_Q(\beta\times l)\bigr)}&\circeq
\epsilon_Q\BAR{\phi(\beta_Q\times lq^{-1})}=
\epsilon_Q\phi\bigl(\beta_Q\times (lq^{-1}n)^{-1}\bigr)=
\epsilon_Q\phi\bigl(\beta_Q\times (lqn)^{-1}\bigr)\circeq\\
&\circeq\phi\bigl(W_Q(\beta\times (ln)^{-1})\bigr)=
\phi\bigl(W_Q(\beta\times m)\bigr).
\end{align*}
So we see that $(\beta\times l)$ can be paired iff $W_Q(\beta\times l)$
can be paired.

We now give the algorithm for finding good~\hbox{$\tau$-representatives.}
The idea to run over all forms $(aN,b,c)$ of discriminant~$D$ with
$a$ small, mapping these via the appropriate Atkin-Lehner involution(s)
to forms with fixed square root~$\beta$, and doing this until the images
cover the class group. Of course, the conjugation action is also considered.
\begin{subalgorithm}
Given $D,N$, find good $\tau$-representatives.
\begin{enumerate}
\item Choose $\beta\in \SS(D,N)$. Set $U=\emptyset$ and $R=\emptyset$.
\item While $\#R\neq\#\Cl(\bigl(Q\sqrt D)\bigr)$ do:
\item \quad Loop over $a$ from 1 to infinity and~$b\in \SS(D,N)$
[lift $b$ from $\Z/2N$ to~$\Z$]:
\item \quad\quad Loop over all solutions~$s$
of $Ns^2+bs+(b^2-D)/4N\equiv 0$ modulo~$a$:
\item \quad\quad\quad Let $f=\bigl(aN,b+2Ns,((b+2Ns)^2-D)/4aN\bigr)$.
\item \quad\quad\quad Loop over all positive divisors~$d$ of~$\gcd(D,N)$
[which is squarefree]:
\item \quad\quad\quad\quad Let $g=W_Q(f)/Q$ where~$Q$
is $d$ times the product of the $p^k\|N$
\break\hphantom0\quad\quad\quad\quad
with \hbox{$b\not\equiv\beta$ mod~$p^k$,}
so that $g\in \H_N^D(\beta)$.
\item \quad\quad\quad\quad
If the reductions of $g$ and $\BAR g$ are both not in~$R$ then
append them\break\hphantom0\quad\quad\quad\quad
to~$R$, and append $f$ to $U$ with weight $\epsilon_Q$
when $g\sim\BAR g$ and with\break\hphantom0\quad\quad\quad\quad
weight $2\epsilon_Q$ when~$g\not\sim\BAR g$.
\end{enumerate}
\end{subalgorithm}
With this subalgorithm,
we get that~$z=\sum_{f\in U} {\rm weight}(f)\phi(\tau_f)$
in Step 2 of the main algorithm.
We expect the maximal~$a$ to be of size $\#\Cl(\Q(\sqrt D))/2\#W$.
This subalgorithm makes
``parameter selection'' fast compared to the computation of the~$\phi(\tau)$.
\subsection{Step 3 of the Algorithm}
We now turn to the last step of our main algorithm,
reconstructing a rational point
on an elliptic curve from a real approximation. The most na\"{\i}ve method
for this is simply to try to recognise the $x$-coordinate as a rational
number. If our height calculation tells us to expect a point whose
\hbox{$x$-coordinate} has a numerator and denominator of about $H$ digits,
the use of continued fractions will recognise it if we do all computations
to about twice the precision, or $2H$~digits. We can note that by using
a \hbox{degree-$n$} map to $\P^{n-1}$ and \hbox{$n$-dimensional} lattice
reduction, this can be reduced to $nH/(n-1)$ digits
for every~$n\ge 3$ --- we will discuss a similar idea later when
we consider combining descent with our Heegner point computations.
But in this case we can do better;
we are able to recognise our rational point with only $H$ digits of
precision due to a trick of Cremona, coming from an idea in a paper
of Silverman~\cite{Silverman}.
The idea is that we know the canonical height of our desired point,
and this height decomposes into local heights; we have
$$h(P)=h_{\infty}(P)+\sum_{p|N} h_p(P)+\log{\rm denominator}(x(P)).$$
The height at infinity $h_{\infty}(P)$ can be approximated from a
real-number approximation to~$P$, and there are finitely many possibilities
for each local height $h_p(P)$ depending the reduction type of~$E$ at~$p$.
We compute the various local heights to $H$ digits of precision, and then
can determine the denominator of $x(P)$ from this, our task being eased
from the fact that it is square. Then from our real-approximation\footnote{
 Elkies tells us that, given the height to precision~$H$,
 the techniques of \cite{elkies-ants4} (see Theorem~4 in particular)
 can reduce the needed precision of the real Heegner approximation
 to $o(H)$ as~$H\rightarrow\infty$.
 The idea is that for a fixed~$C$ the equation
 \hbox{$h_{\infty}\bigl((x,y,z)\bigr)+2\log z=C$}
 defines a {\it transcendental} arc, and thus the use of a sufficiently high
 degree Veronese embedding will reduce the needed precision substantially.
 This method in its entirety might not be that practical, though the
 use of height information in conjunction with the geometry of the curve
 should allow a useful reduction in precision.}
of~$P$ we can recover the \hbox{$x$-coordinate}, and from this
we get~$P$. Note that we need to compute $L'(E,1)$ to a precision
of $H$ digits, but this takes only about $\sqrt N(\log H)^{O(1)}$ time.
In practise, there can be many choices for the sums of local heights,
and if additionally the index is large, then this step can be quite
time-consuming. This can be curtailed a bit by doing the calculations for
the square root of the denominator of the $x$-coordinate to only
about $H/2$~digits, and then not bothering with the elliptic exponential
step unless the result is sufficiently close to an integer.

\subsection{Example}
We now give a complete example. Other explicit descriptions of computations
with Heegner points appear in \cite{BirchStephens,Stephens,Liverance}.
We take the curve given by $[1,-1,0,-751055859,-7922219731979]$
for which the Heegner point has height $139.1747+$. We select $D=-932$,
for which the class number is~12 and the index $l$ is~4.
We have $N=11682$ and choose~$\beta=214$.
Our first form is $(11682,214,1)$ to which we apply~$W_1={\rm id}$.
The reduction of this is $(1,0,233)$, and it pairs with itself
under complex conjugation. Since we have $\gcd(D,N)=2$, we can use~$W_2$
without changing~$\beta$; we get the form
$(206717861394,70769770,6057)$ which reduces
to the self-paired form~$(2,2,117)$.
Our next form is $(11682,2338,117)$ to which we apply~$W_{11}$
to get $(122225810454,230158978,108351)$ which reduces to
$(11,6,22)$ and pairs with~$(11,-6,22)$. Applying~$W_{22}$ gives
a form which reduces to~$(11,-6,22)$, so we ignore it.
Next we have $(11682,2810,169)$ to which we apply~$W_9$,
getting a form that reduces to $(9,2,26)$ and pairs with~$(9,-2,26)$.
Applying~$W_{18}$ gives a form that reduces to $(13,-2,18)$
and pairs with~$(13,2,18)$.
Then we have $(11682,4934,521)$ to which we apply~$W_{99}$,
getting a form that reduces to $(3,-2,78)$ and pairs with~$(3,2,78)$.
And finally applying $W_{198}$ we get a form
that reduces to $(6,-2,39)$ and pairs with~$(6,2,39)$,
and so we have all of our~\hbox{$\tau$-representatives}.
We note that $W_{11}$, $W_9$, and $W_{18}$ switch the sign of the
modular form, and thus the obtained forms get a weighting of~$-2$.
The self-paired forms get a weighting of~$+1$, and the other two forms
get a weighting of~$+2$. For the non-self-paired forms we must remember
to take the real part of the computed~$\phi(\tau)$
when we double it.\footnote{
 The self-paired forms~$f$ have $\phi(f)=\BAR{\phi(f)}$ in $\C/\Lambda$
 but not necessarily in~$\C$ --- the imaginary part cannot be ignored
 when the discriminant of~$E$ is negative and $lm$ is odd.}
The pairing turns is rather simple in this example,
but need not be so perspicacious with respect to the class group.
Note that we use only four distinct forms for our computations.

We need about 60 digits of precision if we use the Cremona-Silverman
method to reconstruct the rational point, which means we must compute
about 20000 terms of the~\hbox{$L$-series}. The curve $E$ has negative
discriminant and no rational torsion points. We compute a real-approximation
to the Heegner point in $\C/\Lambda$ to be
$$z=0.00680702983101357730368201485198918786991251635619740952608094.$$
We have $o=\Im(z)/\Im(\Omega_{\rm im})=0$, and with $l=4$ and $u=2$ we get that
$$\dot z=0.00891152819280235244790996808333469812474933020620405901507952,$$
to which we apply the Cremona-Silverman method of recovery.
The curve~$E$ is annoying for this method,
in that we have many possibilities for~$h_p(P)$.
The height of the Heegner point is given by
$$h(P)=139.174739524758127811521877478222781093487974225206369462318,$$
and the height at infinity\footnote{
 Note that Silverman \cite{Silverman}
 uses a different normalisation of height,
 and his choice of the parameter~$z$ when he computes the height at infinity
 corresponds to $\dot z/\Omega_{\rm re}$ for us.
 Also, his method is only linearly convergent,
 while that given in \cite[\S 7.5.7]{cohen0} is quadratically convergent.}
is given by
$$h_\infty(P)=2.10306651755149369196435189022120441716979687181328497567075.$$
The reduction type at~2 is~$I_{25}$, at~3 it is $I_{13}^\star$,
at~11 it is~$I_1$, and at~59 it is~$I_3$.
Thus we have $13\times 3\times 1\times 2$ choices for the local heights. 
It turns out that we have\footnote{
 This follows {\it a posterori} since $P$ is nonsingular
 modulo these primes of multiplicative reduction.}
$h_p(P)={1\over 6}v_p(\Delta)\log p$ for $p=2,11,59$,
while~$h_3(P)=(13/6)\log 3$.
The denominator of the $x$-coordinate is
$12337088946900997614694947283^2$, and the numerator is
$$5908330434812036124963415912002702659341205917464938175508715.$$

\subsection{Variants}
Next we mention a variant which, for the congruent number curve,
has been investigated in depth by Elkies~\cite{Elkies}.
Here we fix a rank zero curve, say the curve $E: y^2=x^3-x$
of conductor~32, and try to find points on rank~1 quadratic twists~$E_D$
with $D<0$. It can be shown that $E_D$ will have odd functional equation for
\hbox{$|D|\equiv 5,6,7$ (mod~8).} There is not necessarily a Gross-Zagier
theorem in all these cases, and some involve mock Heegner points instead
of Heegner points. However, we still have the prediction that
$h(P)=\alpha_D L(E,1) L'(E_D,1)$ for some~\hbox{$\alpha_D>0$.}
Elkies computes a point $P$ in $\C/\Lambda$ via a method
similar to the above --- however, he generally\footnote{
 Elkies also computes a generator of height $239.6+$
 for $y^2=x^3-1063^2x$ in this manner.}
only attempts to determine if it is non-torsion,
and thus need not worry as much about precision.
There are about $\#\Cl\bigl(\Q(\sqrt D)\bigr)\approx\sqrt{|D|}$
conjugates of~$\tau$ for which $\phi(\tau)$ needs to be computed;
since we have an action of~$\Gamma_0(32)$, computing each $\phi(\tau)$
takes essentially constant time, so we get an algorithm that takes
about time $|D|^{1/2}$ to determine whether the computed point is non-torsion.
Note that we don't obtain $L'(E_D,1)$, which takes about $|D|$ time to
compute, but only whether it is nonzero.
MacLeod~\cite{MacLeod} investigated a similar family of quadratic twists,
those of a curve of conductor~128.
The relevant curves are $y^2=(x+p)(x^2+p^2)$ with $p\equiv 7$ (mod~8);
with $p=3167$ the height is $1022.64+$.
Some additional papers that deal with the theory and constructions in
this case are those of Birch and Monsky \cite{birch,birch2,monsky,monsky2}.

\section{Combination with descent}
To find Heegner points of large height, say 500 or more, it is usually
best first to do a descent on the elliptic curve, as this will tend to
reduce the size of the rational point by a significant factor.\footnote{
 During the mid 1990s, Cremona and Siksek worked
 out a few examples using~\hbox{2-descent}.}
Upon doing a \hbox{2-descent}, we need only $H/3$ digits of precision
if we represent the covering curve as an intersection of quadrics in~$\P^3$
and use \hbox{4-dimensional} lattice reduction,
and if we do a \hbox{4-descent} we need only $H/12$ digits.
We first explain how these lattice reduction methods work,
and then show how to use them in our application.
It might also be prudent to point out that if $E$ has nontrivial
rational isogenies, then one should work with the isogenous curve
for which the height of the generator will be the smallest.\footnote{
 Because $\Sha$ might have different size for the various isogenous curves,
 we cannot always tell beforehand which curve(s) will have a generator of
 smallest height.}

\subsection{Lattice Reduction}
Most of the theory here is due to Elkies~\cite{elkies-ants4}.
We first describe a \hbox{$p$-adic} method --- this is not immediately
relevant to us as we do not know how to approximate the Heegner
point in such a manner, but it helps to understand the idea.
Let $F(W,X,Y)=0$ be a curve in~$\P^2$. We wish to find rational points on~$F$.
Let $(1:x_s:y_s)$ be a (nonsingular) point modulo some prime~$p$,
and lift this to a solution $(1:x_0:y_0)$ modulo~$p^2$.
Then determine $d$ such that any linear combination
of $(1:x_0:y_0)$ and $(0:p:dp)$ will be a solution mod~$p^2$
(computing~$d$ essentially involves taking a derivative).
Then perform lattice reduction on the rows of the matrix
$$\begin{pmatrix} 1&x_0&y_0\\ 0&p&dp\\ 0&0&p^2 \end{pmatrix}.$$
Finally search for global solutions to~$F$ by taking small
linear combinations of the rows of the lattice-reduced matrix.
If we choose $p$ to be around $B$ for some height bound~$B$,
upon looping through all local solutions modulo~$p$ we should
find all global points whose coordinates are of size~$B$;
in general we take $p$ of size $B^{2/n}$ in projective~\hbox{$n$-space}.
This can be used, for instance, to search for points on a cubic model
of an elliptic curve.\footnote{
 This description is due to Elkies and
 is noted by Womack (\cite[Section 2.9]{Womack}).}

Over the real numbers the description is more complicated.
Here we deal with the transformation matrix of the lattice reduction.
If we wanted to do \hbox{2-dimensional} reduction, that is,
continued fractions, on a real number~$x_0$, we would
perform lattice reduction on the rows of the matrix
$$M_2=\begin{pmatrix} 1&-x_0B\\ 0&B \end{pmatrix}$$
to get good rational approximations to~$x_0$.
We can note that $\overrightarrow{(1,x_0)}M_2=\overrightarrow{(1,0)}$
and that the transformation matrix~$T$ for which $TM_2$ is lattice-reduced
has the property that $\overrightarrow{(1,0})T$ is approximately
proportional to~$\overrightarrow{(1,x_0)}$.
In four dimensions we take a point $(1:x_0:y_0:z_0)$ on some curve,
assuming that derivatives of $y$ and $z$ with respect to~$x$
are defined at this point. The matrix we use here is\footnote{
 The \hbox{3-dimensional} version is just the upper-left corner.}
$$M_4=\begin{pmatrix}
1&-x_0B&(y'x_0-y_0)B^2&\bigl(-e(y'x_0-y_0)+z'x_0-z_0\bigr)B^3\\
0&B&y'B^2&(ey'-z')B^3\\ 0&0&B^2&-eB^3\\ 0&0&0&B^3 \end{pmatrix}.$$
Here $e=z''/y''$ and all the derivatives are with respect to~$x$
and are to be evaluated at $(1:x_0:y_0:z_0)$.
Note that if we have computed $(1:x_0:y_0:z_0)$ to $H$ digits of precision,
we must ``lift'' it to precision~$3H$ to use this.
Similar to the \hbox{2-dimensional} case of above we have that
$\overrightarrow{(1,x_0,y_0,z_0)}M_4=\overrightarrow{(1,0,0,0)}$,
and $\overrightarrow{(1,0,0,0)}T$ is approximately proportional
to~$\overrightarrow{(1,x_0,y_0,z_0)}$.

\subsection{Results}
We now combine descent with the Heegner point method.
We assume that we have a cover $C\rightarrow E$, and for each
point $\WP(\dot z)$ given by the above algorithm we compute its
real pre-images on~$C$. For a \hbox{2-covering}~quartic,
the \hbox{$x$-coordinate} has size~$H/4$,
but the \hbox{$y$-coordinate} on the quartic will be of size~$H/2$.
Either continued fractions on the~\hbox{$x$-coordinate}
or \hbox{3-dimensional} lattice reduction on both coordinates and the curve
requires a precision of $H/2$ digits --- however, if
our \hbox{2-cover} is given as an intersection of quadrics in~$\P^3$,
then we only need a precision of~$(H/2)(2/3)$ since the Elkies method
does better in higher dimension. For a \hbox{4-cover} represented as
an intersection of quadrics in~$\P^3$, the coordinates are of size~$H/8$,
and so we need a precision of $(H/8)(2/3)$ to recover our point.

We now give two examples of Heegner points of large height.\footnote{
 These examples exemplify the experimental and heuristic correlation
 between large heights and large cancellation in $c_4^3-c_6^2=1728\Delta$,
 since $\Omega_{\rm re}$ can then be unusually small for a given $|\Delta|$.}
First we consider $E$ given by $[0,1,1,-4912150272,-132513750628709]$,
for which \hbox{$N=421859$}.
Here the Heegner point is of height $3239.048+$.
We refer the reader to \cite{Womack} for how to do a \hbox{4-descent}.
The intersection of quadrics that gives the \hbox{4-cover} is
given by the symmetric matrices
$$\begin{pmatrix}1&3&14&4\\ 3&7&9&8\\ 14&9&-8&19\\ 4&8&19&13\end{pmatrix}
\qquad{\rm and}\qquad
\begin{pmatrix}
16&-10&5&-5\\ -10&29&-3&-5\\ 5&-3&-1&8\\ -5&-5&8&13\end{pmatrix}.$$
We used $D=-795$ for which the index is~4. The class group has size~4;
upon using the pairing from complex conjugation we need only 2 forms,
which we can take to be $(421859,234525,32595)$ and~$(421859,384997,87839)$.
We need to use about $3239/12\log(10)\approx 120$ digits of precision
and take around 1.3 million terms of the~\hbox{$L$-series}.
For our approximation to a generator on $\C/\Lambda$ we get
\vskip0pt\noindent
$$\dot z=\hbox{\small
0.00825831518406814312450985646222558391095207954623175715662897127635126006560626891914983130574212343000780426018430276055.}$$
\vskip0pt\noindent
We find the real pre-images of this on the \hbox{4-cover}
(this can be done via a resultant computation)
and then via \hbox{4-dimensional} lattice reduction
we obtain the point
\vskip-4pt\noindent
$$\hbox{\realsmall
(90585849222350621011339302424932326542192474474854331313031216338204053880670077944701302491852572823731202634266219944146702509489824532529044887987947859472355124939471295729,}$$
\vskip-10pt\noindent
$$\hbox{\realsmall
58207848469468567249250100904745604517491584621654101065649337689496036041318068019646159331652386264579879746727095434743171075008134671399964513924870607157340785327661071757,}$$
\vskip-10pt\noindent
$$\hbox{\realsmall
-52660183473004831875084410642918250532458007522956523515279464248174730240287018424148701587123000269221530600465998289112443045957965377530412500941202059568743656911281766881,}$$
\vskip-10pt\noindent
$$\hbox{\realsmall
120566343955994724443268162651063750286159426913472902595469949397024821835521392662156637479278825638673289873881696613629128450397637909394604102581491799075589326751709650806),}$$
which can then be mapped back to~$E$.
Even though we only used 120 digits of precision in our computation
of a real approximation to the Heegner point, we can recover a point
with approximately 3/2 as many digits.
Note that if we did not use descent, but recovered the point
on the original curve using the Cremona-Silverman method,
we would need 12 times the precision and 12 times as many terms
in the \hbox{$L$-series} --- this could be a total time factor of
as much as~$12^3$, depending upon the efficiency of our high-precision
arithmetic. This computation (including 2-descent and 4-descent which each
take a second) takes less than a minute.

Finally we give a more extreme example --- this is the largest
example which we have computed. The curve is from the database of
Stein and Watkins~\cite{SteinWatkins}.
Let $E$ be given by $[0,0,1,-5115523309,-140826120488927]$,
for which $N=66157667$ and the Heegner point is of height $12557+$.
The intersection of quadrics that gives the \hbox{4-cover} is
given by the two symmetric matrices
$$\begin{pmatrix}0&1&3&3\\ 1&5&-1&-6\\ 3&-1&8&-2\\ 3&-6&-2&16\end{pmatrix}
\qquad{\rm and}\qquad
\begin{pmatrix}
12&-21&-10&-68\\ -21&-13&-7&-27\\ -10&-7&3&-7\\ -68&-27&-7&15\end{pmatrix}.$$
We select $D=-1435$ for which our 2 forms are
$(66157667,2599591,25537)$ and $(66157667,37610323,5345323)$
and the index is~2.
We need 460 digits of precision and 600 million terms of the $L$-series.
This takes less than a day. We list the \hbox{$x$-coordinate} of the
point on the original elliptic curve. It has numerator
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\end{document}

math0506325.extracted_bib
\begin{thebibliography}{99}
\rm % no idea why this is needed
\bibitem{birch}
B. J. Birch, {\it Elliptic curves and modular functions.}
Symposia Mathematica, Vol.~IV (INDAM, Rome, 1968/69), 27--32,
Academic Press, London, 1970.

\bibitem{birch2}
B. J. Birch, {\it Heegner points of elliptic curves.}
Symposia Mathematica, Vol. XV (Convegno di Strutture in Corpi Algebrici,
INDAM, Rome, 1973), 441--445, Academic Press, London, 1975.

\bibitem{BirchStephens}
B. J. Birch, N. M. Stephens, {\it Computation of Heegner points.}
In {\it Modular forms} (Durham 1983),
Papers from the symposium held at the University of Durham, Durham,
June 30--July 10, 1983. Edited by R. A. Rankin.
Ellis Horwood Series in Mathematics and its Applications:
Statistics and Operational Research.
Ellis Horwood Ltd., Chichester;
Halsted Press [John Wiley \& Sons, Inc.], New York (1984), 13--41.

\bibitem{magma}
W. Bosma, J. Cannon, C. Playoust,
\textit{The Magma algebra system. I. The user language.}
In \textit{Computational algebra and number theory}
Proceedings of the 1st MAGMA Conference held at Queen Mary and
Westfield College, London, August 23--27, 1993.
Edited by J. Cannon and D. Holt,
Elsevier Science B.V., Amsterdam (1997), 235--265.
Cross-referenced as J.~Symbolic Comput. {\bf 24} (1997), no. 3-4, 235--265.
Online at \verb+magma.maths.usyd.edu.au+

\bibitem{BSD}
B. J. Birch, H. P. F. Swinnerton-Dyer, {\it Notes on elliptic curves. I. II.}
J. Reine Angew. Math. {\bf 212} (1963), 7--25, {\bf 218} (1965), 79--108.

\bibitem{BCDT}
C. Breuil, B. Conrad, F. Diamond, R. Taylor
{\it On the modularity of elliptic curves over~$\mathbf Q$:
wild 3-adic exercises.} J. Amer. Math. Soc. {\bf 14} (2001), no. 4, 843--939.

\bibitem{BumpFriedbergHoffstein}
D. Bump, S. Friedberg, J. Hoffstein,
{\it Nonvanishing theorems for $L$-functions
of modular forms and their derivatives.}
Invent. Math. {\bf 102} (1990), no. 3, 543--618.

\bibitem{cohen0}
H. Cohen, {\it A Course in Computational Algebraic Number Theory.}
Grad. Texts in Math. {\bf 138}, Springer-Verlag, 1993.

\bibitem{Cohen}
H. Cohen, {\it Diophantine Equations, $p$-adic Numbers, and $L$-functions,}
(to appear).

\bibitem{CDT}
B. Conrad, F. Diamond, R. Taylor,
{\it Modularity of certain potentially Barsotti-Tate Galois representations.}
J.~Amer. Math. Soc. {\bf 12} (1999), no. 2, 521--567.

\bibitem{MSRI}
H. Darmon, S.-W. Zhang, ed., {\it Heegner Points and Rankin $L$-series.}
Math. Sci. Res. Inst. Publ., {\bf 49}, Cambridge Univ. Press, Cambridge, 2004.
Available online from
\verb+www.msri.org/publications/books/Book49/contents.html+

\bibitem{diamond}
F. Diamond, {\it On deformation rings and Hecke rings.}
Ann. of Math. (2) {\bf 144} (1996), no. 1, 137--166.

\bibitem{Elkies}
N. D. Elkies, {\it Heegner point computations.}
In {\it Algorithmic Number Theory},
Proceedings of the First International Symposium (ANTS-I)
held at Cornell University, Ithaca,
New York, May 6--9, 1994. Edited by L. M. Adleman and M.-D. Huang.
Lecture Notes in Computer Science, {\bf 877}.
Springer-Verlag, Berlin (1994), 122--133.
\hfil\break\noindent
N. D. Elkies, {\it Curves $Dy^2=x^3-x$ of odd analytic rank.}
In {\it Algorithmic number theory},
Proceedings of the 5th International Symposium (ANTS-V)
held at the University of Sydney, Sydney, July 7--12, 2002.
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