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\title{Finding all Elliptic Curves over $\Q(\sqrt{5})$\\of Given Conductor}

\author{William Stein\footnote{The work was supported by NSF grants...}}


\begin{document}
\maketitle

\begin{abstract} 
\end{abstract}

\tableofcontents

\section{Introduction}\label{sec:intro}

\subsection{Elliptic Curves over $\Q$}
Tables of elliptic curves over $\Q$ have been of great value in
mathematical research.  The first systematic published tables were in
Antwerp IV \cite{antwerpiv}, which included all elliptic curves over
$\Q$ of conductor up to $200$, and also a table of all elliptic curves
with bad reduction only at $2,3$.

Cremona then wrote a book \cite{cremona:algs} that gave a careful and
detailed description of algorithms that together would
output a list of all elliptic curves over $\Q$ of any specific given
conductor, along with extensive data about each curve.  The proof that
his algorithm output all curves of given conductor had to wait for
\cite{breuil-conrad-diamond-taylor}.  Cremona has subsequently
computed tables \cite{cremona:onlinetables} of all elliptic curves
over $\Q$ of conductor up to $200,000$ with Mordell-Weil groups and
other data about each, and his main current target is reaching
$234,446$, which is the smallest known conductor of a rank $4$ curve.

In a different direction, Stein-Watkins (see
\cite{stein-watkins:ants5, bmsw:bulletins}) created a huge table of
136,832,795 elliptic curves over $\Q$ of conductor $\leq 10^8$, and
another table of 11,378,911 elliptic curves over $\Q$ of prime
conductor $\leq 10^{10}$.  For each curve, these tables contain the
analytic rank numerically computed to some precision, and some other
data, though the actual ranks of all of these curves has not been
determined.  There are also many curves of large discriminant missing
from the Stein-Watkins tables, since these tables are made by
enumerating curves with relatively small defining equations, and
discarding those of large conductor, rather than systematically
finding all curves of given conductor no matter how large the defining
equation.
 
\subsection{Why $\Q(\sqrt{5})$?}

Like $\Q$, the field $F=\Q(\sqrt{5})$ is a totally real field, and
many of the techniques that we have for studying elliptic curves over
$\Q$ generalize, or are conjectured to generalize, to totally real
fields.
As is the case over $\Q$, there is a 
notion of modularity coming from Shimura curve
parametrizations and Hilbert modular forms, which is now backed up by
extensive theoretical \cite{} and some computational \cite{} evidence.
Moreover, extensive work  \cite{} of Zhang has extended many results
of Gross-Zagier \cite{} and Kolyvagin \cite{} to the context of elliptic
curves over totally real fields.

If we order fields by absolute discriminant, then $F=\Q(\sqrt{5})$ is
the next field after $\Q$.  That $5$ divides $\disc(F)=5$ obstructs
attempts to generalize the Taylor-Wiles method to elliptic curves over
$F$, which makes this case even more interesting.  The field $F$ also
has $xx$ CM $j$-invariants, which is far more than any other field of
degree $\leq 2$ [[??]] (see Section~\ref{sec:cm}).  Let
$\vphi=\frac{1+\sqrt{5}}{2}$.  The unit group $\{\pm 1\} \times
\langle \vphi \rangle$ of the ring $R=\cO_F=\Z[\vphi]$ of integers of
$F$ is infinite, which leads to additional complications.  Finally,
the totally real field $F$ has even degree $2$, which makes
computation of Hilbert modular forms and corresponding elliptic curves
more difficult, since the techniques of \cite{} are not available.  In
particular, some curves (e.g., with square level) can have no Shimura
curve parameterization.


\subsection{Modularity conjecture}\label{sec:mod}
The following conjecture is not yet known in general. 
\begin{conjecture}[Modularity]\label{conj:mod}
  The set of $L$-functions of elliptic curves over $F$ equals the set
  of $L$-functions associated to cuspidal Hilbert modular newforms
  over $F$ of parallel weight $2$ with rational Hecke eigenvalues.
\end{conjecture}
Given the huge amount of recent progress on modularity theorems, we
are optimistic that Conjecture~\ref{conj:mod} will be proved in the
near future.  We officially {\em assume} it for the rest of this
paper.


\section{Computing Hilbert modular forms over $F$}\label{sec:hmf}



\section{Strategies for finding an elliptic curve attached to a Hilbert modular form}
Let $f\in S_{(2,2)}(\n)$ be a rational Hilbert newforms as in
Section~\ref{sec:hmf}.  According to Conjecture~\ref{conj:mod}, there
is some elliptic curve $E_f$ over $f$ such that $L(f,s) = L(E_f,s)$.
Note that $E_f$ is only well defined up to isogeny.  Unlike the case
for elliptic curves over $\Q$ (see \cite{cremona:algs}), there appears
to be no fast direct algorithm to find $E_f$.  Nonetheless, there are
numerous approaches.

In all the sections below, we assume that Conjecture~\ref{conj:mod} is
true and that we have computed (as in Section~\ref{sec:hmf}) a
complete list of Hecke eigenvalues $a_\p$ of all rational Hilbert
newforms of some level $\n$, to at least enough precision $B$ to determine
one from another.

\subsection{Naive enumeration}\label{sec:naive}

The most naive strategy is to systematically enumerate pairs $a,b\in
R$, hence elliptic curves $E: y^2 = x^3 + ax + b$ over $F$, and for
each curve compute some $L$-series coefficients $a_\p(E)$ at primes
that do not divide the discriminant of the cubic by counting points
modulo primes.  If all $a_\p(E)$ match with those of the input newform
$f$, up to the bound $B$, we then compute the conductor $\cN_E$, and
if it equals $\n$, we conclude that $L(E_f,s) = L(f,s)$.

Under our hypotheses, naive enumeration is an algorithm to solve our
problem: it will terminate, and when it terminates it outputs a
correct curve $E_f$. Unfortunately, it can be very slow because even
if $\n$ is small, the simplest curve in the isogeny class of $E_f$ may
have very large coefficients, for example, if each is of the form
$x+\vphi y$ with $x$ and $y$ having coefficients around $10^5$, then
we would have to consider $10^{20}$ before finding $E_f$, which is not
feasible.

\subsection{Torsion families}\label{sec:torsion}

We can tell whether or not $\ell\mid \#E(F)_{\tor}$ for some
$E$ in the isogeny class of $E_f$ using the following proposition.

\begin{proposition}
 TODO: Explain how to tell if $\ell\mid \#E(F)_{\tor}$.
\end{proposition}
\begin{proof}
  Use Galois representations, and also Katz's theorem.
\end{proof}

Instead of searching through all curves as in Section~\ref{sec:naive},
...


\subsection{Twisting}

\subsection{Curves with specified $a_\p$}

\subsection{Curves with good reduction outside $S$}

\subsection{Special values of twisted $L$-series}

\subsection{Congruence families}

\section{Enumerating the curves in an isogeny class}

\section{Norm conductor $31$}\label{sec:31}

\section{Related future projects}\label{sec:future}

\section{Tables}\label{sec:tables}

\subsection{Up to norm conductor  $199$}\label{sec:to199}

[[TODO: This will be a short-as-reasonable table of curves of norm
conductor $\leq 199$, hence including the first curve of rank $1$.  It
will probably be about 6 pages long.]]

\subsection{CM elliptic curves over $F$}\label{sec:cm}

[[TODO: A table like in the back of \cite{silverman:aec2}, but over
$\Q(\sqrt{5})$.  We haven't made this yet.  I'm not sure how hard this
is; should be fun.]]

\subsection{Extended version only: up to norm conductor  $1831$}\label{sec:to1831}

[[This will contain a longer version, which includes both curves and
their conjugates, etc., and has all data up to 1831, and will be over
100 pages long.  This will be a special version only available on our
websites; we won't submit this to a journal.]]



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