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\title{Average Ranks of Elliptic Curves: Tension Between Data and Conjecture}
\author{Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins}
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\begin{document}
\begin{abstract}
Rational points on elliptic curves are the gems of arithmetic:
they are, to diophantine geometry, what units in rings of
integers are to algebraic number theory, what algebraic cycles are to
algebraic geometry. A rational point in just the right context, at one
place in the theory, can inhibit and control---thanks to ideas of
Kolyvagin \cite{kolyvagin:subclass}---the existence of rational
points and other mathematical structures elsewhere.
Despite all that we know about these
objects, the initial mystery and excitement that drew mathematicians
to this arena in the first place remains in full force today.
We have a network of heuristics and conjectures regarding rational
points, and we have massive data accumulated to exhibit instances of
the phenomena. Generally, we would expect that our data support our
conjectures; and if not, we lose faith in our conjectures. But here
there is a somewhat more surprising interrelation between data and
conjecture: they are not exactly in open conflict one with the other,
but they are no great comfort to each other either. We discuss
various aspects of this story, including recent heuristics and data
that attempt to resolve this mystery. We shall try to convince the reader
that, despite seeming discrepancy, data and conjecture are, in fact,
in harmony.
\end{abstract}
\maketitle
\pagestyle{myheadings}\markright{AVERAGE RANKS OF ELLIPTIC CURVES}
\section{Introduction}\label{sec:intro}
Suppose you are given an algebraic curve $C$ defined, let us say, as
the locus of zeroes of a polynomial $f(x,y)$ in two variables with
rational coefficients. Suppose you are told that $C$ has at least one
rational point, i.e., there is a pair of rational numbers $(a,b)$ such
that $f(a,b) =0.$ How likely is it that $C$ will have infinitely many
rational points?
Such a question, on the one hand, clearly touches on a fundamental
issue in diophantine geometry, and on the other, is somewhat
meaningless until it is made more precise and appropriately
organized. The question we have just asked has distinctly different
features when considered for each of the three basic ``types'' of
algebraic curves: curves of (geometric) genus $0$, $1$, and $>1$.
Curves of genus~$0$ possessing a rational point {\it always} have
infinitely many rational points (an easy fact; indeed, even known to
the ancient Greeks, since our curve can be written as a conic in this
case); curves of genus $>1$ never do (a hard fact; indeed a theorem
of Faltings \cite{faltings:finiteness}, for which he received the
Fields Medal).
This leaves curves of genus $1$ as the unresolved,
and thus most interesting, case of the problem we posed,
since some elliptic curves, like $$x^3+y^3=1,$$ only have
finitely many rational points (two, in this instance) and others,
like $$y^2+y = x^3-x,$$ have infinitely many, starting
with $(0,0),\,(1,0),\,(-1,-1),\,(2,-3),\,(1/4,-5/8),$\break
$(6,14),\,(-5/9,8/27),\,(21/25,-69/125),\,(-20/49,-435/343),\ldots$.
If we are to try to extract an actual number between~$0$ and~$1$ that
will describe ``the" probability that a curve of genus~$1$ possessing
at least one rational point has infinitely many, we have to be
precise about exactly which curves we want to count, and how we
propose to ``sort" them. Let us agree, then (with details later):
\begin{itemize}
\item to deal only with the smooth projective models of the curves of
genus~$1$ possessing a rational point (these being precisely the {\it
elliptic curves} defined over $\Q$),
\item to count their isomorphism classes over $\Q$, and
\item to list them in order of increasing conductor, banking on the
theorem that tells us that there are only finitely many
isomorphism classes of elliptic curves over $\Q$ with any given
conductor.
\end{itemize}
We can now pose our question.
Does
$$
P(X) =
\frac{\#\{{\rm elliptic\ curves\ of\ conductor}\ \le X\
{\rm with\ infinitely\ many\ rational\ points}\}}
{{\#\{\rm elliptic\ curves\ of\ conductor}\le X\}}
$$
converge as $X$ tends to infinity, and if so, what is the limit
$$
P = \lim_{X \to \infty}P(X)?
$$
In this way we have made our initial question precise:
\vskip8pt
\centerline{\textit{What is the probability~$P$ that an elliptic curve
has infinitely many rational points}?}
\vskip8pt
It is extraordinary how much vacillation there has been in the past
three decades, in the various guesses about the answer to
this---clearly basic---question. The subject of this paper is to
discuss aspects of this drama. Its see-saw history, involving a
network of heuristics and conjectures and massive data that seemed
not to offer much comfort to the conjecturers, comes in four parts.
\begin{enumerate}
\item {\bf The minimalist conjecture.} The ``classical" Birch and
Swinnerton-Dyer conjecture (see Section~\ref{sec:ec}) suggests that
the probability $P$ described by our question is at least $1/2$. The
reason for this is the (heuristic) phenomenon of {\it parity}:
elliptic curves can be sorted into two classes, those of {\bf even parity},
where the ``sign in the functional equation of the $L$-function" is
$+1$, and those of {\bf odd parity}, where the ``sign" is $-1$. The
(conjectural) probability that an elliptic curve is of even parity is $1/2$,
and the same---of course---for odd parity. A consequence of the Birch
and Swinnerton-Dyer conjecture is that {\it all} elliptic curves of
odd parity have infinitely many rational points. This is why no one
doubts that the probability $P$ described above is $\ge 1/2$.
It has long been a folk conjecture that $P$ is {\em exactly}
$1/2$---let us call this the {\bf minimalist conjecture}. Given the
Birch and Swinnerton-Dyer conjecture and the Parity Principle,
an equivalent, and cleaner, way of stating it is as follows:
\begin{conjecture} An elliptic curve of even parity has
probability zero of having infinitely many rational points.
\end{conjecture}
This minimalist conjecture might seem appealing purely on the grounds
that rational points of elliptic curves are accidental gems of
mathematics, and it is hard to imagine that there could be bulk
occurrence of these precious accidents---or at least substantially
more bulk than is already predicted.
It seems that one cannot find such a minimalist conjecture explicitly
in the literature until very recently (see \cite{watkins:rankheuristics} and
Conjecture~\ref{conj:rank2disc}). Nevertheless, for some
particular families of elliptic curves (the ``quadratic twist''
families---see Section~\ref{sec:quad} below)
the conjecture is much older.
Over a quarter of a century ago, Dorian Goldfeld conjectured that
for any elliptic curve~$E$, the probability
$$G(D)=
\frac{\#\{\text{quadratic twists up to~$D$ of $E$
with infinitely many rational points}\}}
{\#\{\text{quadratic twists up to~$D$ of $E$}\}}$$ has $G=1/2$
as its limit as $D\rightarrow\infty$.
\item {\bf Contrary numerical data.} The next phase of our story
involves the accumulation of numerical data regarding this
probability $P$ taken over the entirety of elliptic curves, and also
over various selected families of elliptic curves. The short
description of this data (but see the detailed discussion in the
body of our article) is the following. Over every data set
accumulated so far, about $2/3$ (or sometimes more) of the
curves in the families being considered have had infinitely
many rational points, and rather flatly so over the range
of conductors involved in the computations; these now include a large
set (over 100 million curves) of elliptic curves of conductor~$<10^8$.
\item {\bf A gross heuristic, for special families.}
To get the most precise results we change
the data set, and restrict attention to the probability that a
member of even parity of a {\it quadratic twist family} of elliptic
curves has infinitely many rational points. As a refinement to Goldfeld's
conjecture, Peter Sarnak gave a heuristic that predicts that among
the first $D$ members of such a quadratic twist family
(essentially arranged in order of increasing conductor)
the number of those with even parity and infinitely many
rational points is caught between $D^{3/4-\epsilon}$ and
$D^{3/4+\epsilon}$ for any positive $\epsilon$ and $D$ sufficiently
large. This guess, based on consideration of the size of Fourier
coefficients of modular forms of half-integral weight, revived the
minimalist conjecture: if Sarnak's estimate is correct, we would
indeed have $G=1/2$ in Goldfeld's conjecture, and even-parity members
of a quadratic twist family would have probability zero of having
infinitely many rational points.
At this point in our story, there is decided friction between
accumulated data which suggests something like $2/3$ as the
probability for the general member to have infinitely many
rational points, and a reasoned theoretical expectation, which
suggests exactly $1/2$ for that probability. Generally, the least we
would expect of our data is that they either support our conjectures,
or overthrow them. Here there was a somewhat more surprising
interrelation between data and conjecture: a kind of truce between
them: we believed our guesses, we believed the data, and
acknowledged the apparent gap between them.
\item {\bf A refined heuristic, for special families.}
More recently, another twist to this story has developed.
The work of Katz and Sarnak \cite{katz-sarnak} regarding symmetry groups
of the analogous families of curves over function fields\footnote
{More recently, Kowalski \cite{kow2} has used monodromy results
of Katz to prove upper bounds for average ranks in the function
field analogues.}
gave impetus to the random matrix theory calculations
of Keating and Snaith \cite{keating-snaith} regarding moments of
$L$-functions and their value distribution. This was then combined
with a discretization process by Conrey, Keating, Rubinstein,
and Snaith in \cite{ckrs2002} to give a more
precise guess for the (asymptotic) number of even parity curves
with infinitely many rational points in a given quadratic twist family.
For example, for the
quadratic twist family $y^2=x^3-d^2x$, the prediction is that among
the first $D$ members of this family, the number of those with even
parity and infinitely many rational points is asymptotic to
\begin{equation}\label{eqn:F}
F(D) = c\cdot D^{3/4}\log(D)^{11/8}
\end{equation}
for some (positive) constant~$c$.
\psfrag{0.0}{\raisebox{-5pt}{\tiny $0$}}
\psfrag{1.0}{\raisebox{-5pt}{\tiny $10^8$}}
\psfrag{0.8}{\raisebox{-5pt}{\tiny $8\!\cdot\!10^7$}}
\psfrag{0.6}{\raisebox{-5pt}{\tiny $6\!\cdot\!10^7$}}
\psfrag{0.4}{\raisebox{-5pt}{\tiny $4\!\cdot\!10^7$}}
\psfrag{0.2}{\raisebox{-5pt}{\tiny $2\!\cdot\!10^7$}}
\psfrag{1e7}{\raisebox{-5pt}{\tiny $10^7$}}
\psfrag{x1e8}{}
\graph{xlogx}{Plots of $D^{3/4}\log(D)^{11/8}$ (upper)
and $\Delta^{19/24}(\log \Delta)^{3/8}$\newline
\mbox{}\hspace{4.9em} (lower) up to $10^8$ \label{fig:linear}}
On the one hand, this is a sharpening of the prior heuristic, for
$F(D)$ is comfortably sandwiched between $D^{3/4\pm \epsilon}$. On
the other hand, we may be in for a surprise when we actually plot the
graph of the function $F(D)$. See Figure~\ref{fig:linear}. The
striking aspect of the graph in Figure~\ref{fig:linear} is how
``linear'' it looks. Indeed, if $F(D)$ were replaced by a linear
function with roughly the slope that appears in
Figure~\ref{fig:linear}, it would predict something closer to $2/3$
than $1/2$ for the proportion of curves in the family with infinitely
many rational points.
Similarly---cf. Section~\ref{sec:all} below---if we order all
elliptic curves by discriminant, one of us (see
\cite{watkins:rankheuristics}) has conjectured that the number of
even parity elliptic curves with infinitely many rational points and
absolute discriminant less than $X$ is asymptotically given by
$$
\Phi(X) \ = \ c X^{19/24}(\log X)^{3/8}.
$$
See Figure~\ref{fig:linear}.
\end{enumerate}
Here, roughly speaking, is where the story is at present, as we will
explain in detail in the body of this article. The curious last phase
of it, focussing on {\it special families}, makes it seem now that
data for these families is (a) more closely adhering to the refined
guess than one might expect, even for relatively small values of the
conductor, and (b) a refined guess predicts an asymptotic behavior
that is far from linear, but within the currently attainable range is
so close to linear, that the numerical evidence elucidating these
phenomena (even the very large data sets that computers have amassed)
seem indecisive when it comes to distinguishing convincingly between
such gross questions as: is the probability closer to $1/2$ or to~$2/3$?
It may very well be that until we actually prove our
conjectures, no data that we can accumulate, however massive it may
appear, will give even lukewarm comfort to the conjecturers.\footnote{
We are reminded of the challenge of Shanks \cite[\S 69]{shanks}
regarding Carmichael numbers;
with respect to the conjecture of Erd\H{o}s that, for every $\epsilon>0$,
there are, for sufficiently large~$X$,
at least $X^{1-\epsilon}$ Carmichael numbers up to~$X$,
Shanks (essentially) noted that the data for small $X$
did not remotely conform to this, and proposed giving an
explicit $X$ for which there were at least (say) $\sqrt{X}$
Carmichael numbers up to~$X$, suspecting that exhibiting
such an $X$ would be much beyond the capabilities of computers.}
This conflict raises the question of
whether we as mathematicians may, at times, face a situation where the
substance we study has one shape asymptotically, and yet all
computational evidence elucidating this substance, even up to the very
large numbers that computers today, or in our lifetime, can compute,
seems consistent with the possibility that the data have a different
asymptotic shape.
But, of course, our story will continue. We would hope for
\begin{itemize}
\item a refined heuristic that covers the full set of
elliptic curves, and not just quadratic twist families,
\item an extension of the numerical computation to conductors $<10^{10}$,
which is a range where we may begin to see some significant differences
between the graph of $F(D)$ and a linear function,
\item
a conceptual understanding of how to obtain---by more unified
means--- this impressive bulk of rational points that we see
occurring for even parity elliptic curves, at least for curves
of ``small'' conductor.
\end{itemize}
We find it useful to compare our question {\em what is the probability
that an elliptic curve has infinitely many points} with some of the
other counting problems of current interest. Specifically, consider
the problem of {\it counting quartic fields} and sorting them into
classes corresponding to the isomorphy type of the Galois group of
their Galois closure. We have to be exceedingly
careful when choosing the coefficients of a degree~4 polynomial
if we want a root of that polynomial to generate anything other than a
field whose Galois group is~$S_4$. Hilbert's irreducibility theorem
provides corroboration of this with a proof that if you rank algebraic
numbers of degree 4 by the size of the coefficients of their minimal
polynomial (monic, over~$\Q$) then 100\% of them have Galois group~$S_4$.
But consider the problem of counting quartic fields (rather
than the algebraic numbers that generate them) listed by the size
(absolute~value) of their discriminant. Counting field extensions of a
given field whose Galois closure has its Galois group of a particular
isomorphy type has been the subject of a number of precise conjectures
(initially \cite{cohen-diaz-olivier:disc}, and then successively
refined in \cite{malle:distro, malle:distro2}). Bhargava's remarkable
paper \cite{bhargava:density}, which is further evidence for these
conjectures, proves that when we count quartic fields, nested by
absolute discriminant, we do {\em not} get that 100\% of
them have Galois group $S_4$.
Bhargava regards the problem of counting quartic fields as a problem
purely in the Geometry of Numbers, and {\em proves} the following
theorem:
\begin{theorem}\relax[Bhargava].
When ordered by absolute discriminant, {\em a positive proportion}
(approximately 0.17111) of quartic fields have associated Galois
group~$D_4$ (the dihedral group).
The remaining approximately 0.82889 of quartic fields have
Galois group~$S_4$, and the other three transitive subgroups
occur with probability~0 asymptotically.
\end{theorem}
It should be noted that these are Bhargava's percentages when counting
fields up to isomorphism; when working in a fixed algebraic
closure of the rationals, the percentages are not the same.
We would be more than delighted to see unconditional results of this
precision established for questions such as the one motivating this
survey article.
\vspace{2ex}
\noindent{\bf Acknowledgment.} We thank Armand Brumer, Frank Calegari,
Noam Elkies and Oisin McGuinness for stimulating conversations. We used
PARI \cite{pari} and SAGE \cite{sage} to compute and analyze the data,
and matplotlib \cite{matplotlib} to draw the graphs.
We thank Bob Guralnik for references to work about distributions
of Galois groups. Similar surveys to this are those of Rubin and Silverberg
\cite{rubin-silverberg:ranks} and Kowalski \cite{kowalski}, the latter
of which also relates random matrix theory to the theory of elliptic curves,
and then discusses questions related to the inability of reconciling
experimental data with theoretical asymptotics, particularly with respect
to the work of Heath-Brown \cite{heath-brown}
on the 2-Selmer rank of the congruent number curves.
\section{Elliptic Curves}\label{sec:ec}
An {\em elliptic curve} $E$ over~$\Q$ is a projective nonsingular
curve defined as the projective closure of the zero locus of an
equation of the form
\begin{equation}\label{eqn:weq}
y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4x + a_6,
\end{equation}
with the $a_i$ in $\Q$.
The set $E(\Q)$ of rational points on $E$ is equipped with
an abelian group structure (see \cite{silverman:aec}).
Via completing the square in the $y$-variable, translating to
eliminate the quadratic $x$-term, and then re-scaling,
we find that the equation (\ref{eqn:weq}) is rationally
equivalent to exactly one of the form
\begin{equation}\label{eqn:c4c6}
y^2 = x^3 - 27c_4x - 54c_6,
\end{equation}
with $c_4,c_6,\Delta =(c_4^3-c_6^2)/1728\in\Z$ and for which there is
no prime $p$ with $p^4\mid c_4$ and $p^{12}\mid \Delta$. We call
$\Delta$ the {\em minimal discriminant} of~$E$. (For example, the
minimal discriminant of the curve $y^2+y = x^3-x$ mentioned in
Section~\ref{sec:intro} is $\Delta = 37$; also $c_4=48$ and
$c_6=-216$ for this curve.)
The {\em conductor} of an elliptic curve $E$ over $\Q$ is a positive
integer $N=N_E$ that is a measure of the nature of the reduction of
the elliptic curve modulo the prime divisors of~$\Delta$. For
example, a prime $p\geq 5$ divides the conductor~$N$ only if there is
no way of finding another defining equation (\ref{eqn:weq}) of~$E$ so
that when reduced modulo~$p$ we obtain an equation over the field
$\F_p$ without multiple roots; the maximal power of such a prime $p$
dividing $N$ is 2 and whether it is 1 or 2 is determined by the nature
of the {\em best} reduction of $E$ modulo $p$, i.e., whether its
defining cubic polynomial has a double or a triple root modulo~$p$.
There is a slightly more involved, but elementary, recipe to give
the power of the primes $2$ and $3$ dividing the conductor
(see \cite{tate:antwerpiv}).
Mordell proved in 1922 (see \cite{mordell})
that the {\em Mordell-Weil group} $E(\Q)$ of rational
points on $E$ is a finitely generated abelian group,
so $E(\Q)\approx \Z^r \oplus E(\Q)_{\tor}$. The
integer~$r$ is the {\em rank} of~$E$, and is the main
statistic that we will discuss below; in contrast, the
torsion group is rather well-understood, and is thus of
less interest.
Let $\Delta$ be the minimal discriminant of $E$.
The {\em $L$-function} $L(E,s)$ of $E$ is a Dirichlet series given by a
simple recipe in terms of the number of points $N_p$ of the reduction
of $E$ over $\F_p$ for all primes~$p$. Specifically,
\begin{equation}\label{eqn:eulerprod}
L(E,s)
= \prod_{p \nmid \Delta }\frac{1}{1 - (1+p-N_p)p^{-s}+ p^{1-2s}}
\cdot \prod_{p \mid \Delta}\frac{1}{1 - (1+p-N_p)p^{-s}}
= \sum_{n=1}^{\infty} \frac{a_n}{n^s}.
\end{equation}
The integers $a_n$ are defined by expanding the Euler product;
e.g., $a_p = p+1 - N_p$ and $a_{p^2}=a_p^2-p$ when $p\nmid\Delta$, etc.
As an example, if $E$ is $y^2 + y = x^3 - x$ then
$$
L(E,s) = 1 - \frac{2}{2^{s}}
- \frac{3}{3^{s}} + \frac{2}{4^s} - \frac{2}{5^s} + \frac{6}{6^s}
-\frac{1}{7^s} +\frac{6}{9^s}+\frac{4}{10^s} - \frac{5}{11^s} + \cdots.
$$
For any elliptic curve $E$, the celebrated papers of Wiles
\cite{wiles:fermat} and others \cite{bcdt} imply that $L(E,s)$ extends
to an entire analytic function on the complex plane.
Moreover these results imply that the completed $L$-function
$\Lambda(E, s) = N^{s/2} \cdot (2 \pi)^{-s} \cdot \Gamma(s)\cdot L(E,s)$
satisfies the functional equation
$$
\Lambda(E,s) = u_E\cdot \Lambda(E,2-s),
$$
where $u_E$ is either $-1$ or $1$, and is called the
\emph{sign in the functional equation for $E$}.
Note that $u_E=1$ if and only if $L(E,s)$ vanishes to even
order at $s=1$.
The classical Birch and Swinnerton-Dyer Conjecture \cite{bsd} asserts
that the order of vanishing of $\Lambda(E,s)$ at the point $s=1$
is equal to the rank of the Mordell-Weil group
$E({\bf Q})$. In the data regarding {\it rank} that we will be
reporting below, at times the Mordell-Weil rank $r$ has been computed
directly by finding $r$ rational points of $E$ that are linearly
independent and span a subgroup of finite index in $E({\bf Q})$ and we
will refer to this $r$ as the {\it arithmetic rank} of $E$. At times,
however, what is computed is the apparent order of vanishing of $L(E,s)$
at~$s=1$; we refer to this order of vanishing as the {\it analytic rank}
of $E$. The BSD conjecture asserts that the ranks are in fact equal.
We say a curve has \emph{even parity} if the analytic
rank is even, and \emph{odd parity} if it is odd.
We now state the refined BSD conjecture for curves of rank~0.
When $E$ is given by (\ref{eqn:c4c6}), the real period~$\Omega_{\re}$
is, up to easily determined factors of $2$ and $3$,
equal to the integral~$\int_{E(\R)} dx/y$. For a prime~$p$,
the Tamagawa number $\Omega_p$ is the index in $E(\Q_p)$ of the
subgroup of $p$-adic points that reduce to a nonsingular point in~$E(\F_p)$.
\begin{conjecture}\label{conj:bsd}[Birch and Swinnerton-Dyer].
If $L(E,1)\neq 0$, then
\begin{equation}\label{eqn:bsd0}
L(E,1) = \frac{\Omega(E) \cdot \#\Sha(E)}{\#E(\Q)_{\tor}^2},
\end{equation}
where $\Sha(E)$, the {\em Shafarevich-Tate group of $E$},
is a certain (mysterious) group associated to~$E$
(it measures the failure of a local-global principle), and
$$
\Omega(E) = \Omega_{\re} \cdot \prod_p \Omega_p.
$$
\end{conjecture}
Since $L(E,1)\neq 0$, the group $\Sha(E)$ is known to be finite
\cite{kolyvagin:subclass} of order
a perfect square \cite{cassels:arithmeticiv}.
For any $r = 0, 1, 2, \ldots$ the question we now may ask is: what
percentage of elliptic curves (nested according to size of conductor)
have rank $r$? More correctly, we should ask: do these percentages
exist, and if so what are they?
\section{Conjectures}\label{sec:conjecture}
%\subsection{The parity of the ranks of elliptic curves and
%the minimalist conjecture}
One fairly firm anchor in the study of elliptic curves is a principle
that goes under the heading of parity. This principle is still only
conjectural, but is amply confirmed numerically in our accumulated
data, and we also have theoretical reasons to believe it.\footnote
{In particular, the sign of the functional equation is a product
of local signs for primes $p|\Delta$, each of which is $\pm 1$
with equal proportion.
See the work of Helfgott \cite{HH} for the latest results.}
The parity principle is that 50\% of the members of any of the
sets of elliptic curves we will be considering have even parity,
and 50\% have odd parity (under reasonable orderings).
So by BSD, 50\% should have even rank, and 50\% should have odd rank.
In general terms, the minimalist principle proclaims that from the
rough viewpoint of percentages, there are as few rational points on
elliptic curves as is possible, given the constraint of the parity
principle. That is, 50\% of the members of any of the families of
elliptic curves we will be considering have rank $r=0$, and 50\% have
rank $r=1$, and the remaining ranks $r\geq 2$ account for 0\% of the
family.
As one thing or another things comes to light in the subject, the
minimalist position is sometimes favored, and sometimes not.
For certain special families of
elliptic curves this minimalist conjecture has long been in print, and
has had a wild ride in terms of its being believed, and doubted.
\subsection{The form of the conjectures}\label{sec:form}
There are two types of asymptotic conjectures that we encounter in
discussions regarding rank statistics. The first we might call a {\it
rough conjecture} where it is asserted, or conjectured, for a
certain collection ${\mathcal F}(x)$ of items indexed by a variable
$x$ that there is an {\it exponent} $a$ and a function $x(\epsilon)$
such that the cardinality of ${\mathcal F}(x)$ is bounded above by
$x^{a+\epsilon}$ and below by $x^{a-\epsilon}$ for any positive
$\epsilon$ and any $x \ge x(\epsilon)$.
We also will be discussing {\it fine conjectures} where such
collections ${\mathcal F}(x)$ will be conjectured to have asymptotic
estimates of the form
$$
\#{\mathcal F}(x) \sim x^a\cdot (\log x)^b\cdot c,
$$
for constants $a,b,c$; the delicacy, of course, of these constants is
inversely related to their alphabetical order.
What seems to be a pattern is that the exponent $a$, appearing in both
rough and fine versions, in specific contexts under discussion, can
usually be guessed by more old-fashioned heuristics. But---at present,
at least---the on-going work regarding random matrix eigenvalues is
the only source of heuristics that lead us to formulate specific ``fine
conjectures" regarding ranks, and specifically for guesses regarding
the exponent $b$ of the $\log x$ term. The fact that the graphs of
some of the specific concoctions of the form $ x^a\cdot (\log
x)^b\cdot c$ predicted by random matrix statistics can look
deceivingly like $x^1$ even though $a < 1$ (and for a significant
range of the variable $x$) is one of the curiosities of our story.
\subsection{Random matrix statistics}\label{sec:randmatstat}
Originally developed in mathematical physics, random matrix theory~\cite{mehta}
has now found many applications in number theory, the first being the
oft-told story~\cite{dyson} of Dyson's remark to Montgomery regarding the
pair-correlation of zeros of the Riemann $\zeta$-function.
Based on substantial numerical evidence, random matrix theory appears
to give reasonable models for the distribution of $L$-values in families,
though the issue of what constitutes a proper ``family'' can be delicate.
The work of Katz and Sarnak~\cite{katz-sarnak} regarding families of
curves over function fields implies that for quadratic twists of even
parity, we should expect orthogonal symmetry with even parity.
Though we have no function field analogue when considering all curves
of even parity, we still brazenly assume
(largely from looking at the sign in the functional equation)
that the symmetry is orthogonal with even parity.
What this means is that we want to model properties of the $L$-function
via random matrices taken from $\SO(2M)$ with respect to Haar measure,
for an appropriate value of~$M$.\footnote
{Here we wish the mean density of zeros
of the \hbox{$L$-functions} to match the mean density of eigenvalues of our
matrices, and so, as in~\cite{keating-snaith},
we should take $2M\approx 2\log N$.}
We suspect that the $L$-value distribution is approximately
given by the distribution of the evaluations
at~$1$ of the characteristic polynomials of our random matrices.
In the large, this distribution is determined entirely by
the symmetry type, while finer considerations are distinguished
via arithmetic considerations.
Via the moment conjectures~\cite{keating-snaith}
of random matrix theory and then using Mellin inversion,
we expect that (for some constant~$c>0$)
\begin{equation}\label{eqn:RMT}
\Prob[L(E,1)< t]\sim ct^{1/2}(\log N)^{3/8}
\qquad \text{as}\qquad
t\rightarrow 0,
\end{equation}
when the curves $E$ are taken from a suitable family.
\subsection{Conjectures about twist families}\label{sec:quad}
Let $E$ be an elliptic curve over~$\Q$ defined by an equation $y^2 =
x^3 + ax+b$. The {\em quadratic twist} $E_d$ of $E$ by a nonzero
integer~$d$ is the elliptic curve defined by $y^2 = x^3 + ad^2x +
bd^3$. The twist $E_d$ is isomorphic to $E$ over the field~$\Q(\sqrt{d})$,
and (when $d$ is a fundamental discriminant relatively prime to~$N_E$)
the conductor of $E_d$ is $d^2 \cdot N_E$.
\begin{conjecture}\relax[Goldfeld, \cite{goldfeld:conjectures_quadratic}].
The average rank of the curves $E_d$ is~$\frac{1}{2}$, in the
sense that
$$
\lim_{D\to\infty} \frac{\sum_{|d|0$&\rankzeropos&\rankonepos&\ranktwopos&
\rankthreepos &\rankfourpos&\rankfivepos\\
$\Delta<0$&\rankzeroneg&\rankoneneg&\ranktwoneg&
\rankthreeneg &\rankfourneg&\rankfiveneg\\\hline
{\bf Total \# Curves}&\rankzero&\rankone&\ranktwo&\rankthree
&\rankfour&\rankfive\\
{\bf Proportion} & 0.300 & 0.461 & 0.198 & 0.038 & 0.0026 & 0.00002\\
{\bf Proportion $\Delta > 0$} &
0.279 & 0.455 & 0.217 & 0.046 & 0.0033 & 0.00000\\
{\bf Proportion $\Delta < 0$} &
0.313 & 0.464 & 0.187 & 0.034 & 0.0022 & 0.00003\\
\hline\end{tabular}
\end{center}
\end{table}
In Table~\ref{table:bm}, note that curves with $\Delta > 0$ are more
likely to have large rank. Let $r_\eps(X)$ be the average rank
of elliptic curves in \cite{brumer-mcguinness} with conductor at
most~$X$ and discriminant sign~$\eps$. They observe that in their
data, $r_{+}$ climbs to $1.04$ and $r_{-}$ climbs to $0.94$, and they
remark that \emph{``An interesting phenomenon was the systematic influence
of the discriminant sign on all aspects of the arithmetic of the
curve.''} The more extensive computations do {\em not} always
find this to be the case; see, in particular, Figure~\ref{fig:avgrank}
below, where the graphs split by discriminant cross.
\subsection{The Stein-Watkins Database}\label{sec:ecdb}
Brumer and McGuinness fixed the $a_1$, $a_2$,~$a_3$ invariants (12
total possibilities, as \eqref{eqn:weq} can be modified first to be integral,
and then to ensure that $a_1,a_3\in\{0,1\}$ and $|a_2|\le 1$)
and then searched for $a_4$ and $a_6$ that made
$|\Delta|$ small. Stein and Watkins \cite{stein-watkins:ants5}
broke the $c_4$ and $c_6$ invariants into congruence classes,
and then found small solutions to $c_4^3-c_6^2=1728\Delta$,
with $c_4, c_6$ minimal in the sense of (\ref{eqn:c4c6}).
There is little theoretical advantage in this approach;
more computing power and disk space were the main advances
in~\cite{stein-watkins:ants5}.
Stein and Watkins searched for curves with prime conductor up to~$10^{10}$,
and for composite conductor chose $|\Delta|<10^{12}$ and $N\le 10^8$
as search bounds, and then included isogenous curves and twists
(with $N\le 10^8$) of the curves they found.
\subsection{Completeness of the Databases}
Note that neither the method of Brumer-McGuinness nor Stein-Watkins is
guaranteed to find all curves of prime (absolute) discriminant
up to a given bound
(indeed, it is more likely that they miss a few curves), but we think
that their data sets are reasonable surrogates, and should exhibit
validity when compared to the predictions of the theoretical model.
For curves of composite conductor, the Stein-Watkins database
is much more likely to miss curves. Here the comparison is to
the data set of Cremona~\cite{cremona:onlinetables}, who
used the algorithms of~\cite{cremona:algs} and the modularity
theorem of~\cite{bcdt} to find {\em every} elliptic curve
of conductor up to $120000$. Cremona found $782493$ curves up to
conductor $120000$. In the Stein-Watkins computation, they found
$614442$ curves of conductor up to $120000$, so they found over 78.5\%
of the curves. The first case in which Cremona has a
curve and Stein-Watkins do not is the curve $y^2 + xy + y = x^3 -
7705x + 1226492$ of conductor $174$, which has discriminant
$-621261297432576 =-2^{11} \cdot 3^{21} \cdot 29$, whose absolute
value is substantially larger than $10^{12}$. The conductors up to
$500$ where they miss curves are
$$ 174, 222, 273, 291, 330, 354, 357, 390, 420, 442, 462, 493. $$
Figure~\ref{cremona} shows the proportion of the number of curves in the
Stein-Watkins database to the number of curves in Cremona's database,
as a function of the conductor.
\psfrag{2e4}{\raisebox{-5pt}{\tiny $2\!\cdot\!10^4$}}
\psfrag{4e4}{\raisebox{-5pt}{\tiny $4\!\cdot\!10^4$}}
\psfrag{6e4}{\raisebox{-5pt}{\tiny $6\!\cdot\!10^4$}}
\psfrag{8e4}{\raisebox{-5pt}{\tiny $8\!\cdot\!10^4$}}
\psfrag{1e5}{\raisebox{-5pt}{\tiny $10^5$}}
\psfrag{1.2e5}{\raisebox{-5pt}{\tiny $1.2\!\cdot\!10^5$}}
\graph{proportion_of_cremona}
{Proportion of Cremona's Curves obtained by Stein
\newline\mbox{}\hspace{4.9em}
and Watkins for $N\leq 120000$\label{cremona}}
The rank distribution of Cremona's curves is given in Table~\ref{cremona2}.
The average rank for Cremona's curves is about~$0.688$.
This is smaller than the average rank in other data sets we consider
(and is probably explainable via the real period considerations
of the last section), but we prefer to
highlight the results from other data sets.
\begin{table}[ht]
\caption
{Rank Distribution of All Curves of Conductor $\leq 120000$\label{cremona2}}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
{\bf Rank} & 0 & 1 & 2 & 3\\\hline
%Number &216544& 267564 & 43192 & 355\\\hline
{\bf Proportion} & 0.404 & 0.505 & 0.090 & 0.001\\\hline
{\bf Proportion $\Delta > 0$} & 0.408 & 0.503 & 0.087 & 0.001\\\hline
{\bf Proportion $\Delta < 0$} & 0.401 & 0.506 & 0.092 & 0.001\\\hline
\end{tabular}
\end{center}
\end{table}
As noted above, when ordering by conductor there is presently no
consensus guesstimate for the number of curves up to $X$.
Cremona has commented that there is approximately linear growth
in the number of curves of conductor less than~$120000$,
and extrapolating this gives a prediction close to 650 million
for the number of curves with~$N\le 10^8$.
D.\thinspace{}J. Bernstein suggested that
we try to quantify the completeness of
the Stein-Watkins database by considering what percentage of Cremona's
curves would be obtained by using their search methods with a smaller
discriminant bound. That is, for a parameter~$B$, if we find all
curves with $N\le B^2$, $|\Delta|\le B^3$, and~$c_4\le 100\cdot(12B)^2$,
and then take all isogenous curves and twists of these
with conductor less than~$B^2$, what percentage of Cremona's curves do
we obtain? With~$B=300$, we get 246532 curves, while Cremona has
592519 curves of conductor $\leq 90000=300^2$, so we get about~42\%.
Applying this percentage to the Stein-Watkins database with~$B=10^4$,
this would suggest that there are about 325 million elliptic curves
with conductor less than~$10^8$.
So the two guesses differ by a factor of two,
exemplifying our ignorance on so basic an issue.
\section{Average Ranks: Graphs of Data}
This section contains graphs that at a glance suggest that the
minimalist principle is contradicted by the data for curves of
conductor~$\leq 10^8$; indeed, particularly in
Figure~\ref{fig:avgrank}, we see that {\em the average rank is increasing!}
However, for prime conductor~$\leq 10^{10}$ the average rank drops,
though only slightly from 0.978 to 0.964. With some imagination,
the distribution of rank for prime conductor might appear to support the
minimalist conjecture that the average rank is~$0.5$.
Table~\ref{tbl:avg_ranks}
gives the average rank for various collections of curves that are
described in more detail elsewhere in this paper and
section.
\begin{table}[ht]
\caption{Average Ranks\label{tbl:avg_ranks}}
\begin{tabular}{|l|r|}\hline
Cremona's curves of conductor $\leq 120000$ & 0.688 \\\hline
All Stein-Watkins curves of conductor $\leq 10^8$ & 0.865 \\\hline\hline
Brumer-McGuinness curves of prime conductor $\leq 10^{8}$ & 0.982\\\hline
Stein-Watkins curves of prime conductor $\leq 10^{10}$ & 0.964\\\hline
Selected curves of prime conductor near $10^{14}$ with $\Delta<0$
& 0.869\\\hline
Selected curves of prime conductor near $10^{14}$ with $\Delta>0$
& 0.938\\\hline
\end{tabular}
\end{table}
In this section when we write elliptic curves with
property~$P$, we mean elliptic curves in the Stein-Watkins
database with property~$P$.
\psfrag{Rank 0}{Rank 0}
\psfrag{Rank 1}{Rank 1}
\psfrag{Rank 2}{Rank 2}
\psfrag{Rank 3}{Rank 3}
\psfrag{e11}{\raisebox{-5pt}{\tiny $e^{11}$}}
\psfrag{e12}{\raisebox{-5pt}{\tiny $e^{12}$}}
\psfrag{e13}{\raisebox{-5pt}{\tiny $e^{13}$}}
\psfrag{e14}{\raisebox{-5pt}{\tiny $e^{14}$}}
\psfrag{e15}{\raisebox{-5pt}{\tiny $e^{15}$}}
\psfrag{e16}{\raisebox{-5pt}{\tiny $e^{16}$}}
\psfrag{e17}{\raisebox{-5pt}{\tiny $e^{17}$}}
\psfrag{e18}{\raisebox{-5pt}{\tiny $e^{18}$}}
\psfrag{e19}{\raisebox{-5pt}{\tiny $e^{19}$}}
\psfrag{e20}{\raisebox{-5pt}{\tiny $e^{20}$}}
\psfrag{e21}{\raisebox{-5pt}{\tiny $e^{21}$}}
\psfrag{PosDel}{$\Delta < 0$}
\psfrag{NegDel}{$\Delta > 0$}
\graph{allcurves_rank_all}
{Average Rank of Stein-Watkins
Curves of Conductor $\leq 10^8$\label{fig:avgrank}}
\graph{allcurves_rank_distribution}
{Rank Distribution of Stein-Watkins Curves with
$N\leq 10^8$\label{fig:rank_dist}}
\subsection{Curves Ordered By Conductor}
The average rank of all Stein-Watkins curves
with conductor $\leq 10^8$ is about~$0.87$.
Figure~\ref{fig:avgrank} gives the average rank as a function of
log of the conductor, and also the average rank for curves
of positive and negative discriminant.
We created this graph by computing the average
rank of curves of conductor up to $n\cdot 10^5$ for $1\leq n \leq 1000$.
Figure~\ref{fig:rank_dist} graphs the proportion of curves
with each rank $0$, $1$, $2$, and $3$, as a function of log
of the conductor, all on a single graph.
The overall rank proportions are in Table~\ref{tbl:rankprop}.
\begin{table}[ht]
\caption{Rank Distribution
for Stein-Watkins Curves with $N\leq 10^8$\label{tbl:rankprop}}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}\hline
{\bf Rank} & 0 & 1 & 2 & 3 & $\geq 4$\\\hline
{\bf Proportion} & 0.336 & 0.482 & 0.163 & 0.019 & 0.000\\\hline
{\bf Proportion $\Delta > 0$} & 0.331 & 0.480 & 0.168 & 0.020& 0.000\\\hline
{\bf Proportion $\Delta < 0$} & 0.339 & 0.482 & 0.160 & 0.018& 0.000\\\hline
\end{tabular}
\end{center}
\end{table}
\psfrag{PosDel}{$\Delta > 0$}
\psfrag{NegDel}{$\Delta < 0$}
\graph{prime_conductor_rank_all}
{Average Rank of Curves with prime $N\leq 10^{10}$\label{fig:prime_rank}}
\subsection{Prime Conductor Curves}
The average rank for the curves of prime conductor~$\leq 10^{10}$
is about~$0.964$; see Table~\ref{tbl:optrankprime} for
the rank distribution. Figure~\ref{fig:prime_rank}
plots the average rank of curves of prime conductor $\leq 10^{10}$
as a function of log of the conductor.
Note that here the average ranks are decreasing,
unlike in Figure~\ref{fig:avgrank}.
%
\graph{prime_conductor_rank_distribution_0}
{Rank Distribution of Curves with prime $N\leq 10^{10}$
\label{fig:prime_rank_dist}}
%
\begin{table}[ht]
\caption
{Rank Distribution for Prime Conductor $\leq 10^{10}$\label{tbl:optrankprime}}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}\hline
{\bf Rank} & 0 & 1 & 2 & 3 & $\geq 4$\\\hline
{\bf Proportion} & 0.309 & 0.462 & 0.188 & 0.037 & 0.004 \\\hline
{\bf Proportion $\Delta > 0$} & 0.291 & 0.457 & 0.204 & 0.044 & 0.004 \\\hline
{\bf Proportion $\Delta < 0$} & 0.320 & 0.465 & 0.179 & 0.033 & 0.003 \\\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{An experiment}
The data of \cite{brumer-mcguinness} and \cite{stein-watkins:ants5}
for curves of prime conductor up to $10^8$ and $10^{10}$ show very
little drop in the observed average rank. To investigate the possibility
that the average rank might not decrease much below $0.964$
we chose a selection of curves with
prime conductor of size~$10^{14}$. It is non-trivial to get a good
data set, since we must take congruence conditions on the elliptic
curve coefficients and the variation of the size of the real period
into account; see \cite{watkins:rankheuristics} for more details
on how to account for this.
Our data sets contained 89913 curves of positive prime discriminant,
and 89749 similar curves with negative discriminant, with $|\Delta|$
near $10^{14}$ for all the curves.
It then took a few months to compute the analytic rank for these curves.
We found that for positive discriminant
the average analytic rank is approximately 0.937 and
for negative discriminant it is approximately 0.869 (see
Table~\ref{tbl:rank14} for more details). Note that this is
significantly less than the average rank found in \cite{brumer-mcguinness}
and~\cite{stein-watkins:ants5}. It could be said that this is the strongest
numerical evidence yet for the Minimalist Conjecture, though,
it is still very weak.
Incidentally, the largest rank found in any of these data sets is $6$.
\begin{table}[ht]
\caption{Rank Distribution For a Selection of Curves With
\newline\mbox{}\hspace{4.4em} Prime Conductor Near $10^{14}$
\label{tbl:rank14}}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}\hline
{\bf Rank} & 0 & 1 & 2 & 3 & $\geq 4$ \\\hline
{\bf Proportion $\Delta>0$} & 0.319 & 0.467 & 0.176 & 0.034 & 0.004
\\\hline
{\bf Proportion $\Delta<0$} & 0.343 & 0.475 & 0.154 & 0.025 & 0.002
\\\hline
\end{tabular}
\end{center}
\end{table}
Let $f(\Delta)$ be the ``probability'' that $L(E,1)=0$ for an even
parity curve of discriminant near $\Delta$ for $\Delta$ positive. For
example, Tables~\ref{tbl:optrankprime} and~\ref{tbl:rank14} suggests that
\begin{align*}
f(10^{10}) & \sim \frac{0.204 + 0.004}{0.291 + 0.204 + 0.004}
= 0.417\ldots\\
f(10^{14}) &\sim \frac{0.176 + 0.004}{0.319 + 0.176 + 0.004}
= 0.361\ldots
\end{align*}
(Note that we approximated $f(10^{10})$ using
data for all $|\Delta| < 10^{10}$.)
Motivated by the discussion in Section~\ref{sec:conjecture},
we might heuristically approximate this probability function by
$\hat f(\Delta)=c \cdot (\log\Delta)^{3/8}/\Delta^{1/24}$,
where $\Delta^{1/24}$ comes about as the square root of the
``typical'' real period. The value of
$\hat f(10^{10})/\hat f(10^{14})$ is about $1.29$,
which is not ridiculously far from the observed ratio of
$$ \frac{f(10^{10})}{f(10^{14})}\sim\frac{0.417}{0.361} \sim 1.16.$$
\subsection{Variants}
We also carried out computations similar to the ones described above
when counting isogeny classes instead of isomorphism classes of curves
(isogeny is a coarser equivalence relation than isomorphism,
grouping together curves between which there is a finite degree morphism).
In our data the average size of
isogeny classes for all curves of conductor up to $X$
converges reasonably quickly to $1$ (Duke has shown \cite{duke}
that this is indeed the case under a different ordering).
Thus the data and graphs look almost identical to those presented above.
Table~\ref{tab:variants}
gives rank data for other subsets of the Stein-Watkins database
of curves of conductor $\leq 10^8$.
In the table,
``has CM'' refers to curves that have complex multiplication, i.e.,
whose endomorphism ring (over $\C$) is bigger than $\Z$.
\begin{table}[ht]
\caption{Distribution of Rank in Various Subsets of the\newline
\mbox{}\hspace{4.5em}
Stein-Watkins Database with Conductor $N<10^8$\label{tab:variants}}
\begin{tabular}{|l|r|c|c|c|c|}\hline
{\bf Description} & {\bf Number} & {\bf Rank 0} &
{\bf Rank 1} & {\bf Rank 2} & {\bf Rank $\geq 3$} \\\hline
All Curves & 136832795 &
0.336 & 0.482 & 0.163 & 0.019 \\\hline
All Isogeny Classes & 115821258 &
0.328 & 0.480 & 0.171 & 0.021\\\hline
Has Isogeny & 38599162 &
0.375 & 0.492 & 0.125 & 0.008\\\hline
Has nontrivial torsion & 35249448 &
0.373 & 0.492 & 0.127 & 0.008\\\hline
$N$ squarefree & 21841534 &
0.296 & 0.467 & 0.202 & 0.034\\\hline
Has Full 2-torsion & 1674285 &
0.392 & 0.496 & 0.107 & 0.005\\\hline
$N$ is square & 538558 &
0.416 & 0.496 & 0.084 & 0.004\\\hline
$N$ is prime & 312435 &
0.303 & 0.460 & 0.197 & 0.041\\\hline
Has 3-torsion & 184590 &
0.422 & 0.498 & 0.078 & 0.002\\\hline
Has CM & 135226 &
0.411 & 0.498 & 0.087 & 0.005\\\hline
$N$ is prime squared & 517 &
0.439 & 0.480 & 0.072 & 0.010\\\hline
\end{tabular}
\end{table}
\section{How can we {\em systematically} account
for the Mordell-Weil rank we have already computed?}
Forget all questions of asymptotics. Consider only the curves of prime
conductor up to $10^{10}$ in our data. Is there an argument other
than just computing ranks for each of the elliptic curves in the
databases---is there a pure thought heuristic---that explains why we
are witnessing so much Mordell-Weil rank? In a sense, these rational
points are both analogous, and not analogous, to the physicist's dark
matter.\footnote{
The original idea is due to Zwicky \cite{zwicky};
recent SDSS and WMAP data \cite{WMAP} seem to confirm its existence,
though there are still some doubters (such as \cite{no-dark-matter}).}
This large mass of rational points for elliptic curves of prime
conductor $\leq 10^{10}$ is palpably there. We aren't in the dark about
that. We are merely in the dark about how to give a satisfactory
account of it being there, other than computing instances, one
after another.
We are, in a word, just at the very beginning of this story.
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\end{document}