%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bmw.tex -- Ranks of Elliptic Curves
%
%  (c) 2004, 
%    Baur Bektemirov
%    William Stein
%    Mark Watkins
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\title{\bf\dred Ranks of Elliptic Curves}
\author{William Stein}
\date{\vspace{-2ex}January 27, 2005, Berkeley Colloquium\\
}
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\begin{document}

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\maketitle
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\page{
\section{The Pythagorean Theorem}

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%\section{Babylonians}
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\section{Pythagorean Triples}
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\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
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\vspace{-2ex}\\
( 3, 4, 5 )\\
( 5, 12, 13 )\\
( 7, 24, 25 )\\
( 9, 40, 41 )\\
( 11, 60, 61 )\\
( 13, 84, 85 )\\
( 15, 8, 17 )\\
( 21, 20, 29 )\\
( 33, 56, 65 )\\
( 35, 12, 37 )\\
( 39, 80, 89 )\\
( 45, 28, 53 )\\
( 55, 48, 73 )\\
( 63, 16, 65 )\\
( 65, 72, 97 )\\
( 77, 36, 85 )
\vspace{-1ex}\\\vdots \\
\hline
\end{array}
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\page{
\section{Enumerating Pythagorean Triples}

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%% (Contact: William Stein, http://modular.fas.harvard.edu)
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%\page{
%\section{\dred{Integer} and \dred{Rational} Solutions}
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\page{
\section{Elliptic Curves over the Rational Numbers $\Q$}
\vspace{-3ex}
{\Large An {\dred elliptic curve} is a nonsingular plane cubic curve with
a rational point (possibly ``at infinity'').}
\vspace{1ex}

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\page{
\section{The Secant Process}
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\page{
\section{The Tangent Process}
New rational point from a single rational point.

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\page{
\section{Iterate the Tangent Process\hspace{3in}\mbox{}}
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\page{
\section{The Group Operation}
\vspace{1ex}

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\page{
\section{The First $150$ Multiples of $(0,0)$}
\vspace{1ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\rput[lb](4,-1){{\dgreen Fact:} The group $E(\Q)$ is infinite}
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\rput[lb](4,-3){In contrast, $y^2+y=x^3-x^2$ has}
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\page{
\section{Mordell's Theorem}
\vspace{2ex}
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{\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic
curve is a {\dgreen finitely generated abelian group}, so 
$$
  E(\Q) \cong \Z^r \oplus T,
$$
with $T=E(\Q)_{\rm tor}$ finite. 

\vspace{2ex}


Mazur classified the
possibilities for $T$.  It is conjectured that 
$r$ can be arbitrary, but the biggest $r$ ever
found is (probably) $24$.
} % end page

\page{
\section{The Simplest Solution\hspace{2in}\mbox{}\\Can Be Huge\hspace{2in}\mbox{}}
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Simplest solution to $y^2=x^3+7823$:
\begin{align*}
x &= \frac{2263582143321421502100209233517777}{143560497706190989485475151904721}\\
\\
y &= \frac{186398152584623305624837551485596770028144776655756}{1720094998106353355821008525938727950159777043481}
\end{align*}
\mbox{}
\par
(Found by Michael Stoll in 2002.)

} % end page

\page{
\section{The Central Question\hspace{2em}\mbox{}}

{\dgreen\mbox{}\hspace{1em}\noindent{}Given an elliptic curve,\\
\mbox{}\hspace{1em}what is its rank?}\\
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} % end page

\page{
\section{\dred \mbox{}\hspace{4em}\LARGE Conjectures Proliferated}
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``The subject of this lecture is rather a special one.  I want to
describe some computations undertaken by myself and Swinnerton-Dyer on
EDSAC, by which we have calculated the zeta-functions of certain
elliptic curves.  As a result of these computations we have found an
analogue for an elliptic curve of the Tamagawa number of an algebraic
group; and conjectures have proliferated.  [...] though the associated
theory is both abstract and technically complicated, the objects about
which I intend to talk are usually simply defined and often machine
computable; {\dblue experimentally we have detected certain relations between
different invariants}, but we have been unable to approach proofs of
these relations, which must lie very deep.''
\hfill -- Birch 1965

} % end page

\apage{
\section{Counting Solutions Modulo $p$}
\vspace{-5ex}

$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
$$y^2 + y = x^3 - x \pmod{7}$$
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\rput[bl](-4.5,-1.5){{\tiny Point counting gnomes}}
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} % end page

\page{
\section{The \dred{Error} Term}
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Let
{\LARGE
$$
a_p = p+1 - N(p).
$$
}
Hasse proved that 
{\Huge\dblue
$$
 |a_p| \leq 2\sqrt{p}.
$$}
$$
a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1,
\quad a_{11} = -5,\quad a_{13} = -2,\quad a_{17}=0,$$
$$a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots $$
} % end page

\page{
\section{Guess}
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\vspace{-3ex}

If an elliptic curve $E$ has positive rank, then perhaps $N(p)$ is
on average larger than $p$, for many primes $p$.
Thus maybe
{\Large $$
  f_E(x) = \prod_{p\leq x} \frac{p}{N(p)} \to 0 \text{ as $x\to\infty$}
$$}
exactly when $E$ has positive rank??


}

\page{
\section{Graphs of $f(x)=\prod_{p\leq x} \frac{p}{N(p)}$}
The following are graphs, on a log scale, of $f_E(x)$:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Graph: bsdprod
%% (Contact: William Stein, http://modular.fas.harvard.edu)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}


\page{
\section{Something Better: \dgreen The $L$-Function}
{
{\dred Theorem (Wiles et al., Hecke)} The following
function extends to a holomorphic function on the
whole complex plane:
% and satisfies a functional equation
%that relates $L(E,s)$ to $L(E,2-s)$:
\Large $$
  L(E,s) = \prod_{p\nmid \Delta} 
  \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 
$$}

Note that formally,
$$
  L(E,1) = 
\prod_{p\nmid \Delta} 
  \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right)
 = 
\prod_{p\nmid \Delta} 
  \left(\frac{p}{p-a_p  + 1}\right)
= \prod_{p\nmid \Delta} 
\frac{p}{N_p}
$$
%The proof of analytic continuation uses the proof of
%the Shimura-Taniyama-Weil conjecture, whose proof was
%motivated by Fermat's Last Theorem.
} % end page

%\apage{
%\section{The Riemann Zeta Function}
%The $L$-function of an elliptic curve is analogous to
%the Riemann Zeta function.
%} % end page

\page{
\section{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
\begin{center}
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\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/lser}
\endpspicture
\end{center}

} % end page

\page{
\section{More Graphs of Elliptic Curve $L$-functions}
\vspace{6ex}

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\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/many_lser}
\endpspicture
\end{center}
} % end page

\page{
\section{The Birch and Swinnerton-Dyer Conjecture}
\begin{center}
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\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
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\end{center}
\vspace{-4ex}

{\dred Conjecture:}
Let $E$ be any elliptic curve over~$\Q$.
The order of vanishing of $L(E,s)$ as $s=1$
equals the rank of $E(\Q)$.
} % end page

\page{
\section{The Kolyvagin and Gross-Zagier Theorem}

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\end{center}
\vspace{-4ex}


{\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
then the conjecture is true for $E$.


} % end page

\apage{
\section{What Proportion of Elliptic Curves are Covered by
the Theorem?}
The rest of this talk is about the {\dred state of the art in building
  databases} of elliptic curves, and {\dblue new data about average
ranks of these curves}.  We arrange elliptic curves by either their
conductor, minimal discriminant, or naive height.  Each is an integer
invariant of an elliptic curve, and there are only finitely many
elliptic curves with that invariant. 
\begin{itemize}
\item {\dred Discriminant} of $y^2=x^3+ax+b$ is $-16(4a^3+27b^2)$. 
\item {\dblue Conductor:} Integer divisible by same primes as
  (minimal) discriminant (measures the nature of
  reduction modulo~$p$). 
\item {\dgreen Naive Height:} Measure of size of coefficients of
equation. 
\end{itemize}
} % end page


\page{
\section{What is the Average Rank of All Elliptic Curves?} 
In 1990, Brumer and MicGuinness published a paper in the Bulletins
of the AMS about ranks of elliptic curves of prime conductor. 
Brumer-McGuinness begins:
\begin{quote}
  ``The opinion had been expressed that, in general, an elliptic curve
  might tend to have the smallest possible rank, namely 0 or 1,
  compatible with the rank parity predictions of Birch and
  Swinnerton-Dyer.  We present evidence that this may not be the case. 
  [...] This proportion of rank~$2$ curves seemed too large to conform
  to {\dred the conventional wisdom}.'' 
\end{quote}
} % end page

\page{
\section{Our New Data}
Brumer and McGuinness considered 310,716
curves of prime conductor $\leq 10^8$.  

Mark Watkins, Baur Bektemirov and I consider 136,832,795 curves of 
conductor $\leq 10^8$, and 11,378,911 curves of prime conductor $\leq
10^{10}$.  The results of our rank computations are similar to those
of Brumer and McGuinness, which suggests that if one orders all
elliptic curves over~$\Q$ by their conductor, then {\dred a surprising
  number of curves have rank bigger than one}. 
} % end page

\page{
\section{Brumer and McGuinness}
Brumer and McGuinness found, by thousands
of hours of computer search on numerous Mac II's, 311,219 curves of
prime conductor $\leq 10^8$.  For 310,716 of these curves they
computed the probable rank by a combination of point searches and
computation of apparent order of vanishing of $L$-functions.  
This table summarizes the rank distribution that they found:

% total number found
%
\def\totalfound{311243}
%
% total analysed so far
%
\def\totalnumber{310716}
\def\totalpositive{113969}
\def\totalnegative{196747}
\def\rationegpos{1.726}
\def\totalodd{155658}
\def\totaleven{155058}

\def\rankzero{93337}
\def\rankzeropos{31748}
\def\rankzeroneg{61589}
\def\rankzeropct{30.04} 

\def\ranktwo{61517}
\def\ranktwopos{24706}
\def\ranktwoneg{36811}
\def\ranktwopct{{\dred 19.80}}

\def\rankfour{804}
\def\rankfourpos{377}
\def\rankfourneg{427}
\def\rankfourpct{0.26}

\def\rankevenGtZeroPct{20.06}

\def\rankone{143192}
\def\rankonepos{51871}
\def\rankoneneg{91321}
\def\rankonepct{46.08}

\def\rankthree{11861}
\def\rankthreepos{5267}
\def\rankthreeneg{6594}
\def\rankthreepct{3.82}

\def\rankfive{5}
\def\rankfivepos{0}
\def\rankfiveneg{5}
\def\rankfivepct{}

\def\rankoddGtOnePct{3.83}
{\tiny
$$\vbox{\offinterlineskip
\hrule
\halign{\vrule#&\strut\quad#\hfil\quad&#\vrule\,\vrule&%
&\quad\hfil#\hfil\quad\vrule\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&Rank && 0 & 1 & 2 & 3 & 4 & 5\cr
\noalign{\hrule}
\noalign{\vskip 2pt}
\noalign{\hrule}
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&$\Delta>0$&&\rankzeropos&\rankonepos&\ranktwopos&\rankthreepos%
 &\rankfourpos&\rankfivepos\cr
&$\Delta<0$&&\rankzeroneg&\rankoneneg&\ranktwoneg&\rankthreeneg%
 &\rankfourneg&\rankfiveneg\cr
height2pt&\omit&&&&&&&\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
\noalign{\hrule}
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&Totals&&\rankzero&\rankone&\ranktwo&\rankthree%
 &\rankfour&\rankfive\cr
&Percents&&\rankzeropct&\rankonepct&\ranktwopct&\rankthreepct%
 &\rankfourpct& \rankfivepct\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr}
height2pt&\omit&&&&&&&\cr}
\hrule}
$$}\noindent{}Let $r_\varepsilon(X)$ be the \defn{average rank} of elliptic curves
in the Brumer-McGuinness tables with conductor at most $X$ and
discriminant sign $\varepsilon$.  They observe that in their
data, \dred{$r_{+}$ climbs to $1.04$ and $r_{-}$ climbs to $0.94$}. 
} % end page

\page{
\section{Quadratic Twists}
Let $E$ be an elliptic curve over~$\Q$.  Consider quadratic
twists $Dy^2=x^3+ax+b$ of $E:y^2=x^3+ax+b$ by square-free 
integers~$D$. 
\begin{conjecture}[Goldfeld]
The average rank of the curves $E^D$ is $\frac{1}{2}$, in the
sense that 
$$
\lim_{X\to\infty} \frac{\sum_{|D|<X} \rank(E^D)}{\#\{D : |D| < X\}} = {\dred \frac{1}{2}}. 
$$
(Here the $D$ are squarefree.) 
\end{conjecture}
There are many conditional and unconditional results in the direction
of Goldfeld's conjecture (e.g., work of Katz, Sarnak, et al.).  
For a survey of these results, see Silverberg's PCMI article. 
} % end page

%\page{
%\section{Cubic Twists of $x^3+y^3=1$}
%Kramarz and Zagier considered elliptic curves
%$x^3+y^3=D$ and found that in their data that {\dblue 23.3\%} of the curves
%with even rank have rank at least $2$, and 2.2\% of those with odd
%rank have rank at least $3$.  Mark Watkins followed up on these
%computations, and found that the rank eventually starts {\em decreasing}:\\
%\vspace{3ex}
%
%\mbox{}\hspace{1em} {\dblue 23.3\%} up to $70000$, 
%{\dblue 20.5\%} up to $10^6$, 
%{\dblue 17.7\%} up to $10^7$. 

% Since only $j=0,1728$ have non-quadratic twists, taken together the
% above data and conjectures suggests that the average rank of elliptic
% curves ``ordered by $j$-invariants'' should be $1/2$, since it should
% be an average of infinitely many numbers all but two of which are
% equal to $1/2$.  We will not concern ourselves further with questions
% about twists in this talk. 
} % end page

\page{
\section{The Stein-Watkins Database}
Brumer and McGuinness fixed the $a_1$, $a_2$, $a_3$ invariants
(12 total possibilities) and then searched for
$a_4$ and $a_6$ which made $|\Delta|$ small.  

Instead, Watkins and I
break the $c_4$ and $c_6$ invariants into  congruence classes, and then
find small solutions to $c_4^3-c_6^2=1728\Delta$.  

% Write
% $c_4^\star$ for the least nonnegative residue of $c_4$ modulo~576, and
% $c_6^\star$ for the least nonnegative residue of $c_6$ modulo~1728. 
% Connell has given necessary and sufficient
% conditions on $c_4$ and $c_6$ such
% that an elliptic curve with those
% invariants exists.  We first need that $c_6\equiv 3$ (mod~4) (in which
% case it follows that $c_4$ is odd), or $2^4\mid{}c_4$ and 
% $c_6\equiv 0,8 \pmod{32}$, and secondly we require a local condition at the prime~3,
% namely that $c_6\not\equiv\pm 9 \pmod{27}$.  Using this information and
% the fact that $1728\mid{}\left(c_4^3-c_6^2\right)$, this leads to 288
% possible $\left(c_4^\star,c_6^\star\right)$ pairs. 
} % end page

\page{
\section{Generating Our Table}
\begin{enumerate}
\item
For each of the $288$ possible pairs $\left(c_4\!\mod 576,\, c_6\!\mod 1728\right)$, we
loop over $c_4$ and $c_6$, ``hoping'' to find all curves with $|\Delta|\le
10^{12}$.  We do not know a bound on $c_4$ to guarantee this.  We 
took $c_4\le 1.44\cdot 10^{12}$ in this step.  We throw
away all curves whose conductor is composite and $\geq 10^8$, or prime
and $\geq 10^{10}$. 
\item  Include all curves that are isogenous
to a curve found above. 
\item Include all twists 
with conductor $\leq 10^8$. 
\end{enumerate}
} % end page
\page{

\section{The Number of Curves That We Found}
The following table lists the number of curves in 
our database with various properties:
\begin{center}
\begin{tabular}{|l|r|}\hline
{\dred Type} & {\dred Number} \\\hline\hline
& \vspace{-2ex} \\
Curves with conductor $\leq 10^8$ & {\dred 136,832,795}\\\hline
& \vspace{-2ex} \\
Optimal curves with conductor $\leq 10^8$ & 115,821,258\\\hline
& \vspace{-2ex} \\
Curves with square-free conductor $\leq 10^8$ & 21,826,791\\\hline
\vspace{-2ex} & \\
Optimal curves with square-free conductor $\leq 10^8$ & 19,963,592 \\\hline
\vspace{-2ex} & \\
Curves with prime conductor $\leq 10^{10}$&  11,378,911\\\hline
\vspace{-2ex} & \\
Curves with prime conductor $\leq 10^{8}$&  312,435\\\hline
\end{tabular}
\end{center}
} % end page


\page{
\section{Cremona's Tables}
Cremona has enumerated {\em every} elliptic curve of conductor up to $25,000$. 
He found {\dred $103,174$} isogeny classes of elliptic curves of conductor 
up to $25,000$.  In our computation, we find {\dred $88,700$} isogeny classes
of curves of conductor up to $25,000$, so we miss $14,474$ isogeny
classes.  The first conductor where Cremona has a curve and we don't
is conductor $174$.  The curve 174A1 with
$a$-invariants $[1,0,1,-7705,1226492]$ has discriminant {\dred $-621261297432576$},
which is substantially larger in absolute value than $10^{12}$.\\
{\dred Thus we expect that our table is far from complete for composite
conductor.} 

} % end page

\page{
\section{Cremona's Rank Distribution}
Amongst Cremona's 103,174 isogeny classes, the rank distribution
is as follows:
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
{\dred Rank} & 0 & 1  & 2  & $\geq 3$\\\hline\hline
{\dred Number} & 42165 & 53483 & 7509 & 17\\\hline
{\dred Proportion} & 41\% & 52\% & {\dred 7\%} & 0\%\\\hline
\end{tabular}
\end{center}

} % end page

%\page{
%\section{Completeness of Prime-Conductor Data up to Level 234446}
%In Summer 2003, Andrei Jorza, Jennifer Balakrishna, and I verified
%using the Mestre method of graphs that our table of elliptic curves
%of prime conductor is complete for levels up to $234431$.  %234446
%We did this in order to prove that the {\dred smallest conductor of an
%elliptic curve of rank~$4$ is composite}, in contrast to the case
%for rank $0$ (conductor $11$), rank $1$ (conductor $37$), rank
%$2$ (conductor $389$), and rank $3$ (conductor $5077$), where
%in each case the minimal conductor is prime.   
%} % end page

\page{
\section{Graphs of Ranks}
For the rest of this talk, we give data about the ranks of elliptic
curves in our data.  Thus everywhere hence, when we say
``elliptic curves with property $P$'', we {\bf always} mean
``elliptic curves in the Stein-Watkins database with property $P$''. 
} % end page

\page{
\section{All Curves Ordered By Conductor}
The average rank of all curves of conductor $\leq 10^8$ 
is $0.8664\ldots$.  A graph of the
average rank as a function:
\begin{center}
\vspace{2ex}
\input{graphs2/allcurves-counts}
\vspace{2ex}
\end{center}
We created this graph by computing the average rank of curves
of conductor up to $n\cdot 10^5$ for $1\leq n \leq 1000$. 
} % end page

%\begin{conjecture}
%Suppose $R(X)$ is the average rank of all elliptic curves of conductor 
%$\leq X$.  Then there exists $X'$ such that $R(X')> R(X)$. 
%\end{conjecture}
\page{
\section{Rank Distribution}
The following graph gives the proportion of curves of rank
each rank $0$, $1$, $2$, and $\geq 3$, as a function
of the conductor, all on a single graph.  The top curve
is the proportion of rank~$1$, the second from the top is
the proportion of curves of rank~$0$, the third the proportion
of rank~$2$, and the fourth the proportion of rank $\geq 3$. 
\begin{center}
\vspace{2ex}
\input{graphs2/final_rank_distribution}
\vspace{2ex}
\end{center}
The overall proportion of each rank, i.e., the rightmost point
on each of the above graphs, is as follows:
\begin{center}
\begin{tabular}{|l|l|l|l|l|}\hline
{\bf Rank}        & 0 & 1 & 2 & $\geq 3$\\\hline
{\bf Proportion}  & 33.6\% & 48.2\% & 16.3\% & 1.9\%\\
\hline
\end{tabular}
\end{center}
} % end page

\page{
\section{Optimal Curves  Ordered By Conductor}
The average rank of optimal curves of conductor $\leq 10^8$
is $0.8855\ldots$, and the following is a graph
of the average rank as a function of the conductor:
\begin{center}
\vspace{2ex}
\input{graphs2/curves-counts}
\vspace{2ex}
\end{center}
%We created this graph by computing the average rank of optimal curves
%of conductor up to $n\cdot 10^5$ for $1\leq n \leq 1000$. 
%The data for optimal curves is similar to the data for all curves,
%since the size of an isogeny class is usually $1$ in our data. 
} % end page

\page{
\section{Curves of Squarefree Conductor}
The average rank of all curves of squarefree conductor
$\leq 10^8$ is $0.9756\ldots$.  The following
is a graph of the average rank of squarefree curves
as a function of the conductor:
\begin{center}
\vspace{2ex}
\input{graphs2/allcurves_squarefree-counts}
\vspace{2ex}
\end{center}
The data suggests that the average rank of all elliptic curves of
squarefree conductor tends to~$1$. 
} % end page

\page{
\section{Squarefree Conductor Rank Distribution }
%Proportion of curves with squarefree conductor $\leq 10^8$:
\vspace{-1ex}
\begin{center}
\vspace{2ex}
\input{graphs2/final_squarefree_rank}
\vspace{2ex}
\end{center}
The overall proportion of each rank is as follows:
\begin{center}
\begin{tabular}{|l|l|l|l|l|}\hline
{\bf Rank}        & 0 & 1 & 2 & $\geq 3$\\\hline
{\bf Proportion}  & 29.6\% & 46.7\% & 20.2\% & 3.4\%\\
\hline
\end{tabular}
\end{center}
% The proportion of curves of squarefree conductor of rank~$2$ is over
% 20\%, which is significantly larger than the proportion 16.3\% 
% of rank~$2$ curves of all conductors. 
} % end page

\page{
\section{Comparison of Average Rank Graphs}
The graph below shows the average rank graphs for four different sets
of elliptic curves up to conductor $10^8$ considered above. 
\begin{center}
\vspace{2ex}
\input{graphs2/final_rank_proportion}
\vspace{2ex}
\end{center}
} % end page

\page{
\section{Curves of Prime Conductor  Up to $10^8$}
The average rank of curves of prime conductor $\leq 10^{8}$
is near $1$.  We created the following
graph by computing the average rank up to conductor 
$n\cdot 10^5$ for $1\leq n \leq 1000$. 
\begin{center}
\vspace{2ex}
\input{graphs2/graph_average_rank-_and_is_prime_and_number1}
\vspace{2ex}
\end{center}

} % end page

\page{
\section{Curves of Prime Conductor  Up to $10^{10}$}
The average rank of curves of prime conductor $\leq 10^{10}$
is $0.964\ldots$.  The following graph shows the average
rank graphed on a {\bf logarithmic scale} (the horizontal
axis contains $\log(N)$):
\begin{center}
\vspace{2ex}
\input{graphs2/avg_rank_prime_log-x1_y2}
\vspace{2ex}
\end{center}

} % end page

\page{
\section{All Curves Ordered By Naive Height}
The naive height of $E$ is $\max\{|c_4|^3, |c_6|^2\}$.   The
following graph gives the average rank of all curves up to
a given naive height.% up to $e^{46}\approx 10^{20}$.   

\begin{center}
\vspace{2ex}
\input{graphs2/height_rank_20.txt-sorted_avg-x1_y2.tex}
\vspace{2ex}
\end{center}
Try again with log scale... 
} % end page

\page{
\section{All Curves Ordered By Naive Height}
%The naive height of $E$ is $\max\{|c_4|^3, |c_6|^2\}$.   The
%following graph gives the average rank of all curves up to
%a given naive height.% up to $e^{46}\approx 10^{20}$.   
The horizontal axis is plotted on a {\bf logarithmic scale}, 
and is labeled with the
logarithm of the naive height. 
\begin{center}
\vspace{2ex}
\input{graphs2/height_rank_20.txt-sorted_avg_log-x1_y2}
\vspace{2ex}
\end{center}
Interesting points at $2.5$ million and and $1.7\cdot 10^{14}$.\\
{\dred Theorem (Brumer).} {\em (BSD, RH, etc.) 
imply the limit is $\leq 2.3$.} 
} % end page
\page{

\section{Proportions of Each Rank Ordered By\\Naive Height}
%The naive height of $E$ is $\max\{|c_4|^3, |c_6|^2\}$.   The
%following graph gives the average rank of all curves up to
%a given naive height.% up to $e^{46}\approx 10^{20}$.   
The horizontal axis is plotted on a {\bf logarithmic scale}, 
and is labeled with the
logarithm of the naive height. 
\begin{center}
\vspace{2ex}
\input{graphs2/height_rank_20.txt-sorted_avg_prop_log-x1_y2_3_4_5_6}
\vspace{2ex}
\end{center}
Interesting points at $2.5$ million and and $1.7\cdot 10^{14}$.\\
} % end page

\page{
\section{Open Questions}
\begin{enumerate}
\item What is the average rank ordered by conductor? \vspace{-1ex}
\item What is the average rank ordered by discriminant?\vspace{-1ex}
\item What is the average rank ordered by height?\vspace{-1ex}
\item What is the proportion of curves of rank $2$, rank $3$,
rank $4$, etc.?  Are each $0$ or is every proportion positive? 
\item Why are the so many curves of rank $\geq 2$ in our data?\vspace{-1ex}
\item Is the rank unbounded?  (Standard Conjecture: Yes.) 
\end{enumerate}

Despite our data, Mark Watkins and other people have heuristics that
predict that the average rank in each case is $0.5$ and the proportion
of all curves of rank $\geq 2$ is $0$.  
} % end page


\page{
\section{Acknowledgment.} 

Armand Brumer, Frank Calegari, Keith Conrad, 
Noam Elkies, Benedict Gross, Barry Mazur, Ariel Shwayder. 

} % end page


\end{document}




$$f(10)=0.083\ldots,\quad f(100)=0.032\ldots,\quad f(1000)=0.021\ldots$$
$$f(10000)=0.013\ldots,\quad f(100000)=0.010\ldots.$$
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
$M$&  10&100&1000&10000&1000000\\\hline
$\sim f(M)$ &  0.083 & 0.032 & 0.021 & 0.013 & 0.010\\\hline
\end{tabular}
\end{center}