%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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\\(joint with Baur Bektemirov, Barry Mazur, and Mark Watkins)}
\date{\vspace{-2ex}December 1, 2004, Harvard NTS\\
}
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\begin{document}
\begin{slide}
\maketitle
\end{slide}

\begin{slide}
\section{Part I: Arithmetic of Elliptic Curves}

\end{slide}

\begin{slide}
\section{The Pythagorean Theorem}

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\rput(2.6,0.8){{\LARGE $a^2+b^2=c^2$}}
\rput(1.9,1.8){{\LARGE $c$}}
\rput(4.3,1.3){{\LARGE $a$}}
\rput(2.2,-0.35){{\LARGE $b$}}
\rput(6,1.5){\includegraphics{pics/pythagoras15}}
\rput[cb](6.0,-0.35){Pythagoras}
\rput[cb](6.0,-0.6){Approx 569--475BC}
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\end{slide}

\begin{slide}
\section{Babylonians}

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\rput[lb](0,1){\includegraphics[width=0.53\textwidth]{pics/ybc7289-3-small}}
\rput[lb](3.5,0){1800-1600BC}
\rput[lb](13,1){\includegraphics[width=0.31\textwidth]{pics/lion}}
\rput[lb](13,8.6){\includegraphics[width=0.31\textwidth]{pics/babylon}}
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\end{slide}

\begin{slide}
\section{Pythagorean Triples}
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\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
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\rput[lb](0,0){{$\begin{array}{|c|}\hline
\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}
$}}
\endpspicture
\end{slide}

\begin{slide}
\section{Enumerating Pythagorean Triples}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Graph: param
%% (Contact: William Stein, http://modular.fas.harvard.edu)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\rput[lb](0.4, 0.95){{$(x,y)$}}

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\ds\text{Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\
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  \ds y &=\quad\! \ds\frac{2t}{1+t^2}\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{slide}



\end{document}
\section{Part II: Average Ranks}
\begin{center}
\input{graphs2/final_rank_proportion}
\end{center}
\end{slide}


\begin{slide}
\section{The Conventional Wisdom}
This talk follows up on (Stein-Watkins, Ants 5), which
reported on the computation of a huge dataset of elliptic curves, and
the Bulletins paper of Brumer-McGuinnness 
about Mordell-Weil groups 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{slide}

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

In this talk we consider 136,832,795 curves of any 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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\section{Quadratic Twists}
Let $E$ be an elliptic curve over~$\Q$.  Consider quadratic
twists $E^D$ of $E$ 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{slide}

\begin{slide}
\section{Cubic Twists of $x^3+y^3=1$}
Kramarz and Zagier considered cubic twists of
$x^3+y^3=1$ 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}:\\
\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{slide}

\begin{slide}
\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, we 
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{slide}

\begin{slide}
\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{slide}
\begin{slide}

\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{slide}

%\end{document}

% \begin{slide}
% \section{Comparison to the Brumer-McGuinness  Tables}
% We found $312,435$ curves of prime conductor up to $10^8$, which is
% $1216$ more than Brumer and McGuiness found up to that conductor.
% %\todo{What is the first curve we found but they don't?}
% \end{slide}


\begin{slide}
\section{Comparison to 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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

%\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}
\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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~$1$.  {\bf\dblue\LARGE WHY???}
\end{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}

\begin{slide}
\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{slide}
\begin{slide}

\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{slide}

\begin{slide}
\section{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}
\end{enumerate}
Despite our data, Mark Watkins has 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{slide}


\begin{slide}
\section{A Connection With Visibility?}
% \begin{conjecture}
% Ordered by conductor, pos discriminant, neg discriminant, the
% proportion of rank $2$ is $>0$ and of rank $\geq 3$ is $>0$.
% \end{conjecture}


After learning about the notion of visibility of Shafarevich-Tate
groups, Brumer asked the second author (Stein) if it might ``explain''
why there are so many curves of rank $\geq 2$.  It seems reasonable
that visibility might shed some light on this question. 

If $A$ is an abelian variety over $\Q$, the \defn{Shafarevich-Tate
group} of $A$ is
$$
  \Sha(A) = \Ker\left(\H^1(\Q,A) \to \oplus_{p\leq \infty} \H^1(\Q_p,A)\right)
$$
If $\iota: A\hra B$ is an inclusion of abelian varieties, the visible 
subgroup of $\Sha(A)$ with respect to this inclusion is
$$
 \Vis_B(\Sha(A)) = \Ker(\iota_*: \Sha(A) \to \Sha(B)).
$$
\end{slide}

\begin{slide}
\section{Heuristic One: $\Sha$ Gives Rank}
Whenever there is a nonzero visible $c\in \Sha(A_f)[p]$, for some
$A_f\subset J_0(N)$ of analytic rank~$0$ and odd~$p$, then (I think I
can prove that) there is another factor $A_g$ of $J_0(N)$ with
$\ord_{s=1}L(g,s)\geq 2$.  I think it's very reasonable that
$\Sha(A_f)$ be nontrivial and contain a visible element more than 0\%
of the time.  Also, even if $\Sha(A_f)[p]$ is nontrivial and
invisible, there is the possibility that $\Sha(A_f)[p]$ becomes
visible at infinitely many higher levels.  Moreover, the factors $A_g$
of with $\ord_{s=1}L(g,s)\geq 2$ tend to have small dimension, since
the rank of $A_g$ is unusually large.
\end{slide}

\begin{slide}
\section{Heuristic Two: Rank Gives $\Sha$}
Alternatively, one can view the data we give in this talk as a reason
to believe that there are many visible elements of in groups
$\Sha(A_f)$, for various $A_f\subset J_0(N)$.  This is because if $E$
is an elliptic curve of rank~$\geq 2$, and $p$ is a divisor of the
modular degree $m_E$ of $E$, then in many cases we {\em expect} to
obtain an injection 
$$
(\Z/p\Z)^2 \isom E(\Q)/p E(\Q) \hra
\Vis_{J_0(N)}(\Sha(B)), 
$$
where $E+B = J_0(N)$, $E\cap B = E[m_E]$
and $\Vis_{J_0(N)}(\Sha(B))$ is 
$$
\Ker(\Sha(B) \to \Sha(J_0(N))).  
$$

Thus, at least heuristically, the very unsurprising assertion that
$\Sha(A_f)$ should be nontrivial for many abelian varieties $A_f$ is
closely related to the mysterious and surprising assertion that 
there are many elliptic curves of rank at least~$2$.
\end{slide}


\begin{slide}
\vspace{2ex}
\noindent{\bf Acknowledgment.} Thanks to Armand Brumer, Frank
Calegari, and Noam Elkies for stimulating conversations.
We used PARI, C, MySQL, and Python to compute and analyze the data,
and LaTeX to graph it.  
\end{slide}



% \begin{slide}
% \section{Bibliography}
% {\tiny \bibliography{biblio}}
% \end{slide}
\end{document}