\documentclass[12pt, landscape]{article}
%\documentclass[slides, landscape]{article}
%\documentclass[slides]{article}
\usepackage{fullpage}
\usepackage{fancybox}
\usepackage{graphicx}
\author{William Stein\\
Harvard University\\
{\tt http://modular.fas.harvard.edu/talks/bsd2005/}}
\date{AMS Session: January 8, 2005}
\include{macros}
\renewcommand{\dual}{\vee}
\usepackage[hypertex]{hyperref}
\setlength{\fboxsep}{1em}
\setlength{\parindent}{0cm}
\usepackage{pstricks}
\usepackage{graphicx}
\newrgbcolor{dblue}{0 0 0.8}
\renewcommand{\hd}[1]{\begin{center}\LARGE\bf\dblue #1\vspace{-2.5ex}\end{center}}
\newrgbcolor{dred}{0.7 0 0}
\newrgbcolor{dgreen}{0 0.3 0}
\newcommand{\rd}[1]{{\bf \dred #1}}
\renewcommand{\defn}[1]{{\bf \dblue #1}}
\newrgbcolor{purple}{0.3 0 0.3}
\renewcommand{\magma}{{\purple\sc Magma}}
\newcommand{\bd}[1]{{\bf\dred #1}}
\newcommand{\mysection}[1]{\newpage\begin{center}{
\section{\Huge{\dblue #1}}}\end{center}}
\newcommand{\mysubsection}[1]{{\newpage\subsection{\Huge{\dblue #1}}}}
\theoremstyle{plain}
\newtheorem{listing}[theorem]{Listing}
\title{\Huge\blue Verifying the Birch and
Swinnerton-Dyer Conjecture for\\Specific Elliptic Curves: A Status Report}
\begin{document}
\sf\Large
\maketitle
\begin{abstract}
\vspace{1ex}
\LARGE
\noindent{}%
This 30-minute talk reports on a project to verify the Birch and Swinnerton-Dyer
conjecture for all elliptic curves over~$\Q$ in Cremona's book. \vspace{2ex}
\noindent{\bf Joint Work:} {\Large Stephen Donnelly, Andrei Jorza, Stefan Patrikas,
Michael Stoll. }
\vfill
\noindent{\bf Thanks:} {\Large John Cremona, Ralph Greenberg, Grigor Grigorov,
Barry Mazur, Robert Pollack, Nick Ramsey, and Tony Scholl.}
%\noindent{}\rd{Please interrupt me and ask questions or make comments!!}
\end{abstract}
\mysection{The Birch and Swinnerton-Dyer Conjecture}
\vfill
\bd{BSD Conjecture:}
Let $E$ be an elliptic curve over~$\Q$, and
let $r=r_{\an} = \ord_{s=1} L(E, s)$.
Then $$r_{\an} = \text{rank}\, E(\Q)$$
and
$$
\frac{L^{(r)}(E,1)}{r!}
= \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} c_p }
{\#E(\Q)_{\tor}^2} \cdot \#\Sha(E).
$$
\vfill
Notation:
\begin{center}
\framebox{\begin{minipage}{0.7\textwidth}
\begin{enumerate}
\item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$
\item $\#E(\Q)_{\tor}$ -- \rd{torsion} order
\item $c_p$ -- \rd{Tamagawa numbers}
\item $\Omega_E$ -- \rd{real volume} $\int_{E(\R)} \omega_E$
\item $\Reg_E$ -- \rd{regulator} of $E$
\item $\Sha(E) = \Ker(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E))$ -- \rd{Shafarevich-Tate group}
\end{enumerate}
\end{minipage}
}
\end{center}
\mysection{Birch and Swinnerton-Dyer}
\begin{center}
\includegraphics[height=0.85\textheight]{pics/bsd1}
Birch and Swinnerton-Dyer in Leiden, Netherlands, Summer 2000.
\end{center}
\mysection{Motivating Problem 1}
\bd{Motivating Problem 1.}
Compute every quantity in the
BSD conjecture \rd{\em in practice.}
\vfill
NOTE: 1. This is \rd{not} meant as a theoretical problem about computability,
though by compute we mean ``compute with proof.''\\
2. I am also very interested in the same question but for modular
abelian varieties.
\vfill
\vspace{1ex}
\bd{STATUS:}
\vfill
\begin{enumerate}
\vfill
\item When $r_{\an} =\ord_{s=1}L(E,s) \leq 3$, then we can compute $r_{\an}$.\\
\rd{Open Problem:} Show that $r_{\an}\geq 4$ for some elliptic curve $E$.
\vfill
\item Easy to compute $\#E(\Q)_{\tor}$, $c_p$, $\Omega_E$.
\vfill
\item Computing $\Reg_E$ is same as computing $E(\Q)$;
interesting (sometimes very difficult)
\vfill
\item Computing $\#\Sha(E)$ is very very difficult.\\
\rd{Theorem (Kolyvagin):} $r_{\an}\leq 1 \, \implies$
$\Sha(E)$ is finite (with bounds)\\
\rd{Open Problem:} Prove that $\Sha(E)$ is finite for
some $E$ with $r_{\an}\geq 2$.
\end{enumerate}
\vfill
\mysection{Victor Kolyvagin}
\vfill
\begin{center}
Kolyvagin's work on Euler systems is crucial to our project.
\vfill
\includegraphics[height=0.8\textheight]{pics/kolyvagin-ny}
Kolyvagin in New York's Chinatown, 2003.
\end{center}
\vfill
\mysection{Motivating Problem 2: Cremona's Book}
\vfill
\bd{Motivating Problem 2.} Prove the BSD Conjecture for
every elliptic curve over~$\Q$ of conductor at most $1000$,
i.e., every curve in Cremona's book.
\vfill
\bd{We have:}
\begin{enumerate}
\item
By Tate's isogeny invariance theorem,
it suffices to prove BSD for each $X_0(N)$-\rd{optimal}
elliptic curve of conductor $N\leq 1000$.
\item Rank part
of the conjecture has been verified by
Cremona for all curves with $N\leq 25000$.
\item All of the quantities in
the conjecture, except for $\#\Sha(E/\Q)$, have been computed by
Cremona for all curves of conductor $\leq 25000$.
\item \bd{Cremona (Ch.~4, pg.~106):}
We have
$\Sha(E)[2]=0$ for \rd{all} optimal curves with conductor $\leq 1000$
except 571A, 960D, and 960N.
So we can mostly ignore $2$ henceforth.
\end{enumerate}
\vfill
\mysection{John Cremona}
\begin{center}
John Cremona's software and book are crucial to our project.
\vfill
\includegraphics[height=0.8\textheight]{pics/cremona}
\vfill
Cremona in Nottingham, UK, 2001.
\end{center}
\mysection{The Four Nontrivial $\Sha$'s}
\bd{Conclusion:} In light of Cremona's book, the
problem is to show that $\Sha(E)$ is {\em trivial}
for all but the following four
optimal elliptic curves with conductor at most $1000$:
\vfill
\begin{center}
\begin{tabular}{|c|l|c|}\hline
Curve & $a$-invariants & $\Sha(E)_?$\\\hline
571A& [0,-1,1,-929,-105954] & 4\\
681B&[1,1,0,-1154,-15345] & 9\\
960D& [0,-1,0,-900,-10098] & 4\\
960N& [0,1,0,-20,-42] & 4\\\hline
\end{tabular}
\end{center}
\bd{Divisor of Order:}
\begin{enumerate}
\item Using a $2$-descent we see
that $4\mid \#\Sha(E)$ for 571A, 960D, 960N.
\item For $E=681B$: Using visibility
(or a $3$-descent) we see that $9\mid \#\Sha(E)$.
\end{enumerate}
\vfill
\bd{Multiple of Order:}
\begin{enumerate}
\item For $E=681B$, the mod~$3$ representation is surjective,
and $3\mid\mid [E(K):y_K]$ for $K=\Q(\sqrt{-8})$, so (our refined)
Kolyvagin theorem implies that $\#\Sha(E)=9$, as required.
\item Kolyvagin's theorem and computation $\implies$ $\#\Sha(E) = 4^?$
for 571A, 960D, 960N.
\item
Using MAGMA's FourDescent command,
we compute $\Sel^{(4)}(E/\Q)$ for 571A, 960D, 960N
and deduce that $\#\Sha(E)=4$. (Note: Documentation currently
misleading.)
\end{enumerate}
\vfill
\mysection{The Eighteen Optimal Curves of Big Rank}
There are $18$ curves with conductor $\leq 1000$ and rank $\geq 2$
(all have rank~$2$):
%was@form:~/people/cremona/data$ awk '$5==2 && $1<=1000 {print $1$2" & "$4"\\\\"}' curves.1-8000
\vfill
\begin{center}
389A,
433A,
446D,
563A,
571B,
643A,
655A,
664A,
681C,\\
707A,
709A,
718B,
794A,
817A,
916C,
944E,
997B,
997C
\end{center}
\vfill
For these~$E$ \rd{nobody} currently knows how to show that
$\Sha(E)$ is finite, let alone trivial. (But mention, e.g., Perrin-Riou's
work.)
\vfill
\bd{Motivating Problem 3:}
Prove the BSD Conjecture for all elliptic
curve over~$\Q$ of conductor at most $1000$ and rank $\leq 1$.
\vfill \bd{SECRET MOTIVATION:} Our actual motivation is to
unify and extend results about BSD and
algorithms for elliptic curves. The computational challenge is just
to see what interesting phenomena occur in the data.
\mysection{The Plan}
\bd{The Dataset:}
\begin{itemize}
\item
There are $2463$ optimal curves of conductor at most $1000$.
\item Of these,
$18$ have rank~$2$, which leaves~$2445$ curves.
\item Of these, $2441$ are conjectured to have trivial $\Sha$.
\end{itemize}
%Our goal
%is to prove that $\#\Sha(E)=1$ for $2441$ elliptic curves.
\bd{Strategy:}
\begin{enumerate}
\item{}[Refine] \label{step:refine} Prove a refinement of Kolyvagin's
bound on $\#\Sha(E)$ that is
suitable for computation. Also take
into account refinement of Kato's theorem (Kato assumes $r_{\an}=0$).
\item{}[Algorithm] \label{step:alg} Create an Algorithm:
\begin{quote}
{
\par\noindent{}Input: An elliptic curve over $\Q$ with $r_{\an}\leq 1$.
\par\noindent{}Output: Odd $B \geq 1$ such that if
$p\nmid 2B$, then $p\nmid \#\Sha(E)$.
}
\end{quote}
\item{}[Compute] \label{step:implement} Run the algorithm on our $2441$ curves.
\item{}[Descent] \label{step:analysis}
If $p\mid B$ and $E[p]$ is reducible,
use $p$-descent.
\item{}[New Methods] If $p\mid B$ and $E[p]$ irreducible,
{\purple ????????}. Kato when $r_{\an}=0$. When $r_{\an}=1$,
maybe use Schneider's theorem
and explicit computations with heights and $p$-adic $L$-functions?
Visibility and level lowering? Further refinement of Kolyvagin's
theorem?
\end{enumerate}
\mysection{The Algorithm of Step 2}
INPUT: An elliptic curve~$E$ over $\Q$ with $r_{\an} \leq 1$.\\
OUTPUT: Odd $B\geq 1$ such that if $p\nmid 2B$, then
$\Sha(E/\Q)[p]=0$.
\begin{enumerate}
\item{}{\purple [Choose $K$]} Choose TWO distinct
quadratic imaginary fields $K_1$ and $K_2$ that both
satisfy the Heegner hypothesis and such that $E/K_1$
and $E/K_2$ have analytic rank 1.
\item{}{\purple [Find $p$-torsion]} Decide for which primes $p$ there is a
curve $E'$ that is $\Q$-isogenous to $E$ such that $E'(\Q)[p]\neq 0$.
Let $A$ be the product of these primes.
%\item{}{\purple [Functional Equation]} Compute the sign $w$ in
%the functional equation for $L(E,s)$.
\item{}{\purple [Compute Mordell-Weil]}
%Using Heegner points, $2$-descent, or $4$-descent:
\begin{enumerate}
\item If $r_{\an}=0$, compute a generator $z$ for $E^D(\Q)$ modulo torsion.
\item If $r_{\an}=1$, compute a generator $z$ for $E(\Q)$ modulo torsion.
\end{enumerate}
\item{}{\purple [Height of Heegner point]} Compute the
height $h_K(y_K)$, e.g., using the Gross-Zagier formula.
\item{}{\purple [Index of Heegner point]}
Compute $I_K = \sqrt{h_K(y_K)/h_K(z)} = [E(K)_{/\tor} : \Z y_K].$
\item{}{\purple [Refined Kolyvagin]}
Output $B = A \cdot I_K$.
\end{enumerate}
\vfill
\bd{Theorem (our refinement of Kolyvagin):}
$p\nmid 2B \implies \Sha(E/\Q)[p] = 0.$
\vfill
\mysection{Result of Running the Algorithm}
% It appears that the one case in which $p\mid B$ but there is no
% rational $p$-isogeny and $\Sha(E/\Q)[p]=0$ is when $p$ divides some
% Tamagawa number and $E$ has rank $1$ (when $E$ has rank $0$, a
% theorem of Kato applies).
\vfill
\begin{itemize}
\item Using \magma{} and the MECCAH cluster,
I implemented and ran the algorithm on the
curves of conductor $\leq 1000$, but stopped
runs if they took over 30 minutes.
\item
The computation
for $318$ curves didn't finish. We
do not include them below. Also, I don't trust some of
\magma{}'s elliptic curves functions, since the documentation
is unclear. However, we assume correctness
for the rest of this talk.
\end{itemize}
\vfill
\bd{Results:}
\begin{enumerate}
\item For $1363$ curves we have $B=1$. For these curves
we have proved the full BSD conjecture!
\item There are $94$ curves for which $B\geq 11$. Of
these, only $6$ have rank~$0$.
\item There are $39$ curves for which $B\geq 19$,
for {\em all} of these curves the rank is $1$.
\item The largest $B$ is $77$, for the rank~$1$
curves 618F and 894G.
\item The largest prime divisor of any $B$ is $31$,
for the rank~$1$ curve 674C.
\item When the rank of $E$ is $0$, the algorithm is much more
difficult, so more likely to time out.
\end{enumerate}
\mysection{Major Obstruction: Big Tamagawa Numbers}\label{sec:level}
\vfill
\bd{Serious Issue:} The Gross-Zagier formula and the BSD conjecture
together imply that if
an odd prime $p$ divides a Tamagawa number, then
$p\mid [E(K) : \Z y_K]$.
\vfill
\begin{itemize}
\item
If $E$ has $r_{\an}=0$, and $p\geq 5$, and $\rho_{E,p}$ is surjective,
then Kato's theorem (and Mazur, Rubin, et al.) imply that
$$\ord_p(\#\Sha(E)) \leq \ord_p(L(E,1)/\Omega_E),$$
so squareness of $\#\Sha(E)$ frequently saves us.
\vfill
\item
Unfortunately, in many cases there is a big
Tamagawa number and $r_{\an}=1$.
\end{itemize}
\vfill
\mysection{An Example}
\vfill
The elliptic curve $E$ called
141A and given by $y^2 + y = x^3 + x^2 - 12x + 2$ has rank 1
and $c_3 = 7$.
We know that
$$
\Sha(E) = 49^?.
$$
The representation $\rho_{E,7}$ is surjective, but~$E$ has rank~$1$.\\
\vfill
\begin{itemize}
\item{}[Visibility?]
The Jacobian $J_0(47)$ is of rank
$0$ and is simple of dimension $4$, and we find that $E[7]$ sits in
the old subvariety of $J_0(3\cdot 47)$.
Hope: Proving
something about the Shafarevich-Tate group of the simple rank $0$ abelian
variety $J_0(47)$ will imply something about $\Sha(E)[7]$.
Note that $L(J_0(47),1)/\Omega = 16/23$.
\vfill
\item{}[$p$-Adic Approach?]
Maybe a $p$-adic $L$-function computation will imply that
$7\nmid \#\Sha(E)$???
\end{itemize}
\vfill
\mysection{What Next?}
\vfill
\begin{enumerate}
\vfill
\item{}[Efficiency] Make the algorithm more efficient. The reason we chose two fields
is so we can weaken the surjectivity hypothesis that
Kolyvagin (or at least Gross, in his article) imposed. However,
in many cases one does have surjectivity and could directly use
Kolyvagin's theorem. Also Byungchul Cha's 2003
Johns Hopkins Ph.D. thesis weakens Kolyvagin's
hypothesis in another way. Combining all this should speed up
the algorithm significantly when $r_{\an}=0$.
\vfill
\item{}[Finish!] Run the algorithm to completion on all curves of conductor
up to $1000$. The hard part is finding the full Mordell-Weil group
of rank~$1$ curves of the form $E^D$, where $D$ has $3$ digits (so the conductor
has about $12$ digits).
\vfill
\item{}[New Theory] Find a strategy that works when $r_{\an}=1$ and $E$ has a
Tamagawa number $\geq 5$. Either refine Kolyvagin, use visibility and level lowering,
or Schneider and Kato's results on the $p$-adic main conjecture.
\end{enumerate}
\vfill
\begin{center}
\Huge Questions?
\end{center}
\end{document}
\mysection{More Examples}
\begin{itemize}
\item 190A1: We have $190=2\cdot 5\cdot 19$ and $c_{2}=11$. There
is a $4$-dimensional abelian variety over rank $0$ and level $95$
with $\Sha[11]$ trivial that contains $E[11]$.
\item 214A1: We have $214=2\cdot 107$ and $c_{2}=7$. There is
a rank $0$ simple abelian variety over level $107$ and dimension $7$
that contains $E[7]$.
\item 674C1: We have $214=2\cdot 337$ and $c_{2}=31$. For this one,
there is a rank $0$ simple abelian variety of level $337$ and
dimension $15$ that contains $E[31]$ and according to BSD has
trivial $\Sha[31]$.
\end{itemize}
TODO:
* Add picture of Birch and Swinnerton-Dyer