\page{
\heading{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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\rput[lb](5,2){{\dgreen EXAMPLES}}
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\rput[lb](4,0){{\Large $x^3 + y^3 + 60z^3=0$} (projective)}
\rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}}
\rput[lb](4,-2){{\Large $3x^3+4y^3+5z^3=0$}}
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} % end page

\page{
\heading{The Group Operation}
\vspace{1ex}

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\rput[lb](4,-3){\bf The set of rational points }
\rput[lb](4,-3.7){\bf on $E$ forms an \dred abelian group.}
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} % end page


\page{
\heading{The First $150$ Multiples of $(0,0)$}
\vspace{1ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Graph: group2
%% (Contact: William Stein, http://modular.fas.harvard.edu)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\rput[lb](4,2){(The bluer the point, the}
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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{
\heading{Mordell's Theorem}
\vspace{2ex}
\psset{unit=1.0}
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\endpspicture
{\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 $T$.
\vfill

\rd{Folklore conjecture:}
$r$ can be arbitrary.  (The record is $\geq 24$.)
} % end page


------
\page{
\heading{Motivating Problem}
\vfill
\bd{Motivating Problem.}
Verify the full BSD conjecture for specific curves.

\par\noindent{}(NOTE: I am also very interested in the same question but
for modular abelian varieties.  I don't know a single nontrivial
example of $\dim \geq 3$ that has been done to completion.)
\vfill }



------

\page{
\heading{Our Strategy}
\begin{enumerate}
\item{}[\rd{Kolyvagin}] Study computation of Kolyvagin's bound
(computation of Heegner points)
\item{}[\rd{Refine}] \label{step:refine} Prove a refinement of 
\underline{Kolyvagin's 
bound} on $\#\Sha(E)$ that is 
suitable for computation. 
\item{}[\rd{Kato}] Study computation of Kato's bound (Grigorov)
when $r_\an=0$.
\vspace{-2ex}

  
\end{enumerate}
}

\page{
\heading{What Happens}
I will assume you know about Heegner points now.

Go through bsdalg paper and summarize ``what happens''.

computing images of Galois reps.

computing bounds on heights of Heegner points using search and CPS

refining Kolyvagin in some important cases

$3$-descent

every case gets dealt with one way or another
}




\page{
\heading{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 $c_3 = 7$.
Our computation using Kolyvagin implies that
$$
  \#\Sha(E) \mid 49.
$$
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?]

%\rd{Greenberg's suggestion:} 
%Compute a $p$-adic $L$-function, a $p$-adic regulator, and use
%theorems of Kato and Peter Schneider to show that $7\nmid
%\#\Sha(E)$.  I hope to do this soon.

%\end{itemize}
}


\end{document}


\page{
\heading{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}
}