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%  (c)  William Stein
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\title{\bf\dred The Birch and Swinnerton-Dyer Conjecture}
\author{William Stein}
\date{\vspace{-2ex}February 11, 2005, Dartmouth\\
}
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\begin{document}

\page{
\maketitle
}

\page{
\section{The Pythagorean Theorem}

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\rput(2.6,0.8){{\LARGE $a^2+b^2=c^2$}}
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%\section{Babylonians}
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\page{
\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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% Graph: param
%% (Contact: William Stein, http://modular.fas.harvard.edu)
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\ds\text{Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\
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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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} % end page

\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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\rput[lb](4,-5){\dred What is going on here?}

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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

\page{
\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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} % 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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