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\title{Lecture 34: The Birch and Swinnerton-Dyer Conjecture, Part 1}
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The next three lectures will be about the
Birch and Swinnerton-Dyer conjecture, which is
considered by many people to be the most important
accessible open problem in number theory.  Today I
will guide you through Wiles's Clay Math Institute
paper on the Birch and Swinnerton-Dyer conjecture.

On Friday, I will talk about the following open problem, which is a
frustrating specific case of the Birch and Swinnerton-Dyer conjecture.
Let~$E$ be the elliptic curve defined by
$$
  y^2 + xy = x^3 - x^2 -79x + 289.
$$
Denote by 
$L(E,s)=\sum_{n=1}^{\infty} a_n n^{-s}$
the corresponding $L$-series, which extends to a function
everywhere.  The graph of $L(E,s)$ for $s\in (0,5)$ is
given on the next page.
It can be proved that
$E(\Q) \ncisom \Z^4$
by showing that
$$
  (8,7),\ \left(\frac{120}{27},\frac{29}{27}\right),\,
\left(\frac{70}{8},\frac{81}{8}\right),\, 
\text{ and }\left(\frac{564}{8}, \frac{665}{64}\right)
$$
generate a ``subgroup of finite index'' in $E(\Q)$.
The Birch and Swinnerton-Dyer Conjecture then predicts that
$$\ord_{s=1}L(E,s)=4,$$
which looks plausible from the shape of the graph on the next page. 
It is relatively easy to prove that 
the following is equivalent to showing that
$\ord_{s=1}L(E,s)=4$:

\hd{Open Problem:}
{\sl Prove that $L''(E,1) = 0$.}\vspace{.7em}

If you could solve this open problem, people like Gross, Tate, Mazur,
Zagier, Wiles, me, etc., would be {\bf very} excited.  The related
problem of giving an example of an $L$-series with
$\ord_{s=1}L(E,s)=3$, was solved as a consequence of a very deep
theorem of Gross and Zagier, and resulting in an effective solution to
Gauss's class number problem.

John Tate gave a talk about the BSD conjecture for the Clay Math
Institute.  I strongly encourage you to watch it online at\vspace{0.5em}

\hspace{-3.5em}
\begin{minipage}{1.1\textwidth}
\begin{verbatim}
http://www.msri.org/publications/ln/hosted/cmi/2000/cmiparis/index-tate.html
\end{verbatim}
\end{minipage}



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\vspace{1.5em}

The $L$-series of the ``simplest'' known elliptic curve of rank~$4$.
\end{center}



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