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\title{Exercise for Section 2:\\The Birch and Swinnerton-Dyer Conjecture}
\author{Math 582e, Winter 2009, University of Washington}
\date{\bf Due Wednesday January 14, 2009}
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\begin{enumerate}
\item Let $E$ be an elliptic curve over $\QQ$. For $P\in E(\QQ)$,
prove that if $P \in E(\QQ)_{\tor}$, then $\hat{h}(P) = 0$, where
$\hat{h}$ is the N\'eron-Tate canonical height. *
(Note: The converse is also true, but more difficult to prove.)
\item Using a computer, compute every quantity in the BSD
formula, except $\#\Sha(E/\QQ)$ for each of the following
elliptic curves:
\begin{enumerate}
\item $y^2 + xy + y = x^3 + 4x - 6 $
\item $y^2 + y = x^3 + x^2$
\item $y^2 + xy = x^3 +1$
\item $y^2 + xy = x^3 + x^2 - 1154x - 15345$
\end{enumerate}
\item The congruent number problem has been called the oldest specific
open problem in mathematics (over 1000 years old).
{\bf Congruent Number Problem:}{\em Give an algorithm to decide whether or not
a given integer $n$ is the area of a right triangle with rational
side lengths.}
\begin{enumerate}
\item
Explain why if the rank part of the Birch and Swinnerton-Dyer
conjecture were known (i.e., that $\rank(E(\QQ)) = \ord_{s=1}L(E,s)$),
then we would also have a solution to the congruent number problem. [Hint:
I want a few paragraphs summary as an answer. You can find everything
you need to easily answer this question by looking, e.g., in Koblitz's
book {\em Introduction to Elliptic Curves and Modular Forms}.
Alternatively, see the end of section 1 of
Andrew Wiles' article on the Clay Math Website about
the Birch and Swinnerton-Dyer Conjecture.]
\item Is $2009$ a congruent number?
\end{enumerate}
\end{enumerate}
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