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\title{Math 583: Project Set 1}
\author{Spring 2007, University of Washington}
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\begin{enumerate}

\item {\bf Robert Bradshaw:} {\em $2$-descent
on elliptic curves.}  Do the following, in
some order (not necessarily linearly).
\begin{enumerate}
\item Learn something about how $2$-descent works, e.g., by reading
Silverman, Cremona, etc., or even by looking at this Harvard senior
thesis that I advised:\\{\tt http://www.wstein.org/projects/danielle\_li.pdf/}.
\item SAGE contains two distinct approaches to descent -- the
invariants method described in Cremona's book and implemented in mwrank, 
and the algebraic method as implement by Denis Simone.  
Look at the algebraic method's code in \sage: 
{\tt SAGE\_ROOT/data/extcode/pari/simon/} and \\
{\tt SAGE\_ROOT/devel/sage/sage/schemes/elliptic\_curves/gp\_simon.py}.
\item Finish the wrapper of Simon 2-descent so it works over number
fields instead of just over $\Q$.  (Your knowledge of French will help.)
\end{enumerate}

\item {\bf Robert Miller:} {\em Graphs associated to elliptic curves.}
  There are at least two very different extremely interesting graphs
  associated to elliptic curves.  I'll tell you about one in this
  project, and about the other, which involves congruences between
  modular forms, and is much more subtle -- but still easy to
  describe, in a later project.
\begin{enumerate}
\item Read the definition of {\em isogeny} in a book
on elliptic curves. (It's just a homomorphism of finite degree
between elliptic curves.)   By the way, it's a deep theorem
of John Tate that the BSD conjecture is true for a curve $E$
if and only if it is true for any curve $F$ that is isogenous
to $E$.
\item Try this in SAGE:
\begin{verbatim}
sage: e = EllipticCurve('11a')
sage: e.isogeny_class()
(..., 
[0 5 5]
[5 0 0]
[5 0 0])
\end{verbatim}
  Note that the second part of the output is a labeled graph (with
  three vertices), which desribes the {\em isogeny class} of the
  elliptic curve 11a.  The labels are the degrees of the isogenies.
\item Create a command in SAGE (in the {\tt ell\_rational\_field.py}
file) called {\tt isogeny\_graph()} that calls isogeny class
and uses it to construct the isogeny graph and outputs that.
\item Conjecturally classify all possible graphs that occur with and
  without the labelings. Also, what automorphism groups appear,
with and without labelings?  You can enumerate the first few
thousand elliptic curves like this:
\begin{verbatim}
sage: for e in cremona_optimal_curves(range(1,50)):
...   print e
\end{verbatim}
Important Note: Tom Boothby did this last summer, so you
should compare notes with him.
\end{enumerate}


\item {\bf Dustin Moody:} {\em Computing $\#E(\F_p)$.}
\begin{enumerate}
\item Learn something about algorithms for computing $\#E(\F_p)$ for
  $p$ {\em small}, e.g., from Cohen's number theory book. Small means
  $p < 10^{15}$, say.
\item Setup and start a distributed computation that creates a table,
  easily usable from SAGE (or any software), of the Fourier
  coefficients $a_p$ for $p<10^7$, for every elliptic curve over $\Q$
  of conductor up to $5077$.  Such a table currently doesn't exist,
  would be very useful for some computations I'm doing with Barry
  Mazur right now, will be useful for the second graph project that
  Robert Miller will do, and is generally useful for investigations
  into the BSD conjecture.  This table should have rows like
\begin{verbatim}
11a -2 -1 1 -2 1 4 -2 0 -1 0 ...
\end{verbatim}
  I hope computing all $a_p$ for $p<10^7$ is reasonable.
This computation could be done almost entirely using the
gp command {\tt ellan}, or the SAGE command 
{\tt anlist(n, pari\_ints=True)}.  However, you'll need to break
the computation up into groups in order to take advantage of parallel
computation (e.g., that {\tt sage.math.washington.edu} has 16
CPU cores).
\end{enumerate}

\item {\bf Ralph Greenberg:}
I used David Harvey's amazingly fast implementation of computation of $p$-padic
regulators to compute the regulator of the rank $2$ curve
389a for each good ordinary
prime $p \leq 62591$.  There is exactly one prime $p$ such that
$\ord_p(\Reg_p(E)) > 0$, and it's $p = 16231$, where the regulator is
$$
        1930 \cdot 16231 + 2051 \cdot 16231^2 + \cdots
$$
Can you think of any interesting questions related to the BSD
conjecture (or its $p$-adic analogues) that one might investigate in
this case?



\end{enumerate}
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