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\title{Homework 9: Elliptic Curves\\
{\sc\Large Due Wednesday, November 28}}
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\begin{document}
\maketitle 
{\sl There are 5 problems.  Choose 4 of the 5 problems and
{\em clearly} indicate which ones you will be graded on (as usual,
your score will be a fraction between $0$ and $1$).  
As usual, you may use PARI for any of them, as long as 
you explain what you are doing.  Work in
groups.}

\begin{problems}
\item (10 points) Let $\Phi$ be the set of the $15$ possible groups of
the form $E(\Q)_{\tor}$ for~$E$ an elliptic curve over~$\Q$ (see
Lecture~27).  For each group $G\in\Phi$, if possible, find a finite
field $k = \Z/p\Z$ and an elliptic curve~$E$ over~$k$ such that $E(k)
\ncisom G$.  (Hint: It is a fact that
$|p + 1 - \#E(\Z/p\Z))| \leq 2\sqrt{p}$, so you only have to try 
finitely many~$p$ to show that a group~$G$ does not occur as the group
of points on an elliptic curve over a finite field.)

%\item (5 points) Demonstrate how to use the Lutz-Nagell theorem
%(Lecture~27) to compute the torsion subgroup of the elliptic
%curve defined by the equation
%$$y^2 +xy = x^3 -8369487776175x + 9319575518172005625.$$

\item (6 points) Many number theorists, such as myself one week ago,
incorrectly think that Lutz-Nagell works well in practice.  
Describe the steps you {\em would}
take if you were to use the Lutz-Nagell theorem
(Lecture~27) to compute the torsion subgroup of the elliptic
curve~$E$ defined by the equation
$$y^2 +xy = x^3 -8369487776175x + 9319575518172005625,$$
then tell me why it would be {\em very} time consuming to actually
carry these steps out.  Find the torsion subgroup of~$E$
using the {\tt elltors} command in PARI.  Does {\tt elltors}
use the Lutz-Nagell algorithm by default?

\item (6 points) Let $E$ be the elliptic curve
defined by the equation $y^2 = x^3 +1$.
\begin{subprob}
\item For each prime $p$ with $5\leq p<30$, describe the group
of points on this curve having coordinates in the finite field
$\Z/p\Z$. (You can just give the order of each group.)
\item For each prime in (i), let $N_p$ be the number of
points in the group.  (Don't forget the point infinity.)
For the set of primes satisfying $p\con 2\pmod{3}$,
can you see a pattern for the values of $N_p$?
Make a general conjecture for the value of $N_p$
when $p\con 2\pmod{3}$.
\item Prove your conjecture.
\end{subprob}

\item (6 points)
Let~$p$ be a prime and let $E$ be the elliptic curve
defined by the equation $y^2 = x^3 +px$.
Use Lutz-Nagel to find all points of finite order in $E(\Q)$.

\item (4 points)
\begin{subprob}
\item 
Let~$E$ be an elliptic curve over the real numbers~$\R$.
Prove that $E(\R)$ is not a finitely generated abelian group.
\item 
Let~$E$ be an elliptic curve over a finite field $k=\Z/p\Z$.
Prove that $E(k)$ is a finitely generated abelian group.
\end{subprob}





\end{problems}

\end{document}