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\begin{document}

\begin{center}
{\bf The University of Oxford}\\[5mm]

{\bf MSc (Mathematics and Foundations of Computer Science)}\\[5mm]

{\large\bf Elliptic Curves}\\[3mm]

{\bf Hilary Term 2006} 

\end{center}

\medskip

\noindent {\em Answer as many problems as you can.}

\medskip
\noindent {\em Please write on one side of the paper only.}\\[3ex]
\smallskip
\noindent
{\bf Question 1.}
\begin{itemize}
\item[(i)]
Find an $x\in\Z$ such that
$| x^2 + 6 |_5 < 5^{-3}$.
%\hfill {\bf [2~marks]}
\item[(ii)]
Let $\alpha 
%= 12,\overline{14}
= 1\cdot 5^{-1} + 2\cdot 5^0 + 1\cdot 5^1
+ 4\cdot 5^2 + 1\cdot 5^3
+ 4\cdot 5^4 + \ldots \in \Q_5$. Express~$\alpha$ in the
form~$a/b$, where $a,b\in\Z$.
%\hfill {\bf [2~marks]}
\item[(iii)]
Let $p\not= 2$ be prime and let $b\in\Z_p$, with $|b|_p = 1$.
Show that there exist $x,y\in \Z_p$ such that
$x^2 + y^2 = b$.
%\hfill {\bf [4~marks]} 
\item[(iv)]
Determine the primes~$p$ for which
there exist $x,y \in \Z_p$ such that\par $y^2 =  x^3 + 2 x + 2$. 
Are there
$x,y\in \Z$ such that $y^2 =  x^3 + 2 x + 2$?
%\hfill {\bf [5~marks]} 
\item[(v)]
Construct examples of elliptic curves $y^2 = x^3 + f_2 x^2 + f_1 x + f_0$,
where $f_0,f_1,f_2\in \Z$,
which are satisfied by some $x,y\in \Z_p$ with $|y|_p = 1$,
for all primes $p$ except
$3,5,7$. Find other examples of similar types, and explain what
method you are using to construct your examples.
%\par\hfill {\bf [7~marks]} 
\end{itemize}
\medskip

\noindent {\bf Question 2.}
\begin{itemize}
\item[(i)]
Find the torsion group over~$\Q$
for the elliptic curve:\par
$Y^2 = X(X+1)(X+4)$.
%\par\hfill {\bf [6~marks]}
\item[(ii)]
Find the torsion group over~$\Q$
for the elliptic curve:\par
$Y^2 = X(X-1)(X-4)$.\par
Are you able to compute this using only reductions modulo~$p$?
Are you able to compute this using only the fact that the
order of ${\mathcal E}_\mathrm{tors}(\Q)$
divides the order of every 
${\widetilde {\mathcal E} ({\mathcal F}_p)}$?
Find other examples of a similar type, explaining how you
construct them.
%\hfill {\bf [14~marks]}
\end{itemize}
\medskip

\noindent {\bf Question 3.}
\begin{itemize}
\item[(i)]
Show that any elliptic curve over $\Q$ with
a rational point of order~$5$ is birationally equivalent to:
\ $Y^2 + (1+v)XY + v Y = X^3 + v X^2$,
for some $v\in \Q.$
%\hfill {\bf [7~marks]}
\item[(ii)]
Discuss variations of the same idea, for different values
of order~$N$. Find elliptic curves over~$\Q$ with a point
of order~$N$, for various choices of~$N$; for each
such curve, compute the whole of the torsion
group over~$\Q$.
%\hfill {\bf [13~marks]}
\end{itemize}
\medskip

\noindent {\bf Question 4.}
\begin{itemize}
\item[(i)]
Find the rank of the elliptic curve $Y^2 = X(X^2 + 6 X + 1)$.
%\hfill {\bf [7~marks]} 
\item[(ii)]
Compute the rank, of an elliptic curve $Y^2 = X(X^2 + aX + b)$,
with $a,b\in\Z$,
of your own choosing, but where $b$ is divisible by at
least three distinct primes. Also compute the rank of an example
for which the rank is at least~$2$.
%\hfill {\bf [9~marks]} 
\item[(iii)]
For an elliptic curve $Y^2 = X(X^2 + aX + b)$, suppose that
$b(a^2 - 4b)$ is divisible by precisely $k$ distinct primes.
Give an upper bound on the rank. Are there any conditions
on the sign of $a,b$ which allow this bound to be improved?
Can you describe any other conditions on $a,b$ which allow
the bound to be improved?
%\hfill {\bf [9~marks]}
\end{itemize}

%% I might consider removing this one and redistributing the marks.
%{\bf Question 5.}
%\begin{itemize}
%\item[(i)]
%Let ${\mathcal C} : Y^2 = X(X + \lambda_1)(X + \lambda_2)$,
%where $\lambda_1,\lambda_2 \in \Q^*$ and $\lambda_1 \not= \lambda_2$.
%Determine 
%$\mu_1,\mu_2 \in \Q(\sqrt{\lambda_1}, \sqrt{\lambda_2})$
%such that there is a $2$-isogeny defined over $\Q$ from
%${\mathcal C}$ to the curve $ {\mathcal D} : Y^2 = X(X - \mu_1)(X - \mu_2)$.
%Suppose that $(0,0)\in 2{\mathcal C}(\Q)$; show
%that $\lambda_1,\lambda_2,\mu_1,\mu_2 \in {({\Q}^*)^2}$.
%%\hfill {\bf [7~marks]} 
%\item[(ii)]
%Let ${\mathcal E}$ be any elliptic curve, defined over~$\Q$.
%Find the best bound you can on the possible number of $\Q$-rational
%points of order~$4$ on~${\mathcal E}$.
%%\hfill {\bf [8~marks]} 
%\end{itemize}

\end{document}