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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 2016} 

\end{center}

\medskip

\noindent {\em The steps of miniproject are for your guidance; if
you wish 
to take an alternative route to the desired goal, you are free to do so.
But, if 
you follow the suggested route and find yourself unable to carry out any
particular 
step, you may simply assume it so that you can continue with the
miniproject, 
but should make this assumption clear in your presentation.}


\medskip
\noindent {\em Please write or print on one side of the paper only.}\\[3ex]
\medskip

\smallskip
\noindent
{\bf Question 1.}
\begin{itemize}
\item[(i)]
Find an $x\in\Z$ such that
$| x^2 + 6 |_5 < 5^{-3}$.
%\hfill {\bf [3~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 [3~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[(iii)]
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 [6~marks]} 
\item[(iv)]
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 [8~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
$Y^2 = X(X+1)(X-8)$.\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}
