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\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 2023} 

\end{center}

\medskip

\noindent {\em The steps of the 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 the torsion group over~$\Q$ of the elliptic curve 
$$ Y^2 = X(X + 1)(X - 3).$$
% C_2 x C_2, using 5 and at least one other prime.
\item[(ii)]
Let $\mathcal{E}$ be the elliptic curve $Y^2 = X^3 + 9$.
Show that $\mathcal{E}(\Q)$ has a point of order~$3$
and a point of infinite order.
% (0,3) is of order 3, and (3,6) is of infinite order.
\item[(iii)]
Find other elliptic curves $\mathcal{E}$ for which
$\mathcal{E}(\Q)$ has a point of order $N > 1$ and
a point of infinite order.
\end{itemize}
\medskip

\noindent {\bf Question 2.}
For any elliptic curve~$\mathcal{E}$ and any $(s,t) \in \mathcal{E}(\Q)$ 
with $s,t\in \Q$ and $s = \frac{c}{d},\ \gcd(c,d) = 1$, let
the height function $h_x(s,t)$ be defined, as usual, by: 
$$ h_x\bigl( (s,t) \bigr) = \log \max \bigl( | c |, | d | \bigr),$$
and define $h_x( {\bf o} ) = 0$.
\begin{itemize}
\item[(i)] Let:
$$ \mathcal{E}_B : Y^2 = X^3 + B,$$
where $B \in \Z$
and $B\not= 0$, be an elliptic curve. Find a constant $C_1$,
which depends only on~$B \in \Z$ and $Q\in \mathcal{E}_B(\Q)$, such that
$h_x(P+Q) \le 2h_x(P) + C_1$ for all $P\in \mathcal{E}_B(\Q)$.
Find a constant $C_2$, which depends only on~$B\in\Z$, such that
$h_x(2P) \ge 4h_x(P) - C_2$ for all $P\in \mathcal{E}_B(\Q)$.
\item[(ii)] Find examples of elliptic curves of the form
$Y^2 = X^3 + B$ of rank $>0$, where you are able to find
generators for $\mathcal{E}(\Q)$.
% Involves digging through heights and resultants.
\end{itemize}
\medskip

\noindent {\bf Question 3.}
\begin{itemize}
\item[(i)] Find the rank of the elliptic curve $Y^2 = X(X^2 + 3X + 7)$.
% I believe the following has rank 0.
\item[(ii)] What can you say about the rank of the elliptic curve 
$Y^2 = X(X^2 - p)$, when~$p$ is prime and $p \equiv 3$~(mod~$8$)?
% See 2006.
\item[(iii)] Find infinite families of elliptic curves where
you are able to determine the rank. Find other infinite families of
elliptic curves where you are able to show the rank is at most~1.
\end{itemize}
\medskip

%\noindent {\bf Question 4.}
%For any point $P = (x,y)$, let $x(P)$ denote
%the $x$-coordinate of~$P$. Define
%$$ {\mathcal{C}} : Y^2 = X(X^2 + aX + b),\ \
%{\mathcal{D}} : Y^2 = X(X^2 + a_1 X + b_1),$$
%where $a,b\in \Z$ with $b(a^2-4b)\not= 0$ and $a_1 = -2a$,
%$b_1 = a^2 - 4b$. Let the map~$\phi$ [which you may assume to
%be a homomorphism] be defined as usual by 
%$$ \phi : {\mathcal{C}} (\Q ) 
%      \rightarrow {\mathcal{D}} (\Q) : (x,y)
%\mapsto \bigl( \frac{x^2 + ax + b}{x} , y - \frac{by}{x^2} \bigr).$$
%\noindent Let~$q$ be defined as usual by
%$$ q : {\mathcal{D}} (\Q ) \rightarrow \Q^* / (\Q^*)^2 : (u,v) \mapsto u
%\hbox{ when }u\not= 0,$$
%$$ q : (0,0) \mapsto b_1,\,\,\, q: {\bf o} \mapsto 1. $$
%Also define:
%$$ \phi_x : x \mapsto \frac{x^2 + ax + b}{x}.\ \
%T_{(0,0)} : x(P) \mapsto x\bigl( P + (0,0) \bigr) : x \mapsto b/x.$$
%\begin{itemize}
%\item[(i)] By finding the fixed points of $T_{(0,0)}$ or otherwise,
%express $\phi_x$ in the form $\phi_x = M \tau N$, where
%$\tau : x \mapsto x^2$ and $M,N$ are fractional linear transformations
%defined over $\Z[\sqrt{b}]$; that is to say, $M,N$ are of
%the form $(q_{11} x + q_{12})/(q_{21} x + q_{22})$, where
%$q_{11}, q_{12}, q_{21}, q_{22} \in \Z[\sqrt{b}]$.
%\item[(ii)] Find examples of curves $\mathcal{C}$, where $b,b_1\in\Q$,
%and where part~(i) (together with the similar result for
%the dual isogeny) helps to reduce the size of the constant $C_2$,
%such that $h_x(2P) \ge 4h_x(P) - C_2$ for all $P\in \mathcal{C}(\Q)$.
%\item[(iii)] Show, for any elliptic curve defined over~$\C$,
%the duplication map $[2]$ can be expressed in the form
%$[2] = M_1 \tau M_2 \tau M_3$, where $M_1, M_2, M_3$ are
%fractional linear transformations over~$\C$.
%\end{itemize}
%\medskip

\noindent {\bf Question 4.} For any point $P = (x,y)$, let $x(P)$ denote
the $x$-coordinate of~$P$. Find elliptic curves
$Y^2 = X(X^2 + aX + b)$, where $a,b\in\Z$, $a^2 - 4b \in (\Q^*)^2$,
$b \in (\Q^*)^2$,
such that the map $[2]_x : x(P) \mapsto x(2P)$
can be written in the form $[2]_x = M_1 \tau M_2 \tau M_3$,
where $\tau : x \mapsto x^2$ and $M_1, M_2, M_3$ are
fractional linear transformations defined over~$\Z$; 
that is to say, $M_1,M_2,M_3$ are of
the form $(q_{11} x + q_{12})/(q_{21} x + q_{22})$, where
$q_{11}, q_{12}, q_{21}, q_{22} \in \Z$.
% See 1993. Also see problem sheet 2, qn 3.
\medskip

\noindent {\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 $\lambda_1,\lambda_2,\mu_1,\mu_2 \in {({\Q}^*)^2}$.
\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~${\mathcal E}$.
% See commented-out part of solns to mfocs 2006.
\end{itemize}
\medskip

\noindent {\bf Question 6.} Construct several examples where the
elliptic curve method is used to factorise 5-digit numbers,
using small multiples of a point on an elliptic curve.
Try to construct instances where the same 5-digit number
can be factorised using different elliptic curves.

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