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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 Michaelmas Term 2025} 

\end{center}

\medskip

\noindent {\em The steps of (each) mini project 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
mini project, but should make this assumption clear in your presentation.}\\

\smallskip
\noindent


\noindent {\em The different parts of this miniproject are largely independent, and are likely to be of
quite different lengths. Give clear and precise references to any results you use from the 
lecture notes, example sheets, books, on-line resources, or elsewhere, and justify all answers. Please
include your own examples in addition to those gathered from other sources.}\\

\smallskip
\noindent

\noindent {\em The following question parts test understanding of the course material without an open-ended component: 1(i),(ii); 3(i),(ii); 5(i).
	
	\noindent Good answers to these parts will be sufficient to achieve a passing mark.}\\

\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

\newpage
\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~$P$. 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 4.}
\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

\newpage
\noindent {\bf Question 5.} \begin{itemize}\item[(i)] 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.
\item[(ii)] Try to construct instances where the same 5-digit number
can be factorised using different elliptic curves.
\end{itemize}
\end{document}