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

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

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

{\bf Hilary Term 2010} 

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\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.}


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

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This miniproject consists of two independent parts.  You should
attempt both.
\medskip

\noindent {\bf Part 1}
\medskip

This section of the miniproject is designed to investigate the
question --- If $\mathcal{E}$ is an elliptic curve defined over
$\mathbb{Q}$, to what extent is $\mathcal{E}(\mathbb{Q})$ dense 
in $\mathcal{E}(\mathbb{R})$ (with the usual topology of
$\mathbb{R}^2$)?  If there are parts of your analysis that you are
unable to make completely rigorous then you should give a partly
rigorous argument.

Begin by taking $\mathcal{E}$ in the form $w=z^3+Aw^2z+Bw^3$.  Draw 
sketches of $\mathcal{E}(\mathbb{R})$ in the cases $A=1,B=0$ and
$A=-1,B=0$. Describe the group law geometrically, taking
$\mathbf{o}=(0,0)$. How do you find twice a point, and 
the inverse of a point?  What can
you say about points of order 2?

Given $P\in\mathcal{E}(\mathbb{Q})$, when can you say that the
sequence $P,2P,3P,\ldots$ must
contain a convergent subsequence in $\mathcal{E}(\mathbb{R})$?  If
there is a subsequence tending to one of the points at infinity, show
that there is a subsequence converging to $\mathbf{o}=(0,0)$. Equally, if
there is a subsequence tending to a finite point of
$\mathcal{E}(\mathbb{R})$, show that there is a subsequence converging 
to $\mathbf{o}$.

Now suppose a large value of $H$ is chosen.  Show that there is a
small value $\eta=\eta(H)>0$ such that the map $(Q,R)\mapsto Q+R$
from
\[\left(\mathcal{E}(\mathbb{R})\cap[-H,H]^2\right)\times
\left(\mathcal{E}(\mathbb{R})\cap[-\eta,\eta]^2\right)\]
to $\mathcal{E}(\mathbb{R})$ is continuous, and deduce that for any
$\varepsilon>0$ there is a corresponding $\delta>0$ so that if 
$Q\in\mathcal{E}(\mathbb{R})\cap[-H,H]^2$ and
$R\in\mathcal{E}(\mathbb{R})\cap[-\delta,\delta]$ then the distance
from $Q+R$ to $Q$ (in the real metric) is at most $\varepsilon$.  Show
that if in addition $R\not=\mathbf{o}$, then there is a value $\varpi(R)>0$
such that the distance from $Q+R$ to $Q$ is at least $\varpi(R)$.
 
Show that, for suitable $Q$ and $R$, there must be a multiple of $R$
within $\varepsilon$ of $Q$.

What do you conclude about the density of rational points on
$\mathcal{E}: y^2=x^3+Ax+B$ ? Give explicit examples where the
different cases hold.
\bigskip\bigskip\bigskip


\noindent {\bf Part 2}
\bigskip

In this part of the miniproject you will investigate  integral points
on the elliptic curve $\mathcal{E}: y^2=x^3+Ax+B$, where
$A,B\in\mathbb{Z}$.  

Show that if $P\in\mathcal{E}(\mathbb{Q})$ and
$nP\in\mathcal{E}(\mathbb{Z})$ for some non-zero integer $n$, then
$P\in\mathcal{E}(\mathbb{Z})$. If $n=2$, show that if $P=(x,y)$ with
$y\not=0$ then $y^2|\Delta$, where $\Delta=4A^3+27B^2$.

Suppose now that $x^3+Ax+B$ has three real roots $e_1<e_2<e_3$.  Show
that if $P$ and $Q$ are points on $\cal{E}$ with $x(P),x(Q)\ge e_3$ then
$x(P+Q)\ge e_3$.  Show similarly that if $e_1\le x(P),x(Q)\le e_2$ then
$x(P+Q)\ge e_3$.  Suppose in addition that $\mathcal{E}(\mathbb{Q})$
has rank 1. Under what additional assumption can you deduce that if
$P\in\mathcal{E}(\mathbb{Q})$ has $x(P)\ge e_3$ then $P=2Q$ for some
$P\in\mathcal{E}(\mathbb{Q})$?

Formulate conditions on $\cal{E}$ arising from the above
considerations, under which you can give an algorithm to determine all
integer solutions to $y^2=x^3+Ax+B$, and describe your algorithm.  
How would it help if you knew
that $A\equiv 2\pmod{5}$ and $B\equiv 0\pmod{5}$?

Find all integers $a$ and $b$ for which $a(a+1)=b(b+1)(b+2)$. (You may
assume that the relevant elliptic curve has rank 1).


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Show rank \le omega(D), and examples with rk=0, omega growing.