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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 2012} 

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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 {\em Any queries about this miniproject, about possible
  misprints for example, should be addressed to Wendy Adams
{\tt <wendy.adams@comlab.ox.ac.uk>}}.\\[3ex]
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\bigskip
This miniproject explores properties of elliptic curves of the form
\[\mathcal{E}_B : Y^2 = X^3 + B\]
where $B\in\mathbb{Z}$ with $B\not=0$.
\bigskip 


\begin{enumerate}
\item[(a)] 
For each admissible value of $B$ describe (with proofs) the torsion
group of $\mathcal{E}_B(\mathbb{Q})$.
\item[(b)]
Let $N(x)$ denote the number of integers $B\in\mathbb{N}$ with $B\le
x$, such that $\mathcal{E}_B(\mathbb{Q})$ has positive rank.  By
constructing appropriate numbers $B$, show
that
\[\liminf_{x\rightarrow\infty}\frac{N(x)}{x^{1/2}}>0.\]
Can you prove such a statement with an exponent larger than $1/2$~?
How much larger~? (Take care to ensure that the numbers $B$ you
construct are distinct.)
\item[(c)] Suppose that $(x,y)$ lies on $\mathcal{E}_B$.
By considering $x'=x^{-2}(x^3+4B)$ find a point on
  $\mathcal{E}_{-27B}$.  What can
  you deduce about the ranks of $\mathcal{E}_B(\mathbb{Q})$ and
$\mathcal{E}_{-27B}(\mathbb{Q})$ ?
\item[(d)] Given a solution of $u^3+v^3=2k$ show how to construct a
point $(x,y)$ on $\mathcal{E}_{-27k^2}$ with $x=6k/(u+v)$. What can
  you deduce about the number of solutions $s,t\in\mathbb{Q}$ of an
  equation of the type $s^3+t^3=A$, for a given integer $A$?
\item[(e)] Show that if $a+b=2$ with $a,b\in\mathbb{Q}$ then there is
  a rational point
  $(x,y)$ on $\mathcal{E}_{(ab)^2}$ with $x=-ab$.  What do you
  deduce about representations of $2ab$ as a sum of two cubes of
  rational numbers? Find a solution $s,t\in\mathbb{Q}$ of $s^3+t^3=6$.
\item[(f)]
For any $(s,t) \in 
\mathcal{E}_B(\mathbb{Q})$ with $s,t\in\mathbb{Q}$ 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 $h_x( {\bf o} ) = 0$.
Find a constant $C_1$,
which depends only on~$B$ and $Q\in \mathcal{E}_B(\mathbb{Q})$, such that
$h_x(P+Q) \le 2h_x(P) + C_1$ for all $P\in \mathcal{E}_B(\mathbb{Q})$.
Find a constant $C_2$, which depends only on~$B$, such that
$h_x(2P) \ge 4h_x(P) - C_2$ for all $P\in \mathcal{E}_B(\mathbb{Q})$.
\item[(g)] Give an example of an elliptic curve $\mathcal{E}_B$ of
  positive rank, and for
  which you are able to find
generators for $\mathcal{E}_B(\mathbb{Q})$.

\end{enumerate}



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