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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 2024} 

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\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




\begin{enumerate}
\item[(a)] If $\mathcal{E}$ is an elliptic curve defined over
$\mathbb{Q}$, we begin by asking whether one can have
$\mathcal{E}(\mathbb{F}_p)\cong\mathcal{E}(\mathbb{F}_q)$ for distinct
primes $p$ and $q$.  Show by example that this can happen when $p=5$
and $q=7$.  Prove in contrast that it can never happen for $p=101$ and $q=151$.
\bigskip

\item[(b)] Suppose that the elliptic curve $\mathcal{E}_{a,b}$ is
  given  by $y^2=x^3+ax+b$ with $a,b\in\mathbb{Z}$.  Give an example
  of congruence
  conditions on $a$ and $b$ which will ensure that $\mathcal{E}_{a,b}$
  has only trivial torsion (over $\mathbb{Q}$).  Deduce that a
  positive proportion of elliptic curves over $\mathbb{Q}$ have
  trivial torsion, in the sense that if
\[N(X):=\#\{(a,b)\in\mathbb{Z}:|a|,|b|\leq X,\,\mathcal{E}_{a,b}\mbox{
  has trivial torsion}\}\]
then
\[\liminf_{X\rightarrow\infty} X^{-2}N(X)>0.\]
Do you expect the proportion of curves with non-trivial torsion groups
to be positive, or zero?  You should discuss your
answer, {\bf but you are not expected to prove anything}.
\bigskip

\item[(c)] For any integer $n\in\mathbb{N}$ we write $\omega(n)$ to
  denote the number of distinct prime factors of $n$, so that
  $\omega(36)=2$, for example.  Let $\mathcal{E}: y^2=x(x^2+ax+b)$ 
be an elliptic curve with $a,b\in\mathbb{Z}$ satisfying $b>0$ and $a^2-4b>0$.
Show how to give a bound for the rank of $\mathcal{E}$ in terms of
$\omega(b)$ and $\omega(a^2-4b)$.

If $b$ and $a^2-4b$ are distinct primes, under what circumstances can
you show that $\mathcal{E}$ must have rank zero? Can you say anything
about the rank being zero in the case in which $b=p_1p_2$ and 
$a^2-4b=p_3$, for distinct primes $p_1,p_2,p_3$? 

Prove that if
$b$ and $a^2-4b$ are distinct primes, then the torsion group must 
have size at most 
100, and say to what extent this bound can be improved.
\bigskip

\item[(d)] 
Let $E_n$ be the elliptic curve
$$y^2=x^3-nx.$$
where $n$ is a non-zero integer. 

For a non-zero point $P \in E_n(\Q)$, with $x$-coordinate $x(P) = \frac{p}{q}$ written in lowest terms, we define $h(P) = \log(\max(|p|,|q|))$. We also define $h(O) = 0$.

\medskip


(i) Let $Q\in E_n(\Q)$. Write $Q=(a/d^2, b/d^3)$, where $a,b,d$ are coprime integers. Find explicitly a constant $C_1$ that depends only on $n$ and $Q$ such that
$$h(P+Q)\leq 2h(P)+C_1$$
for all $P \in E_n(\Q)$.
\medskip

(ii) Find explicitly a constant $C_2$ that depends only on $n$ such that
$$h(2P)\geq 4h(P)-C_2$$
for all $P\in E_n(\Q)$.
\medskip

(iii) Find as many $n$ as possible such that $E_n(\Q)$ has positive rank, and for which you can determine completely a set of generators for $E_n(\Q)$.
\end{enumerate}


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