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\begin{document}
% Uncomment the next line.
\exam{Section C}

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\topictitle{Elliptic Curves}% Put your topic title here

% Uncomment the next line.
\author{Victor Flynn}% Put your name here
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\date{2/6/2006}% Put date of submission of this draft here

%\message{Please be sure to mark each draft submitted with the correct
%  date.}

\maketitle

% Uncomment the next line.
\topiccode{C9.1b}% Put your topic code here

\begin{question}{1}
\begin{ritem}
\itm
Let~$K$ be a field, complete with respect to a non-Archimedean
valuation~$|\ \, |$, with valuation ring~$R = \{ x\in K : |x| \leqslant 1\}$.
Prove Hensel's Lemma, that if $f(x) \in R[x]$ and $a_0\in R$ satisfies
$| f(a_0) | < | f'(a_0) |^2$, then there exists a unique $a\in R$ such that
$f(a) = 0$ and $| a - a_0 | \leqslant | f(a_0) |/ | f'(a_0) |$.
%\hfill {\bf [10~marks]}

\itm
For which primes~$p$ do there exist
$x,y\in \Z_p$ such that $3y^2 = 4 x^3 - 10$?
%\hfill {\bf [7~marks]}

\itm
For prime $p\not= 2$, determine how many elements there are
in the set ${\mathbb Q}_p^* / \bigl( {\mathbb Q}_p^* \bigr)^2$.
Determine how many elements there are
in the set ${\mathbb Q}_2^* / \bigl( {\mathbb Q}_2^* \bigr)^2$.
%\hfill {\bf [8~marks]} 
\end{ritem}
\end{question}

\begin{question}{2}
Let~$R$ be any ring (commutative, with~1), and let~$F,G$ be
formal groups over~$R$.

\begin{ritem}
\itm
Show that there exists a unique
normalised invariant differential for~$F$, which is given by
$\omega = F_X(0, T)^{-1}\hbox{d} T \in R[[T]] \hbox{d} T$,
and that every invariant differential for~$F$ is of the form $a\omega$
for some $a\in R$.
%\hfill {\bf [8~marks]}

\itm
Let~$f$ be a homomorphism over~$R$ from~$F$
to~$G$. Let~$\omega_F, \omega_G$ be
the normalised invariant differentials on~$F,G$, respectively.
Show that $\omega_G \circ f = f'(0)\ \omega_F$.
Deduce that, for any prime~$p$, there
exist $f,g\in R[[T]]$
such that $[p](T) = p f(T) + g(T^p)$ [where~$[p]$ represents
the multiplication-by-$p$ map on~$F$].
%\hfill {\bf [9~marks]}

\itm
Let $m,n\in \Z$, with $n\not= 0$.
Show that the curve $Y^2 = X^3 - (m^2+1)^2 X + 9 n^2$ has infinitely
many $\Q$-rational points.
%\hfill {\bf [8~marks]}
\end{ritem}
\end{question}

\begin{question}{3}
\begin{ritem}
\itm
Find a proper factor of $N=1517$ 
%[that is, $d | N$ and $1 < d < N$]
by applying the Elliptic Curve Method,
using the curve $Y^2 = X^3 + 7 X - 7$ and~$4P$, where~$P=(1,1)$.
%\hfill {\bf [8~marks]}

\itm
Find the torsion group over~$\Q$
of the elliptic curve $Y^2 = X^3 - 2X$.
%\hfill {\bf [7~marks]}

\itm
Let~${\mathcal E}_k$ be the elliptic curve~$Y^2 = X^3 + k$,
where~$k\in\Q$ and~$k\not= 0$. 
%Show that, for any point~$(x,y) \not= {\bf o}$
%on~${\mathcal E}_k$, 
%$$
%3(x,y) = {\bf o} \iff \frac{x(x^3 - 8k)}{4(x^3 + k)} = x.
%$$
Show that there is always a point of order~$3$ 
in~${\mathcal E}_k({\mathbb C})$
which is not in~${\mathcal E}_k(\Q)$.
%\hfill {\bf [10~marks]} % I might consider including the hint.
\end{ritem}
\end{question}

\begin{question}{4}
\begin{ritem}
\itm
Find the rank of the elliptic curve $Y^2 = X(X^2 + 3X + 5)$.
%\hfill {\bf [13~marks]} 

\itm
For any prime $p \equiv 5$~(mod~$8$), show that
the elliptic curve $Y^2 = X^3 + p^2 X$ has rank~$0$.
%\hfill {\bf [12~marks]}
\end{ritem}
\end{question}

\tidy
\end{document}