\documentclass[11pt,twoside]{article}
\usepackage{amsmath,amssymb,amsthm,bm}
\usepackage{oxmathex}

\usepackage{graphicx}

\newcommand{\topictitle}[1]{\title{#1}\def\t@pic{#1}\setcounter{equation}{0}}
\newcommand{\topiccode}[1]{\topic{#1}{\t@pic}}
\renewcommand{\author}[1]{\def\auth@r{#1}}
\renewcommand{\date}[1]{\time{\auth@r\quad{#1}}}
\newcommand{\ve}[1]{\boldsymbol{\mathrm{#1}}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\diff}{\mathrm{d}}
\renewcommand{\d}[1]{\,\mathrm{d}{#1}}
\newcommand{\ee}{\mathrm{e}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\pdhfrac}[2]{\mathchoice{\frac{#1}{#2}}{#1/#2}{#1/#2}{#1/#2}}
\newcommand{\dd}[2]{\pdhfrac{\mathrm{d}#1}{\mathrm{d}#2}}
\newcommand{\sdd}[2]{\pdhfrac{\mathrm{d}^2#1}{\mathrm{d}#2^2}}
\newcommand{\pd}[2]{\pdhfrac{{\partial}#1}{{\partial}#2}}
\newcommand{\spd}[2]{\pdhfrac{\partial^2#1}{{\partial}#2^2}}
\newcommand{\mpd}[3]{\pdhfrac{\partial^2#1}{{\partial}#2{\partial}#3}}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\theequation}{\topicmark\arabic{equation}}
\def\o{\underline{\bf o}}

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

% Uncomment the next line.
\topictitle{Elliptic Curves}% Put your topic title here

% Uncomment the next line.
\author{Victor Flynn}% Put your name here
% Uncomment the next line.
\date{3/3/2007}% 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
Find an $x\in \Z$ such that $| x^2 + 2 |_3 < 3^{-2}$. Show that
there does not exist $x\in \Z$ such that $| x^2 + 3 |_3 < 3^{-2}$.
%\hfill {\bf [6~marks]}

\itm
Let $p \not= 2$ be prime. Find the $p$-adic expansion
of $\frac{1 + 2p}{p - p^3}$.
%\hfill {\bf [5~marks]}

\itm
Does there exist a prime~$p$ such that $p = p^p$
in ${\mathbb Q}_p^* / \bigl( {\mathbb Q}_p^* \bigr)^p$?
%\hfill {\bf [5~marks]} 

\itm
Let $q,r$ be distinct primes, and
let $\alpha,\beta \in {\mathbb Q}$.
Show that there exists a sequence $x_n \in \Q$ such
that $x_n \rightarrow \alpha$ with respect to $|\ \ |_q$,
and $x_n \rightarrow \beta$ with respect to $|\ \ |_r$,
as $n \rightarrow \infty$.
Does there exist a sequence $y_n \in \Q^*$ such that
$y_n \rightarrow 0$ with respect to $|\ \ |_p$ for all
primes~$p$, and $y_n \rightarrow 0$ with respect to $|\ \ |_\infty$,
as $n\rightarrow \infty$?
%\hfill {\bf [9~marks]} 

\end{ritem}
\end{question}

\begin{question}{2}
\begin{ritem}
\itm
Let $K$ be field, complete with respect to a discrete
non-Archimedean valuation, $R = \{ x\in K : |x| \leqslant 1\}$,
${\mathcal{M}} = \{ x\in K : |x| < 1\}$,
and assume that $R/{\mathcal{M}}$ is of characteristic~$p$,
for some prime~$p$.
Let~$F(X,Y)$ be a formal group defined over~$R$ and suppose
that~$z\in {\mathcal{M}}$ has exact order~$p^n$, for some~$n\geqslant 1$,
with respect to the group operation $x \oplus y = F(x,y)$ 
on~$\mathcal{M}$. Show that: 
$$ | z | \geqslant | p |^{\frac{1}{p^n - p^{n-1}}}.$$
[You may assume the result that, for any prime~$p$,
the multiplication by~$p$ map $[p](T)$ can be written as
$[p](T) = p f(T) + g(T^p)$, for some
$f(T) = T + \ldots \in R[[T]]$ and $g(T) \in R[[T]]$.]
%\hfill {\bf [9~marks]}

\itm
Let~${\mathcal{E}} : y^2 = x^3 + A x + B$, be an elliptic curve,
where~$A,B\in \Z_p$, and let~${\widetilde {\mathcal{E}}}$
denote the reduction of~${\mathcal{E}}$ modulo~$p$.
Show that any $(x,y) \in {\mathcal{E}}_{\mathrm{tors}}(\Q_p)$ 
satisfies $|x|_p\leqslant 1, |y|_p\leqslant 1$. 
\par\noindent
When~${\widetilde {\mathcal{E}}}$ is non-singular, 
show that ${\mathcal{E}}_{\mathrm{tors}}(\Q_p)$
is isomorphic to a subgroup of ${\widetilde {\mathcal{E}}}(\F_p)$.
%\hfill {\bf [8~marks]}

\itm
Let $D \in \Z$, $D > 0$, $D \equiv 2$~(mod~$3$). Describe the
torsion group over~$\Q$ of the elliptic curve $Y^2 = X^3 + D X$.
%\hfill {\bf [8~marks]}
\end{ritem}
\end{question}

\begin{question}{3}
Let ${\mathcal{C}} : Y^2 = X(X^2 + aX + b)$
and ${\mathcal{D}} : Y^2 = X(X^2 + a_1 X + b_1)$,
where $a,b\in \Z$ with $b(a^2-4b)\not= 0$ and $a_1 = -2a$,
$b_1 = a^2 - 4b$. Let the map~$\phi$ [which you may assume to
be a homomorphism] be defined as usual by 
$$ \phi : {\mathcal{C}} (\Q ) 
      \rightarrow {\mathcal{D}} (\Q) : (x,y)
\mapsto \Bigl( \frac{y^2}{x^2} ,\ y - \frac{by}{x^2} \Bigr) 
= \Bigl( \frac{x^2 + ax + b}{x},\ y - \frac{by}{x^2} \Bigr).$$
\noindent Let~$q$ be defined as usual by
$$ q : {\mathcal{D}} (\Q ) \rightarrow \Q^* / (\Q^*)^2 : (u,v) \mapsto u
\hbox{ when }u\not= 0,$$
$$ q : (0,0) \mapsto b_1,\,\,\, q: {\o} \mapsto 1. $$

\begin{ritem}
\itm
Show that $q$ is a homomorphism.
\par\noindent
[You are only required to show that $q(P+Q) = q(P)q(Q)$ in the typical case 
when none of $P,Q,P+Q$ are $(0,0)$ or {\o}.]
%\hfill {\bf [6~marks]}

\itm
Show that $q$ has kernel $\phi ({\mathcal{C}} (\Q))$.
%\hfill {\bf [6~marks]}

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

\begin{question}{4}
\begin{ritem}
\itm
Find a proper factor of $N=10573$ 
%[that is, $d | N$ and $1 < d < N$]
by applying the Elliptic Curve Method,
using the curve $Y^2 = X^3 - X - 5$ and~$3P$, where~$P=(2,1)$.
%\hfill {\bf [8~marks]} 

\itm For any elliptic curve~$\mathcal{E}$ and $(s,t) \in \mathcal{E}(\Q)$ 
with $s,t\in \Q$ and $s = \frac{c}{d}$, $c,d \in \Z$, $\gcd(c,d) = 1$, 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 $h_x( {\o} ) = 0$.
\par
Let $\mathcal{C}$, $\mathcal{D}, \phi$ be as defined 
in the previous question. Find a constant~$k$, which depends
only on~$a,b$, such that $h_x\bigl( \phi(P) \bigr) \leqslant
2 \bigl( h_x(P) \bigr) + k$, for all $P \in \mathcal{C} (\Q)$.
Find a constant~$\ell$, which depends
only on~$a,b$, such that $h_x\bigl( 2P \bigr) \leqslant 
4 \bigl( h_x(P) \bigr) + \ell$, for all $P \in \mathcal{C} (\Q)$.
%\hfill {\bf [8~marks]} 

\itm
Let $p \not= 2$ be prime, let $m \in \F_p^*$
and let $\mathcal{E}$ be the elliptic curve $Y^2 = X(X^2 + m^2)$,
defined over~$\F_p$. Show that $\# {\mathcal{E}}(\F_p)$
is always divisible by~$4$.
\par\noindent
[You may wish to consider separately the cases
$p\equiv 1$~(mod~$4$) and $p\equiv 3$~(mod~$4$).]
%\hfill {\bf [9~marks]}
\end{ritem}
\end{question}

\tidy
\end{document}
