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\begin{document}
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\exam{Section C}

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

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\author{Victor Flynn}% Put your name here
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\date{3/3/2008}% Put date of submission of this draft here

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

\maketitle

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\topiccode{C9.1b}% Put your topic code here

\begin{question}{1}
\begin{ritem}
\itm
Decide whether there exists $x\in {\mathbb Q}_p$ such that
$x^3 = 5$ for each of: $p=3,5,13$.
%\hfill {\bf [7~marks]}

\itm
Find the $3$-adic expansion of~$-\frac{13}{8}$. For any prime~$p$,
and $a_0,a_1,a_2 \in \{0,\ldots ,p-1\}$,
express 
$a_0,\overline{a_1 a_2} 
= a_0 + a_1 p + a_2 p^2 + a_1 p^3 + a_2 p^4 + \ldots$
in the form $m/n$, where $m,n\in {\mathbb Z}$.
%\hfill {\bf [6~marks]}

\itm
Let ${\mathcal{E}} : x^3 + y^3 = p$, defined over~${\mathbb Q}_p$,
and let $\widetilde{\mathcal{E}} : x^3 + y^3 = 0$, defined
over~${\mathbb F}_p$, be the reduction of~${\mathcal{E}}$
modulo~$p$. Show that~$(0,0)$ is a singular point
on~$\widetilde{\mathcal{E}}({\mathbb F}_p)$ and that it
does not lift to a point on~${\mathcal{E}}({\mathbb Q}_p)$.
Find a curve~$\mathcal{D}$, nonsingular and defined 
over~${\mathbb Q}_p$, such 
that~$\widetilde{\mathcal{D}} = \widetilde{\mathcal{E}}$
and such that $(0,0)\in \widetilde{\mathcal{D}}({\mathbb F}_p)$
does lift to a point on~${\mathcal{D}}({\mathbb Q}_p)$. 
%\hfill {\bf [6~marks]} 

\itm
Let $p\not= 2$ be prime and let $a,b,c \in {\mathbb Z}_p$
satisfy $|a|_p = |b|_p = |c|_p = 1$. Show that there
exist $x,y\in {\mathbb Z}_p$ such that $ax^2 + by^2 = c$.
\par [You might first wish to consider, for any
$\alpha, \beta, \gamma \in {\mathbb F}_p\backslash \{ 0 \}$,
the sizes of the sets $\{ \alpha x^2 : x \in {\mathbb F}_p \}$ 
and $\{ \gamma - \beta y^2 : y \in {\mathbb F}_p \}$.]
%\hfill {\bf [6~marks]} 

\end{ritem}
\end{question}

\begin{question}{2}
\begin{ritem}
\itm
State and prove the Nagell-Lutz Theorem
for $\mathbb Q$-rational torsion points on 
the elliptic curve $y^2 = x^3 + Ax + B$, 
with $A,B\in {\mathbb Z}$.
[You may assume the result that any $\mathbb Q$-rational 
torsion point $(x,y)$
on such a curve satisfies $x,y\in {\mathbb Z}$. You may also use the
polynomial identity:
$\phi_1(X) \psi_1(X) + \phi_2(X) \psi_2(X) = 4A^3 + 27B^2$,
where $\phi_1(X)= 3X^2+4A$, $\psi_1(X) = (3X^2+A)^2$,
$\phi_2(X)= -27(X^3 + AX - B)$ and $\psi_2(X) = X^3 + AX + B$.]
%\hfill {\bf [9~marks]}

\itm
Find the torsion group over~$\Q$
for the elliptic curve $y^2 = x^3 + x + 1$,
and deduce that this curve has infinitely many rational
points.
%\hfill {\bf [5~marks]}

\itm
Let $D\in {\mathbb Z}$ satisfy $D \not\equiv 0$~(mod~5)
and $D \equiv 2$~(mod~7).
Suppose also that there exists a prime $p \equiv 1$~(mod~3)
such that~$D$ is not a quadratic residue mod~$p$.
Find the torsion group over~$\mathbb Q$
for the elliptic curve $y^2 = x^3 + D$.
%\hfill {\bf [5~marks]}

\itm
Let ${\mathcal E} : y^2 = x(x-1)(x-4)$. 
Show that $(2,2i), (1 - i\sqrt{3}, 3 + i\sqrt{3}),
(4 + 2\sqrt{3}, 6 + 4\sqrt{3})$ each
have order~4 in ${\mathcal E}({\mathbb C})$.
Show that $\# \widetilde {\mathcal E} ({\mathbb F}_p)$
is divisible by~8,
for all primes $p \not= 2,3$.
Show that
the torsion group of~${\mathcal E}({\mathbb Q})$ has order~4.
%\hfill {\bf [6~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 the image of~$q$ is a subset
of the finite set
$$
\{ r : r \hbox{ is a square free integer and } r | b_1 \}.
$$
%\hfill {\bf [12~marks]}

\itm
% See 2005.
Find the rank of the elliptic curve $Y^2 = X(X^2 + 3X - 3)$.
[Standard results may be used without proof, provided they
are accurately stated.]
%\hfill {\bf [13~marks]}
\end{ritem}
\end{question}

\begin{question}{4}
\begin{aitem}
\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 $h_x( {\o} ) = 0$. 
\begin{ritem}
\itm
Show that, for any~$m\ge 1$, there is a constant~$C_m$,
independent of~$P$, such that 
$ | h_x( mP ) - m^2 h_x(P) | \le C_m$, for
all $P \in \mathcal{E}(\Q)$.
\itm
Show that, for any $P\in \mathcal{E}(\Q)$, the sequence
$4^{-n} h_x( 2^n P)$ is Cauchy and therefore
convergent in~$\R$ as~$n \rightarrow\infty$.
\end{ritem}
\par [You may use the result that there
exists a constant~$C$, independent of~$P,Q$, such that
$ | h_x(P+Q) + h_x(P-Q) - 2 h_x(P) - 2 h_x(Q) | \le C $,
for all $P,Q \in \mathcal{E}(\Q)$.]
\par
%\hfill {\bf [11~marks]} 

\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 let $F({\mathcal{M}})$ denote
the set~${\mathcal{M}}$ together with the
group operation: $x \oplus y = F(x,y)$. For any~$m\ge 1$,
let~$[m](x)$ denote, as usual, $x \oplus \ldots \oplus x$ [$m$ times].
Show that, for any~$x\in {\mathcal{M}}$, the sequence
$[p^n](x) \rightarrow 0$ as $n\rightarrow \infty$.
%\hfill {\bf [6~marks]}.

\itm
Let $p \equiv 1 \hbox{ (mod 12)}$ 
and $q \equiv 5 \hbox{ (mod 12)}$ be primes. Show that there exists 
an elliptic curve of the form $y^2 = x^3 + a x - a$,
with~$a\in\Z$,
for which the Elliptic Curve Method, using~$3(1,1)$,
successfully factorises $N = pq$.
%\hfill {\bf [8~marks]}
\end{aitem}
\end{question}

\tidy
\end{document}
