% N.B. Double check ranks and torsions in Magma, both for
% the numerical examples and for the families!
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\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{23/2/2009}% 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{aitem}
\itm
% Bookwork from Section 1.
Let ${\mathcal C}$ be a non-singular cubic curve, defined
over a field~$K$, with a $K$-rational point~{\bf o}. Describe
the standard law for adding two points on~${\mathcal C}$, and 
prove that it is associative.
%\hfill {\bf [10~marks]}

\itm
% See 2006 exam (ellcurves/old06/ex06.pdf), qn 1(b).
For which primes~$p$ do there exist
$x,y\in \Z_p$ such that $y^2 = x^3 - x - 1$?
[You may find it helpful first to compute $h(-2),h(0),h(3)$
for $h(x) = x^3 - x - 1$.]
%\hfill {\bf [6~marks]}

\itm
% Similar to 2007, 4(i).
Find a proper factor of $N=221$
%[that is, $d | N$ and $1 < d < N$]
by applying the Elliptic Curve Method,
using the curve $Y^2 = X^3 + 5 X - 5$ and~$3P$, where~$P=(1,1)$.
%\hfill {\bf [6~marks]}
%
%\itm
%% Qn 1(d) removed from 2006/07 Sheet3 (old07/ex.pdf)
%Let $K$ be a field with non-Archimedean valuation $|\ |$.
%Assume that $\sum_{n=1}^\infty x_n$ is convergent (in $K, |\ |$)
%to $x \in K$. Show that there exists $n_0 \in \N$
%such that $|x_n| \leq |x_{n_0}|$ for all $n\in\N$.
%Show also that $|x| \leq |x_{n_0}|$.
%%\hfill {\bf [5~marks]} 

\itm
% New
Find a sequence $x_n\in\Q_p$ such that $\sum_{n=1}^\infty x_n$
is convergent in~$\Q_p$ but $\sum_{n=1}^\infty |x_n|_p$ 
is not convergent in~$\R$.
%\hfill {\bf [3~marks]} 

\end{aitem}
\end{question}

\begin{question}{2}
\begin{aitem}
\itm
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 [7~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 [7~marks]}
\end{ritem}

\itm
Find the torsion group over~$\Q$ for the elliptic curve $Y^2 = X^3 - 9$.
%\hfill {\bf [5~marks]}

\itm
Let $n\in\Z$ be not divisible by~$15$, let~$n$ and~$-n$ 
both be nonsquare, and let ${\mathcal E}, {\mathcal F}$
be the elliptic curves ${\mathcal E} : Y^2 = X^3 + nX$ and 
${\mathcal F} : Y^2 = X^3 - nX$. Show that at least one of
${\mathcal E}(\Q)$ or ${\mathcal F}(\Q)$ has torsion group
given by $\{ {\bf o}, (0,0) \}$. 
%\hfill {\bf [6~marks]}

\end{aitem}
\end{question}

\begin{question}{3}
\begin{aitem}
\itm
% Removed from 2006/07 Sheet 7 (ellcurves/old07/ex.pdf)
Find the rank of the elliptic curve $Y^2 = X(X^2 + 5X - 5)$.
% Not so sure about the following, since this can always
% be assumed (and is used) for all questions, so why state
% is only here; they might think it not applicable elsewhere?
%[Standard results may be used without proof, provided they
%are accurately stated.]
%\hfill {\bf [13 marks]}

\itm
% New
Let~$p$ be prime. Show that the elliptic curve $Y^2 = X^3 + pX$
has rank at most~$2$.
%\hfill {\bf [6~marks]}

\itm
% New.
Let $\alpha\in\Z$, $\alpha\not= 0,-4$, and let ${\mathcal C}$ be
the elliptic curve $Y^2 = X^3 + \alpha X^2 - \alpha X$.
Show that $(1,1) \in 2{\mathcal C}(\Q) \iff 
\alpha \hbox{ or } \alpha+4 \hbox{ is the square of an integer.}$
[You may find it helpful first to construct a curve~${\mathcal D}$
with $2$-isogenies $\phi : {\mathcal C} \rightarrow {\mathcal D}$,
$\hat\phi : {\mathcal D} \rightarrow {\mathcal C}$,
and find the preimages of~$(1,1)$ under~$\hat\phi$.]
%\hfill {\bf [6~marks]}
\end{aitem}
\end{question}

%\begin{question}{4}
%\begin{aitem}
%\itm 
%% Similar to 2007, 4(i).
%Find a proper factor of $N=221$ 
%%[that is, $d | N$ and $1 < d < N$]
%by applying the Elliptic Curve Method,
%using the curve $Y^2 = X^3 + 5 X - 5$ and~$3P$, where~$P=(1,1)$.
%%\hfill {\bf [10~marks]}
%
%\itm
%For any elliptic curve~$\mathcal{E}$ and $P = (s,t) \in \mathcal{E}(\Q)$ 
%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: 
%$$ h_x(P) = h_x\bigl( (s,t) \bigr) = \log \max \bigl( | c |, | d | \bigr),$$
%and define $h_x( {\o} ) = 0$. Also, define ${\hat h}(P)
%= \hbox{lim}_{n\rightarrow \infty}\ 4^{-n} h_x( 2^n P)$, 
%which you may assume to be convergent.
%Show the following, for any $P,Q \in \mathcal{E}(\Q)$ and any $n\in\N$.
%\begin{ritem}
%\itm
%${\hat h}(P + Q) + {\hat h}(P - Q) = 2 {\hat h}(P) + 2 {\hat h}(Q)$.
%%\hfill {\bf [3~marks]}
%\itm
%${\hat h}(nP) = n^2 {\hat h}(P)$.
%%\hfill {\bf [5~marks]}
%\itm
%$P \in {\mathcal E}_\mathrm{tors}(\Q) \iff {\hat h}(P) = 0$.
%%\hfill {\bf [7~marks]}
%\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 $P,Q \in \mathcal{E}(\Q)$; you may also use the result
%that, for any~$B$, the set $\{ P \in {\mathcal E}(\Q) : h_x(P) \leq B \}$
%is finite.]
%
%\end{aitem}
%
%\end{question}

\tidy
\end{document}