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\begin{document}

\noindent
\centerline{\bf Elliptic Curves. MT 2025. Sheet 4.}
\rm

\bigskip
%\par\noindent {\bf (d) [optional].} $Y^2 = X^3 - 219X + 1654$.
%\par\noindent Note: (d) requires significant computation
%(you should do an initial search for points with $x$-coordinate
%in the range $|x| \leqslant 20$, with $x\in \Z$)
%and is of course much more time consuming than anything which
%would be asked in a timed exam. It is mainly intended to demonstrate
%that interesting torsion groups can occur.
%\par\noindent {\bf (a).} $Y^2 = X^3 + 1$.
%\par\noindent {\bf (b).} $Y^2 = X^3 - 219X + 1654$.
%%\par\noindent {\bf (c).} $Y^2 = X(X+1)(X+4)$.
%\par\noindent {\bf (c).} $Y^2 = X(X+81)(X+256)$.
%\par\noindent {\bf (d).} $Y^2 = X(X-1)(X-2)$.
%\par\noindent {\bf (e).} $Y^2 = X^3 + 1/3^6$.
%\par\noindent {\bf (f).} $Y^2 + Y = X^3 - X + 13$.
%%\par\noindent {\bf (g).} $Y^2 = X^3 - X^2 + 1/4$.

\noindent{\it Section A}
\medskip

\begin{enumerate}\item Let, as usual, $\c : Y^2 = X(X^2 + aX + b)$
and $\d : Y^2 = X(X^2 + a_1X + b_1)$, where $a,b\in\Z$, $a_1 = -2a,
b_1 = a^2 - 4b$ and $b(a^2-4b)\not= 0$. 
Let $\cotq$ denote the set of torsion elements of $\c (\Q)$ which have
odd order, and let $\dotq$ denote the set of torsion elements of $\d (\Q)$
which have odd order. Show that $\cotq$ and $\dotq$ are isomorphic.

\item Let $\c$ and $\d$ be as in Question 1.
Let the homomorphisms~$\phi, \hat\phi$ be defined as usual by 
\[\phi : \c \rightarrow \d : (x,y)
\mapsto \Bigl( \bigl( {y\over x}\bigr)^2 , 
y - {by\over x^2} \Bigr),\]
\[\hat\phi : \d \rightarrow \c : (u,v)
\mapsto \Bigl( {1\over 4} \bigl( {v\over u} \bigr)^2,
{1\over 8} \bigl( v - {b_1 v\over u^2}\bigr) \Bigr).\]


What are the preimages of $(0,0)$ under $\hat\phi$?

Show that $(0,0) \in 2\c (\Q)$ if and only if there
exist~$m,n\in \Z$ such that $b = m^2$ and $a+2m = n^2$. 
\end{enumerate}
\medskip

\noindent{\it Section B}
\begin{enumerate}[resume] 
	\item As in lectures, write $\g = \c(\Q)$ and $\h = \d(\Q)$. Let $a$ and $b$ be the
integers so that
\[ \g / \hat{\phi}(\h) \cong C_2^a \mbox{ and } \h /\phi(\g) \cong C_2^b. \]

\par\noindent {(a)} Explain why the map
\[ \hat{\phi}:  \h /\phi(\g) \rightarrow \hat{\phi}(\h) / 2 \g,\quad P  + \phi(\g) \mapsto \hat{\phi}(P) + 2 \g \]
is well-defined and surjective. 

Prove that the kernel of this map is generated by the class $[(0,0)] \in   \h /\phi(\g)$
of $(0,0) \in \h$. 
%% Deduce that 
%% \[ \hat{\phi}(\h) / 2 \g \cong C_2^b \mbox{ or } C_2^{b-1} \]
%% according to whether $(0,0) \in \phi(\g)$ or $(0,0) \not \in \phi(\g)$.
\par\noindent {(b)} Hence show that 
\[ \g /  2 \g \cong C_2^{a + b} \mbox{ or } C_2^{a + b - 1}\]
according to whether $(0,0) \in \phi(\g)$ or $(0,0) \not \in \phi(\g)$.
\par\noindent {(c)} Show that $(0,0) \in \phi(\g)$ if and only if the $2$-torsion of
$\g$ is isomorphic to $C_2^2$. Deduce that the rank of $\g$ is
$a + b - 2$.



\item Find the ranks of the
following elliptic curves.
\par\noindent {(a)} $Y^2 = X(X^2 + 2X + 3)$.
\par\noindent {(b)} $Y^2 = X(X^2 + 14X + 1)$.
%%\par\noindent {\bf (a).} $Y^2 = X(X^2 + 3X + 5)$.
%\par\noindent {\bf (a).} $Y^2 = X(X^2 + 5X - 5)$.
%\par\noindent {\bf (b).} $Y^2 = X(X^2 + 14X + 1)$.
%\par\noindent {\bf (c).} $Y^2 = X(X^2 + 2X + 3)$.
%%\par\noindent {\bf (d).} $Y^2 = X(X^2 + 2X + 9)$.  
%%\par\noindent {\bf (e).} $Y^2 = X(X^2 + 9X - 1)$.    
%%\par\noindent {\bf (f).} $Y^2 = X(X-12)(X-36)$. 


\item (\textit{2006 paper, Q 8(ii)}) For any prime $p \equiv 5 \bmod{8}$ show that the elliptic
curve $Y^2 = X^3 + p^2 X$ has rank $0$.

\item 
Let $A,+$ be an Abelian group.
Let $h : A \rightarrow \R_{\ge 0}$ satisfy:
\par\noindent \ \ \ \ (I) There exists a constant~$C$, 
independent of~$P,Q$, such that
\par \ \ \ \ $ | h(P+Q) + h(P-Q) - 2 h(P) - 2 h(Q) | \le C $, 
for all $P,Q \in A$,
\par\noindent \ \ \ \ (II) For any~$B\in\R$, 
the set $\{P\in A:h(P)\le B\}$ is finite.
\par\noindent Show that~$h$ is a height function on~$A$. 
Show also that there exists 
a constant~$C_3$, independent of~$P$, such that $|h(3P) - 9 h(P)|\le C_3$, 
for all $P \in A$. 
\par (\textit{$\R_{\ge 0}$ denotes $\{ x\in\R : x \ge 0\}$}).
\end{enumerate}
\medskip

\noindent {\it Part (b) of the next question is {\it Section B}, and the others parts {\it Section C}}

\begin{enumerate}[resume]
\item
A four-letter word $L_1L_2L_3L_4$ has been divided
into two pairs: $L_1L_2$ and $L_3L_4$.
Each of these pairs has been converted
into an integer (of at most 4 digits)
via the standard map: $A \mapsto 01 , B \mapsto 02, \ldots ,
Z \mapsto 26$. These integers have been encoded by taking each to the
power of $d=4085$, modulo $N=10481$. The encoded message reads:
{\bf 6012},\ {\bf 3236}. 
\noindent You may assume that $N$ is the product of two primes. 
You should show, in your calculations, how you are only using
numbers of length at most~$9$ digits.
\par\noindent
{(a)} Find a proper factor of~$N$ (that is, a factor~$d$
of~$N$ satisfying~$1 < d < N$) by 
applying Pollard's ``$p-1$'' method,
using base~$2$ and exponent~$46$.
\par\noindent
{(b)} Factorise $N$ by applying the Elliptic Curve Method,
using the curve $\e : Y^2 = X^3 - X + 1$ and~$3P$, where~$P=(5,11)$.
\par\noindent
{(c)} Use the factorisation of~$N$ to decode the message
(which is the name of the town famous for being the country
music capital of New Zealand).
\end{enumerate}
\end{document}

\medskip
\hrule
\medskip
{\it The following question is compulsory for students taking
the MSc in MFoCS (Mathematics and the Foundations of Computer
Science). For everyone else, it is optional.}
\medskip\noindent
{\bf 7.} Show that any elliptic curve over $\Q$ with
a rational point of order~$4$ is birationally equivalent to:
$Y^2 + XY + v Y = X^3 + v X^2$, for some $v\in \Q$.
\vfil \eject \end

