% N.B. Do not issue this sheet to the students!!

% MARKS ALLOCATED AS FOLLOWS:
% 1: 20. 2: 20. 3: 30. 4: 30. Total: 100.
% In more detail:
% 1: 20. 2: 20. 3: each part 10. 4: 30.
% By the way, delete 3(c) (and soln). Refer them also to exams qn. 6.
% Also, refer them after 4 to exams qn. 7.
% Possible qn 6 for the future: y^2 = x*(x^2 + x + 7)?
\input amssym.def
\input amssym.tex
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\null
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%
\chaptitle
\noindent
\centerline{Elliptic Curves.}
\centerline{Some exam-style questions.}
\rm
\bigskip
\bigskip
The following are some exam-style questions,
to supplement the actual exams available. 
\bigskip
\hrule
\bigskip
\bigskip
\chaptitle
\centerline{Elliptic Curves. Some Exam-style Questions.}
\rm
\bigskip
\bigskip
\noindent
\bigskip\noindent {\bf Question 1.}
\medskip\par\noindent{\bf (i)} Find an $x\in\Z$ such that
$| x^2 + 3 |_7 < 7^{-1}$.
\medskip\par\noindent{\bf (ii)} For what prime~$p$ does $-{9\over 8}$ have
$p$-adic expansion $1,\overline{2} = 1 + 2p + 2p^2 + 2p^3 + \ldots$?
%For what~$p$ does $-{1\over 110}$ have $p$-adic expansion
%$11,\overline{1} = p^{-1} + 1 + p + p^2 + p^3 + \ldots$?
\medskip\par\noindent{\bf (iii)} Let~$p \equiv 1$~(mod~$3$) be prime.
Show that~$-3$ is a quadratic residue mod~$p$
[{\it Hint: consider separately the cases $p\equiv 1$~$($mod~$4)$
and $p\equiv 3$~$($mod~$4)$}].
Use Hensel's Lemma to
deduce that~$-3$ is a square in~$\Q_p^*$.
%\par\noindent{\bf (iv)} Let~$p \equiv 2$~(mod~$3$) be a prime.
%Show that $\phi : \F_p \rightarrow \F_p$,
%defined by $\phi (x) = x^3$, is an injection and therefore a bijection.
%Let~$d \in \Z$ be not divisible by~3.
%Show that~$d$ is a cube in~$\Q_p^*$.
\medskip\par\noindent{\bf (iv)} Let $q\equiv 1$~(mod~27) be prime.
Show that $(X^2 + 3)(X^3 - q) = 0$
has solutions in $\R$ and every $\Q_p$.
\bigskip\noindent{\bf Question 2.}
\medskip\par\noindent{\bf (i)} Find the torsion group over~$\Q$
of the elliptic curve $Y^2 = X^3 + 3$.
\medskip\par\noindent{\bf (ii)}
Find the torsion group over~$\Q$ of the elliptic curve
$Y^2 = X^3 + 4 X$.
\medskip\par\noindent{\bf (iii)} Let $n\in \Z$ satisfy~$n \not= 0,\pm 1$.
Show that~$\bigl( n , \pm n(n+1) \bigr)$, 
$\bigl( -n , \pm n(n-1) \bigr)$ are points of
order~$4$ on the elliptic curve~$Y^2 = X(X+1)(X+n^2)$.
Find the torsion group over~$\Q$ when~$n \equiv 2$~(modulo~$5$).
\medskip\par\noindent{\bf (iv)} Let $k \in \Z$, $k \not= 0$, 
let $\e$ be the elliptic curve
$Y^2 = X^3 - k^2 X + k^3$, and let $(x,y)$ be a point of finite
order in ~$\e (\Q)$. Show that $ | y | \le 5 |k|^3$
and $| x | \le 3 |k|^2$.
\bigskip\noindent{\bf Question 3.}
Let $\c : Y^2 = X(X^2 + aX + b)$
and $\d : Y^2 = X(X^2 + a_1 X + b_1)$,
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 
$$ \phi : \c (\Q ) \rightarrow \d (\Q) : (x,y)
\mapsto ( {y^2 \over x^2} , y - {by\over x^2} ) .$$
\noindent Let~$q$ be defined as usual by
$$ q : \d (\Q ) \rightarrow \Q^* / (\Q^*)^2 : (u,v) \mapsto u
\hbox{ when }u\not= 0,$$
$$ q : (0,0) \mapsto b_1,\,\,\, q: {\underline {\bf o}} \mapsto 1, $$
where ${\underline {\bf o}}$ denotes the point at infinity on~$\d$.
\medskip\par\noindent{\bf (i)} Show that the image of~$q$ is a subset
of the finite set 
$$
\{ r : r \hbox{ is a square free integer and } r | b_1 \}.
$$
\medskip\par\noindent{\bf (ii)} Find the rank of the elliptic 
curve $Y^2 = X(X^2 + 2X + 3)$.
\bigskip\noindent{\bf Question 4.}
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 $d=4451$, modulo $N=10001$. The encoded message reads:
$$ 6847,\, 2577.$$ 
\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.
\medskip\par\noindent{\bf (i)}
Factorise $N$ by applying Pollard's ``$p-1$'' method,
using base~$2$ and exponent~$68$.
\medskip\par\noindent{\bf (ii)} 
Use the factorisation of~$N$ to decode the message
(which is the name of the animal used as the mascot for the sports
teams at the University of California at Santa Cruz).
\medskip\par\noindent{\bf (iii)} Let~$A$ be an Abelian group with
group operation~$+$, and let
$h : A \longrightarrow \R$ satisfy:
\par
(1) For any $Q\in A$, there exists $C_1 = C_1(Q)$
such that $h(P+Q) \le 2h(P) + C_1$ for all
\par\ \ \ \ \ $P\in A$.
\par
(2) There exists $C_2$, independent of~$P$, such that
$h(2P) \ge 4h(P) - C_2$ for all $P\in A$.
\par
(3) For any~$C_3$, the set $\{ P\in A : h(P) \le C_3\}$ is finite.
\par\noindent
Suppose also that $A/2A$ is finite. Prove that $A$ is
finitely generated.
\medskip\par\noindent{\bf (iv)} Let $A$ and~$h$ be as in~(c).
Suppose that $P$ is a torsion element of~$A$
[that is: there exists an integer $N > 0$ such that
$NP$ is the identity element of~$A$]. Show that
$h(P) \le {1\over 3}{C_2}$.
\bigskip
\bigskip
\hrule
\vfil \eject\end