% 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
%\def\Bbb{\bf}
\nopagenumbers
\magnification=\magstep1
%\hoffset=1truecm
%\voffset=2truecm
\baselineskip = 5.2 true mm
\font\frkkk=eufm10
\font\twelverm=cmr12
\font\tenrm=cmr10
\font\ninerm=cmr9
\font\ninebf=cmbx9
\font\eightrm=cmr8
\font\sevrm=cmr7
\font\sixrm=cmr6
\font\scrpp=eusm10
\font\frkk=eufm10
\font\deffont=cmssi10
\font\chaptitle=cmbx10 at 14 pt
\tolerance=10000
\def\sqr{\ifmmode\square\else{$\square$}\fi}
\def\square{\vcenter{
\hrule height.1mm
\hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm}
\hrule height.1mm}}                  % This is a slimmer sqr.
\null
\def\le{\leqslant}
\def\ge{\geqslant}
\def\etq{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q})}
\def\etqp{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q}_p)}
\def\c{{\cal C}}
\def\d{{\cal D}}
\def\e{{\cal E}}
\def\pk{\phi _\kappa}
\def\im{{\hbox{\sl im}}}
\def\hs{H_{\varsigma}}
\def\hpk{\hat \phi _\kappa}
\font\sc=cmssqi8
\def\scc#1{\hbox{\sc #1}}
\def\sf{{\scc F}}
\def\pnbq{{\Bbb P}^n(\overline {\Bbb Q} )}
\def\hk{{\hat \kappa}}
\def\bq{{\overline {\Bbb Q}}}
\def\hq{{\hat q}}
\def\pv{\prod\limits_v }
\def\pnk{{\Bbb P}^n(K)}
\def\mnkvw{{\Bbb M}^n(K[{\bf v}^2,{\bf w}^2])}
\def\pnkv{{\Bbb P}^n(K[{\bf v}^2])}
\def\kj{\kappa (J)}
\def \qmods {{\Bbb Q}^*/({\Bbb Q}^*)^2}
\def \qmodss { {\Bbb Q}^*/({\Bbb Q}^*)^2 \times
{\Bbb Q}^*/({\Bbb Q}^*)^2 }
\def \qs{{\Bbb Q}^*}
\def \qss{({\Bbb Q}^*)^2}
\def\bbQ{\Bbb Q}
\def\bbF{\Bbb F}
\def\bbZ{\Bbb Z}
\def\bbR{\Bbb R}
\def\bbC{\Bbb C}
\def\notdiv{{\not\hskip-.5pt |\ }}
\def\Q{{\Bbb Q}}
\def\F{{\Bbb F}}
\def\Z{{\Bbb Z}}
\def\R{{\Bbb R}}
\def\C{{\Bbb C}}
%
\chaptitle
\noindent
\centerline{Elliptic Curves. Sheet 8.}
\centerline{(Some exam-style questions and the 2006,2007,2008 Exams).}
\rm
\bigskip
\bigskip
The following exam-style questions and the 2006,2007,2008 Exams
need not be handed in for classes; they are
intended primarily to help you with your revision in
Trinity Term (when I shall have consultation sessions,
in case you have any questions during your Trinity Term revision). 
Of course, there are no guarantees
about whether specific topics in the exam-style questions
and the 2006,2007,2008 Exams
will be asked in this year's exam.
Note that the parts (i),(ii),$\ldots$
of a question may or may not be related to each other.
Note that, in the 2006,2007,2008 Exams, two of the
questions have portions that are `bookwork' (reproducing proofs
from lectures). It will also be true of this year's exam
that two of the questions will include some bookwork.
I bring to your attention the new 2008/09 Part C Examination
Conventions (which have been posted at:
www.maths.ox.ac.uk/notices/undergrad), in particular note
the change this year to the number of questions available
(described in 3.1 {\it Structure of Papers}).
You will probably also find it helpful to have a calculator
in the exam; please see the examination regulations for
a description of the types of calculators allowed.
\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 
\chaptitle
\noindent
\centerline{Elliptic Curves Questions from the 2006 Examination.}
\rm
\bigskip
\bigskip
\noindent
\bigskip\noindent {\bf Question 1.}
\medskip\par\noindent{\bf (i)}
Let~$K$ be a field, complete with respect to a non-Archimedean
valuation~$|\ \, |$, with valuation ring~$R = \{ x\in K : |x| \leqslant 1\}$.
Prove Hensel's Lemma, that if $f(x) \in R[x]$ and $a_0\in R$ satisfies
$| f(a_0) | < | f'(a_0) |^2$, then there exists a unique $a\in R$ such that
$f(a) = 0$ and $| a - a_0 | \leqslant | f(a_0) |/ | f'(a_0) |$.
%\hfill {\bf [10~marks]}
\medskip\par\noindent{\bf (ii)}
For which primes~$p$ do there exist
$x,y\in \Z_p$ such that $3y^2 = 4 x^3 - 10$?
%\hfill {\bf [7~marks]}
\medskip\par\noindent{\bf (iii)}
For prime $p\not= 2$, determine how many elements there are
in the set ${\bbQ}_p^* / \bigl( {\bbQ}_p^* \bigr)^2$.
Determine how many elements there are
in the set ${\bbQ}_2^* / \bigl( {\bbQ}_2^* \bigr)^2$.
%\hfill {\bf [8~marks]} 
\bigskip\noindent {\bf Question 2.}
Let~$R$ be any ring (commutative, with~1), and let~$F,G$ be
formal groups over~$R$.
\medskip\par\noindent{\bf (i)}
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 [8~marks]}
\medskip\par\noindent{\bf (ii)}
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 [9~marks]}
\medskip\par\noindent{\bf (iii)}
Let $m,n\in \Z$, with $n\not= 0$.
Show that the curve $Y^2 = X^3 - (m^2+1)^2 X + 9 n^2$ has infinitely
many $\Q$-rational points.
%\hfill {\bf [8~marks]}
\bigskip\noindent {\bf Question 3.}
\medskip\par\noindent{\bf (i)}
Find a proper factor of $N=1517$ 
%[that is, $d | N$ and $1 < d < N$]
by applying the Elliptic Curve Method,
using the curve $Y^2 = X^3 + 7 X - 7$ and~$4P$, where~$P=(1,1)$.
%\hfill {\bf [8~marks]}
\medskip\par\noindent{\bf (ii)}
Find the torsion group over~$\Q$
of the elliptic curve $Y^2 = X^3 - 2X$.
%\hfill {\bf [7~marks]}
\medskip\par\noindent{\bf (iii)}
Let~${\cal E}_k$ be the elliptic curve~$Y^2 = X^3 + k$,
where~$k\in\Q$ and~$k\not= 0$. 
%Show that, for any point~$(x,y) \not= {\underline {\bf o}}$
%on~${\cal E}_k$, 
%$$
%3(x,y) = {\underline {\bf o}} \iff 
%{{x(x^3 - 8k)}\over {4(x^3 + k)}} = x.
%$$
Show that there is always a point of order~$3$ 
in~${\cal E}_k({\bbC})$
which is not in~${\cal E}_k(\Q)$.
%\hfill {\bf [10~marks]} % I might consider including the hint.
\bigskip\noindent {\bf Question 4.}
\medskip\par\noindent{\bf (i)}
Find the rank of the elliptic curve $Y^2 = X(X^2 + 3X + 5)$.
%\hfill {\bf [13~marks]} 
\medskip\par\noindent{\bf (ii)}
For any prime $p \equiv 5$~(mod~$8$), show that
the elliptic curve $Y^2 = X^3 + p^2 X$ has rank~$0$.
%\hfill {\bf [12~marks]}
\bigskip
\bigskip
\hrule
\vfil\eject
\chaptitle
\noindent
\centerline{Elliptic Curves Questions from the 2007 Examination.}
\rm
\bigskip
\bigskip
\noindent
\bigskip\noindent {\bf Question 1.}
\medskip\par\noindent{\bf (i)}
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]}
\medskip\par\noindent{\bf (ii)}
Let $p \not= 2$ be prime. Find the $p$-adic expansion
of ${1 + 2p}\over {p - p^3}$.
%\hfill {\bf [5~marks]}
\medskip\par\noindent{\bf (iii)}
Does there exist a prime~$p$ such that $p = p^p$
in ${\Bbb Q}_p^* / \bigl( {\Bbb Q}_p^* \bigr)^p$?
%\hfill {\bf [5~marks]} 
\medskip\par\noindent{\bf (iv)}
Let $q,r$ be distinct primes, and
let $\alpha,\beta \in {\Bbb Q}$.
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]} 
\bigskip\noindent {\bf Question 2.}
\medskip\par\noindent{\bf (i)}
Let $K$ be field, complete with respect to a discrete
non-Archimedean valuation, $R = \{ x\in K : |x| \leqslant 1\}$,
${\cal M} = \{ x\in K : |x| < 1\}$,
and assume that $R/{\cal M}$ is of characteristic~$p$,
for some prime~$p$.
Let~$F(X,Y)$ be a formal group defined over~$R$ and suppose
that~$z\in {\cal M}$ has exact order~$p^n$, for some~$n\geqslant 1$,
with respect to the group operation $x \oplus y = F(x,y)$ 
on~$\cal M$. Show that: 
$$ | z | \geqslant | p |^{{1}\over {p^n - p^{n-1}}}.$$
[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]}
\medskip\par\noindent{\bf (ii)}
Let~${\cal E} : y^2 = x^3 + A x + B$, be an elliptic curve,
where~$A,B\in \Z_p$, and let~${\widetilde {\cal E}}$
denote the reduction of~${\cal E}$ modulo~$p$.
Show that any $(x,y) \in \etqp$ 
satisfies $|x|_p\leqslant 1, |y|_p\leqslant 1$. 
\par\noindent
When~${\widetilde {\cal E}}$ is non-singular, 
show that $\etqp$
is isomorphic to a subgroup of ${\widetilde {\cal E}}(\F_p)$.
%\hfill {\bf [8~marks]}
\medskip\par\noindent{\bf (iii)}
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]}
\bigskip\noindent {\bf Question 3.}
Let ${\cal C} : Y^2 = X(X^2 + aX + b)$
and ${\cal D} : Y^2 = X(X^2 + a_1 X + b_1)$,
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 
$$ \phi : {\cal C} (\Q ) 
\rightarrow {\cal D} (\Q) : (x,y)
\mapsto \Bigl( {{y^2}\over {x^2}} ,\ y - {{by}\over {x^2}} \Bigr) 
= \Bigl( {{x^2 + ax + b}\over {x}},\ y - {{by}\over {x^2}} \Bigr).$$
\noindent Let~$q$ be defined as usual by
$$ q : {\cal D} (\Q ) \rightarrow \Q^* / (\Q^*)^2 : (u,v) \mapsto u
\hbox{ when }u\not= 0,$$
$$ q : (0,0) \mapsto b_1,\,\,\, q: {\underline {\bf o}} \mapsto 1. $$
\medskip\par\noindent{\bf (i)}
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 ${\underline {\bf o}}$.]
%\hfill {\bf [6~marks]}
\medskip\par\noindent{\bf (ii)}
Show that $q$ has kernel $\phi ({\cal C} (\Q))$.
%\hfill {\bf [6~marks]}
\medskip\par\noindent{\bf (iii)}
Find the rank of the elliptic curve $Y^2 = X(X^2 + X - 2)$.
%\hfill {\bf [13~marks]}
\bigskip\noindent {\bf Question 4.}
\medskip\par\noindent{\bf (i)}
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]} 
\medskip\par\noindent{\bf (ii)}
For any elliptic curve~$\cal E$ and 
\par $(s,t) \in {\cal E}(\Q)$ 
with $s,t\in \Q$ and $s = {c\over d}$, $c,d \in \Z$, $\gcd(c,d) = 1$, 
\par\noindent 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 $h_x( {\underline {\bf o}} ) = 0$.
\par
Let $\cal C$, ${\cal D}, \phi$ 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 $P \in {\cal C} (\Q)$.
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 $P \in {\cal C} (\Q)$.
%\hfill {\bf [8~marks]} 
\medskip\par\noindent{\bf (iii)}
Let $p \not= 2$ be prime, let $m \in \F_p^*$
and let $\cal E$ be the elliptic curve $Y^2 = X(X^2 + m^2)$,
defined over~$\F_p$. Show that $\# {\cal E}(\F_p)$
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~$4$).]
\bigskip
\bigskip
\hrule
\vfil\eject
\chaptitle
\noindent
\centerline{Elliptic Curves Questions from the 2008 Examination.}
\rm
\bigskip
\bigskip
\noindent
\bigskip\noindent {\bf Question 1.}
\medskip\par\noindent{\bf (i)}
% See Problem Sheet 4, Question 1.
Decide whether there exists $x\in {\Bbb Q}_p$ such that
$x^3 = 5$ for each of: $p=3,5,13$.
%\hfill {\bf [7~marks]}
\medskip\par\noindent{\bf (ii)}
% Note: $-{{13}\over {8}}$ should have 3-adic 
% expansion 1,121212...  
Find the $3$-adic expansion 
of~$-{{13}\over {8}}$. For any prime~$p$,
and $a_0,a_1,a_2 \in \{0,\ldots ,p-1\}$,
express 
%the $p$-adic expansion
$a_0,\overline{a_1 a_2} 
= a_0 + a_1 p + a_2 p^2 + a_1 p^3 + a_2 p^4 + \ldots$
in the form $m/n$, where $m,n\in {\Bbb Z}$.
%\hfill {\bf [6~marks]}
\medskip\par\noindent{\bf (iii)}
Let ${\cal E} : x^3 + y^3 = p$, defined 
over~${\Bbb Q}_p$, and let $\widetilde{\cal E} : 
x^3 + y^3 = 0$, defined over~${\Bbb F}_p$, be the 
reduction of~${\cal E}$ modulo~$p$. Show that~$(0,0)$ 
is a singular point on~$\widetilde{\cal E}({\Bbb F}_p)$ 
and that it does not lift to a point 
on~${\cal E}({\Bbb Q}_p)$.  Find a curve~$\cal D$, 
nonsingular and defined over~${\Bbb Q}_p$, such 
that~$\widetilde{\cal D} = \widetilde{\cal E}$
and such that $(0,0)\in \widetilde{\cal D}({\Bbb F}_p)$
does lift to a point on~${\cal D}({\Bbb Q}_p)$. 
%\hfill {\bf [6~marks]} 
\medskip\par\noindent{\bf (iv)}
% See 1997, Qns 3(c),(d). I might give a hint.
Let $p\not= 2$ be prime and let $a,b,c \in {\Bbb Z}_p$
satisfy $|a|_p = |b|_p = |c|_p = 1$. Show that there
exist $x,y\in {\Bbb Z}_p$ such that $ax^2 + by^2 = c$.
\par [You might first wish to consider, for any
$\alpha, \beta, \gamma \in {\Bbb F}_p\backslash \{ 0 \}$,
the sizes of the sets $\{ \alpha x^2 : x \in {\Bbb F}_p \}$ 
and $\{ \gamma - \beta y^2 : y \in {\Bbb F}_p \}$.]
%\hfill {\bf [6~marks]} 
\bigskip\noindent{\bf Question 2.}
\medskip\par\noindent{\bf (i)}
% Bookwork from lectures.
State and prove the Nagell-Lutz Theorem
for $\Bbb Q$-rational torsion points on 
the elliptic curve $y^2 = x^3 + Ax + B$, 
with $A,B\in {\Bbb Z}$.
[You may assume the result that any $\Bbb Q$-rational 
torsion point $(x,y)$
on such a curve satisfies $x,y\in {\Bbb Z}$. 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]}
\medskip\par\noindent{\bf (ii)}
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]}
\medskip\par\noindent{\bf (iii)}
Let $D\in {\Bbb Z}$ 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~$\Bbb Q$
for the elliptic curve $y^2 = x^3 + D$.
% ... alternatively: 
% Let $D\in {\Bbb Z}$ satisfy $D \not\equiv 0$~(mod~5)
% and $D \equiv 3$~(mod~7).
% Find the torsion group over~$\Bbb Q$
% for the elliptic curve $y^2 = x^3 + D$.
%\hfill {\bf [5~marks]}
\medskip\par\noindent{\bf (iv)}
Let ${\cal E} : y^2 = x(x-1)(x-4)$. 
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 ${\cal E}({\Bbb C})$.
Show that $\# \widetilde {\cal E} ({\Bbb F}_p)$
is divisible by~8,
for all primes $p \not= 2,3$.
Show that
the torsion group of~${\cal E}({\Bbb Q})$ has order~4.
%\hfill {\bf [6~marks]}
\bigskip\noindent{\bf Question 3.}
Let ${\cal C} : Y^2 = X(X^2 + aX + b)$
and ${\cal D} : Y^2 = X(X^2 + a_1 X + b_1)$,
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 
$$ \phi : {\cal C} (\Q ) 
\rightarrow {\cal D} (\Q) : (x,y)
\mapsto\Bigl( {{y^2}\over {x^2}},\ y - {{by}\over {x^2}}\Bigr) 
=\Bigl( {{x^2 + ax + b}\over {x}},\ y - {{by}\over {x^2}}
\Bigr).$$
\noindent Let~$q$ be defined as usual by
$$ q : {\cal D} (\Q ) \rightarrow \Q^* / (\Q^*)^2 : 
(u,v) \mapsto u \hbox{ when }u\not= 0,$$
$$ q : (0,0) \mapsto b_1,\,\,\, q: {\underline {\bf o}}
\mapsto 1. $$
\medskip\par\noindent{\bf (i)}
% See Mock Exam.
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
%% I might water down (just use first part) or even remove this
%% part. Also, I might consider putting the marks for each
%% part on the exam this year!
%Show that $ | {\cal C} (\Q ) / 2 {\cal C} (\Q ) |
%= k | {\cal D} (\Q ) / 2 {\cal D} (\Q ) |$,
%for some $k \in \{ {{1}\over {2}}, 1, 2 \}$.
%\par
%When $b,b_1 \not\in \bigl( {\Bbb Q}^* \bigr)^2$,
%show that $ | {\cal C} (\Q ) / 2 {\cal C} (\Q ) |
%= | {\cal D} (\Q ) / 2 {\cal D} (\Q ) |$.
%% Also true when b,b_1 are both squares, which might be included?
%%\hfill {\bf [4~marks]}
\medskip\par\noindent{\bf (ii)}
% 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]}
\bigskip\noindent{\bf Question 4.}
\medskip\par\noindent{\bf (a)}
% 1994, qn 3.
For any elliptic curve~$\cal E$ and $(s,t) \in 
{\cal E}(\Q)$ with $s,t\in \Q$ and $s = {{c}\over {d}}$, 
$c,d \in \Z$, $\gcd(c,d) = 1$, 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 $h_x( {\underline {\bf o}} ) = 0$.  
\medskip\par\noindent{\bf (i)}
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 $P \in {\cal E}(\Q)$.
\medskip\par\noindent{\bf (ii)}
Show that, for any $P\in {\cal E}(\Q)$, the sequence
$4^{-n} h_x( 2^n P)$ is Cauchy and therefore
convergent in~$\R$ as~$n \rightarrow\infty$.
\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 $P,Q \in {\cal E}(\Q)$.]
\par
% [use induction for the first part; for the second part,
% use the m=2 case of the first part, together with p.228,
% where one first shows Cauchy].
% Include here a portion of 1994, qn 3 and/or cgce of the
% limit which gives the canon ht.
% Remind them that they may use the fact from lectures, 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 $P,Q \in {\cal E}(\Q)$.
%\hfill {\bf [11~marks]} 
\medskip\par\noindent{\bf (b)}
Let $K$ be field, complete with respect to a discrete
non-Archimedean valuation, $R = \{ x\in K : |x| \leqslant 1\}$,
${\cal M} = \{ x\in K : |x| < 1\}$,
and assume that $R/{\cal M}$ is of characteristic~$p$,
for some prime~$p$.
Let~$F(X,Y)$ be a formal group defined over~$R$,
and let $F({\cal M})$ denote
the set~${\cal M}$ 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~$x\in {\cal M}$, the 
sequence $[p^n](x) \rightarrow 0$ as $n\rightarrow \infty$.
% Include here Silverman, p.129 (a) [and/or (b)] on formal groups.
%\hfill {\bf [6~marks]}. Uses [p](x) = pf(x) + g(x^p).
\medskip\par\noindent{\bf (c)}
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]}
\bigskip
\bigskip
\hrule
\vfil \eject\end