\input amssym.def
\input amssym.tex
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\hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm}
\hrule height.1mm}}                  % This is a slimmer sqr.
\null
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\def\etqp{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q}_p)}
\def\c{{\cal C}}
\def\d{{\cal D}}
\def\e{{\cal E}}
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\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}
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%
\chaptitle
\noindent
\centerline{Elliptic Curves. Sheet 8.}
\centerline{(Some exam-style questions}
\centerline{and the 2006,2007,2008,2009 Exams).}
\rm
\bigskip
\bigskip
The following exam-style questions and the 2006,2007,2008,2009 Exams
need not be handed in for classes; they are
intended primarily to help you with your revision in
Trinity Term (when there will be consultation sessions in Trinity Term,
in case you have any questions during your Trinity Term revision). 
\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
\chaptitle
\noindent
\centerline{Elliptic Curves Questions from the 2009 Examination.}
\rm
\bigskip
\bigskip
\noindent
\bigskip\noindent {\bf Question 1.}
\medskip\par\noindent{\bf (a)}
Let ${\cal C}$ 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~${\cal C}$, and 
prove that it is associative.
%\hfill {\bf [10~marks]}
\medskip\par\noindent{\bf (b)}
For which primes~$p$ do there exist
$x,y\in {\Bbb Z}_p$ 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]}
\medskip\par\noindent{\bf (c)}
Find a proper factor of $N=221$
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]}
\medskip\par\noindent{\bf (d)}
Find a sequence $x_n\in{\Bbb Q}_p$ such that $\sum_{n=1}^\infty x_n$
is convergent in~${\Bbb Q}_p$ but $\sum_{n=1}^\infty |x_n|_p$ 
is not convergent in~${\Bbb R}$.
%\hfill {\bf [3~marks]} 

\bigskip\noindent {\bf Question 2.}
\medskip\par\noindent{\bf (a)}
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 [7~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 [7~marks]}
\medskip\par\noindent{\bf (b)}
Find the torsion group over~${\Bbb Q}$ for the elliptic curve $Y^2 = X^3 - 9$.
%\hfill {\bf [5~marks]}
\medskip\par\noindent{\bf (c)}
Let $n\in{\Bbb Z}$ be not divisible by~$15$, let~$n$ and~$-n$ 
both be nonsquare, and let ${\cal E}, {\cal F}$
be the elliptic curves ${\cal E} : Y^2 = X^3 + nX$ and 
${\cal F} : Y^2 = X^3 - nX$. Show that at least one of
${\cal E}({\Bbb Q})$ or ${\cal F}({\Bbb Q})$ has torsion group
given by $\{ {\bf o}, (0,0) \}$. 
%\hfill {\bf [6~marks]}

\bigskip\noindent {\bf Question 3.}
\medskip\par\noindent{\bf (a)}
Find the rank of the elliptic curve $Y^2 = X(X^2 + 5X - 5)$.
%\hfill {\bf [13 marks]}
\medskip\par\noindent{\bf (b)}
Let~$p$ be prime. Show that the elliptic curve $Y^2 = X^3 + pX$
has rank at most~$2$.
%\hfill {\bf [6~marks]}
\medskip\par\noindent{\bf (c)}
Let $\alpha\in{\Bbb Z}$, $\alpha\not= 0,-4$, and let ${\cal C}$ be
the elliptic curve $Y^2 = X^3 + \alpha X^2 - \alpha X$.
Show that $(1,1) \in 2{\cal C}({\Bbb Q}) \iff 
\alpha \hbox{ or } \alpha+4 \hbox{ is the square of an integer.}$
[You may find it helpful first to construct a curve~${\cal D}$
with $2$-isogenies $\phi : {\cal C} \rightarrow {\cal D}$,
$\hat\phi : {\cal D} \rightarrow {\cal C}$,
and find the preimages of~$(1,1)$ under~$\hat\phi$.]
%\hfill {\bf [6~marks]}
\bigskip
\bigskip
\hrule
\vfil \eject\end