\input amssym.def
\input amssym.tex
%\def\Bbb{\bf}
\nopagenumbers
\magnification=\magstep1
%\hoffset=1truecm
%\voffset=2truecm
\baselineskip = 5 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. Mock exam-style questions.}
\rm
\medskip
The following exam-style questions are not to be handed
in for intercollegiate classes; they are merely in case
you would like extra revision practice in addition to
the actual past exams (from 2006 onwards). 
\medskip\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 $\Q_p$.
\medskip\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 $| x | \le 3 |k|^2$.
\medskip\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 $Y^2 = X(X^2 + 2X + 3)$.
%\medskip\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
%\hrule
\vfil\eject\end