\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 0.}
\rm
\bigskip
{\it This sheet is not intended to be handed in. It is merely
for you to use (as you please) to reinforce the preliminary
reading in Section~0 ``Background Material in Algebra,
Number Theory and Geometry''.}
\bigskip
\bigskip
\noindent {\bf 1.} Determine whether the following are groups.
\par\noindent {\bf (a).} The set of all $2\times 2$ matrices
under matrix multiplication. 
\smallskip
\par\noindent {\bf (b).} The set of all $2\times 2$ matrices
under matrix addition. 
\medskip
\noindent {\bf 2.} For each of the following, decide whether
$\phi$ is a homomorphism. When $\phi$ is a homomorphism,
decide whether~$\phi$ is injective, surjective, bijective, and
find the kernel of~$\phi$.
\par\noindent {\bf (a).} $\phi : \bbZ , + \rightarrow \qs , \times
: x \mapsto x^2+1$.
\smallskip
\par\noindent {\bf (b).} $\phi : \bbQ , + \rightarrow \bbR , +
: w \mapsto \sqrt{2}\, w$.
\smallskip
\par\noindent {\bf (c).} $\phi : \bbZ  , +  \rightarrow
\bbZ / 3\bbZ , + : x \mapsto 2x$.
\medskip
\noindent {\bf 3.} 
\par\noindent {\bf (a).} In $\qmods$, decide whether
the following are
true or false: $3=1/27$, $-4=4$, $3=5/6$.
\smallskip
\par\noindent {\bf (b).} In $\qmods$, write each of the following as
a square free integer: $-2/27$, $16$, $12$, $1/3$.
\smallskip
\par\noindent {\bf (c).} Perform each of the following in $\qmods$,
writing your answer as a square free integer:
$6\times 10$, $10 / 21$, $15^{101}$, $3^{-1}$.
\smallskip
\par\noindent {\bf (d).} How many elements are in each of
the groups: $\qmods$,
${\Bbb R}^*/({\Bbb R}^*)^2$,
${\Bbb C}^*/({\Bbb C}^*)^2$?
\medskip
\noindent {\bf 4.}
\par\noindent {\bf (a).} Find all singular points on the curve
$$ \c : f(X,Y) = X^4 + Y^3 - 3 X^2 Y = 0. $$
\noindent Find all tangents to $\c$ at the point $(0,0)$.
\smallskip
\par\noindent {\bf (b).} Find all singular points on the curve
$$ \c : f(X,Y) = Y^2 - X(X^2-1)^2 = 0.$$
\noindent Find all tangents to $\c$ at the points $(0,0)$ and $(1,0)$.
\medskip \noindent {\bf 5.} Show that $\c : Y^2 = X^3 + AX + B$ is smooth
if $4A^3 + 27B^2 \not= 0$.
\medskip\noindent {\bf 6.} For each of the following curves,
find the irreducible components over~$\bbQ$ and the irreducible
components over~$\bbC$.
\par\noindent {\bf (a).} $\c : Y^2 = X^5$.
\smallskip
\par\noindent {\bf (b).} $\c : Y^3 = X^3$.
\smallskip
\par\noindent {\bf (c).} $\c : Y^2 = X^3 + 1$.
\medskip\noindent{\bf 7.}
\par\noindent {\bf (a).} Find a birational transformation over~$\bbQ$
between the curves $2X^2 - Y^2 = 1$ and $X^2 + Y^2 - 6XY = 1$.
\smallskip
\par\noindent {\bf (b).} Find a birational transformation over~$\bbQ$
between the curves $Y^2=(X+2)^6(X^3+1)$ and $Y^2 = X^3 + 1$.
\smallskip
\par\noindent {\bf (c).} Find a birational transformation over~$\bbC$
between the curves~$Y^2=2X^2$ and~$Y^2=X^2$. Is there a birational
transformation over~$\bbQ$?
\medskip
\noindent {\bf 8.}
\par\noindent {\bf (a).} Find the discriminant of~$X^4-2$.
\smallskip
\par\noindent {\bf (b).} Find the resultant of $X^3 - a$ and $X^2 - b$,
where $a,b$ are constants.
\medskip\noindent {\bf 9.} Find all intersection points
(with multiplicities) over~$\Bbb C$ of the curves:
$X^3 + Y^3 = Z^3$ and $X^2 + Y^2 = Z^2$.
\medskip
\noindent {\bf 10.}
\par\noindent {\bf (a).}
Decide whether each of 
$2,3,5,10,15$
are quadratic residues modulo~1009 (if you use quadratic reciprocity,
this should not involve any lengthy computations).
\smallskip
\par\noindent {\bf (b).} Describe all primes~$p$ such that $3$
is a quadratic residue modulo~$p$.   
Describe all primes~$p$ such that $5$
is a quadratic residue modulo~$p$. 
Describe all primes~$p$ such that $10$
is a quadratic residue modulo~$p$. 
\medskip
\noindent {\bf 11.} Are there integers $a,b,c$, not all~$0$,
such that $2a^2 + 5b^2 = c^2$?
\medskip
\noindent {\bf 12.} For any $n\in{\Bbb N}$ define, as usual, Euler's
$\phi$-function by: 
$$ \phi(n) = \# \{ x :
1 \leqslant x \leqslant n \hbox{ and gcd} (x,n) = 1 \}.
$$
For any prime~$p$, what is $\phi(p^r)$?
For any distinct primes $p_1,p_2$, what is $\phi(p_1 p_2)$?
\par
For each of the following examples of
the type $a^b \ (\hbox{mod }n)$, reduce $a^b \ (\hbox{mod }n)$ to a member
of $\{ 0, \ldots , n-1 \}$.
\par\noindent
$2^{12} \ (\hbox{mod }13)$,
$3^{12} \ (\hbox{mod }13)$,
$3^{24} \ (\hbox{mod }13)$,
$3^{12000} \ (\hbox{mod }13)$,
$3^{12002} \ (\hbox{mod }13)$,
\hfill\par\noindent
$4^{24} \ (\hbox{mod }35)$,
$4^{48} \ (\hbox{mod }35)$,
%$4^{48000} \ (\hbox{mod }35)$,
$4^{48000001} \ (\hbox{mod }35)$,
\hfill\par\noindent
$7^{24} \ (\hbox{mod }35)$,
$7^{48} \ (\hbox{mod }35)$,
%$7^{48000} \ (\hbox{mod }35)$,
$7^{48000001} \ (\hbox{mod }35)$.
\vfil \eject %\end
\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 1. To be handed in during 2nd Week.}
\rm
\bigskip
\noindent
\noindent {\bf 1.} For each of the following elliptic curves,
find all the points (including, as always,
the point at infinity) over ~$\bbF_5$.
Draw a complete
group table in each case and describe each group as a product of
cyclic groups.
\par\noindent
{\bf (a).} $Y^2 = X^3 + 2 X$.
\smallskip
\par\noindent
{\bf (b).} $Y^2 = X^3 + 1$.
\medskip
\noindent {\bf 2.} Show that the point~$(2,4)$ is of order~4
on $Y^2 = X^3 + 4X$, defined over~$\bbQ$.
\medskip
\par\noindent {\bf 3.}
\par\noindent {\bf (a).} Let $m\in {\Bbb N}$ be odd or $f_m\in \qss$ (or both).
Show that the curve
\par
\centerline{$Y^2 = f_mX^m + f_{m-1}X^{m-1} + \ldots + f_0$, where
all $f_i\in \bbQ$ and $f_m\not= 0$,}
\par\noindent
can be birationally transformed over~$\bbQ$ to 
a curve of the form
\par
\centerline{$Y^2 = X^m + g_{m-1}X^{m-1} + \ldots+ g_0$, with
all $g_i\in \bbZ$.}
\smallskip
\par\noindent {\bf (b).} Birationally
transform over $\bbQ$ the curve $Y^2 = {1\over 5}X^3 + 3 X^2
+ 1$ to a curve of the form~$Y^2 = X^3 + AX + B$, where $A,B\in \bbZ$.
\medskip
\par\noindent {\bf 4.}
\par\noindent {\bf (a).} Let $p \equiv 2$~(mod~$3$) be prime
and let $A \in \bbF_p^*$.
Show that the number of points (including the point
at infinity) on the curve $Y^2 = X^3 + A$ over $\bbF_p$
is exactly $p+1$.
\smallskip
\par\noindent {\bf (b).} Let $p \equiv 3$~(mod~$4$) be prime
and let $B \in \bbF_p^*$.
Show that the number of points (including the point
at infinity) on the curve $Y^2 = X(X^2 + B)$ over $\bbF_p$
is exactly $p+1$.
\bigskip
\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
\par\noindent {\bf 5.}
\par\noindent {\bf (a).} Let $\c : Y^2 = Q(X) = X^4 + f_3X^3
+ f_2X^2 + f_1X + f_0$, where all $f_i\in \bbQ$. Show that
the curve $\c$ can be birationally transformed over $\bbQ$
to a curve of the form $Y^2 = X^3 + AX + B$, where $A,B\in \bbZ$.
[Begin by finding $G(X)$ and $H(X)$ such that $Q(X) = G(X)^2 + H(X)$,
with $G$ quadratic and $H$ linear, and let $T=Y+G(X)$,
$S=X\bigl(Y+G(X)\bigr)$.]
\smallskip
\par\noindent {\bf (b).} Birationally
transform over $\bbQ$ the curve $\c_1: Y^2 = 2X^4 + 9$
to the standard form $Y^2 = X^3 + AX + B$,
with $A,B\in \bbZ$. Show that $\c_2: Y^2 = 2X^4 + 7$ can
also be birationally
transformed over $\bbQ$ to the same form.
[Hint: In each case, first transform to the
form $Y^2 = Q(X)$ of part (a).]
\smallskip
\par\noindent {\bf (c).} Birationally transform over $\bbQ (i)$ the
curve $Y^2 = -X^4 - 1$ to the standard form $Y^2 = X^3 + AX + B$,
with $A,B\in \bbZ$. Is this possible over $\bbR$? Is this
possible over $\bbQ (\sqrt{-2})$?
\medskip
\vfil \eject %\end
\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 2. To be handed in during 3rd Week.}
\rm
\bigskip
\noindent {\bf 1.}
\par\noindent {\bf (a).} Show that the point $(2,0)$ is of
order 2 on $Y^2 = (X-2)(X^2 + X + 1)$.
\par\noindent {\bf (b).} Find all $\bbQ$-rational points of order~2
and all $\bbC$-rational points of order~2
on each of the following elliptic curves: $Y^2 = X(X^2-3)$,
$Y^2 = X^3 - 7$ and $Y^2 = X(X-1)(X-7)$. In each case, find the
group structure (expressed as a product of cyclic groups)
of the $\bbQ$-rational 2-torsion group (that is, the group
of all $\bbQ$-rational points~$P$ such that $2P = {\bf o}$).
\medskip
\par\noindent {\bf 2.} Show that the point $(0,2)$ is of
order 3 on $Y^2 = X^3 + 4$. 
\medskip
\noindent
{\bf 3.}
\par\noindent {\bf (a).} Let $Y^2 = (X-\alpha)(X^2 + aX + b)$
be an elliptic curve with $a,b,\alpha\in K$ (characteristic $\not= 2$),
and  ${\bf o} =$ point at infinity, as usual. Show that $(\alpha , 0)$
is a point of order~2. Let $x',y'$ be defined by: $(x',y')
= (x,y) + (\alpha , 0)$, and define $T: K\rightarrow K : x\mapsto x'$.
Find $t_{11},t_{12},t_{21},t_{22}$ in terms of $a,b,\alpha$ such that:
$x' = \mu(x) = (t_{11} x + t_{12})/(t_{21} x + t_{22})$.
Check that $\mu^2 : x\mapsto x$.
\smallskip
\par\noindent {\bf (b).} Consider~$Y^2 = (X-\alpha_1)(X-\alpha_2)
(X-\alpha_3)$, with $\alpha_1,\alpha_2,\alpha_3$ distinct,
and let $T_1,T_2,T_3$ be as in (a), but with $\alpha$ replaced
by $\alpha_1,\alpha_2,\alpha_3$, respectively. 
Express each $T_i$ in terms of $x,\alpha_1,\alpha_2,\alpha_3$.
Show, directly from expressions, that $T_1,T_2,T_3$
commute (i.e.\ $T_1 T_2 = T_2 T_1$, $T_1 T_2 = T_2 T_1$ and
$T_2 T_3 = T_3 T_2$), and that $T_1 T_2 T_3 : x\mapsto x$.
Find the fixed points of $T_1$
and show that they are permuted by $T_2$.
\medskip
\noindent {\bf 4.}
Let $K$ be any field with $\hbox{Char }K \not= 2,3$,
and let 
$${\cal E}: F(X_0,X_1,X_2)
= X_1^2 X_2 - (X_0^3 + A X_0 X_2^2 + B X_2^3), \hbox{ with }A,B \in K,$$
be an elliptic curve (N.B. This is just the standard projective form,
but with $X,Y,Z$ replaced by $X_0,X_1,X_2$). Let~$P$ be a
point on~$\cal E$.
\smallskip
\par\noindent {\bf (a).} Show that $3P = {\bf o}$ iff.\ the tangent line
to $\cal E$ at~$P$ intersects $\cal E$ only at~$P$.
\smallskip
\par\noindent {\bf (b).} Show that if $3P={\bf o}$ then the $3\times 3$
matrix
$\bigl( \partial^2 F / \partial X_i \partial X_j (P) \bigr)$
has determinant~$0$. [This matrix is called the Hessian matrix].
\smallskip
\par\noindent {\bf (c).} Show that there are at most nine
$3$-torsion points over~$K$.
%\smallskip
%\par\noindent {\bf (b).} Show that $3P={\bf o}$ iff.\ the $3\times 3$
%matrix
%$\bigl( \partial^2 F / \partial X_i \partial X_j (P) \bigr)$
%has determinant~$0$. [This matrix is called the Hessian matrix].
%\smallskip
%\par\noindent {\bf (c).} Show that there are exactly nine
%$3$-torsion points over $\overline K$, the algebraic
%closure of~$K$.
\medskip
\vfil \eject %\end
\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 3. To be handed in during 4th Week.}
\rm
\bigskip
\noindent
{\bf 1.} Let $K$ be a field with
non-Archimedean valuation $|\ |$.
\smallskip
\par\noindent {\bf (a).} For any $x,y\in K$ show that, if $|x| \not= |y|$
then $|x \pm y | = \hbox{max}( |x|, |y| )$.
\par\noindent {\bf (b).}
If $x_1, \ldots , x_n \in K$ and
if there exists $\ell$ such that
$|x_\ell| > |x_i|$ for all $i\not= \ell$,
then show that $|x_1 + \ldots + x_n| = |x_\ell|$.
\smallskip
\par\noindent {\bf (c).} Suppose that $s_n \rightarrow s$
in $K,|\ |$. Show that $|s_n| \rightarrow |s|$
in $\bbR, |\ |_\infty$. 
%Suppose
%that $s_n \rightarrow s \not= 0$ in~$\bbQ_p$; show
%that there exists $N$ such that, for all $n > N$, $|s_n|_p = |s|_p$.
When $s \not= 0$, show
that there exists $N$ such that, for all $n > N$, $|s_n| = |s|$.
%\smallskip
%\par\noindent {\bf (d).}
%Show that if $\sum_{n=1}^\infty x_n$
%converges to $x \in K, |\ |$, then
%the set $\{ | x_i | : i \geqslant 1\} \subset \bbR$ 
%has a maximum element, and $|x| \leqslant \hbox{max}_i |x_i|$.
%Show that, if there exists $\ell$ such that
%$|x_\ell| > |x_i|$ for all $i\not= \ell$,
%then $\sum_{n=1}^\infty x_n$ does not converge to~$0$.
\medskip
\noindent {\bf 2.} 
\par\noindent {\bf (a).} Find: $| 3/50 |_5$, $| 3/50 |_3$, $|3/50 |_7$,
$d_5(2/3 , 1/5)$, $d_7(2/3 , 1/5)$, $d_{11}(2/3, 1/5)$.
\smallskip
\par\noindent {\bf (b).} Describe $| 3/7 |_p$ for all~$p$. What
is the product $\prod | 3/7 |_i$, taken over $i=p$, for all primes $p$,
and $i=\infty$? Given any $x\in\Q$ ($x\not= 0$), what is $\prod | x |_i$?
\medskip
\noindent {\bf 3.} Which of the following are convergent in $\Q_5$?
\par\ \ \ \ \ \ \
${\bf (a).}\ 1/5^n.\ \ {\bf (b).}\ n.\ \ {\bf (c).}\ n!
\ \ {\bf (d).}\ 3 + 10^n.
\ \ {\bf (e).}\ \sum_0^\infty 10^n.\ \ {\bf (f).}\ \sum_0^\infty 7^n.$
%$ a_n = 1/5^n,\ \ a_n = n,\ \ a_n = n!,\ \ a_n = 3 + 10^n.$
%\medskip
%\noindent {\bf 4.} Which of the following are convergent in $\Q_5$?
%$\sum_0^\infty 10^n , \ \ \sum_0^\infty 7^n.$
%\medskip
%\noindent {\bf 5.} For each $p,m,r$, either find an $x\in \Z$ such that
%$|x-r|_p \leqslant p^{-m}$ or show that no such~$x$ exists.
%\smallskip
%\par\noindent
%{\bf (a).} $p=257, r=1/ 2, m=1$.\ \
%{\bf (b).} $p=3, r=7/ 9, m=7$.\ \
%{\bf (c).} $p=5, r=1/ 4, m=4$.
\medskip
\noindent {\bf 4.} For each $p,m,r$, either find an $x\in \Z$ such that
$|x^2-r|_p \leqslant p^{-m}$ or show that no such~$x$ exists. 
\smallskip
\par\noindent
{\bf (a).} $p=5, r=-1, m=4$.\ \
{\bf (b).} $p=3, r=7/8, m=7$.\ \
{\bf (c).} $p=5, r=5/4, m=4$.\ \
\medskip 
\noindent {\bf 5.} Find the $7$-adic expansion of each of: $200$ and $3/14$.
Determine the member of~$\Q$ expressed by
the $5$-adic expansion $2,\overline{34}$.
\medskip\noindent {\bf 6.} Let $x\in \Q$. Show that
$x\in \Z \iff \bigl( x\in \Z_p \hbox{ for all }p\bigr)$. 
%\medskip 
%\noindent {\bf 7.} Find the $7$-adic expansion of each of: $200$ and $3/14$.
%\medskip 
%\noindent {\bf 8.} What member of~$\Q$ is expressed by each of
%the following two $5$-adic expansions? $23,4$ and $2,\overline{34}$.
%\medskip\noindent {\bf 9.} Let $x\in \Q$. Show that
%$x\in \Z \iff \bigl( x\in \Z_p \hbox{ for all }p\bigr)$. 
\bigskip
\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 $|n!|_p = p^{-M}$ where $M = \sum_{i=1}^\infty 
\bigl[ {n\over p^i} \bigr]$ (where $[ x ]$ denotes the
greatest integer $\leqslant x$).
Let $K$ be a field containing $\Q_p$,
let $|\ |$ be a non-Archimedean valuation on~$K$ which extends
$|\ |_p$, and assume that $K$ is complete with respect
to this valuation. For any $x\in K$, show that 
$\hbox{exp}_p(x) = \sum_{n=0}^\infty {x^n\over n!}$
converges if and only if
$|x| < p^{-{1\over p-1}}$. 
When $K = \Q_p$ ($p\not= 2$), show that $\hbox{exp}_p(x)$ converges
if any only if $|x|_p < 1$. When $K = \Q_2$, show that
$\hbox{exp}_2(x)$ converges
if any only if $|x|_2 < {1\over 2}$.
\vfil \eject %\end
% MARKS ALLOCATED AS FOLLOWS:
% 1: 10. 2: 10. 3: 10. 4: 10. 5: 10. 6: 20. 7: 30. Total: 100.
% In more detail:
% 1: 10. 2: 10. 3: 10. 4: 10. 5: 10. 6: each part 5. 7: each part 5.
\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 4. To be handed in during 5th Week.}
\rm
\bigskip
\noindent {\bf 1.} Decide whether there exists $x\in \Q_p$ such that
$x^2 = -28$ for each of: $p=2,3,5,7,11$.
%\medskip\noindent {\bf 2.} Show that, for all~$p$, there
%exist~$x,y\in \Z_p$ such that
%$y^2 = x^3 + x - 3$.
\medskip\noindent {\bf 2.} Show that $(X^2 - 2)(X^2-17)(X^2-34)$
has a root in $\R$ and in every $\Q_p$, but not in $\Q$.  
\medskip\noindent {\bf 3.} Is~$4$ a cube in~$\Q_3$? Is~$28$ a cube
in~$\Q_3$? Is~$13$ a cube in~$\Q_7$?
\medskip\noindent {\bf 4.} Show that the curve $2 Y^2 = X^4 - 17$
has points in $\R$ and every $\Q_p$, but not in~$\Q$. 
\par\noindent
[Hint: For showing that there are points in every $\Q_p$,
it is helpful to use Theorem~1.15 (note also that
the curve is birationally equivalent to $V^2 = 2 X^4 - 34$,
where $V = 2Y$). For showing there are
no points in~$\Q$,
first show that, if there were points in~$\Q$, then there would exist
$r,s,t\in \Z$ with $\hbox{gcd}(r,t) = 1$ such
that $2 s^2 = t^4 - 17 r^4$, and then show that any prime dividing
$s$ is a quadratic residue modulo~$17$].
\medskip\noindent {\bf 5.} Let $p\equiv 2$ mod~$3$. For any
$a\in \Z$ such that $p\notdiv a$, show that there exists
$x\in \Z_p$ with $x^3 = a$.
\medskip\noindent {\bf 6.} Let $K$ be any field with a
non-Archimedean valuation $|\ |$, and
let $ R = \{ x\in K : |x| \leqslant 1\}$.
Let $f(X) \in R[x]$ have discriminant~$D$, and let
$a_0 \in R$ satisfy $|f(a_0)| < |D|^2$. Show that $f(X)$
has a root $a\in R$.
\bigskip
\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
\medskip\noindent {\bf 7.} Show that $f(X) = 5 X^3 - 7 X^2 + 3 X + 6$
has a root $\alpha \in \Z_7$ with $|\alpha - 1|_7 < 1$.
Find $a\in\Z$ such that $|\alpha - a|_7 \leqslant 7^{-4}$.
\vfil\eject
\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 5. To be handed in during 6th Week.}
\rm
\medskip\noindent {\bf 1.} Prove that, if $d\in\Z_p$
is non-square, then 
$$ | a + b\sqrt{d} |_p = | a^2 - b^2 d |_p^{1/2}, 
\hbox{ for any } a,b \in \Q_p, $$
defines a non-Archimedean valuation on $\Q_p(\sqrt{d})$ which extends
the usual $|\ |_p$ on $\Q_p$. 
\smallskip\par\noindent [Hint: First show that, for
any $\alpha \in \Q_p(\sqrt{d})$, $|\alpha|_p \leqslant 1
\Rightarrow |\alpha + 1 |_p \leqslant 1$].
\medskip\noindent {\bf 2.} Let $\e : Y^2 = X^3 + 17$, defined
over~$\Q$, and $\widetilde \e : Y^2 = X^3 + 2$, defined over
$\F_5$. What does $(-64/25 , 59/125) \in \e (\Q)$ map to
under the reduction map modulo~$5$?
\medskip\noindent {\bf 3.} Let $\e : Y^2  = X^3 + p$, defined
over~$\Q_p$, and $\widetilde \e : Y^2 = X^3$, defined over~$\F_p$,
where~$p\not= 2$. Show that~$(0,0)$ on~$\widetilde \e$ does not
lift to a point in~$\e (\Q_p)$.
\medskip\noindent {\bf 4.} Give examples of elliptic curves
defined over~$\Z_p$ ($p\not= 2$) such that $\widetilde \e$,
defined over~$\F_p$, has:
\par \noindent {\bf (a).} A cusp which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (b).} A cusp which does not lift to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (c).} A node which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (d).} A node which does not lift to a point
in~$\e (\Q_p)$.
\medskip\noindent {\bf 5.} A {\it non-commutative formal group}
over a ring~$R$ is a power series $F(X,Y) \in R[[X,Y]]$
which satisfies:
\par $F(X,Y) = X + Y +$ terms of degree~$\geqslant 2$,
\par and 
\par $F(X, F(Y,Z)) = F(F(X,Y),Z)$ [associativity],
\par\noindent but not $F(X,Y) = F(Y,X)$ [commutativity].
Let $R = \F_p[t]/I$, where $I = t^2 \F_p[t]$.
Find a non-commutative formal group over~$R$.
%of the form $F(X,Y) = X + Y + c X^i Y^j$, where $c$ is a constant.
\bigskip
\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
\medskip\noindent {\bf 6.} Let $\e : Y^2 = X^3 + AX$, where
$A \in \Z$ and $A \not= 0$. Let $F(X,Y)$ be the formal group
associated to~$\e$ [as in lectures] and let $F(X,Y) = \sum F_n(X,Y)$,
where each $F_n(X,Y)$ is homogeneous of degree~$n$.
Show that $F_n(X,Y) = 0$ unless $n\equiv 1$~(mod~$4$).
What is the similar result when the elliptic curve
is of the form $\e : Y^2 = X^3 + B$?
\vfil\eject
\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\cotq{{\cal C}_{\lower 1pt\hbox{\eightrm oddtors}}({\Bbb Q})}
\def\dotq{{\cal D}_{\lower 1pt\hbox{\eightrm oddtors}}({\Bbb Q})}
\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 6. To be handed in during 7th Week.}
\rm
\bigskip
\medskip\noindent {\bf 1.} Find the torsion group over~$\Q$
for each of:
\par\noindent {\bf (a).} $Y^2 = X^3 + 1$.
\par\noindent {\bf (b).} $Y^2 = X(X-1)(X-2)$.
\par\noindent {\bf (c).} $Y^2 = X^3 + 1/3^6$.
\par\noindent {\bf (d) [optional].} $Y^2 = X^3 - 219X + 1654$.
%\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).} $Y^2 = X^3 - X^2 + 1/4$.
\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.
\medskip\noindent {\bf 2.} 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.
\medskip\noindent {\bf 3.} Let $\c$ and $\d$ be as in question~2.
Let the homomorphism~$\phi, \hat\phi$ be defined as usual by 
$$\phi : \c (\Q ) \rightarrow \d (\Q) : (x,y)
\mapsto \Bigl( \bigl( {y\over x}\bigr)^2 , 
y - {by\over x^2} \Bigr),$$
$$
\hat\phi : \d (\Q) \rightarrow \c (\Q) : (u,v)
\mapsto \Bigl( {1\over 4} \bigl( {v\over u} \bigr)^2,
{1\over 8} \bigl( v - {b_1 v\over u^2}\bigr) \Bigr).
$$
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 $a+2m = n^2$. 
\bigskip
\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\medskip\noindent
{\bf 4.} 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
% 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 7. To be handed in during 8th Week.}
\rm
\bigskip
\noindent
{\bf 1.} Find the ranks of the
following elliptic curves.
\par\noindent {\bf (a).} $Y^2 = X(X^2 + 2X + 3)$.
\par\noindent {\bf (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).} $Y^2 = X(X-12)(X-36)$. 
\medskip\noindent {\bf 2.}
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 [$\R_{\ge 0}$ denotes $\{ x\in\R : x \ge 0\}$].
\medskip\noindent {\bf 3.}
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:
$$ 6012,\, 3236.$$ 
\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
{\bf (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
{\bf (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
{\bf (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).
%\medskip\noindent {\bf 4.} A 16-letter message (including any spaces)
%has been split into 8 pairs
%of letters. Each pair of letters has been encoded into 4 digits,
%using the usual map $A\rightarrow 01,\ldots ,Z\rightarrow 26$
%and $\hbox{space}\rightarrow 00$. Each of these 4-digit blocks
%has been further encoded using the map $X \rightarrow X^d$~(mod~$N$),
%where $d=4903$, $N=8777$, resulting in:
%\par
%$4195\vert 7645\vert 1876\vert 3549\vert 7864\vert 3057\vert 72\vert 3654$.
%\par\noindent Working mod~$N$, find the multiple $k\cdot P$
%on ${\cal E}:Y^2=X^3 + X - 1$, where $P=(1,1)$ and
%$k=2^6\cdot 3^4 \cdot 5$ (explain the way
%that you have efficiently computed $k\cdot P$). Use this to factor~$N$
%(you may assume that $N$ is the product of two primes). Also, factor
%$N$ using Pollard's $p-1$ method, using base~$2$.
%Use the factorisation of~$N$ to deduce the decoding exponent~$e$
%such that $X \rightarrow X^e$~(mod~$N$) reverses the map
%$X\rightarrow X^d$~(mod~$N$). Hence decode the message. You should
%show in your working how you have done all of the above
%computations using only an eight-digit calculator.
\medskip
\hrule
%\bigskip
\bigskip
\bigskip
\bigskip
\centerline{\bf A Few Pieces of Computational Advice}
%\par
%I have the impression that some of you are making
%cryptography questions unduly time-consuming, due to
%very slow calculator methods for performing some of the
%basic steps. 
\medskip
\sevrm
\baselineskip = 3.4 true mm
If you want to perform something like: 2046 $\cdot$ 8018 mod~8777
on a pocket calculator, then the fast way is as follows. 
First, 2046 $\cdot$ 8018 = 16404828. Now divide by 8777
to get the decimal~1869.070069; now subtract off the integer
part 1869 to get .070069; now multiply by 8777 to
get the decimal 614.99561; this is guaranteed to be almost
exactly an integer, and the nearest integer (namely: 615) will 
be 16404828 mod~8777. If you want to check it 
be 100\% sure, then you can verify it by:
(2046 $\cdot$ 8018 - 615)/8777 and seeing that the result is
an exact integer. [N.B. This is much faster than, for example,
repeatedly subtracting 8777 from 16404828 until getting
a number less than 8777; in this case that approach would
require 1869 subtractions!].
This same idea can also make quicker steps
of Euclid's Algorithm. 
\par
If you want to write a number, such as k=25920,
in base~2, then a fast way is as follows. Type 25920
into the calculator. At each step, we reduce the size of our
current number either by the step [divide-by-2]
(if our current number is even) or  
by the step [subtract-1-and-then-divide-by-2]
(if our current number is odd). This
allows us to write down the base~2 digits from right to left, where
we write down a~0 if we've done the first of the above,
and a~1 if we've done the second. For example,
with k=25920, we first perform [divide-by-2] and write
down~0 as our rightmost digit (and the calculator display
now reads 12960). After doing the [divide-by-2] 5 more
times, we have now written a total of 000000 as the
six rightmost digits, and the calculator reads: 405.
Now, perform [subtract-1-and-then-divide-by-2], and write
down a~1 on the left, so that your piece of paper
currently reads: 1000000, and your calculator
display reads: 202. Now perform [divide-by-2], so that
you piece of paper reads: 01000000 and your calculator
reads: 101. Continuing until your calculator reads 0
will make your final piece of paper read:
110010101000000; that is:
k = $\hbox{2}^{\hbox{\fiverm 6}}$ +
$\hbox{2}^{\hbox{\fiverm 8}}$ +
$\hbox{2}^{\hbox{\fiverm 10}}$ +
$\hbox{2}^{\hbox{\fiverm 13}}$ +
$\hbox{2}^{\hbox{\fiverm 14}}$ [take care to remember
that the last digit in 110010101000000 is the coefficient
of~$\hbox{2}^{\hbox{\fiverm 0}}$.]
\vfil \eject %\end
% 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
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%\voffset=2truecm
\baselineskip = 5.2 true mm
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\font\sevrm=cmr7
\font\sixrm=cmr6
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\font\deffont=cmssi10
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\tolerance=10000
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\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~${\cal E}_k$, 
%$$
%3(x,y) = {\underline {\bf o}} \iff \frac{x(x^3 - 8k)}{4(x^3 + k)} = x.
%$$
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
